SSgr A ∗ and General Relativity Tim Johannsen
Perimeter Institute for Theoretical Physics, Waterloo, ON, N2L 2Y5, CanadaDepartment of Physics and Astronomy, University of Waterloo, Waterloo, ON, N2L 3G1, CanadaE-mail: [email protected]
Abstract.
General relativity has been widely tested in weak gravitational fields but still stands largelyuntested in the strong-field regime. According to the no-hair theorem, black holes in general relativitydepend only on their masses and spins and are described by the Kerr metric. Mass and spin arethe first two multipole moments of the Kerr spacetime and completely determine all higher-ordermoments. The no-hair theorem and, hence, general relativity can be tested by measuring potentialdeviations from the Kerr metric affecting such higher-order moments. Sagittarius A ∗ (Sgr A ∗ ), thesupermassive black hole at the center of the Milky Way, is a prime target for precision tests of generalrelativity with several experiments across the electromagnetic spectrum. First, near-infrared (NIR)monitoring of stars orbiting around Sgr A ∗ with current and new instruments is expected to resolvetheir orbital precessions. Second, timing observations of radio pulsars near the Galactic center maydetect characteristic residuals induced by the spin and quadrupole moment of Sgr A ∗ . Third, theEvent Horizon Telescope, a global network of mm and sub-mm telescopes, aims to study Sgr A ∗ onhorizon scales and to image the silhouette of its shadow cast against the surrounding accretion flowusing very-long baseline interferometric (VLBI) techniques. Both NIR and VLBI observations mayalso detect quasiperiodic variability of the emission from the accretion flow of Sgr A ∗ . In this review,I discuss our current understanding of the spacetime of Sgr A ∗ and the prospects of NIR, timing, andVLBI observations to test its Kerr nature in the near future.
1. Introduction
General relativity has been tested and confirmed by a variety of different experiments ranging fromEddington’s solar eclipse expedition of 1919 to modern observations of double neutron stars [1]. Thesetests place tight limits on the properties of theories of gravity in the weak-field regime and leave littleroom for deviations from general relativity. Seldom, however, have these tests probed settings of strongspacetime curvature, where two theories of gravity can differ significantly in their predictions, eventhough both of them satisfy the current experimental constraints. In fact, general relativity still standspractically untested in the strong-field regime and it is incumbent to expand the scope of the currenttests of general relativity [2].This can be illustrated by defining a parameter space spanned by the gravitational potential ε ≡ GMrc (1)and the spacetime curvature ξ ≡ GMr c (2)of an object with mass M , where the coordinate r measures the radial distance from the source andwhere G and c are the gravitational constant and the speed of light, respectively [2]. While the fullspacetime of the object is characterized by a metric g αβ with a corresponding Riemann tensor R αβγδ a r X i v : . [ a s t r o - ph . GA ] M a y nd not simply by the potential ε and the curvature ξ defined above, the latter two quantities allow foran intuitive characterization of strong and weak gravitational fields using dimensional arguments.Figure 1 shows the regimes probed by a wide range of astrophysical and cosmological systems inthis parameter space as well as a number of corresponding experiments. Only a small fraction of thisparameter space has been accessed so far. In the solar system, where most tests of general relativityhave been performed to date, typical values of the potential and the curvature can be vastly differentfrom those found near compact objects or on cosmological scales [2, 3]. To date, there have been onlya few strong-field tests of general relativity in the context of either neutron stars (see Ref. [4]; [2, 5–8])or black holes [9–17]. Figure 1.
Left: A parameter space for gravitational fields, spanned by the gravitational potential ε and curvature ξ , showing the regimes probed by a wide range of astrophysical and cosmologicalsystems. Right: The experimental version of the parameter space. Some of the label abbreviationsare: SS = planets of the Solar System, MS = Main Sequence stars, WD = white dwarfs, PSRs = binarypulsars, NS = individual neutron stars, BH = stellar mass black holes, MW = Milky Way, SMBH =supermassive black holes, BBN = Big Bang nucleosynthesis, PPN = parameterized post-Newtonianregion, Inv. Sq. = laboratory tests of the 1 /r behavior of the gravitational force law, Atom = atominterferometry experiments to probe screening mechanisms, EHT = Event Horizon Telescope, ELT =Extremely Large Telescope, DETF4 = a hypothetical ‘stage 4’ Dark Energy experiment, Facility = afuturistic large radio telescope such as the Square Kilometre Array. Taken from [3]. There are two basic approaches to test general relativity. Either, one assumes a particular theoryof gravity and obtains constraints on potential deviations from general relativity within that theory.Almost all of the current strong-field tests fall into this category. Alternatively, one searches for genericdeviations from general relativity and, thereby, hopes to gain insight into the underlying (but usuallyunknown) theory of gravity.Both approaches are valid and have their place. However, tests of a specific theory of gravity are apriori limited by the fact that observations are interpreted within the narrow confines of that theory. Our2ack of knowledge of a quantum theory of gravity as part of a grand unified theory of all fundamentalinteractions forces us to consider many different alternatives to general relativity. It is, therefore,advisable to search for deviations from general relativity using a broader setting that encompasses asmany modified theories of gravity as possible. In this phenomenological approach, stars and compactobjects have properties that differ from those in general relativity. These deviations can be expressed interms of free parameters, which can, in principle, be determined by observations. The confirmation orexclusion of the predictions of different gravity theories can then lead to a greater understanding of thefundamental theory of gravity [20]. The latter approach also has the practical advantage that differenttheories of gravity can be constrained at once without having to analyze (potentially large) data setsfor each individual theory of gravity.In this article, I focus on the regime probed by compact objects and by Sgr A ∗ in particular; seeRef. [21] for a recent summary of cosmological tests of gravity. Sgr A ∗ is a prime target for near-future tests of general relativity for several reasons. First, the highly-relativistic environment of Sgr A ∗ provides a laboratory to study some of the most extreme gravitational field strengths in the universe.The gravitational potential of Sgr A ∗ is ∼ ∗ have been accurately measured by NIR observations ofstars on orbits around Sgr A ∗ [22–25] and in the Galactic nuclear star cluster [26–28]. The distanceof Sgr A ∗ has also been inferred from parallax and proper motion measurements of masers throughoutthe Milky Way [29]. These measurements robustly determine a mass of ∼ × M (cid:12) and a distanceof ∼ ∗ .Third, the shadow of Sgr A ∗ has the largest opening angle of any black hole in the sky [30] makingthis source resolvable with mm/sub-mm VLBI observations. In addition, the emission of Sgr A ∗ becomesoptically thin at wavelengths ∼ ∗ . Reference [35] resolved structures of Sgr A ∗ on scales of only 8 r g , where r g ≡ GM/c is the gravitational radius of Sgr A ∗ , and Refs. [36] and [37]detected variability and polarized emission on event horizon scales, respectively. Similar observationsof the supermassive black hole at the center of M87 have also detected structure on the order of ≈ . ∗ in the near future. All of these operate at frequencies within atmospheric absorption windows (see, e.g.,Ref. [40]) and are ground-based. Orbital precessions of stars sufficiently close to Sgr A ∗ [41] may bedetected by continued NIR monitoring with current instruments and with the forthcoming instrumentGRAVITY for the Very Large Telescope (VLT) [42]. GRAVITY is also expected to detect quasi-periodicvariability originating from density inhomogeneities orbiting in the accretion flow of Sgr A ∗ [42]. Future30m-class optical telescopes such as the Thirty Meter Telescope (TMT [43]) and the European ExtremelyLarge Telescope (E-ELT [44]) will possess further improved resolutions and sensitivities. Likewise, radiopulsars are thought to populate the stellar cluster at the Galactic center [45, 46], spurred by the recentdiscovery of a magnetar at a distance of only ∼ . ∗ [47–51]. High-precision timingobservations of such pulsars with exisiting 100m-class radio telescopes or future facilities such as theSquare Kilometre Array (SKA [52]) may detect characteristic residuals in their spectra which dependon the general-relativistic properties of Sgr A ∗ [53–55]. Finally, the Event Horizon Telescope (EHT),a global very-long baseline interferometer comprised by mm and sub-mm telescopes, is expected toperform detailed space- and time-resolved studies of the accretion flow of Sgr A ∗ and to take thefirst-ever image of a black hole [56–58].Each of these tests of general relativity requires an appropriate theoretical framework. Observationsof stars and (for the most part) pulsars probe Sgr A ∗ in the weak-field regime, i.e., at radii r (cid:29) r g . In3 .1110100 A n g u l a r D i a m e t e r o f R i n g ( µ a r c s e c ) Distance (pc) λ =1mm, D=2R Earth λ =0.5mm, D=0.3D Moon
Sgr A* M31 M87 λ =0.4mm, D=300000km Figure 2.
Left: Angular shadow diameters and distances of several supermassive black holes. Theshadow of Sgr A ∗ has the largest angular diameter, closely followed by the one of the supermassiveblack hole at the center of M87 due to its high mass, making these sources ideal targets for the EHT.Taken from Ref. [30]. Right: The long-wavelength spectrum of Sgr A ∗ . The gray filled areas show theenvelopes of emission models fitting the data. Taken from Ref. [31]. this regime, it is sufficient to employ a parameterized post-Newtonian framework within which suitablecorrections to Newtonian gravity in flat space can be calculated [59]. In the strong-field regime, however,i.e., at radii r ∼ r g , which are targeted by observations of the accretion flow of Sgr A ∗ with the EHTand NIR instruments such as GRAVITY, the parameterized post-Newtonian formalism can no longerbe applied. Instead, a careful modeling of the underlying spacetime in terms of a Kerr-like metric(e.g., [60–68]) is required.Strong-field tests of general relativity with black holes have also been proposed using gravitational-wave observations of extreme mass-ratio inspirals (EMRIs) [61–63, 65, 69–80] and of gravitationalringdown radiation of perturbed black holes after a merger with another object [81–83]. See Refs. [84,85]for recent reviews. Likewise, strong-field tests of general relativity have been suggested using otherelectromagnetic observations of accretion flows in terms of their continuum spectra [12, 68, 86–95],relativistically-broadened iron lines [93,94,96–103], variability [101,104–109], X-ray polarization [91,95,110], jets [111], and other accretion properties [12, 13, 112–116]. See Ref. [117] for a review.All three approaches to test general relativity with observations of Sgr A ∗ in the electromagneticspectrum are based on tests of the (general-relativistic) no-hair theorem, which I briefly review in Sec. 2.In Secs. 3 and 4, I discuss tests of general relativity with observations of stars and pulsars around Sgr A ∗ ,respectively. In Sec. 5, I review Kerr-like metrics and some of their properties. Table 2 at the end ofthis section contains a list of the parameters and some of the essential properties of several Kerr-likemetrics. In Sec. 6, I discuss tests of general relativity with EHT observations of Sgr A ∗ . Quasiperiodicvariability of the emission from the accretion flow of Sgr A ∗ may be probed by both NIR and VLBIobservations, which I discuss in Sec. 7. Sec. 8 contains my conclusions.
2. The No-Hair Theorem
According to the general-relativistic no-hair theorem, isolated and stationary black holes are uniquelycharacterized by their masses M , spins J , and electric charges Q and are described by the Kerr-4ewman metric [118], which reduces to the Kerr metric [119] in the case of electrically neutral blackholes. This metric is the unique stationary, axisymmetric, asymptotically flat, vacuum solution of theEinstein field equations which contains an event horizon but no closed timelike curves in the exteriordomain [120–126]. The no-hair theorem relies on the cosmic censorship conjecture [127] as well as onthe physically reasonable assumption that the exterior metric is free of closed timelike curves. SeeRefs. [128, 129] for reviews. If these requirements are met, then all astrophysical black holes should bedescribed by the Kerr metric ‡ .Black holes are commonly believed to be the final states of the evolution of sufficiently massivestars at the end of their lifecycle. The gravitational collapse of such stars leads to the formation ofa black hole [130, 131], where any residual signature of the progenitor other than its mass and spin isradiated away by gravitational radiation [132, 133]. This scenario provides an astrophysical mechanismwith which black holes can be generated. Almost all nearby galaxies harbor dark objects of high massand compactness at their centers [134] including our own galaxy [22, 23] providing strong evidence thatblack holes are realized in nature. In addition, the measurement of the orbital parameters of manyGalactic binaries supports the claim that they contain stellar-mass black holes (e.g., [135]).Despite the large amount of circumstantial evidence, there has been no direct proof, so far, for theexistence of an actual event horizon. An event horizon, one of the most striking predictions of generalrelativity, is a virtual boundary that causally disconnects the interior of a black hole from the exterioruniverse. The presence of an event horizon in black-hole candidates has only been inferred indirectlyfrom either the lack of observations of Type I X-ray bursts [136–139] (see the discussion in Ref. [140])or, in the case of the supermassives black holes at the centers of the Milky Way and the galaxy M87, thefact that these supermassive compact objects are greatly underluminous [141, 142]. These observationsindicate the absence of a hard (stellar) surface which most likely identifies the compact objects as blackholes. In addition, the recent first direct observation of gravitational waves detected a waveform thatis consistent with the inspiral of two black holes with masses of ∼ M (cid:12) and ∼ M (cid:12) [18, 19] (but seeRef. [143]).Astrophysical black holes, however, will not be perfectly stationary nor exist in perfect vacuumbecause of the presence of other objects or fields such as stars, accretion disks, or dark matter, whichcould alter the Kerr nature of the black hole (see, e.g., Ref. [144]). Nonetheless, under the assumptionthat such perturbations are so small to be practically unobservable, one can argue that astrophysicalblack holes are indeed described by the Kerr metric. This is the assumption I make throughout thisarticle.In Boyer-Lindquist coordinates, the Kerr metric g K µν is given by the expressions (setting G = c = 1) g K tt = − (cid:18) − M r Σ (cid:19) ,g K tφ = − M ar sin θ Σ ,g K rr = Σ∆ ,g K θθ = Σ ,g K φφ = (cid:18) r + a + 2 M a r sin θ Σ (cid:19) sin θ, (3)where ∆ ≡ r − M r + a , (4) ‡ Note that astrophysical black holes are thought to be essentially electrically neutral, because any residual electric chargewould quickly neutralize. Also note that the mathematical status of the no-hair theorem is not without controversy,principally in relation to the assumption (in the classical proof) of analyticity; see Sec. 3.4 in Ref. [128] for a discussion. ≡ r + a cos θ (5)and where a ≡ J/M is the spin parameter. For values of the spin | a | ≤ M, (6)the event horizon of the black hole is located at the radius r K+ = M + (cid:112) M − a . (7)The alternative hypothesis that these dark compact objects are not described by the Kerr metricbut perhaps by a solution of the Einstein equations with a naked singularity (e.g., [60]) and, therefore,violate the no-hair theorem is still possible within general relativity. Alternatively, these dark objectsmight be stable stellar configurations consisting of exotic fields (e.g., boson stars [145], gravastars [146],black stars [147]) or black holes surrounded by a stable scalar field [148] (c.f., Refs. [149, 150]).Finally, the fundamental theory of gravity may be different from general relativity in the strong-field regime, and the vacuum black-hole solution might not be described by the Kerr metric atall. In fact, black hole solutions in several theories of gravity other than general relativity havealready been found including exact solutions of rotating black holes in Randall-Sundrum-type (RS2)braneworld gravity [151] and modified gravity (MOG) [152] as well as perturbative (i.e., black holesolutions valid for small deviations from the Kerr metric) and numerical solutions of rotating blackholes in Einstein-dilaton-Gauss-Bonnet gravity (EdGB; [153–159]; see Ref. [160] for an asymptoticexpansion of a nonperturbative solution), dynamical Chern-Simons gravity (dCS; [157, 161, 162]),massive gravity [163, 164], Einstein-Æther gravity [165–167], Ho˘rava-Lifshitz gravity [165, 166], as wellas Horndeski gravity [168]. See Refs. [169–172] for reviews on this topic.As a result, testing the no-hair theorem allows us not only to verify the identification of darkcompact objects in the universe with Kerr black holes but to test the strong-field predictions of generalrelativity, as well. Unfortunately, such tests are slightly complicated by the fact that the Kerr metric isnot unique to general relativity but also the most general black hole solution in a large class of scalar-tensor theories of gravity [173–175] (a similar property also holds for the Kerr-Newman metric [176]).Linear systems in flat space are often best described by a set of multipole moments. In theoriesof gravity like general relativity, however, space is curved due to the presence of stress-energy, and theresulting field equations are highly non-linear. It is, therefore, not immediately obvious that such aspacetime can actually be characterized by a set of multipole moments.In Newtonian gravity, the potential Φ satisfies the Laplace equation ∇ Φ = (cid:26) πGρ (interior)0 (exterior) , (8)where ρ is the mass density. Therefore, the potential Φ can always be expanded in spherical harmonics Y lm as Φ = − G ∞ (cid:88) l =0 π l + 1 l (cid:88) m = − l M lm Y lm r l +1 (9)with mass multipole moments M lm = (cid:90) r r (cid:48) l +2 dr (cid:48) (cid:73) d Ω (cid:48) Y ∗ lm (Ω (cid:48) ) ρ ( r (cid:48) , Ω (cid:48) ) . (10)In the curved space of general relativity, however, the vacuum Einstein equations R µν − g µν R = 0 (11)have to be solved for the spacetime metric g µν with the corresponding Ricci tensor R µν and Ricciscalar R . The Einstein equations are nonlinear and, therefore, cannot always be solved in terms of an6xpansion over orthonormal polynomials. Nonetheless, it can be shown that a multipole expansion ofcurved spacetime does indeed exist in certain cases (see Ref. [177] for a review).For an asymptotically flat vacuum solution of the Einstein equations, (tensor) multipole momentscan be defined based on a conformal compactification of 3-space, if the spacetime is also static [178] or,more generally, if it is stationary [179]. In both cases, such a set of multipole moments characterizesthe spacetime uniquely [180, 181] (see, also, Ref. [182] and references therein) and obeys an appropriateconvergence condition [183]. If the spacetime is also axisymmetric, the multipole moments are given bya bi-infinite series of scalars M l and S l , which are interpreted as mass and current multipole moments,respectively. The mass multipole moments are analogous to the multipole moments in Newtoniangravity given by expression (10) and are nonzero only for even l . The current multipole moments arenonzero only for odd l and arise from the fact that, in general relativity, all forms of stress-energygravitate [179]. Stationary axisymmetric vacuum solutions of the Einstein equations can likewise begenerated from a given set of multipole moments [184, 185].In general relativity, black-hole spacetimes are asymptotically flat vacuum solutions of the Einsteinequations. If these spacetimes are stationary, they must also be axisymmetric [123] and can, therefore,be described by a sequence of (Geroch-Hansen) multipole moments { M l , S l } . As a consequence of theno-hair theorem, the Kerr spacetime is the unique such black-hole solution within general relativity, andall multipole moments of order l ≥ M = M and the spin J = S . This fact can be expressed mathematically with the relation [178, 179] M l + i S l = M (i a ) l . (12)In the astrophysical context, the fact that the no-hair theorem requires the multipole moments ofa stationary black hole to be locked by expression (12) allows for it to be tested quantitatively usingobservations of such black holes. Since the first two multipole moments (i.e., the mass and spin) alreadyspecify the entire spacetime, a promising strategy for testing the no-hair theorem, then, is to measure(at least) three multipole moments of the spacetime of a black hole [69].
3. NIR Monitoring of Stellar Orbits
Observations of the Galactic center region over several decades have established the fact that its massdistribution can be well described by a compact supermassive object at the center, Sgr A ∗ , surroundedby a dense nuclear star cluster with an extent of several parsecs. The most common explanation of thehigh mass and compactness of Sgr A ∗ is that this object is indeed a black hole. See Refs. [186–192] forreviews.To date, numerous observations have led to precise measurements of the mass M and distance R of Sgr A ∗ . References [22–25] and Refs. [26–28] inferred the mass and distance of Sgr A ∗ from NIR monitoring stars on orbits around Sgr A ∗ , the so-called S-stars, and in the old Galacticnuclear star cluster, respectively. References [22, 25] obtained the measurements M = (4 . ± . × M (cid:12) , R = 7 . ± . M = (4 . ± . × M (cid:12) , R = 8 . ± .
11 kpc [28]. Figure 3 shows the orbits of 20 S-stars aroundSgr A ∗ including the orbit of the star S2 which has been observed over the full span of its orbitalperiod. The S-star orbits are consistent with Keplerian ellipses within the remaining uncertainties inthe NIR coordinate system (c.f., Ref. [194]). In addition, the distance of Sgr A ∗ has been obtainedfrom parallax and proper motion measurements of masers throughout the Milky Way by Ref. [29]finding R = 8 . ± .
16 kpc. Although the stellar-orbit measurements presently disagree at an almoststatistically-significant level, primarily due to the uncertainties in the NIR coordinate system, the abovemeasurements robustly determine a mass of ∼ × M (cid:12) and a distance of ∼ ∗ .The constraints on the mass and distance from the observations of stellar orbits will be improved inthe near future by continued monitoring and by the use of the second-generation instrument GRAVITY7 - - - - H " L D ec H " L (cid:45) (cid:45) (cid:45) (cid:72) '' (cid:76) D e c (cid:72) '' (cid:76) (cid:45) (cid:45) (cid:72) yr (cid:76) v r ad (cid:72) k m (cid:144) s (cid:76) Figure 3.
The left panel shows the orbits of 20 S-stars around Sgr A ∗ obtained from fits of NIRobservations in 1992–2008 (taken from Ref. [23]). The center and right panels show the orbit ofthe star S2 around Sgr A ∗ on the sky (black circle) and in radial velocity, respectively (taken fromRefs. [193]). Blue, filled circles denote the observations of Refs. [23, 24] (updated to 2012) with theNew Technology Telescope (NTT) and the VLT, red circles denote the Keck observations of Ref. [22],and grey crosses are the positions of infrared flares (see Sec. 7). The orbit of S2 is not a closed ellipsedue to the proper motion of Sgr A ∗ , which, however, is consistent with the uncertainties of the NIRreference frame. for the VLT, which is expected to achieve astrometry with a precision of ∼ µ as and imaging with a ∼ ∗ are likely to be determined with a precision of ∼ .
1% [195].The monitoring of stellar orbits may also measure the spin and, perhaps, even the quadrupolemoment of Sgr A ∗ , thus testing the no-hair theorem via the relation in Eq. (12). A star that issufficiently close Sgr A ∗ experiences an acceleration a star which is given by the equation (see, e.g.,Ref. [59]) a star = − M x r + M x r (cid:18) Mr − v (cid:19) + 4 M ˙ rr v − Jr (cid:34) v × ˆJ − r n × ˆJ − n ( h · ˆJ ) r (cid:35) + 32 Qr (cid:104) n ( n · ˆJ ) − n · ˆJ ) ˆJ − n (cid:105) , (13)where x and v are the position and velocity of the star, n = x /r , ˙ r = n · v , h = x × v , ˆJ = J / | J | , and J and Q are the angular momentum vector and the quadrupole moment of Sgr A ∗ , respectively. Thefirst three terms of the acceleration in Eq. (13) correspond to the Schwarzschild part of the metric atfirst post-Newtonian order, the next term is the frame-dragging effect induced by the spin of Sgr A ∗ ,and the final term is the effect of the quadrupole moment at Newtonian order. There are additionalquadrupolar corrections to the acceleration in Eq. (13) at first post-Newtonian order, but these will bemuch smaller, because they have a stronger dependence on the distance of the star from Sgr A ∗ .The corrections to the Newtonian gravitational potential of Sgr A ∗ cause the orbit of the star toprecess. The Schwarzschild-type corrections in Eq. (13) lead to a precession in the orbital plane of thestar, while the corrections induced by the spin and quadrupole moment of Sgr A ∗ cause the orbit to8 Fig. 1.—
Schematic diagram for the star-MBH system and the coordinate systems. Right panel: a pseudo-Cartesian coordinate ( x ′ , y ′ , z ′ )is defined in the rest frame of a distant observer (i.e., the observer’s frame), where the x ′ y ′ plane represents the sky plane of the observerand is taken as the reference plane, the z ′ axis represents the line of sight, and the x ′ axis is taken as the reference direction on the sky plane.Another pseudo-Cartesian coordinate xyz is defined to relate the Boyer-Lindquist coordinates ( r, θ, φ ) to orthogonal coordinates ( x, y, z ),where the z axis represents the spin direction of the MBH and the y axis represents the intersection line of the MBH equatorial plane withthe observer’s sky plane. The direction of the MBH spin in the observer’s frame is therefore described by two angles, i.e., the angle the z axis and the z ′ axis ( i ) and the angle between the projection vector of the x axis on the x ′ y ′ plane and the x ′ axis ( ǫ ). The directions ofthe R.A. ( − ~y ′ ) and Dec. ( − ~x ′ ) are marked in this figure. Left panel: the orbit of the star may be approximated as a Newtonian ellipse inthe observer’s frame, especially when the semimajor axis of the star is large, which is described by six orbital elements, i.e., the semimajoraxis a orb , eccentricity e , the longitude of ascending node Ω ′ , argument of periapsis Υ ′ , true anomaly υ ′ and inclination I ′ , respectively. In the Newtonian case, the orbital motion of a starrotating around a massive object is determined by sixorbital elements, i.e., the semimajor axis a orb , the eccen-tricity e orb , the longitude of the ascending node Ω ′ , theargument of the pericenter Υ ′ , the true anomaly υ ′ , andthe orbital inclination angle I ′ (see Figure 1). For con-venience in comparison with the Newtonian orbits cur-rently determined for a few GC S-stars with the smallestsemimajor axis, in our following simulations of the or-bital motions of some (example) stars, we set their initialconditions by fixing the six orbital elements at a givenmoment in the distant observer’s frame. Then, the val-ues for the four-position r ⋆, and the tetrad-velocity u ⋆, of a star can be approximately obtained by the followingprocedures.1. The initial position ( x ′ ⋆, , y ′ ⋆, , z ′ ⋆, ) and the threevelocity ( v ′ x⋆, , v ′ y⋆, , v ′ z⋆, ) of the star in the distantobserver’s frame are first obtained from the six or-bital elements initially set.2. Transforming the position and the velocity of thestar in the distant observer’s frame into those inthe LNRF frame, i.e., ~r ⋆, = ( x ⋆, , y ⋆, , z ⋆, ) and ~v ⋆, = ( v x⋆ , v y⋆, , v z⋆, ), by the rotation given inEquation (18), and then transforming these twovectors from that using the xyz coordinates intothat using the rθφ coordinates.3. Transforming ~r ⋆, and ~v ⋆, to the four-position r ⋆, and the tetrad-velocity u ⋆, in the Boyer-Lindquistcoordinates according to Equations (9)-(11). The orbital elements set here are used in a simple way to gen-erate the initial conditions of a star moving in the Boyer-Lindquistcoordinates. The elliptic orbit defined by these orbital elementsis only taken as a Newtonian approximation to the real orbit ofa star in the Boyer-Lindquist coordinates. These orbital elementsmay lose their original meaning in the curved spacetime around aKerr MBH.
Once the initial conditions are given for a star, the mo-tion parameters λ , ξ , and q can be obtained by Equations(12), (13), and (16), respectively. With those constantsof motion, the orbital motion of a star, in principle, canbe obtained by integrating the motion Equations (3)-(6).However, it is required to frequently judge the sign “ ± ”at the left side of the motion Equations (3) and (4) in thenumerical integrations. To avoid this complication, onemay use the following equations to replace the motionEquations (3) and (4). Σ ˙ r = ∆ p r , (19)Σ ˙ θ = p θ , (20)Σ ˙ p r = 12∆ ∂R∂r − R ∂ Θ ∂θ , (21)Σ ˙ p θ = 12 ∂ Θ ∂θ . (22)We use the code DORPI5 based on the explicit fifth(fourth)-order Runge Kutta method (Dormand & Prince1980; Hairer et al. 1993) to integrate the motion Equa-tions (19)-(22) and (5) and (6) to obtain the orbit motionof a star. We set the relative integration errors to be ≤ − for all the position and momentum quantities( r, θ, φ, p r , p θ , p φ ) in those equations, which are suffi-cient for the convergence of the numerical results andthe required accuracy in this study. Orbits of Example Stars
Observations in the past two decades have revealed anumber of GC S-stars rotating around the MBH withsemimajor axes in the range of ∼ −50050 R.A. (mas) D e c ( m a s ) S0-2/S2 −50050100
R.A. (mas) −200−1000 D e c ( m a s ) S0-102 −20−1001020
R.A. (mas) −500 D e c ( m a s ) Ea−15−10−505
R.A. (mas) −500 D e c ( m a s ) Eb −505
R.A. (mas) −20−100 D e c ( m a s ) Ec −10−50510
R.A. (mas) −10010 D e c ( m a s ) Ed Fig. 3.—
Evolution of the apparent position of each example star on the observer’s sky plane. The apparent position of an examplestar is described by the R.A. and Dec. Panels from left to right, top to bottom, represent the example stars S0-2/S2, S0-102, Ea, Eb, Ec,and Ed, respectively. The initial settings of the orbital parameters for these stars are listed in Table 1. Three full orbits are shown foreach example star. Open and solid circles mark the locations of the apoapsides and periapsides, respectively. In each panel, the dashedand dotted lines represent the eccentric vectors obtained by approximating the star orbit to a Newtonian ellipse for the first and the thirdorbit, respectively. As seen from the figure, the apparent orbital precession is most significant ( ∼ . ure 5 is obtained in a way slightly different from the δ apo estimated from Equation (31). The values of δ are cal-culated for the projected distances at any given time be-tween the star rotating around a rapidly spinning MBH( a = 0 .
99) and a star with the same initial orbital ele-ments but rotating around a non-spinning MBH ( a = 0);while the value of δ apo is estimated by considering thedifference between the position differences of two adja-cent apoapsides of a star rotating around an MBH with a = 0 .
99 and those of a star with the same initial orbitalelements rotating around a non-spinning MBH. As theorbital periods of these two stars are slightly different,the shift of δ at two adjacent apoapsides shown in Fig-ure 5 should therefore be roughly the same as, but differ-ent from, the analytical estimates δ apo , and the differenceis most significant for those with small eccentricity.For a consistency check between the numerical resultsobtained in this study and the analytical ones, as an ex-ample, we show the apoapsis shifts numerically estimatedin two different ways for those stars with the same semi-major axis of 300 AU (or 80 AU) but various eccentricitiesafter one full orbit motion in Figure 6. The other initialorbital elements of those stars are the same as those ofEa (or Ec). The top panel of Figure 6 shows the numer-ical results obtained in the same way as δ apo , which arewell consistent with the analytical estimates at e < ∼ . e > ∼ .
9. Thereare three reasons for the difference at high eccentrici- ties: (1) the Newtonian approximation to the orbit of astar for the initial conditions becomes inaccurate if thestar is close to the MBH (e.g., for those orbits with ex-tremely high eccentricities); (2) for an Ea-like star with e orb higher than 0 .
98, its orbital precession can be largerthan ∼ ◦ per orbit, therefore, the first order approx-imation (for the projection) to obtain Equation (31) isinaccurate; (3) the contributions from the higher orderprecessions are ignored in the analytical estimates. Asshowed in Figure 6, the combinations of the above threefactors can almost lead to 20% to a factor of two dif-ference when e orb > ∼ .
98. The bottom panel of Figure 6shows the numerical results of δ obtained in the same wayas those shown in Figure 5, which apparently differ signif-icantly from the analytical estimates at both e orb < ∼ . e orb > ∼ . δ at apoapsis and δ apo are definedand obtained in a slightly different way.According to Figure 5, we conclude that, if the ac-curacy in determining the apparent positions of any ofthose example stars (except S0-102) on the sky planecan reach < ∼ µ as, the spin of the GC MBH can bethen constrained by fitting the evolution of its positionsover several orbits as demonstrated by a Bayesian fittingmethod in Section 6 (see also Tab. 2).For a star with given initial orbital elements, the dis-placement of apoapsis after one full orbit due to the spineffects is proportional to the absolute value of the spin,and it also depends on the spin direction (see eqs. B5-B8). Figure 4.
The left panel shows the geometry of a star on an orbit around Sgr A ∗ , which is describedby six orbital elements, the semi-major axis ˜ a (not shown), eccentricity e (not shown), longitude ofascending node Ω (cid:48) , longitude of pericenter Υ (cid:48) , true anomaly v (cid:48) , and inclination I (cid:48) . The right panelshows the evolution of the apparent position in the sky (in terms of the right ascension “R.A.” anddeclination “Dec”) of a hypothetical star over three full orbits around Sgr A ∗ (located at the origin)with a semi-major axis of 10 mas, an orbital eccentricity of 0 .
88, and an inclination of 45 ◦ . The stellartrajectory includes the effects of the Schwarzschild corrections to the Newtonian potential of Sgr A ∗ aswell as of frame dragging (assuming a spin value χ = 0 . precess both in and out of the orbital plane of the star § . Using standard orbital perturbation theory,Ref. [41] (see, also, Refs. [198–201]) calculated the precessions per orbit of the pericenter angle Υ, nodalangle Ω, and inclination i (see the left panel of Fig. 4, identifying Υ = Υ (cid:48) , Ω = Ω (cid:48) , and i = I (cid:48) ) whichare given by the expressions∆Υ = A S − A J cos α − A Q (1 − α ) , (14)sin i ∆Ω = sin α sin β ( A J − A Q cos α ) , (15)∆ i = sin α cos β ( A J − A Q cos α ) . (16)Here [41, 202] A S = 6 πc GM (1 − e )˜ a ≈ . (cid:48) (1 − e ) − (cid:18) M × M (cid:12) (cid:19) (cid:18) ˜ a mpc (cid:19) − , (17) A J = 4 πχc (cid:20) GM (1 − e )˜ a (cid:21) / ≈ . (cid:48) (1 − e ) − / χ (cid:18) M × M (cid:12) (cid:19) / (cid:18) ˜ a mpc (cid:19) − / , (18) A Q = 3 πχ c (cid:20) GM (1 − e )˜ a (cid:21) ≈ . (cid:48) × − (1 − e ) − χ (cid:18) M × M (cid:12) (cid:19) (cid:18) ˜ a mpc (cid:19) − , (19)where χ ≡ cJ/GM (20)is the dimensionless spin of Sgr A ∗ , α and β are the polar angles of the angular momentum vector J with respect to the orbital plane of the star, and e and ˜ a are the eccentricity and the semi-major axis ofthe orbit, respectively. In Eq. (19), the quadrupole moment of Sgr A ∗ is assumed to be the quadrupolemoment of a Kerr black hole, − χ G M /c [c.f., Eq. (12)]. Note that the expressions A S , A J , and A Q § The spin-induced precession of the stellar orbit is commonly referred to as Lense-Thirring precession [197], c.f., Eqs. (37)–(39). A J and A Q are proportional to χ and χ , respectively. Therefore, orbital precessions are easier to detect for stars on highly eccentricorbits that approach Sgr A ∗ closely and for high values of the spin. The right panel of Fig. 4 illustratesthe apparent position of a hypothetical star orbiting around Sgr A ∗ with a semi-major axis of 10 mas,an orbital eccentricity of 0 .
88, and an inclination of 45 ◦ . The simulated stellar orbit is affected by theSchwarzschild and spin-induced corrections to the Newtonian potential of Sgr A ∗ (assuming a value ofthe spin χ = 0 .
99) which cause the orbit to precess. -20 -16 -12 -8 -4 a cc e l e r a t i o n ( c m s - ) Distance from the Black Hole (rc /GM BH ) a N a J a Q a dw a d Figure 5.
The left panel (taken from Ref. [202]) shows the characteristic timescales of the precessionof orbital planes about Sgr A ∗ due to frame-dragging ( t J ), quadrupolar torque ( t Q ), and Newtonianperturbations from nearby 1 M (cid:12) stars ( t N ) assuming that Sgr A ∗ is maximally spinning. For thetimescales t J and t Q , the line thickness corresponds to the orbital eccentricity ranging from e = 0 . e = 0 . e = 0 . t N , the line thickness denotes the totaldistributed mass of perturbing stars within 1 mpc from Sgr A ∗ ranging from 10 M (cid:12) (thickest) to1 M (cid:12) (thinnest), assuming that their density falls off as r − . The shaded (green) region correspondsto a time interval of observations over 1–10 years. The right panel (taken from Ref. [203]) shows thecorresponding accelerations of a 10 M (cid:12) star with a 10 R (cid:12) radius including the accelerations due tohydrodynamic drag ( a d ; red line) and the gravitational interaction of the star with its wake ( a dw ;green line) assuming a spin of χ = 0 . ∗ . Although the orbits of S-stars are predominently affected by the gravitational potential of theGalactic center, they may also be perturbed by other stars or astrophysical effects. In order to assessthe magnitude of these effects, it is instructive to compare the timescales on which they contribute tothe acceleration of the star. Ref. [202] defined characteristic timescales corresponding to the precessionof the orbital plane of the star, which are given by the equations t S ≡ (cid:20) A S πP (cid:21) − = P ˜ ac GM (1 − e ) ≈ . × (cid:0) − e (cid:1) (cid:18) M × M (cid:12) (cid:19) − / (cid:18) ˜ a mpc (cid:19) / yr , (21) t J = P χ (cid:20) c ˜ a (1 − e ) GM (cid:21) / ≈ . × (cid:0) − e (cid:1) / χ − (cid:18) M × M (cid:12) (cid:19) − (cid:18) ˜ a mpc (cid:19) yr , (22)10 Q = P χ (cid:20) c ˜ a (1 − e ) GM (cid:21) ≈ . × (cid:0) − e (cid:1) χ − (cid:18) M × M (cid:12) (cid:19) − / (cid:18) ˜ a mpc (cid:19) / yr , (23)where P = 2 π ˜ a / √ GM ≈ . (cid:18) M × M (cid:12) (cid:19) − / (cid:18) ˜ a mpc (cid:19) / yr (24)is the orbital period.The left panel of Fig. 5 shows these timescales together with the characteristic timescale ofNewtonian perturbations of nearby stars for different eccentricities of the stellar orbit assuming that thespin of Sgr A ∗ is maximal. The right panel of Fig. 5 shows the corresponding accelerations experiencedby a star with a mass of 10 M (cid:12) and a radius of 10 R (cid:12) of an orbit around Sgr A ∗ assuming a black-hole spin of χ = 0 .
1, together with the accelerations of the star due to hydrodynamic drag and thegravitational interaction of the star with its wake. The latter two effects are much weaker than theeffect of the quadrupole moment out to ∼ Schwarzschild radii and, therefore, can be neglected forstars sufficiently close to Sgr A ∗ [203]. t i m e s c a l e ( s ) Semi-Major Axis (ac /GM BH ) 10 -2 t i m e s c a l e ( y r ) t S t J t Q
10 M t w,-7 χ =0.3 Period t i m e s c a l e ( s ) Semi-Major Axis (ac /GM BH ) 10 -2 t i m e s c a l e ( y r ) t S t J t Q M S =10 M R S = 6 R χ BH =0.3 e=0.5 e=0.8disruption Figure 6.
Characteristic timescales similar to the ones shown in the left panel of Fig. 5 but assuminga black-hole spin χ = 0 . e = 0 . e = 0 . M (cid:12) stardue to the presence of a stellar wind at a mass loss rate of 10 − M (cid:12) yr − . The red lines in the rightpanel show the characteristic timescale for orbital evolution due to the tidal dissipation of the orbitalenergy. For stars sufficiently close to Sgr A ∗ , the effect of stellar winds is negligible, while the tidaldissipation of their orbital energies occurs at timescales comparable to the timescale of precession dueto the quadrupole moment of Sgr A ∗ . Taken from Ref. [204]. Figure 6 shows the characteristic timescales for the orbital evolution of a 10 M (cid:12) star including theeffects of its stellar wind and the tidal dissipation of its orbital energy. For stars sufficiently close toSgr A ∗ , the former effect is negligible, while the latter effect occurs at timescales comparable to thetimescale of orbital-plane precession induced by the quadrupole moment of Sgr A ∗ . Thus, the tidaldissipation of stars on orbits close to Sgr A ∗ are a potential source of systematic uncertainty for testsof the no-hair theorem with the observations of such stars [204].Reference [202] performed extensive N-body simulations of a populations of stars orbiting aroundSgr A ∗ and assessed their impact on the precessions of the orbital plane of a star caused by the spinand quadrupole moment of Sgr A ∗ . Reference [202] showed that the effect of the quadrupole moment11 igure 7. Simulated 10 yr evolution of the orbital angular momenta of stars around Sgr A ∗ dueto frame-dragging (dashed lines) and stellar perturbations (dotted lines) as measured by (top panels)the nodel angle ∆Ω and (bottom panels) the angle ∆ θ between the inital and final orbital angularmomentum as a function of the orbital semi-major axis a . The three panels correspond to a Kerr blackhole with a dimensionless spin (a) χ = 1, (b) χ = 0 .
1, and (c) χ = 0. The blue and red dots correspondto 30 M (cid:12) stars with initial orbital eccentricities 0 ≤ e ≤ . . < e ≤
1, respectively. In the lowerpanels, the dashed and dotted lines correspond to the precessions induced by frame-dragging (setting e = 2 /
3) and stellar perturbations, respectively. Taken from Ref. [202]. on the orbit of such a star is masked by the effect of the spin for the group of stars known to orbitSgr A ∗ . However, if a star can be detected within ∼ ∗ and if it canbe monitored over a sufficiently long period of time, this technique may also measure the spin and eventhe quadrupole moment of Sgr A ∗ (see Fig. 7). Reference [205] found similar results for the effect ofsuch perturbations using methods of orbital perturbation theory. References [206, 207] derived “cross-term” expressions to be added to the right-hand-side of Eq. (13) describing the coupling between thepotential of the central black hole and the potential due to other stars at the first post-Newtonian order,which may need to be incorporated in long-term N-body simulations of stars orbiting around Sgr A ∗ .Reference [208] calculated the perturbing effects of a distribution of dark matter around Sgr A ∗ anddemonstrated that its effect is small compared to the effects of the spin and quadrupole moment ofSgr A ∗ . References [209, 210] analyzed tidal disruption events of S-stars in N-body simulations findinga (cid:46)
1% probability for such an event to take place over the full lifetime of an S-star.Figure 8 shows the range of the orbital parameters of S-stars for which the stars follow nearlytest-particle orbits. This region is bounded by two curves along which the timescale of orbital-planeprecession due to the quadrupole moment of the black hole ( t Q ) is equal to the orbital evolution timescaledue to stellar winds and due to tidal dissipation, respectively. Inside the blue shaded region, GRAVITYcan detect the effect of frame-dragging assuming a signal-to-noise ratio of five and a range of astrometricaccuracies between 10 − µ as [204]. Figure 8 also shows the fractional contribution to the mass,spin, and quadrupole moment of Sgr A ∗ inside the orbit of a star due to the enclosed distribution ofother stars and objects as estimated by Ref. [211]. These fractional contributions represent the limitingaccuracies to which the corresponding properties of Sgr A ∗ can be inferred using observations of orbitsof stars (and pulsars; see Sec. 4).In addition to the precession of the orbit of the star, photons emitted from the star may alsobe Doppler-shifted and gravitationally lensed by Sgr A ∗ and experience a corresponding time delay.12 S e m i - M a j o r A x i s ( a c / G M B H ) P e r i o d ( y r ) S2 Tidal Disruption
S0-104 t Q =t d t Q =t w,-7 GRAVITY
S14 -5 -4 -3 -2 -1 F r a c t i o n a l S t e ll a r C o n t r i b u t i o n Semi-Major Axis (ac /GM BH ) Period (yr)
Mass Angular Momentum Quadrupole
Figure 8.
The left panel (taken from Ref. [204]) shows the range of the orbital parameters of stars onorbits around Sgr A ∗ for which the stars follow nearly test-particle trajectories, bounded by the twoblue curves showing the loci of orbital parameters at which the timescale of orbital-plane precessiondue to the quadrupole moment of the black hole ( t Q ) is equal to the orbital evolution timescale due tostellar winds ( t w, − ) and due to tidal dissipation ( t d ), respectively. The blue shaded area shows therange of orbital parameters for which frame dragging will be detectable with GRAVITY at a signal-to-noise ratio of five, assuming a range of astrometric accuracies between 10 − µ as. In the red shadedarea, such stars are tidally disrupted at pericenter. All curves are for a black-hole spin of χ = 0 . M (cid:12) star. The three filled circles show the orbital parameters of the three stars nearest to Sgr A ∗ that are presently known. The right panel (taken from Ref. [211]) shows the fractional contributionto the mass, spin, and quadrupole moment of Sgr A ∗ inside the orbit of a star due to the encloseddistribution of other stars and objects. These fractional contributions represent the limiting accuraciesto which the corresponding properties of Sgr A ∗ can be inferred using observations of orbits of stars.The solid and dahsed lines correspond to stellar distributions with a density profile n ∝ ˜ a − and n ∝ ˜ a − / , respectively. R ed s h i ft ( k m s − ) Arrival Time (days)Classical fit Time dilation fit R ed s h i ft ( k m s − ) Arrival Time (days) Space curvature fit R ed s h i ft ( k m s − ) Arrival Time (days)
Figure 9.
Simulated radial velocity data for a full orbit of an S2-like star with a semi-majoraxis 100 times shorter than the one of S2. The three panels show data fits which include differentgravitational effects: Keplerian orbit only (left panel), gravitational time delay included (center panel),Schwarzschild spacetime curvature included (right panel). Taken from Ref. [212]. ∗ , and the Schwarzschild spacetime curvature [c.f., Eq. (13)].The relativistic effects, including those induced by the spin and the quadrupole moment of Sgr A ∗ ,are strongest near the pericenter of the stellar orbit (see Refs. [198, 199, 213–216]). In the next fewyears, already existing instruments (e.g., the spectrograph SINFONI at the VLT [217, 218]) will likelydetect at least the redshift corrections due to the special-relativistic Doppler effect and the gravitationalredshift [199] and the pericenter precession due to the Schwarzschild term in Eq. (13) [198] for the starS2, in particular during its next pericenter passage in 2018 [22–25]. Reference [219] computed the timevariations of S-stars averaged over one orbit up to the quadrupole order as well as in the presence ofdark matter. i ( ◦ ) (cid:2) ( ◦ ) −303 ǫ M • ( M ⊙ ) a −303 ǫ R G C ( p c )
50 100 150 i ( ◦ ) (cid:2) ( ◦ ) −3 0 3 ǫM • (10 M ⊙ ) P P P P −3 0 3 ǫR GC (pc) P i ( ◦ ) (cid:2) ( ◦ ) −15015 ǫ M • ( M ⊙ ) a −606 ǫ R G C ( p c )
42 45 48 i ( ◦ )
177 180 183 (cid:2) ( ◦ ) −20−10 0 10 20 ǫM • (10 M ⊙ ) P P P P −6 0 6 ǫR GC (pc) P Figure 10.
Simulated probability densities of the mass δM (relative to a mass M = 4 × M (cid:12) ),distance δR (relative to a distance R = 8 kpc), and spin of Sgr A ∗ , as well as of the orientation angles i and ε of the spin axis with respect to the plane of the sky and the line of sight, respectively, for (leftpanel) the star S2 and (right panel) a hypothetical star with a semi-major axis of 37 . .
98, and an inclination of 45 ◦ . The simulations consist of a set of 120 observationsfor each star over two to three full orbits ( ∼
45 yrs) with astrometric and radial velocity precisions of10 µ as and 1 km / s, respectively, and with a cadence ∝ r − . . Sgr A ∗ is assumed to have a spin value χ = 0 .
99 in each case. The uncertainties of the mass and distance are slightly lower for the star S2,while the uncertainty of the spin is slightly lower for the hypothetical star on a highly eccentric orbit.Taken from Ref. [196].
Reference [196] calculated the positions of stars orbiting around Sgr A ∗ using a ray-tracingalgorithm which includes all (general-)relativistic effects. Reference [196] simulated the precision withwhich the mass, distance, and spin of Sgr A ∗ as well as the orientation of the orbits of such stars canbe determined assuming a set of 120 observations for each star over two to three full orbits ( ∼
45 yrs)with astrometric and radial velocity precisions of 10 µ as and 1 km / s, respectively, and with a cadence ∝ r − . . Figure 10 shows two triangle plots of the probability densities of the mass δM (relative to14 mass M = 4 × M (cid:12) ), distance δR (relative to a distance R = 8 kpc), and spin of Sgr A ∗ ,as well as of the orientation angles i and ε of the spin axis with respect to the plane of the sky andthe line of sight, respectively. These triangle plots correspond to the star S2 and a hypothetical starwith a semi-major axis of 37 . .
98, and an inclination of 45 ◦ for ablack-hole spin χ = 0 .
99. In this simulation, Ref. [196] obtain the values χ = 0 . +0 . − . , i = 37 ◦ +39 ◦ − ◦ , ε = 156 ◦ +74 ◦ − ◦ , δR = − . +0 . − . pc, and δM = − . +1 . − . × M (cid:12) for S2 and χ = 0 . +0 . − . , i = 45 ◦ +1 ◦ − ◦ , ε = 181 ◦ +1 ◦ − ◦ , δR = − . +2 . − . pc, and δM = − . +3 . − . × M (cid:12) for the hypotheicalstar quoting 2 σ uncertainties. In this scenario, the mass and distance of Sgr A ∗ can be measured veryprecisely with observations of either star. The precision of the spin measurement depends significantlyon the eccentricity of the stellar orbit. Interestingly, the uncertainties of the mass and distance areslightly lower for the star S2 (eccentricity e = 0 .
88 [22–25]), while the uncertainty of the spin is slightlylower for the hypothetical star on a highly eccentric orbit. B l a c k H o l e Q u a d r u p o l e M o m e n t σ θ =10 µ as B l a c k H o l e Q u a d r u p o l e M o m e n t σ θ =100 µ as Figure 11.
Posterior likelihood of measuring the spin and quadrupole moment of Sgr A ∗ by tracing N = 40 orbits of two stars with GRAVITY, assuming an astrometric precision of (left panel) 10 µ asand (right panel) 100 µ as. The dashed curves show the 68% and 95% confidence limits, while thesolid curve shows the expected relation between these two quantities in the Kerr metric. The filledcircle marks the assumed spin and quadrupole moment of a Kerr black hole with a value of the spin a = 0 . r g . The two stars are assumed to have orbital separations equal to 800 r g and 1000 r g andeccentricities of 0.9 and 0.8, respectively. Even at these relatively small orbital separations, tracingthe orbits of stars primarily measures the spin of the black hole, unless a very high level of astrometricprecision is achieved. Taken from Ref. [211]. Reference [211] estimated the precision with which the spin and quadrupole moment of Sgr A ∗ canbe measured with GRAVITY observations of the nodal and apsidal precessions of two stars with semi-major axes of 800 r g and 1000 r g and eccentricities of 0.9 and 0.8, respectively. Figure 11 shows the 68%and 95% confidence contours of the probability density of measuring the spin and quadrupole momentof Sgr A ∗ for GRAVITY observations of such stars over N = 40 orbits with astrometric precisionsof 10 µ as and 100 µ as, respectively, assuming that Sgr A ∗ is a Kerr black hole with a value of thespin a = 0 . r g . Even at these relatively small orbital separations, tracing the orbits of stars primarilymeasures the spin of the black hole, unless a very high level of astrometric precision is achieved [211].For one S-star, Ref. [211] estimated that GRAVITY observations can measure its spin with a15recision σ χ ∼ . (cid:18) σ θ µ as (cid:19) (cid:18) N (cid:19) − / (cid:18) ˜ a r g (cid:19) / (cid:18) r g /D . µas (cid:19) − (cid:20) (1 − e )(1 − e ) / . (cid:21) (cid:18) cos i . (cid:19) − , (25)where σ θ is the astrometric precision of GRAVITY observations over N orbits, D is the distance ofSgr A ∗ , and where ˜ a , e , and i are the semi-major axis, eccentricity, and orbital inclination of the S-star.Relativistic effects on the orbits of S-stars may also be imprinted on potential gravitational lensingevents caused by the deflection of light rays by Sgr A ∗ , which would result in the presence of twoor more images of the same S-star. The position and magnification of images of gravitationally-lensed S-stars depend primarily on the mass and distance of Sgr A ∗ , but may also be affected by theSchwarzschild part of the potential sourced by Sgr A ∗ or even its spin and quadrupole moment [220–233].Gravitational lensing events of S-stars may be resolvable with instruments such as GRAVITY [234] andcould potentially reveal deviations from the Kerr metric [235–237]. See Ref. [238] for a review. Starssuch as S2 may also be used as a probe for intermediate-mass black holes [239]. GRAVITY may alsomeasure the spin and quadrupole moment of Sgr A ∗ by observing localized NIR flares in the accretionflow surrounding Sgr A ∗ over the course of several orbits (see Sec. 7).
4. Pulsar Timing
Radio pulsars emit regular, steady beams of electromagnetic radiation, the periods of which can oftenbe measured very precisely. Several binary pulsars have been discovered to date (see Ref. [240] for areview) including the “Hulse–Taylor” pulsar PSR B1913+16 [241] and the double pulsar PSR J0737–3039 [242, 243]. Due to the compactness of such systems with semi-major axes ∼ R (cid:12) and orbitalperiods of only several hours, timing observations of double pulsars can be used for measurements ofthe parameters of the binary system including the masses of both stars and for consistency tests ofgeneral relativity with great precision [244, 245].As such, double pulsars can provide a very good testing ground for weak-field general relativity(see Refs. [246–248] for reviews) and, in some cases, even for strong-field tests of particular theoriesof gravity such as Brans-Dicke gravity [249] and the second-order scalar-tensor theory by Damourand Esposito-Far`ese [250, 251] (see Ref. [4]; [2, 5, 8]) as well as certain Lorentz-violating theories ofgravity [6, 7]. In these alternative theories of gravity, modifications in the strong-field regime such asthe predicted existence of dipolar gravitational radiation lead to observable effects even in the weak-fieldregime. The orbital evolution of pulsars in binaries or triple systems may also be used to test the strongequivalence principle with high precision [252, 253]. A pulsar in a binary with a stellar-mass black holecould also reveal the presence of a “large” extra dimension in RS2-type braneworld gravity [254] atthe ∼ µ m level through its orbital evolution induced by mass loss of the black hole into the extradimension [255, 256] (c.f., Refs. [257, 258]).Although a large number ( ∼ (cid:48) of Sgr A ∗ to date, the closest of which has a distance of ∼ (cid:48)(cid:48) ( ∼ ∗ is most likely caused bythe strong scattering of radio waves in this region at typical observing frequencies ∼ − ωΨΦ iS λ pericenter p l ane o f t he sky σ T O A [ m s ] Frequency [GHz]
Figure 12.
The left panel shows the geometry of a pulsar orbiting around Sgr A ∗ as characterized bya set of angles. The angles i and ω are the orbital inclination and longitude of pericenter as measuredfrom the ascending node in the plane of the sky. The pulsar orbit with respect to the equatorial planeof Sgr A ∗ is determined by the inclination θ , the equatorial longitude of the ascending node Φ, andthe equatorial longitude of pericenter Ψ, while λ denotes the angle between the line of sight and thespin axis (cid:126)S of Sgr A ∗ (c.f., Fig. 4). The right panel shows the predicted uncertainty of the pulsearrival time for a pulsar arbiting around Sgr A ∗ for two different spectral indices α , which includes theuncertainties from pulse phase jitter intrinsic to the pulsar, pulse broadening from scattering alongthe line of sight, and from interstellar scintillation. The curves assume a four-hour integration timefor a 100 m radio telescope and a one-hour integration time for an SKA-like telescope, each with abandwidth of 1 GHz. Observational frequencies above ≈
15 GHz are favored and precisions of ∼ µ sseem achievable with an SKA-like telescope. Taken from Ref. [55]. -20 -16 -12 -8 -4 a cc e l e r a t i o n ( c m s - ) Distance from the Black Hole (rc /GM BH ) a N a J a Q a dw a d o pulsar P r ece ss i on ti m e s ca l e [ y r] P b [yr] QSMP a [mpc]
Figure 13.
The left panel (taken from Ref. [203]) shows the accelerations of a 2 M (cid:12) pulsar dueto the Newtonian potential ( a N ) (without the Schwarzschild corrections), frame-dragging ( a J ), andquadrupolar torque ( a Q ), as well as the accelerations due to hydrodynamic drag ( a d ; red line) andthe gravitational interaction of the star with its wake ( a dw ; green line) assuming a spin of χ = 0 . ∗ . The right panel (taken from Ref. [55]) shows the characteristic timescales of such apulsar assuming an orbital eccentricity of 0.5 and 10 M (cid:12) perturbing stars within 1 mpc of Sgr A ∗ ,where the letters M, S, Q and P correspond to the contributions from the mass monopole (includingthe Schwarzschild corrections), spin (frame dragging), quadrupole moment, and stellar perturbations,respectively. ∗ is discovered, pulsar timing could provideanother means to measure the mass, spin, and quadrupole moment of Sgr A ∗ and, thereby, test theno-hair theorem. Such a pulsar experiences the same accelerations as an S-star; see Eq. (13). Figure 12shows the basic geometry of the binary with the corresponding definitions of angles which characterizethe orbital motion of the pulsar around Sgr A ∗ . Figure 12 also shows the uncertainties σ TOA of thepulse arrival time (estimated by Ref. [55]) for a pulsar with two different spectral indices as a function ofobserving frequency assuming a four-hour integration time for a 100 m radio telescope and a one-hourintegration time for an SKA-like telescope, each with a bandwidth of 1 GHz. The TOA uncertaintyincludes three contributions according to the relation σ ≡ σ + σ + σ , (26)where σ rn , σ j , and σ scint correspond to the uncertainties due to radiometer noise, intrinsic pulse phasejitter, and interstellar scintillation (see Refs. [272, 273]), respectively. Timing observations are favoredat observing frequencies above ≈
15 GHZ and precisions of ∼ µ s seem achievable with an SKA-liketelescope [55].Figure 13 shows the accelerations of a 2 M (cid:12) pulsar due to the Newtonian potential ( a N ), frame-dragging ( a J ), and quadrupolar torque ( a Q ) assuming a (dimensionless) spin χ = 0 . ∗ [c.f.,Eq. (20)]. Such a pulsar will also experience accelerations due to hydrodynamic drag ( a d ) and thegravitational interaction of the star with its wake ( a dw ) as calculated by Ref. [203]; c.f., Fig. 5. Figure 13likewise shows the timescales defined in Eqs. (21)–(23) assuming an orbital eccentricity of 0.5, as wellas the timescale of stellar perturbations for a distribution of 10 M (cid:12) stars within 1 mpc of Sgr A ∗ .In addition to the spin and quadrupole moment of Sgr A ∗ , such a system has three groups ofparameters: non-orbital parameters such as the pulse period, the rates of change of this period, andtheir positions in the sky; five “Keplerian” parameters such as the eccentricity e , the orbital period P b ,and the semi-major axis; as well as five “post-Keplerian” parameters such as the mean rate of pericenteradvance (cid:104) ˙ ω (cid:105) , the Einstein delay γ E of the emitted radio pulse (a combination of the relativistic Dopplereffect and the gravitational redshift), and the orbital period derivative ˙ P b . Assuming general relativity,the post-Keplerian parameters can be expressed in terms of the Keplerian parameters and the masses m and m of the pulsar and Sgr A ∗ , respectively, according to the expressions [274–276] (cid:104) ˙ ω (cid:105) = 6 πP b (cid:18) πGmc P b (cid:19) / (1 − e ) − , (27) γ E = e (cid:18) πP b (cid:19) − (cid:18) πGmc P b (cid:19) / m m (cid:16) m m (cid:17) , (28)˙ P b = − π (cid:18) π M P b (cid:19) / F ( e ) , (29)(30)where m ≡ m + m and M ≡ GM (cid:12) c ( m m ) / ( m + m ) / , (31) F ( e ) ≡ (cid:18) e + 3796 e (cid:19) (1 − e ) − / . (32)Since M BH ≡ m (cid:29) m and, therefore, M BH ≈ m , the pulsar mass can be neglected in theseequations. For Sgr A ∗ , the equations for the mean pericenter advance and for the Einstein delay canthen be written as (see Ref. [55]) (cid:104) ˙ ω (cid:105) (cid:39) − e (cid:18) πP b (cid:19) / (cid:18) GM BH c (cid:19) / . ◦ − e (cid:18) P b (cid:19) − / (cid:18) M BH × M (cid:12) (cid:19) / yr − , (33) γ E (cid:39) e (cid:18) P b π (cid:19) / (cid:18) GM BH c (cid:19) / (cid:39) e (cid:18) P b (cid:19) / (cid:18) M BH × M (cid:12) (cid:19) / s . (34)In addition, radio pulses experience a (Shapiro) time delay when passing through the gravitationalpotential of Sgr A ∗ , which is given by the expression ( [275]; see Ref. [55])∆ S (cid:39) GM BH c ln (cid:18) e cos ϕ − sin i sin( ω + ϕ ) (cid:19) (cid:39) . (cid:18) M BH × M (cid:12) (cid:19) ln (cid:18) e cos ϕ − sin i sin( ω + ϕ ) (cid:19) s , (35)where ω and ϕ are the angular distance of the pericenter in the orbital plane and the orbital phase ofthe pulsar, respectively, and i is the inclination of the orbital plane with respect to the observer’s lineof sight.Potential measurements of the mass, spin, and quadrupole moment of Sgr A ∗ with pulsar timingobservations were discussed in detail by Ref. [55]. The Keplerian parameters of the orbit of the pulsarcould be measured relatively easily. Consequently, a measurement of either the pericenter advance, theEinstein delay, or the Shapiro delay would suffice to infer the mass of Sgr A ∗ . However, the pericenteradvance and the Shapiro delay are also affected by the spin of Sgr A ∗ and the Einstein delay cannot beseparated from the Roemer delay which describes the contribution of the proper motion of the pulsarto the observed time delay (see the discussion in, e.g., Ref. [55]). However, the mass of Sgr A ∗ and theinclination i are also coupled via Kepler’s third law, which defines the so-called mass function GM BH (cid:39) (cid:16) cx sin i (cid:17) (cid:18) πP b (cid:19) , (36)where x is the projected semi-major axis of the pulsar orbit (in light seconds), which is an observableKeplerian parameter.The contributions of the spin and the quadrupole moments can, then, be separated from the effectof the mass alone through a fit [53, 277]. In practice, the mass is obtained within a model for the pulsearrival time, where the pericenter advance as well as the Shapiro, Roemer, and Einstein time delaysare inferred simultaneously; see Ref. [55]. Reference [55] simulated the fractional precision of a massmeasurement of Sgr A ∗ for the pericenter precession, Einstein delay, and Shapiro delay as a functionof the orbital period of the pulsar. The left panel of Fig. 14 shows the simulated fractional precision ofsuch a mass measurement of Sgr A ∗ in the case that Sgr A ∗ does not rotate. This simulation assumesweekly measurements of the pulse arrival time with an uncertainty of 100 µ s over a time span of fiveyears as well as an orbital eccentricity e = 0 . i = 60 ◦ of the pulsar. Precision levels of10 − − seem achievable [55].In order to infer the spin of Sgr A ∗ , the precession of the orbital plane of the pulsar induced byframe-dragging has to be taken into account. For one orbit, the corresponding precession rates of theangles Φ and Ψ (see Fig. 12) are given by the expressions [278] [c.f., Eqs. (14)–(15) and Ref. [279]]˙Φ = Ω LT (37)˙Ψ = − LT cos θ, (38)where Ω LT ≡ π GMc P (1 − e ) − / χ (39)19 M / M -8 -7 -6 -5 -4 P b [yr] γ E ∆ S ω . χ λ = χ c o s ( λ ) -1-0.500.51 χ θ = χ cos(θ) -1 -0.5 0 0.5 1 x ωω .... . x .. Figure 14.
The left panel shows the simulated fractional precision of a mass measurement of Sgr A ∗ asa function of the orbital period P b for the pericenter precession of the orbit of the pulsar ( ˙ ω ), Einsteindelay ( γ E ), and Shapiro delay (∆ S ) assuming weekly measurements of the pulse arrival time with anuncertainty of 100 µ s over a time span of five years for a pulsar orbiting around a non-rotating blackhole with an orbital eccentricity e = 0 . i = 60 ◦ . Precision levels of 10 − − seemrealistic. The right panel illustrates the (simulated) determination of the spin orientation of Sgr A ∗ in the plane spanned by χ θ ≡ χ cos θ and χ λ ≡ χ cos λ (c.f., Fig. 12) assuming an orbital period of0.3 yr, an eccentricity e = 0 .
5, a spin magnitude χ = 1, and angles Φ = Ψ = 45 ◦ . The inferredspin magnitude of Sgr A ∗ in this simulation has a value χ = 0 . ± . is the Lense-Thirring frequency. The longitude of pericenter ω and the projected semi-major axis x canthen be expressed in terms of a Taylor expansion, ω = ω + ˙ ω ( t − t ) + 12 ¨ ω ( t − t ) + . . . , (40) x = x + ˙ x ( t − t ) + 12 ¨ x ( t − t ) + . . . , (41)where the coefficients of these expansions and, thereby, the spin magnitude and orientation are obtainedfrom a fit of the timing data [53, 280] (c.f., Ref. [55]).Reference [55] also simulated the precision with which the spin and the quadrupole moment ofSgr A ∗ can be determined with timing observations of a pulsar orbiting around the Galactic center.The right panel of Fig. 14 illustrates the determination of the spin orientation of Sgr A ∗ in the planespanned by χ θ ≡ χ cos θ and χ λ ≡ χ cos λ as simulated for a pulsar with an orbital period of 0.3 yr, aneccentricity of e = 0 .
5, angles Φ = Ψ = 45 ◦ , and a spin magnitude χ = 1 of Sgr A ∗ . The inferred spinmagnitude of Sgr A ∗ in this simulation has a value χ = 0 . ± . ω , ¨ ω , and ¨ x of the expansions in Eqs. (40)–(41)have to intersect in one point if the orbit of a pulsar is unperturbed by other effects, which can serveas an independent test for the presence of such perturbations [55].The precession of the orbit of the pulsar induced by the quadrupole moment of Sgr A ∗ leads to avariation in the Roemer delay which can be identified from the timing residuals of the pulse arrival timesonce the effects of the mass monopole and frame-dragging have been subtracted [53]. Figure 15 showssuch characteristic timing residuals over the span of two orbits as simulated by Ref. [55] assuming thesame parameters as in the simulated determination of the spin of Sgr A ∗ (see the left panel of Fig. 14)and a spin value χ = 1 of Sgr A ∗ . Figure 15 also shows the precision of a measurement of the quadrupole20 [ m s ] -4-2024 time [d] δ q [ % C . L . ] P b [yr] e = 0.9e = 0.8e = 0.5 Figure 15.
The left panel shows the simulated timing residuals due to the presence of the quadrupolemoment of Sgr A ∗ assuming a maximally-rotating Kerr black hole. The other parameters of thesimulation are the same as in the one for the determination of the spin; see the left panel of Fig. 14.The right panel shows the simulated precision of a measurement of the quadrupole moment for threedifferent eccentricities of the orbit of the pulsar as a function of its orbital period obtainable over afive-year time span in the absence of perturbations. Taken from Ref. [55]. moment as a function of the orbital period of the pulsar for different values of the eccentricity assuming(on average) weekly observations over a five-year baseline with a higher cadence around the time ofpericenter passage. Thus, pulsar timing may test the no-hair theorem with high precision, especially ifSgr A ∗ has a high value of the spin [55]. Figure 16.
Fractional precision (2 σ ) for a measurement of the spin χ as a function of pericenterpassages, based on a dense timing campaign, neglecting (left panel) and including (right panel) theeffects of external perturbations. The pulsar has an assumed orbital period of 0.5 yr and an eccentricityof 0.8, while Sgr A ∗ has an assumed value of the spin χ = 0 .
6. The three curves correspond to timingprecisions of 100 µ s (black), 10 µ s (red), and 1 µ s (blue), respectively. The orientation of the spin isthe same as in Fig. 14. Taken from Ref. [211]. Reference [211] refined the timing model used in Ref. [55] by including higher-order post-Newtonianterms derived by Ref. [279]. Figure 16 shows the fractional precision of a spin measurement as a21 igure 17.
Quadrupolar timing residuals around pericenter passage observed over a few years forthe case of a Kerr black hole with values of the spin χ = 1 (left panel) and χ = 0 . µ s. Thequadrupole moment can be inferred with high precision for high values of the spin, but can still bemeasured accurately even for low values of the spin. Taken from Ref. [211]. function of pericenter passages for a pulsar with an orbital period of 0.5 yr and an eccentricity of 0.8orbiting around a Kerr black hole with a value of the spin χ = 0 .
6, based on a dense timing campaign.The fractional precisions shown in the left and right panels neglect and include the effects of externalperturbations, respectively. Figure 17 shows the quadrupolar timing residuals around pericenter passageobserved over a few years for different values of the black-hole spin, assuming the same pulsar and spinorientation as in Fig. 16 and a timing precision of 10 s. The quadrupole moment can be inferred withhigher precision for higher values of the spin.Figure 18 shows the corresponding posterior likelihoods of measuring the spin and quadrupolemoment of Sgr A ∗ for different observing campaigns assuming a timing precision of 100 µ s and a Kerrblack hole with a value of the spin χ = 0 .
6. Even in the case of a comparably low timing precision of100 µ s and the presence of external perturbations, a quantitative test of the no-hair theorem is possibleafter only a few pericenter passages and the spin and quadrupole moment of Sgr A ∗ can be measuredwith high precision after a few orbits [211].Since the orbital parallax of the pulsar also makes a significant contribution to the observed timingsignals, the distance of Sgr A ∗ can likewise be measured using pulsar timing. For N equally distributedtime-of-arrival measurements with an uncertainty σ TOA , the distance can be inferred with a fractionalprecision given by the equation [211] δD ∼ cσ TOA √ N (cid:18) Da (cid:19) ∼
20 pc (cid:18) σ TOA µ s (cid:19) (cid:18) N (cid:19) − / (cid:18) D . (cid:19) (cid:16) a au (cid:17) − . (42)Reference [281] calculated the Shapiro time delay experienced by photons emitted from a pulsaron an orbit around a black hole to second parameterized post-Newtonian order which also depends onthe spin and quadrupole moment of the black hole (see, also, Refs. [282, 283]). Reference [281] usedthe metric of Butterworth and Ipser [284,285] as the underlying spacetime which describes a stationaryand axisymmetric rotating fluid body in general relativity up to order ( GM/rc ) in quasi-isotropiccoordinates ( t, r, θ, φ ) and depends on the mass M , spin a , and quadrupolar parameter β of the fluidbody. 22 .00.80.60.40.20.0 B l a c k H o l e Q u a d r u p o l e M o m e n t KerrPeriapsisonlyFullorbit B l a c k H o l e Q u a d r u p o l e M o m e n t Figure 18.
Simulated posterior likelihood of measuring the spin and quadrupole moment of Sgr A ∗ assuming a Kerr black hole with a value of the spin χ = 0 .
6. In the left panel, the dashedcurves show the 68% and 95% confidence contours, while, in the right panel, the solid curves showthe 95% confidence contours. The solid curve shows the expected relation between the spin andquadrupole moment of a Kerr black hole. The pulsar is assumed to have an orbital period of 0.5 yr(corresponding to an orbital separation of ≈ r g ) and an eccentricity of 0.8, while three time-of-arrival measurements per day with equal timing uncertainty of 100 µ s have been simulated. The leftpanel compares the uncertainties in the measurement when only three pericenter passages have beenconsidered in the timing solution to those when the three full orbits are taken into account. The rightpanel shows the increase in the precision of the measurement when the number of pericenter passagesis increased from three to five. Taken from Ref. [211]. Figure 19 shows the second-order contributions to the Shapiro delay as a function of orbital phasedue to the mass, spin, and quadrupole moment of the black hole, as well as the corresponding amplitudesas a function of the closest approach distance. Although all three second-order effects are muchsmaller than the first-order Shapiro time delay, they are much larger than the expected measurementuncertainties for observations of pulsars around Sgr A ∗ with 100m-class radio telescopes or the SKA.However, these effects will primarily introduce a small bias to the measurement of the quadrupolemoment discussed in Ref. [211], because the quadrupole-order time-delay and orbital effects have verydifferent signatures on the time-of-arrival measurements [281].Reference [286] showed that a binary pulsar orbiting around Sgr A ∗ could also be used as a probe ofthe distribution of dark matter at the Galactic center. Such a pulsar and its companion would experiencea wind of dark-matter particles that can aect the orbital motion through dynamical friction leading toa characteristic seasonal modulation of the orbit and a secular change of the orbital period [286]. Thestrong gravitational lensing of a pulsar orbiting around Sgr A ∗ could potentially also be used as a probeof certain quantum gravity effects [287].
5. A Framework for Strong-Field Tests
By defintion, strong-field tests of the no-hair theorem cannot rely on the parameterized post-Newtonianformalism and a careful modeling of the underlying spacetime is required instead. Constructing asuitable spacetime for this purpose is a highly nontrivial task. Since it is unclear at present whethergeneral relativity is modified in the strong-field regime and, if so, in what manner, an efficient approach isto test the no-hair theorem using a model-independent framework. Such a phenomenological framework23 ∆ ( ) m a ss ( M ) -6 -4 -2 0 2 4Orbital Phase -0.02-0.010.000.010.02 ∆ ( ) s p i n ( M ) -6 -4 -2 0 2 4Orbital Phasea=1.0a=0.5a=-1.0-0.0010-0.00050.00000.00050.0010 ∆ ( ) q u a d ( M ) -6 -4 -2 0 2 4Orbital Phase β=0.2β=0.5β=−0.5 A m p li t u d e ( M ) c (M) 0.1110 A m p li t u d e ( s ) M Spin Quad
Figure 19.
Second-order contributions to the (Shapiro) light travel time delay for a pulsar on a circularorbit around a spinning black hole as a function of orbital phase due to the mass (top left panel), spin(top right panel), and quadrupole moment (bottom left panel) of the black hole. The orbit of the pulsarhas a radius of 1000 r g and an inclination of 80 ◦ , and the superior conjunction occurs at an orbitalphase of π/
2. The top right and bottom left panels show the second-order contributions for differentvalues of the spin a and the (dimensionless) quadrupole parameter β , respectively. The bottom rightpanel shows the corresponding amplitudes of the second-order Shapiro delay for a maximally spinningblack hole but as a function of the closest approach distance r c . The axis on the right assumes ablack hole mass M = 4 . × M (cid:12) . Although all three second-order effects are much smaller than the(first-order) Shapiro time delay, they are much larger than the expected measurement uncertaintiesfor observations of pulsars around Sgr A ∗ with 100m-class radio telescopes or the SKA. Taken fromRef. [281]. is provided by a parametrically deformed Kerr-like spacetime which encompasses many different theoriesof gravity at once. Kerr-like metrics generally do not derive from the action of any particular gravitytheory. The underlying theory is usually unknown and insight into this theory is hoped to be gainedthrough observations.Kerr-like metrics are a class of so-called metric theories of gravity [288] typically obeying thefull Einstein equivalence principle (EEP) [1]. The EEP is the foundation of a theory of gravity andis comprised of three fundamental principles, the weak equivalence principle (WEP), local Lorentzinvariance (LLI), and local position invariance (LPI). The WEP postulates that the trajectory of a24reely falling “test” body, i.e., a body that is not affected by forces such as electromagnetism or tidalgravitational forces, is independent of its internal structure and composition. In Newtonian gravity, thisstatement is equivalent to the equality of the inertial and gravitational mass of such a test body. TheLLI states that the outcome of any local non-gravitational experiment is independent of the velocityof the freely-falling reference frame in which it is performed. The LPI postulates that the outcome ofsuch an experiment is independent of its position and the time of its performance. It then follows fromthe EEP that gravitation can be described by the curvature of a spacetime (e.g., [59]).The only theories of gravity that are consistent with the EEP are metric theories of gravity. In thesemetric theories, the spacetime is endowed with a symmetric metric, the trajectories of freely falling testbodies are geodesics of that metric, and in local freely falling reference frames, the non-gravitationallaws of physics are those of special relativity. This setup, then, allows for the calculation and predictionof possible observable signatures of the theory. The three components of the EEP have been thoroughlytested by many different experiments, at least in the weak-field regime [1]. Other requirements of theEEP, however, can be relaxed, such as the LLI for black holes in Lorentz-violating theories [157,161,162].In this section, I describe Kerr-like metrics and some of their properties, focusing primarily onthree particular Kerr-like metrics. Throughout this section, I use geometric units, where G = c = 1. Following the discovery of the Schwarzschild [289] and Kerr [119] metrics in 1916 and 1963, respectively,Hartle and Thorne [290, 291] constructed a metric for slowly rotating neutron stars with arbitrary (butsmall) quadrupole moments in the late 1960s (see Ref. [292] and references therein for alternative formsof this metric). Tomimatsu and Sato [293, 294] found a discrete family of spacetimes in 1972 thatcontains the Kerr metric as a special case. In 1985, Quevedo and Mashhoon [295] constructed a metricof a rotating mass with an arbitrary quadrupole moment building on the static metric found by Erezand Rosen [296–298] in 1959. After a full decade of research, Manko and Novikov [60] found two classesof metrics in 1992 that are characterized by an arbitrary set of multipole moments.Many exact solutions of the Einstein field equations are now known [299]. Of particular interestis the subclass of stationary, axisymmetric, vacuum (SAV) solutions of the Einstein equations, andespecially those metrics within this class that are also asymptotically flat. Once an explicit SAV hasbeen found, all SAVs can in principle be generated by a series of HKX-transformations ( [300, 301]and references therein), which form an infinite-dimensional Lie group [302, 303]. Each SAV is fully anduniquely specified by a set of scalar multipole moments [180,181] and can also be generated from a givenset of multipole moments [184,185]. These solutions, however, are generally very complicated and oftenunphysical. For some astrophysical applications, such as the study of neutron stars, it is oftentimesmore convenient to resort to a numerical solution of the field equations [284, 304–307].Kerr-like metrics focus on parameteric deviations from the Kerr metric and need not be vacuumsolutions in general relativity. Several Kerr-like metrics have been constructed thus far (e.g., [60–68]),which depend on one or more free parameters that measure potential deviations from the Kerr metricand which include the Kerr metric as the special case when all deviations vanish. Observations can thenbe used to measure these deviations, should they exist, and, thereby, infer properties of the underlyingtheory of gravity. If no deviations are detected, the compact object is verified to be a Kerr black hole. If,on the other hand, nonzero deviations are measured, there are two possible interpretations. If generalrelativity still holds, the object is not a black hole but, instead, another stable stellar configurationor, perhaps, an exotic object [308]. Otherwise, the no-hair theorem would be falsified. Alternatively,within general relativity, the deviation parameters may also be interpreted as a measure of the systematicuncertainties affecting the measurement so that their effects can be treated in a quantitative manner.The Kerr metric is the only stationary, axisymmetric, asymptotically flat, vacuum solution tothe Einstein equations that possesses an event horizon and is free of timelike curves outside of the25orizon. Hence, it uniqely describes black holes in general relativity [120–125, 127, 132, 133]. Due toHawking’s rigidity theorem [123], stationary (asymptotically flat, vacuum) black holes are automaticallyaxisymmetric, and, thus, axisymmetry is not a requirement in general relativity. This, however, neednot be the case outside of general relativity. In addition, the Kerr metric possesses a third constant ofmotion, the Carter constant, making geodesic motion integrable in this spacetime [309].A Kerr-like metric necessarily has to differ from the Kerr metric in at least one of above properties(whether or not it admits the existence of a Carter-like constant). The many proposed metrics in theliterature can be divided into two subclasses: those that are Ricci flat, i.e., R µν = 0, and those thatare not. In the former case, the metric in the far field satisfies the Laplace equation, and thus, whenin asymptotically Cartesian and mass-centered coordinates, it can be expressed as a sum of mass andcurrent multipole moments (see, e.g., Ref. [177]). For small deviations from the Kerr metric, one canrelate these moments to each other via [61, 63, 310] M (cid:96) + i S (cid:96) = M (i a ) (cid:96) + δM (cid:96) + i δS (cid:96) , (43)where δM (cid:96) and δS (cid:96) are mass and current multipole deformations.When the metric is not Ricci flat, the above parameterization of the metric in the far field (as a sumover mass and current multipole moments that depend only on the (cid:96) harmonic number) is not valid.Such metrics generically arise from explicit or implicit modifications to the Einstein-Hilbert action. Inthese cases, it is not clear what the general structure of a modification of Eq. (12) would look like.A second important distinction between different Kerr-like metrics is the degree of nonlinearityof their deviations from the Kerr metric. Some Kerr-like metrics have been defined as small (and,therefore, linear) perturbations away from the Kerr metric, while other metrics are “exact,” i.e., theyare considered exact (and often nonlinear) solutions to (usually unknown) sets of field equations. Thisis an important difference, because deviations from the Kerr metric, should they exist, could be largeand still satisfy the current observational constraints. Thus, there is no need to a priori limit Kerr-likespacetimes to the description of only small deviations.In addition, it is sometimes useful in practice to compute the properties of Kerr-like metrics withlinear deviations from the Kerr metric to all orders in the deviation parameters, i.e., without expandingthe results of such computations to linear order in the deviation parameters. While an expansion in smalldeviation parameters can always be performed in analytic calculations, it is a lot more difficult and, insome cases, even impossible to enforce in other settings such as the ones involving magnetohydrodynamicsimulations, which numerically solve the (nonlinear) geodesic equations. In this interpretation, Kerr-like metrics containing small perturbations from the Kerr metric also have to be considered exact.Similarly, it can be instructive to study nonlinear Kerr-like metrics also in the limit of small deviations,expanding these metrics to first order in the deviation parameters and treating the resulting metrics asperturbative.Examples of Kerr-like metrics defined as linear deviations from the Kerr metric include the bumpyKerr metric [61, 63], the quasi-Kerr metric [62], and the modified-gravity bumpy Kerr metric [65]. TheManko-Novikov metric [60] and the metrics of Refs. [64,66] are examples of nonlinear Kerr-like metrics.The Manko-Novikov metric is Ricci flat, the quasi-Kerr metric is Ricci flat up to terms containingthe quadrupole moment, and the bumpy Kerr metric is a vacuum solution of the linearized Einsteinequations if the spin vanishes. The modified-gravity bumpy Kerr metric and the metrics of Refs. [64,66]are not Ricci flat.On the other hand, the quasi-Kerr, the bumpy Kerr, and the Manko-Novikov metrics, as wellas the metric of Ref. [64] are stationary, axisymmetric, and do not possess a Carter-like constant,while the modified-gravity bumpy Kerr metric also admits an approximate Carter-like constant. Themetric of Ref. [66] possess an exact Carter-like constant. All of these metrics are asymptotically flat.The quasi-Kerr, the bumpy Kerr, and the Manko-Novikov metrics harbor naked singularities, whilethe modified gravity bumpy Kerr metric and the metric of Ref. [66] describe black holes. The metric26f Ref. [64] generally harbors a naked singularity, which is located at the Killing horizon and canhave either spherical or disjoint topology, but describes a black hole for small values of the deviationparameter when it is linearized in that parameter. See Ref. [311] and references therein for a detaileddiscussion. When linearized in its deviation parameters, the metric of Ref. [66] can be mapped to themodified-gravity bumpy Kerr metric in at least certain cases [66]. Figure 20.
The left panel shows the null surface and regions with Lorentz violations and closedtimelike curves (denoted “CTCs”) in the quasi-Kerr metric [62] for values of the spin a = 0 . M andthe deviation parameter (cid:15) = 0 .
1. This metric harbors a naked singularity located at the null surface andLorentz violations as well as closed timelike curves exist around the poles. Such pathological regionshave to be excluded by the introduction of a suitable cutoff radius which shields outside observers fromtheir adverse effects. The center and right panels show the location of the naked singularity harboredby the metric of Ref. [64] for values of the spin | a | = 0 . M and different values of the deviationparameter (cid:15) . At this value of the spin, the naked singularity is of spherical topology if (cid:15) (cid:46) .
32 andof disjoint topology otherwise. Taken from Ref. [311].
By construction, however, Kerr-like metrics often contain pathological regions of space wheresingularities, closed timelike curves, or violations of Lorentz symmetry exist, such as outside of thecentral objects of the quasi-Kerr, bumpy Kerr, and Manko-Novikov metrics (see Ref. [311]). The leftpanel of Fig. 20 shows an example of such regions in the quasi-Kerr metric. These regions are unphysicaland have to be excised by introducing a cutoff radius, which acts as an artificial event horizon. Allphotons and matter particles that pass through this horizon are considered “captured” and are excludedfrom the domain outside of the horizon. the presence of a cutoff radius, therefore, limits the abilityof these metrics to serve as a framework for observational tests of the no-hair theorem. They impactboth EMRI observations in the gravitational-wave spectrum, as well as electromagnetic observations ofaccretion flows, since both depend sensitively on the behavior of the metric near the innermost stablecircular orbit (ISCO); see the discussion in Ref. [64]. Note that causality is violated everywhere inthe Kerr metric if the spin exceeds the Kerr bound in Eq. (6), because, in that case, any event inthat spacetime can be connected to any other event by both a future and a past directed timelikecurve [124, 309] (see, also, Ref. [312]). This property usually also restricts the applicability of Kerr-likemetrics to values of the spin for which the Kerr metric harbors a black hole.The modified-gravity bumpy Kerr metric and the metrics of Refs. [64,66] are free of such pathologiesexterior to the central object making them particularly suited for tests of the no-hair theorem. In thecase of the metric of Ref. [64], a cutoff radius has to be introduced just outside of the central nakedsingularity. This, however, does not limit the applicability of this metric in practice, because the cutoffradius can always be chosen so that the ISCO still lies in the domain exterior to the cutoff [64]. Thismetric was later generalized by Ref. [67] to include two independent types of deviations.27ote, however, that the metrics of Refs. [63,64,67] have been constructed by the use of the Newman-Janis algorithm [313, 314] to generate rotating solutions from static seeds and it is not guaranteed thatthis procedure can be applied consistently to general metrics which are not solutions of the Einsteinfield equations. This is not a surprise, because it is still not fully clear why the Newman-Janis algorithmworks even in general relativity and what the necessary conditions are so that a static metric can be usedas a seed in this method [315–320]. However, there are at least several known examples of black holesolutions other than the Kerr solution for which this is indeed the case [321–325]. Recently, Ref. [326]constructed a much simpler form of the Newman-Janis algorithm in Kerr-Schild coordinates.While the existence of a Carter-like constant in the modified-gravity bumpy Kerr metric and themetric of Ref. [66] necessarily restricts the scope of these metrics to include only black hole metrics thatadmit a Carter-like constant, it allows for the separation of the geodesic equations, which can greatlyfacilitate the study of observables in these spacetimes. Nonetheless, it would be desirable to employ aneven more general Kerr-like metric which contains all Kerr-like metrics with at least two constants ofmotion. At present, however, no such metric is known. In its most general form, the modified-gravitybumpy Kerr metric covers the entire class of stationary, axisymmetric black hole metrics with smalldeviations from the Kerr metric which admit three constants of motion and have a Carter-like constantthat is quadratic in the momentum [65]. Whether or not the corresponding property holds for thenonlinear metric of Ref. [66] is unclear; see the discussion in Ref. [66].Recently, Ref. [68] proposed a Kerr-like metric in the form of a Kerr metric for which the mass M isreplaced by two deviation functions m ( r ) and m ( r ) which reduce to the mass M if all deviations fromthe Kerr metric vanish. This metric harbors a black hole and is free of curvature singularities outsideof the event horizon [68]. As can easily be seen from the form of the metric elements, for certain rangesof the deviation parameters the exterior domain in this metric is also free of pathological regions.This metric can be mapped to the static black hole solution found by Bardeen [327] (which describesblack holes in general relativity with a magnetic monopole coupled to a nonlinear electromagneticfield [328]) and a corresponding rotating solution constructed by Ref. [329] based on the Newman-Janis algorithm [313, 314]. Whether or not this stationary solution belongs to the same theory andphysical setup as the static solution by Bardeen is unclear (c.f., the discussion on the applicability ofthe Newman-Janis algorithm above). The spacetime of Ref. [68] can also be mapped to several metricsin certain quantum-gravity inspired scenarios [330–333]. See Ref. [68] for these mappings. I discussa generalization of this metric together with a mapping to the metric of Ref. [66] in Appendix A.4.References [334, 335] expressed generic deviations from the Kerr metric in terms of a continued-fractionexpansion.Here, I focus on three particular Kerr-like metrics, which have been used frequently in the contextof no-hair tests with electromagnetic observations of Sgr A ∗ and of black holes in general: the quasi-Kerrmetric [62], the metric of Ref. [64], and the metric of Ref. [66].The quasi-Kerr metric derives from the Hartle-Thorne metric [290,291] and contains an independentquadrupole moment which is not assumed to depend on mass and spin through Eq. (12). The quasi-Kerrmetric modifies the quadrupole moment of the Kerr metric by the amount δM = − (cid:15)M , (44)where the parameter (cid:15) measures deviations from the Kerr metric. The full quadrupole moment is then M = − M (cid:0) a + (cid:15)M (cid:1) . (45)In Boyer-Lindquist-like coordinates, i.e., in spherical-like coordinates that reduce to Boyer-Lindquistcoordinates in the Kerr limit, the quasi-Kerr metric g µν is given by the expression g µν = g K µν + h µν , (46)28here the correction h µν to the Kerr metric g K µν in Eq. (3) is diagonal with the components (of thecontravariant metric) h tt = (cid:18) − Mr (cid:19) − (cid:2)(cid:0) − θ (cid:1) F ( r ) (cid:3) ,h rr = (cid:18) − Mr (cid:19) (cid:2)(cid:0) − θ (cid:1) F ( r ) (cid:3) ,h θθ = − r (cid:2)(cid:0) − θ (cid:1) F ( r ) (cid:3) ,h φφ = − r sin θ (cid:2)(cid:0) − θ (cid:1) F ( r ) (cid:3) . (47)The functions F , ( r ) are given in Appendix A of Ref. [62]. Recently, the Hartle-Thorne metric wasextended to include terms that are of higher order in the quadrupole moment [336, 337].While the quasi-Kerr metric has the advantage of being of a relativily simple form, it dependson only one deviation parameter and, strictly speaking, can only be applied to slowly to moderatelyspinning compact objects. The metric of Ref. [64] depends on one infinite set of deviation parameters,which are the coefficients of a series expansion of a deviation function h ( r, θ ) (which could also be of amore general form [64]). The nonvanishing components of this metric can be written as g tt = − [1 + h ( r, θ )] (cid:18) − M r Σ (cid:19) ,g rr = Σ[1 + h ( r, θ )]∆ + a sin θh ( r, θ ) ,g θθ = Σ ,g φφ = (cid:20) sin θ (cid:18) r + a + 2 a M r sin θ Σ (cid:19) + h ( r, θ ) a (Σ + 2 M r ) sin θ Σ (cid:21) ,g tφ = − aM r sin θ Σ [1 + h ( r, θ )] , (48)where h ( r, θ ) ≡ ∞ (cid:88) k =1 (cid:18) (cid:15) k + (cid:15) k +1 M r Σ (cid:19) (cid:18) M Σ (cid:19) k (49)and where (cid:15) is usually set to zero in order to be consistent with the parameterized post-Newtonianconstraints on deviations from general relativity (c.f., Ref. [1]). The lower-order coefficients (cid:15) and (cid:15) areneglected here. The coefficient (cid:15) vanishes due to the requirement that the metric be asymptoticallyflat, and the coefficient (cid:15) is likewise strongly constrained by weak-field tests of gravity. The lattercoefficient can also be absorbed into a trivial rescaling of the mass and, thus, plays no role [66]. Thegeneralization of this metric by Ref. [67] decouples the set of coefficients (cid:15) k into two independent infinitesets of deviation parameters (cid:15) tk , (cid:15) rk and reduces to the metric of Ref. [64] when (cid:15) tk = (cid:15) rk for all k [67].The metric of Ref. [64] harbors a naked singularity unless it is expanded to linear order in thedeviation parameters in which case it describes a black hole for small deviations from the Kerr metric.Focusing only on the lowest-order nonvanishing deviation parameter (cid:15) , the event horizon is then loctedat the radius r H = r K+ (cid:34) − (cid:15) a M sin θ √ M − a (cid:0) M r K+ − a sin θ (cid:1) (cid:35) , (50)where r K+ is the event horizon of a Kerr black hole, see Eq. (7). If no event horizon exists, the nakedsingularity is located at the Killing horizon which can be of either spherical or disjoint topology. Again,29etting all deviation parameters other than (cid:15) to zero, the Killing horizon is of spherical topology forvalues of the deviation parameter (cid:15) ≤ (cid:15) bound3 and of disjoint topology otherwise, where (cid:15) bound3 ≡ a/M ) (cid:20) (cid:16) (cid:112) − a/M ) (cid:17) − a/M ) (cid:16)
40 + 7 (cid:112) − a/M ) (cid:17) + 150( a/M ) (cid:16)
15 + (cid:112) − a/M ) (cid:17) (cid:21) (51)assuming | a | ≤ M [311]. The center and left panels of Fig. 20 show the location of the naked singularitywith spherical and disjoint topology, respectively, for a value of the spin | a | = 0 . M . The metric ofRef. [67] has similar properties [67].Still, even with the expanded scope of this metric compared to the one of the quasi-Kerr metric,it is possible to introduce additional degrees of freedom with suitably chosen deviation functions inmore general Kerr-like metrics. In general relativity, stationary, axisymmetric, and asymptoticallyflat metrics that admit the existence of integrable two-dimensional hypersurfaces generally depend ononly four functions. As a consequence of Frobenius’s theorem, such hypersurfaces are automaticallyguaranteed to exist if the spacetime is also vacuum. Such metrics can be written in the form of thePapapetrou line element where the metric is expressed with respect to the Weyl-Papapetrou coordinatesand has only three metric functions. Out of these functions only two are independent, while the thirdone can be derived from the other two.This happens for three reasons. First, the symmetries imposed, the assumption of asymptoticflatness and the vanishing of the Ricci tensor allow for the spacetime to have integrable two-dimensionalhypersurfaces that are orthogonal to the two Killing fields and on which one can define coordinates thatcan be carried along integral curves of these Killing fields to the rest of the spacetime. Thus the metriccan be written in a 2 × t, φ )part of the metric, then the metric can be written in a form that has only four independent functions.Second, the field equations imply, by the vanishing of the Ricci tensor (i.e., the vacuum assumption),that the coordinate ρ which is defined by the determinant of the ( t, φ ) part of the metric is a harmonicfunction and thus one can define the second coordinate on the two-dimensional hypersurfaces as theharmonic conjugate of ρ and absorb one of the functions in the process, reducing the independentfunctions to three. Finally the vacuum field equations imply that the third of the functions is relatedto the other two and can be determined up to the addition of a constant [338].In alternative theories of gravity, however, the vacuum assumption does not necessarily imply theexistence of two-dimensional integrable hypersurfaces. Therefore, one would expect that metrics whichdescribe black holes in alternative theories of gravity depend on at least four independent functions.This motivated the construction of the Kerr-like metric of Ref. [66] (and, earlier, of the modified-gravitybumpy Kerr metric [65]), which has the nonvanishing components g tt = − ˜Σ[ ¯∆ − a A ( r ) sin θ ][( r + a ) A ( r ) − a A ( r ) sin θ ] ,g tφ = − a [( r + a ) A ( r ) A ( r ) − ¯∆] ˜Σ sin θ [( r + a ) A ( r ) − a A ( r ) sin θ ] ,g rr = ˜Σ¯∆ A ( r ) ,g θθ = ˜Σ ,g φφ = ˜Σ sin θ (cid:2) ( r + a ) A ( r ) − a ¯∆ sin θ (cid:3) [( r + a ) A ( r ) − a A ( r ) sin θ ] , (52)30here ¯∆ ≡ ∆ + βM , (53) A ( r ) = 1 + ∞ (cid:88) n =3 α n (cid:18) Mr (cid:19) n , (54) A ( r ) = 1 + ∞ (cid:88) n =2 α n (cid:18) Mr (cid:19) n , (55) A ( r ) = 1 + ∞ (cid:88) n =2 α n (cid:18) Mr (cid:19) n , (56)˜Σ = Σ + f ( r ) , (57) f ( r ) = ∞ (cid:88) n =3 (cid:15) n M n r n − . (58)The metric of Ref. [66] contains the four free functions f ( r ), A ( r ), A ( r ), and A ( r ) that dependon four sets of parameters which measure potential deviations from the Kerr metric. In addition, thismetric depends on the deviation parameter β . In the case when all deviation parameters vanish, i.e.,when f ( r ) = 0, A ( r ) = A ( r ) = A ( r ) = 1, β = 0, this metric reduces to the Kerr metric in Eq. (3).Formally, the parametrization also includes the Kerr-Newman metric and potential deviations from it if β = Q /M , where Q is the electric charge of the black hole. However, astrophysical black holes shouldbe electrically neutral, because any residual electric charge is expected to neutralize quickly. Therefore,I will treat the parameter β primarily as a pure deviation from the Kerr metric.The deviation functions in Eqs. (54)–(58) are written as power series in M/r (but could also beof a more general form [66]). The lowest-order coefficients of these series vanish so that the deviationsfrom the Kerr metric are consistent with all current weak-field tests of general relativity (c.f., Ref. [1])as in the case of the metric of Ref. [64] discussed above. However, certain restrictions on these functionsand on the deviation parameter β exist which are determined by the properties of the event horizon.The event horizon itself is independent of all deviation parameters except for the parameter β andis located at the radius r + ≡ M + (cid:112) M − a − βM . (59)Thus, the event horizon coincides with the event horizon of a Kerr black hole if the parameter β vanishes.In order for an event horizon exist, the parameter β must obey the usual relation βM ≤ M − a (60)and the functions ˜Σ, A ( r ), A ( r ), and A ( r ) have to be positive everywhere on and outside of theevent horizon. In the case of the lowest-order metric, i.e., when this metric is truncated at the lowestnonvanishing order in the deviation parameters, the latter requirement can be rewritten as the relations (cid:15) > B , α > B ,α > B , α > B , (61)where B ≡ − (cid:0) M + √ M − a (cid:1) M ,B ≡ − (cid:0) M + √ M − a (cid:1) M . (62)Otherwise, this metric harbors a naked singularity instead of a black hole [66]. Figure 21 shows theallowed values of the five deviation parameters in the lowest-order metric as a function of the spin.31 igure 21. Allowed values of the deviation parameters (cid:15) , α , α , α , and β as a function of thespin neglecting higher-order terms in the deviation functions. The purple line shows the maximumvalues of the parameter β , while the blue and red lines show the minimum values of the parameters α , α and (cid:15) , α , respectively. Reference [109] defined and computed multipole moments of the Kerr-Newman metric as a vacuumsolution in f ( R ) gravity theories finding that the relation of the Kerr multipole moments in Eq. (12)is preserved in a modified form with the simple substitution M → − (cid:112) M − Q . Consequently, themultipole moments of the metric of Ref. [66] in the case when β is the only nonvanishing deviationparameter are given by the relation M l + i S l = M (cid:112) − β (i a ) l , (63)at least as long as this metric is interpreted as a vacuum solution in f ( R ) gravity. In particular, thefirst three multipole moments are: M = M √ − β , S = M √ − βa , and M = − M √ − βa . Notethat I use a different sign convention for the multipole moments in Eq. (63) compared to the one usedby Ref. [109] so that the relation in Eq. (12) is recovered in the limit β → β can bemapped to the tidal charge β tidal in the RS2 model and to the coupling constant α in MOG via theequations β = β tidal /M and β = α (1 + α ) , respectively. In both cases, the parameter β can be eitherpositive or negative as long as Eq. (60) is fulfilled. The metric of Ref. [66] can also be mapped to theblack hole solutions of EdGB gravity [153, 154, 156–159] and dCS gravity [157, 161, 162] up to linearorder in the spin, the Bardeen metric [327, 329], as well as to other Kerr-like metrics; see AppendixA for the detailed mappings. Table 5.1 summarizes the known mappings of the metric of Ref. [66] tospecific black-hole solutions. The properties of the spacetime of a black hole play an important role in the characteristics ofastrophysical observables such as the electromagnetic radiation emitted from a surrounding accretionflow. Kerr-like spacetimes can have properties that differ significantly from the properties of the Kerrspacetime leading to modified observed fluxes and spectra. These signals, then, encode properties ofthe underlying spacetime which may be inferred by observations.32 lack Hole Metric Nonvanishing Deviation Parameters ValidityKerr —Kerr-Newman β = Q /M RS2 β = β tidal /M MOG β = α (1 + α ) EdGB α = − ζ EdGB , α = − ζ EdGB , α = − ζ EdGB , . . . O ( a ), O ( ζ EdGB ) α = − ζ EdGB , α = − ζ EdGB , α = − ζ EdGB , . . .α = ζ EdGB , α = 3 ζ EdGB , α = ζ EdGB , . . .dCS α = ζ dCS , α = ζ dCS , α = ζ dCS O ( a ), O ( ζ dCS )Bardeen α = − g M , α = 2 α , α = g ( a +5 g − M ) M , . . . at least O ( g ) α = α , α = α , α = α , . . .α = − α , α = − α , α = − α , . . . Table 1.
Mappings of the metric of Ref. [66] to known black hole solutions. The mappings to theEdGB and dCS metrics are only valid up to linear order in the spin and for small values of therespective deviation parameters, while the mapping to the Bardeen metric is valid for small values ofthe parameter g at least up to O ( g ). There are no such restrictions on the other mappings. Notethat Bardeen metric for nonzero values of the spin constructed by Ref. [329] may not belong to thesame theory and physical setup as the static solution by Bardeen [327]. Many of these effects are somewhat generic to Kerr-like spacetimes but can vary in magnitude.Here, I demonstrate some of these effects for nonzero values of the deviation parameter α in themetric of Ref. [66]; see Ref. [66] and Table 2 regarding the other parameters of this metric. Adescription of the corresponding effects in the quasi-Kerr metric and the metric of Ref. [64] can befound in Refs. [62, 104, 340] and Refs. [64, 92, 339], respectively. Properties of other Kerr-like metricswere analyzed in Refs. [61,63,65,75–77,112,113,160,311,341–346]. Reference [347] studied the possibilityof spinning up certain classes of Kerr-like metrics past extremality with point particles or accretion disks,which would, thereby, violate the cosmic censorship conjecture [127].For nonzero values of the parameter α the coordinate locations of the circular photon orbit andof the ISCO are shifted compared to their coordinate locations in the Kerr metric. Similarly, photonsexperience either stronger or weaker lightbending near the black hole and the orbital frequencies of testparticles are altered. Figure 22 shows the dependence of the ISCO radius on the spin and the deviationparameter α . Figure 22 also shows the modified lightbending of photons around a black hole with aspin a = 0 . M and the Keplerian frequency of a particle on a circular equatorial orbit around a blackhole with spin a = 0 . M as a function of radius for different values of the deviation parameter α .Relativistic boosting and beaming as well as the gravitational redshift of photons propagatingthrough such a black-hole spacetime likewise are important factors in the observed radiation. Figure 23illustrates the combination of these effects (as well as of the ISCO shift and the modified lightbending)using direct images of geometrically thin Novikov-Thorne [348] accretion disks around black holes withvalues of the spin a = 0 . M and the parameter α = 0 and α = −
1, respectively. The thermalradiation emitted by the disk is radially symmetric, but the observed disk flux has a much morecomplicated structure.Since all of these effects typically depend on both the spin and deviation parameters of a givenKerr-like metric, there is an inherent degeneracy between the spin and the deviation parameters whichcan complicate the detection of a potential deviation from the Kerr metric. This is the case especially forthe ISCO radius (see, e.g., the left panel of Fig. 22), the location of which allows for spin measurementsof Kerr black holes based on, e.g., their continuum x-ray emission [349] or relativistically broadenediron lines [350]. For non-Kerr black holes, however, the dependence of the ISCO radius on the spin andthe deviation parameters leads to a strong correlation between all of these parameters if the location of33 igure 22.
Effects of the deviation parameter α on the ISCO radius, the amount of lightbending,and the Keplerian frequency of a particle on a circular euqatorial orbit. The left panel (taken fromRef. [66]) shows contours of constant ISCO radius as a function of the spin and the deviation parameter.At a fixed value of the spin, the location of the ISCO increases for increasing values of the deviationparameter. In the green shaded region, the energy has two local minima and the ISCO is located atthe outer radius where these minima occur. In the red shaded region, circular equatorial orbits donot exist at radii r ∼ . M and the ISCO is located at the outer boundary of this radial interval.The black shaded region marks the excluded part of the parameter space. The center panel showstrajectories of photons lensed by a black hole with a (counterclockwise) spin a = 0 . M for severalvalues of the parameter α . The shaded region corresponds to the event horizon. The right panel(taken from Ref. [66]) shows the Keplerian frequency ν φ = c Ω φ / πGM as a function of radius for ablack hole with mass M = 10 M (cid:12) and spin a = 0 . M for different values of the parameter α . At agiven radius, the Keplerian frequency increases for increasing values of the parameter α . The dotdenotes the location of the ISCO. Figure 23.
Direct images of geometrically thin accretion disks around (left panel) a Kerr blackhole with spin a = 0 . M and (right panel) a Kerr-like black hole with the same spin and avalue of the deviation parameter α = −
1. Both panels show the observed number flux densityof (radially-symmetric) thermal disk emission at 1 keV in units of keV − cm − s − . The highestemission originates from a strongly localized region near the ISCO on the side of the black hole thatis approaching the observer where Doppler boosting and beaming are particularly high. This regionshifts toward the black hole and emits a significantly higher flux for negative values of the parameter α . Taken from Ref. [93]. the ISCO is the primary quantity being measured, either directly or indirectly [91–93, 96, 97, 104].A property of the metric of Ref. [66] which is not generic to Kerr-like metrics is the fact that itcan be written in a Kerr-Schild-like form which removes all coordinate singularities at the location ofthe event horizon [66]. This allows for a consistent treatment of accretion flows in fully relativistic34agnetohydrodynamic simulations; c.f., e.g., Refs. [351, 352].Table 2 summarizes several important properties of the quasi-Kerr metric [62] and the metricsof Refs. [64, 66]. Table 2 lists the type of the compact object harbored by each of these metrics,their respective deviation parameters together with the metric elements affected by them in the Boyer-Lindquist-like forms given in Eqs. (46), (48), and (52), respectively, and the magnitude of the effect ofthe (lowest-order) deviation parameters on the location of the ISCO, the lightbending, and the orbitalfrequency ν φ of a particle on a circular equatorial orbit around the compact object. Metric Object Parameters Modified Metric Elements ISCO Lightbending Frequency ν φ Ref. [62] NS [311] (cid:15) ( t, t ), ( r, r ), ( θ, θ ), ( φ, φ ) strong [340] strong [340] weak [104]Ref. [64] NS a [311] (cid:15) , (cid:15) , . . . ( t, t ), ( r, r ), ( φ, φ ), ( t, φ ) strong [64] strong [339] weak [97]Ref. [66] BH [66] (cid:15) , (cid:15) , . . . ( t, t ), ( r, r ), ( θ, θ ), ( φ, φ ), ( t, φ ) weak [66] strong weak [66] α , α , . . . ( t, t ), ( φ, φ ), ( t, φ ) strong [66] strong weak [66] α , α , . . . ( t, t ), ( φ, φ ), ( t, φ ) strong [66] strong weak [66] α , α , . . . ( r, r ) none [66] strong none [66] β ( t, t ), ( r, r ), ( φ, φ ), ( t, φ ) strong strong weak a BH for small deviations at linear order.
Table 2.
Properties of the Kerr-like metrics of Refs. [62,64,66]. The table lists the type of the compactobject (NS – naked singularity, BH – black hole), nonvanishing deviation parameters, modified metricelements, and the effects of these parameters on the location of the ISCO, lightbending, and theKeplerian frequency ν φ of matter particles on circular equatorial orbits around the compact object.
6. Very-Long Baseline Interferometric Observations of the Accretion Flow
Sgr A ∗ is a prime target of high-resolution VLBI observations with the EHT [56–58]. Initial VLBIobservations of Sgr A ∗ in 2007–2009 at 230 GHz with a three-station array comprised by the James ClerkMaxwell Telescope (JCMT) and Sub-Millimeter Array (SMA) in Hawaii, the Submillimeter TelescopeObservatory (SMTO) in Arizona, and several dishes of the Combined Array for Research in Millimeter-wave Astronomy (CARMA) in California resolved structures on scales of only 4 r S [35], where r S ≡ r g is the Schwarzschild radius of Sgr A ∗ . Similar observations also detected time variability on thesescales in Sgr A ∗ and measured a closure phase along the Hawaii–SMA–SMTO triangle [36]. In 2009–2013, follow-up observations with the same telescope array [also including the Caltech SubmillimeterObservatory (CSO) in Hawaii] have led to an increased data set including numerous closure phasemeasurements [353] and the detection of polarized emission originating from within a few Schwarzschildradii [37]. Such measurements have demonstrated the feasibility of VLBI imaging of Sgr A ∗ with theEHT on event horizon scales.In 2015, the existing three-station EHT array has been expanded to include ALMA in Chile, theLarge Millimeter Telescope (LMT) in Mexico, the South Pole Telescope (SPT), the Plateau de BureInterferometer (PdB) in France, and the Pico Veleta Observatory (PV) in Spain; see Ref. [354] fora recent description of the EHT. Simulations based on such enlarged telescope arrays support thepossibility of probing the accretion flow of Sgr A ∗ in greater detail and of directly imaging the shadowof Sgr A ∗ [56]. At around 230 GHz, the emission from Sgr A ∗ becomes optically thin (see the right panelof Fig. 2) and the shadow will be (at least partially) unobscured by the surrounding accretion flow (seeRef. [31] and references therein). The sensitivity and resolution of this enlarged array will be greatlyincreased, caused primarily by ALMA which will have a sensitivity that is about 50 times greater thanthe sensitivity of the other stations and the long baselines from the stations in the Northern hemisphereto the SPT. In addition, this array allows for the measurement of closure phases along many differenttelescope triangles, some of which depend very sensitively on the parameters of Sgr A ∗ [355], as well as35f closure amplitudes along telescope quadrangles.In this section, I first review the importance of the shadow for tests of the no-hair theorem. Second, Ifocus on studies of the accretion flow of Sgr A ∗ as a means for such tests. Third, I discuss other potentialtests of the no-hair theorem based on orbiting hot spots in the accretion flow. ∗ The shadow is a prominent feature of resolved accretion flow images of supermassive black holes. Sucha shadow is the projection of the circular photon orbit onto the sky along null geodesics. For a Kerrblack hole, the circular photon orbit is located at a coordinate radius that ranges from 9 r g in the caseof a maximally counterrotating black hole to 1 r g in the case of a maximally corotating black hole [356].The shadow is expected to be surrounded by a bright ring corresponding to photon trajectories thatwind around the black hole many times. Thanks to the long path length through the emitting mediumof the photons that comprise the ring, these photons can make a significantly larger contribution to theobserved flux than individual photons outside of the ring.Images of shadows of Kerr black holes, either with or without optically and geometrically thinaccretion disks, have been calculated by a number of different authors [94,356–371]. Images of shadowsand accretion flows around non-Kerr black holes and exotic objects in general relativity or other theoriesof gravity were analyzed by Refs. [95, 101, 152, 367, 372–388]. References [389, 390] studied the stronggravitational lensing near Kerr-like compact objects. Black hole shadows are also clearly visible inseveral (three-dimensional) general-relativistic magnetohydrodynamic simulations (GRMHD) reportedto date [391–394].Since the shape of the shadow of a black hole is determined only by the geometry of the underlyingspacetime, it is independent of the complicated structure of the accretion flow making it an excellenttarget of imaging observations with the EHT. For a Kerr black hole, the shape of the shadow dependsuniquely on the mass, spin, and inclination of the black hole (e.g., [363]). For a Schwarzschild blackhole, the shadow is exactly circular and centered on the black hole. For Kerr black holes with nonzerovalues of the spin and the inclination, the shadow is displaced off center and retains a nearly circularshape [364, 365, 367], except for extremely high spin values a (cid:38) . r g and large inclinations, in whichcase the shape of the shadow becomes asymmetric [367, 368].However, images of black hole shadows can be significantly altered if the no-hair theorem isviolated. For black holes that are described by a Kerr-like metric, the shape of the shadow can becomeasymmetric [367,378,379,381–385,387,388] and its size can vary significantly [375,377,378,381–383,385,387, 388]. Figure 24 shows several examples of shadows of Kerr and Kerr-like black holes for differentvalues of the spin a , the inclination i , and the deviation parameter α in the metric of Ref. [66]. Kerrnaked singularities, i.e., compact objects described by the Kerr metric with values of the spin exceedingthe Kerr bound in Eq. (6), do not have a well-defined shadow making them easily distinguishable fromthe shadows of black holes [396, 397].Several authors have quantified the effects of the spin and inclination as well as of potentialdeviations from the Kerr metric on the position and shape of the shadow. Reference [364] showedthat the displacement of the shadow occurs in the direction perpendicular to the spin axis of the blackhole with an approximately linear dependence on the spin and characterized the shape of the shadowin terms of the maximum and minimum width of the shadow. Reference [367] defined the displacement D as the mean of the maximum and minimum abscissae x (cid:48) max and x (cid:48) min on the axis perpendicular tothe spin ( y (cid:48) ) axis, as well as the diameter L and asymmetry A of the shadow as an angular average ofits radius and the root mean square of its radius, respectively, which is easier to measure in practice.These expressions are given by the equations D ≡ | x (cid:48) max + x (cid:48) min | , (64)36 α -coordinate ( M )-505 β - c oo r d i n a t e ( M ) i = 30 ° a = 0 a = 0.999 -5 0 5 α -coordinate ( M )-505 i = 60 ° a = 0 a = 0.999 -5 0 5 α -coordinate ( M )-505 i = 90 ° a = 0 a = 0.999 Figure 24.
Shadows of Kerr (top row; taken from Ref. [368]) and non-Kerr (bottom row; taken fromRef. [377]) black holes with different spins a , inclinations i , and deviations from the Kerr metric α .In the top panels, different colors represent different spins ranging from a = 0 (black) to a = 0 . r g (red). For each inclination, the size of the shadow is determined primarily by the black hole mass M (in gravitational units) and depends only weakly on the black hole spin and the inclination. Thedisplacement of the shadow along the x -axis is a function of the spin and the inclination. Moreover,the shadow retains its nearly circular shape unless the black hole spin is very high and the inclinationis large. For nonzero values of the parameter α , however, the size of the shadow is altered and itsshape can be significantly aymmetric. L ≡ π (cid:90) π ¯ Rdϑ, (65) A ≡ (cid:115) (cid:82) π (cid:0) ¯ R − (cid:10) ¯ R (cid:11)(cid:1) dϑ π , (66)where ¯ R ≡ (cid:112) ( x (cid:48) − D ) + y (cid:48) (67)is the average radius andtan ϑ ≡ y (cid:48) x (cid:48) . (68)The displacement of the shadow around Kerr black holes is reminiscent of the location of thecaustics in the Kerr spacetime [366, 398]. References [367, 368, 377] computed approximate expressionsof the displacement, size, and asymmetry of the shadow of a Kerr black hole. Reference [367] andRef. [377] found approximate expressions of the displacement, diameter, and asymmetry for the shadow37 igure 25. Diameter (left column), displacement (center column), and asymmetry (right column) ofthe shadows of Kerr-like black holes with a spin a = 0 . M as a function of the inclination for valuesof the deviation parameters α = − , − , , , α = − , − , , , β = − , − , , . , .
18 (bottom row) in the metric of Ref. [66]. The shadow diameter depends onlyweakly on the parameter α , while it is practically constant for fixed values of the parameters α and β . The shadow displacement is affected by all three deviation parameters but depends primarilyon the spin. Negative values of the parameter α and positive values of the parameters α and β can cause the shadow shape to be significantly more asymmetric than the shadow of a Kerr black holewith the same spin. The size and asymmetry of the shadow are direct measures of the degree to whichthe no-hair theorem is violated.
38f the compact object described by the quasi-Kerr metric [62] and of the Kerr-like black hole describedby the metric of Ref. [66] (in terms of the parameters α and α ), respectively. See Appendix Bfor a list of these expressions, including approximate expressions of the displacement, diameter, andasymmetry of the shadow in terms of the parameter β in the metric of Ref. [66]. Figure 25 shows thedisplacement, diameter, and asymmetry of the shadow of Kerr-like black holes with spin a = 0 . r g as afunction of the inclination for different values of the deviation parameters α , α , and β in the metricof Ref. [66]. Reference [334] expressed the polar curve R ( ϑ ) as an expansion in Legendre polynomials. FIG. 5: From left to right: shadows of (top) configurationI, Kerr =ADM and Kerr =H ; (2 nd row) configuration II andKerr =ADM ; (3 rd row) transition configurations between II andIII (detail); (bottom) configuration III and Kerr =ADM . is Kerr-like ( J H /M < =ADM BH, which is exhibited in the top mid-dle panel. The latter is slightly larger and more D -like – acharacteristic of extremal Kerr BHs. Shadow I turns outto be closer to the one of the Kerr =H BH, exhibited in thetop right panel. This observation can be quantitativelychecked: σ K = 4 . / . =ADM /Kerr =H for the comparable BH, cf . Table I [35].New types of BH shadows, quite distinct from those ofKerr =ADM
BHs, appear on the left 2 nd and 4 th row pan-els of Fig. 5, corresponding to shadow II and III. In bothcases, the central BH is non-Kerr-like ( J BH /M > ∼ =ADM BH. It is also more ‘square’, with alarger normalized deviation from sphericity. Shadow IIIis remarkably distinct. Its central BH has J H /M ∼ D C D x D y ¯ r σ r σ r ¯ r (%) σ K (%)Shadow I 2.07 8.48 9.33 4.48 0.170 3.8 4.81/0.52Kerr =ADM =ADM A 2.38 8.66 9.86 4.70 0.260 5.54 0Kerr =H A 2.07 8.48 9.36 4.50 0.180 3.99 0Shadow II 2.39 7.14 6.93 3.60 0.118 3.29 25.5Kerr =ADM
A 1.79 9.32 9.86 4.82 0.103 2.15 0Shadow III 1.79 5.30 4.67 1.63 0.838 51.3 68.1Kerr =ADM
A 1.92 9.22 9.86 4.80 0.125 2.60 0TABLE I: Parameters for Kerr shadows with ‘ A ’ are com-puted from the analytic solution [3]; σ K is always computedwith respect to such solution. The second line in the table,computed for a Kerr BH generated numerically and using thesame ray tracing code as for KBHsSH, estimates the numeri-cal error ( ∼ × , allowed by the ‘heavy’ hair that it is dragging ( cf . the discussion in [25]). The lensing of this hair resemblesclosely that of the ultra-compact BS on the bottom leftpanel of Fig. 4. Interestingly, multiple disconnected shad-ows of the (single) BH occur: the largest ones (besidesthe main ‘hammer-like’ shadow) are two eyebrows [26], atsymmetric positions above and below the main shadow;but we have detected many other smaller shadows, hint-ing again at a self-similar structure. On the 3 rd row ofFig. 5, the shadows of four solutions in between configu-rations II and III illustrate the transition between them.Finally, we remark that the shadows of KBHsSH can havearbitrarily small sizes by considering solutions arbitrarilyclose to the BS curve in Fig. 1. Remarks.
KBHsSH can lead to qualitatively novel typesof shadows in GR, as shown by shadows II and III. Evenfor KBHsSH close to Kerr, their shadows are distinguish-able from the latter, with the same asymptotic quantities,as illustrated by shadow I. Regardless of the astrophysi-cal relevance of these solutions – which is unclear – theycan yield new templates with small or large deviationsfrom the Kerr shadows, hopefully of use for VLBI ob-servations. An exhaustive analysis of KBHsSH shadowsspanning the space of solutions in Fig. 1, and at differ-ent observation angles, will be presented elsewhere, forproducing such templates [27]. But the examples hereinalready raise a challenge to the parameterizations of de-viations from Kerr suggested in the literature [7–11]: canthey describe shadows with such large deviations?Besides the peculiar shape of some of the shadows ex-hibited, this model has one general prediction: smal lerobserved shadows than those expected for Kerr BHs withthe same asymptotic charges. Indeed, a ‘smaller’ centralBH seems a natural consequence of the existence of hair,carrying part of the total energy.Finally, for the setup herein, the redshift, which de-pends only on the source’s and camera’s positions, isconstant throughout the image and has been neglected.
Figure 26.
Images of shadows of Kerr black holes surrounded by a toroidal-shaped scalar fieldwhich can carry a significant fraction of the total mass and angular momentum. The imagescorrespond to objects with values (in units of 1 /µ where µ is the mass of the boson particle and G = c = 1) of the Arnowitt–Deser–Misner (ADM) mass M ADM ≈ . , . , . M H ≈ . , . , . J ADM ≈ . , . , .
85, and black-holeangular momentum J H ≈ . , . , . ◦ [385]. Thedeformations of the shadows deviate greatly from a nearly circular shape (not shown) for values of thefraction J H /M >
1. In this case, the shadow can even be disconnected as shown in the right panel(corresponding to the fraction J H /M ≈ . × ) and include two “eyebrow“ shadows above andbelow the central “hammer-like“ shadow. The different colors (red, blue, yellow, green) illustrate thegravitational lensing by the object. The extreme deformations of the shadow cannot be modeled byany presently known Kerr-like metric, but should be easily distinguishable from the deformed shadowsshown in Fig. 24 with EHT observations. Taken from Ref. [385]. Recently, Ref. [385] analyzed the shapes of shadows of Kerr black holes surrounded by a stabletoroidal-shaped scalar field which can carry a significant fraction of the total mass and angularmomentum of the system [148] (c.f., Refs. [149,150]). For the case in which the angular momentum of theblack hole exceeds the Kerr bound in Eq. (6) (i.e., J H /M > J H /M ≈ . , . , . × ,respectively, which cannot be modeled by any presently known (vacuum) Kerr-like metric satisfyingEq. (6). However, such extreme deformations of the shadow should be easily distinguishable from thedeformed shadows shown in Fig. 24 with EHT observations (c.f., Fig. 31). Note that the (numerical)analysis of Ref. [385] finds no singularities or pathological regions on or outside of the event horizonimplying that the central object in the presence of the scalar field remains a black hole even if the Kerrbound is violated. Reference [395] showed that the exterior domain of a Kerr black hole coupled toother matter sources with a more general configuration can likewise be free of singularities in that case.39 .2. The Accretion Flow of Sgr A ∗ In addition to the shadow, the accretion flow that surrounds Sgr A ∗ probes the innermost region nearthe event horizon outside of the black hole and can reveal important characteristics of the underlyingspacetime. Unlike active galactic nuclei, Sgr A ∗ is underluminous, with a bolometric luminosity of 10 in Eddington units [399]. While the detailed morphology of the emitting region remains uncertain,the existing spectral and polarization data across the electromagnetic spectrum have provided insightinto some of the properties of the accretion flow which include a peaked (often approximated by aMaxwellian) electron distribution function with a power-law high-energy tail (e.g., [400–409]; see theleft panel of Fig. 2), nearly equipartition magnetic fields [50], and variability (see Sec. 7). See Ref. [192]for a review.Several plausible models for the accretion flow of Sgr A ∗ exist (e.g., [410–417]), many of whichare categorized as radiatively inefficient accretion flows (RIAFs). Assuming that Sgr A ∗ is a Kerrblack hole, Refs. [418, 419] combined the early EHT observations of Sgr A ∗ in 2007–2009 [35, 36] withmeasurements of its spectral energy distribution and obtained values of the spin magnitude and directionemploying the RIAF model of Refs. [414, 420]. Likewise, Ref. [421] analyzed the early EHT data ofSgr A ∗ using an accretion flow model with plasma wave heating for several different values of the spinand disk inclination. References [391, 392, 422, 423] fitted the early EHT and spectral data to sets ofimages obtained from GRMHD simulations [424–427]. In the future, the determination of the spin andorientation of the black hole can be complemented with a multiwavelength study of polarization [428](c.f., Refs. [429–431]). See Refs. [192, 432] for reviews on accretion flow models of Sgr A ∗ . The outerextent of the accretion flow could be constrained by the observation (or lack thereof) of X-ray flaresoriginating from shock waves caused by the interaction of S-stars and their winds with the accretionflow [433].References [394, 434] computed images and spectra for a set of six GRMHD simulations withdifferent magnetic field configurations, black-hole spins, and thermodynamic properties and showedthat the combination of current spectral and early EHT observations rules out all models with strongfunnel emission. Reference [435] showed that GRMHD simulations for disk-dominated models produceshort timescale variability in accordance with current observationations, while GRMHD simulations forjet-dominated models generate only slow variability, at lower flux levels. Neither set of models show anyX-ray flares, which most likely indicate that additional physics, such as particle acceleration mechanisms,need to be incorporated into the GRMHD simulations. A similar analysis by Ref. [436] showed thatcurrent observations favor models with ordered magnetic fields near the black hole event horizon,although both disk- and jet-dominated emission can satisfactorily explain most of the current EHTdata. Reference [436] also showed that stronger model constraints should be possible with upcomingcircular polarization and higher frequency (349 GHz) measurements.Reference [437] argued that the angular momentum vector of the accretion flow (and perhaps ofthe black hole itself) is aligned with the angular momentum vector of the inner disk of stars within ∼ (cid:48)(cid:48) of Sgr A ∗ . Reference [438] inferred the spin orientation of Sgr A ∗ from the 2007–09 EHT data using aBayesian estimator based on different GRMHD simulations reported in Refs. [394, 434]. These resultsare broadly consistent with the spin orientation obtained by Refs. [418, 419] but have a larger overalluncertainty [438]. Reference [439] showed that different disk and jet models in GRMHD simulationsbased on those by Ref. [391] are consistent with the closure phase measurement by Ref. [36] and tendto favor higher inclinations, while the spin magnitude and orientation are only poorly constrained bythe same measurement.The follow-up observations with the EHT in 2009–2013 measured a number of closure phases alongthe SMTO–CARMA–Hawaii triangle which have a nonzero (positive) mean [353]. This implies thatthe millimeter emission from Sgr A ∗ is asymmetric on scales of a few Schwarzschild radii and can beused to break the 180 ◦ rotational degeneracy of amplitude data alone. Since the sign of these closure40hase measurements remained stable over most observing nights, the implied asymmetry in the imageof Sgr A ∗ is likely persistent and unobscured by refraction due to interstellar electrons along the line ofsight [353].Ref. [440] updated the RIAF analysis of Refs. [418,419] including the EHT data of Ref. [353] findingan improvement of the constraints on the spin magnitude and orientation as well as on the inclinationby about a factor of two. While the 180 ◦ degeneracy of the spin orientation angle is now removed, areflection degeneracy in the inclination remains. One of these inclinations is in remarkable agreementwith the orbital angular momentum of the infrared gas cloud G2 [441, 442] and the clockwise disk ofyoung stars surrounding Sgr A ∗ [191,443], possibly suggesting a relationship between the accretion flowof Sgr A ∗ and these features [440]. Figure 27.
Examples of (top row) black hole shadows for non-rotating black holes viewed at aninclination θ = 90 ◦ with illustrative values of the quadrupolar parameter (cid:15) and (bottom row) spectrallyfit RIAF model images at 230 GHz for the same parameter values. For negative values of the deviationparameter (cid:15) (left panels), the shadow has a more prolate shape than the shadow of a Kerr black holewith the same spin (central panels), while for positive values of the deviation parameter (right panels),the shadow has a more oblate shape. In all cases, the shadow is clearly visible in the model images eventhough it is partially obscured by the accretion flow on the left side of the shadow due to relativisticboosting and beaming. Nonzero values of the deviation parameter (cid:15) modify the morphology andmeasured intensity of the crescent and, for sufficiently negative values of the deviation parameter, thecrescent acquires a more pronounced tongue-like flux feature in the equatorial plane of the black hole.Taken from Ref. [16]. Reference [16] performed an analysis of RIAF images of Sgr A ∗ similar to the ones of Refs. [418,419],but using the quasi-Kerr metric as the underlying spacetime. For nonzero values of the deviation41 igure 28. RIAF images and visibility magnitudes of Sgr A ∗ assuming values of the spin magnitude a = 0 r g , spin orientation ξ = 127 ◦ , and inclination θ = 65 ◦ for (top to bottom rows) different values ofthe parameter (cid:15) . The panels in columns 1-2 and 3-4, respectively, show the corresponding intrinsic andscatter-broadened images. Even though these images look similar, existing EHT data (blue points inthe visibility magnitude plots) can already distinguish them assigning each image at different likelihood p ( a, θ, ξ, (cid:15) ) of being consistent with the data. Taken from Ref. [16]. parameter (cid:15) , the shadow becomes asymmetric [367] (see the discussion in Sec. 6.1). Reference [16]showed that images of accretion flows in the quasi-Kerr spacetime can be significantly different fromimages of accretion flows around Kerr black holes revealing the asymmetric distortions of the shadow.Figure 27 shows a set of black hole shadows and the corresponding RIAF images for values of the spin a = 0 r g , inclination θ = 90 ◦ , and different values of the deviation parameter (cid:15) .Reference [16] also showed that such differences in the RIAF images can be distinguished alreadyby early EHT data [35,36]. Figure 28 shows RIAF images and visibility magnitudes of Sgr A ∗ with andwithout the effect of the observed scatter-broadening of such images assuming a Schwarzschild blackhole with inclination θ = 65 ◦ and orientation ξ = 127 ◦ for different values of the parameter (cid:15) . Eventhough the images look similar, their corresponding likelihoods of being consistent with the EHT datavary by about a factor of two.Reference [16] only considered values of the spin and the deviation parameter (cid:15) for which the ISCOlies at a radius r ≥ r g and neglected all radiation passing through a cutoff radius located at r = 3 r g inorder to avert the adverse impact of the naked singularity harbored by this metric. The cutoff radius42cts as an artificial event horizon and effectively turns the compact object into a black hole for thepurposes of the simulated images. Thereby, Ref. [16] actually underestimate the effects of the spinand the deviation parameter on the images, which are strongest near the compact object. However,with this choice the quasi-Kerr metric can also be applied to “rapidly” spinning black holes, althoughit is, strictly speaking, only valid for slowly to moderately spinning black holes (see the discussion inRef. [311] and Sec. 5.1). Figure 29.
2D posterior probability densities as a function of (top row, left panel) dimensionlessspin magnitude a ∗ and inclination θ , (top row, right panel) spin magnitude and quadrupolar deviation (cid:15) , (bottom row, left panel) inclination and quadrupolar deviation, (bottom row, center panel) spinorientation and quadrupolar deviation, and (bottom row, right panel) inclination and spin orientation,respectively marginalized over all other quantities. In each panel, the solid, dashed, and dotted linesshow the 1 σ , 2 σ , and 3 σ confidence regions, respectively. In the top right panel, lines of constantISCO radius are shown as dashed gray lines, corresponding to 6 r g , 5 r g , and 4 r g from top to bottom,while the gray region in the lower right is excluded. Taken from Ref. [16]. Fitting the early EHT data to a library of RIAF images, Ref. [16] showed that previousmeasurements of the inclination and spin position angle in the same RIAF model [418, 419] are robustto the inclusion of a quadrupolar deviation from the Kerr metric. Figure 29 shows the 2D posteriorprobability densities of various combinations of the spin magnitude, spin orientation, inclination, andquadrupolar deviation, each marginalized over the remaining two parameters not shown. The spinmagnitude and the quadrupolar deviation are strongly correlated, roughly along lines of constant ISCOradius as shown in Fig. 29, while the spin and the inclination are only modestly correlated. The spinorientation could be determined only up to a 180 ◦ degeneracy. Reference [16] obtained constraints (with1 σ errors) on the spin magnitude a ∗ = 0 +0 . , spin orientation ξ = 127 ◦ +17 ◦ − ◦ (up to a 180 ◦ degeneracy),43nd inclination θ = 65 ◦ +21 ◦ − ◦ , while constraints on the deviation parameter (cid:15) remained weak. However,such constraints within a specific RIAF model will improve dramatically with EHT observations usinglarger telescope arrays [444]. ν [ Hz ] -20 -18 -16 -14 -12 -10 ν F ν [ e r g s − c m − ] † = − † =0 † =2 † =5 ν [ Hz ] -20 -18 -16 -14 -12 -10 ν F ν [ e r g s − c m − ] † = − † =0 † =2 † =5 Figure 30.
Simulated spectra emitted by a toroidal accretion flow surrounding Sgr A ∗ in the modelof Refs. [416,417,445–447] for different values of the deviation parameter (cid:15) in the metric of Ref. [64].All spectra were computed for fixed values of the spin a = 0 . r g , inclination θ = 60 ◦ , specific angularmomentum of the fluid particles (left panel) λ = 0 . λ = 0 .
6, magnetic to totalpressure ratio β = 0 .
1, polytropic index n = 3 /
2, central energy density ρ c = 10 − g/cm , andcentral electron temperature T c = 0 . T v where T v is the virial temperature. The spectra show asignificent dependence on the deviation parameter. Taken from Ref. [448]. Building on the work of Refs. [416, 417] who calculated images and spectra for a toroidal accretionflow around a Kerr black hole in the model of Refs. [445–447], Reference [448] calculated spectra for thesame torus model in the background of the Kerr-like metric of Ref. [64] and showed that such spectracan depend significantly on deviations from the Kerr metric. Figure 30 shows two sets of spectra fordifferent values of the deviation parameter, where the black hole has fixed values of the spin a = 0 . r g and inclination θ = 60 ◦ and the torus has fixed values of the specific angular momentum of the fluidparticles λ = 0 . λ = 0 . β = 0 .
1, electron to ion temperature ratio ξ = 0 .
1, polytropic index n = 3 /
2, central energy density ρ c = 10 − g/cm , and central electron temperature T c = 0 . T v , where T v is the virial temperature.Reference [449] simulated 1.3 mm images of boson stars [149, 150, 450–452] surrounded by such atoroidal accretion flow with a fixed inner radius (motivated by the accretion flows surrounding Kerrblack holes) producing an “effective” shadow. Reference [449] pointed out that the apparent sizes ofthe shadows in the latter setup are very similar to the sizes of the shadows of Kerr black holes withthe same mass and spin leading to a potential confusion problem (see, also, Ref. [385]). Figure 31shows simulated images for two such configurations with different values of the ADM mass (measuredin units of m /m , where m p is the Planck mass and m is the mass of one boson with a typical valuecorresponding to ∼ − eV) and spin. However, at least for extreme modifications of the shadowwhich is comprised by multiple shadows, these hammer-like features reveal clearly visible symmetricstructures across the equatorial plane of the object which should be easily detectable with the EHT.References [453, 454] argued that certain properties of boson stars are also consistent other observedcharacteristics of Sgr A ∗ such as its low accretion rate. ∗ Given the complexities of the accretion flow, a key question is how accurately the shadow can bedetected with the EHT. Since the shape of the shadow can reveal potential deviations from the Kerr44
50 0 50−50050 x ( µ as) y ( µ a s ) k=1, ω =0.7 −50 0 50−50050 x ( µ as) y ( µ a s ) ω =0.77 Figure 31.
Images of boson stars with (left panel) an ADM mass M = 1 . m /m and ADM spin a = 0 . r g and (right panel) an ADM mass M ≈ . m /m (corresponding to the maximum of thecurve M ( ω ), where ω is the frequency of the scalar field; see Ref. [449]) and ADM spin a = 0 . r g using the accretion flow model of Refs. [416,417] with a fixed inner radius. These images are computedat a wavelength of 1.3 mm and an inclination of 85 ◦ . Although the size of the shadow in each imageis similar to the size of the shadow of a Kerr black hole with the same mass and spin, the imagesshow hammer-like features (c.f., Fig. 26) which should allow for these images to be distinguishedfrom the corresponding images of Kerr black holes. In each image, the dotted circles show the 1 σ confidence limits on the angular size of the emitting region imposed by the EHT measurement ofRef. [35], centered on the maximum of the intensity distribution. The solid black contour encompassesthe region emitting 50% of the total flux. Taken from Ref. [449]. metric directly (see Sec. 6.1), such a measurement can, at least in principle, evade the systematicuncertainties that arise from the unknown details of the accretion flow. N o r m a li z e d L i k e li h ood MPE/VLT+S2 starUCLA/KeckStellar clusterUCLA/Keck+S0-2 and S0-38
Figure 32.
Posterior likelihood of the angular size of one gravitational radius (i.e.,
GM/c D ) forSgr A ∗ , as inferred from fitting Keplerian orbits to astrometric observations of S-stars [22, 23, 455].The posterior likelihood in the analysis of Ref. [455] corresponds to an angular size of one gravitationalradius of 5 . ± . µ arcsec. Taken from Ref. [456]. Since the size of the shadow is determined primarily by the mass-distance ratio
M/D , the existing45ass and distance measurements, for which mass and distance are correlated either roughly as M ∼ D in the case of observations of stellar orbits [22,23] or as D ∼ M in the case of the maser observations byRef. [29], can be improved by measurements of this ratio with the EHT [30]. If Sgr A ∗ is indeed a Kerrblack hole, then its angular radius measured by upcoming EHT observations has to coincide with theangular radius inferred from existing measurements of the mass and distance of Sgr A ∗ which constitutesa null test of general relativity [456]. Figure 32 shows the posterior likelihoods of the angular size of onegravitational radius for Sgr A ∗ obtained from two different sets of observations of the S-stars orbitingaround the Galactic center [22, 23]. The posterior likelihood in the analysis of Ref. [23] corresponds toan angular size of one gravitational radius of 5 . ± . µ arcsec.Reference [30] used simple scaling arguments to estimate the precision for a measurement of thesize of the shadow at a wavelength λ with an EHT array comprised of five to six stations. Assumingthat the photon ring surrounding the shadow contributes ∼ /
15 to the total flux and that thesignal-to-noise ratio of such a measurement scales linearly with the uncertainty reported in early EHTobservations [35, 58], Ref. [30] found an uncertainty of σ (cid:39) . × (cid:18) λ (cid:19) − (cid:34) (cid:18) λ (cid:19) − − (cid:35) − µ as . (69) B l a c k H o l e Q u a d r u p o l e M o m e n t Kerr B l a c k H o l e Q u a d r u p o l e M o m e n t Kerr PulsarsStars EHTShadow
Figure 33.
The left panel shows the 68% and 95% confidence contours of the posterior likelihood for anEHT measurement of the asymmetry of the shadow as a function of the spin a and the (dimensionless)quadrupole moment q of Sgr A ∗ . The solid curve shows the expected relation between spin andquadrupole moment for a Kerr black hole, while the filled circle marks the assumed spin and quadrupolemoment ( a = 0 . r g , q = 0 . Based on this estimate, Ref. [211] argued that the EHT can measure the asymmetry of the shadowas defined in Eq. (66) with a precision of σ A = 0 . µ as. Assuming a Gaussian distribution of theasymmetry with a width σ A and a dependence of the asymmetry on the spin and quadrupole momentof Sgr A ∗ as found in Ref. [367] [see Eq. (B.6)], Ref. [211] obtained a Bayesian likelihood of such a46easurement. Figure 33 shows this likelihood as a function of the spin and the quadrupole moment fora Kerr black hole with a value of the spin a = 0 . r g . Figure 33 also shows the corresponding likelihoodsof their simulated measurements of the spin and quadrupole moment using GRAVITY observations oftwo stars and pulsar-timing observations of three periase passages of a low-precision pulsar (see Fig. 18).The contours of the GRAVITY and pulsar-timing observations are nearly orthogonal to the contoursof the EHT measurement reducing the uncertainty of a combined measurement significantly [211]. I n t e n s i t y ( a r b i t r a r y ) -10 -5 0 5 10Displacement (M) Black Hole Shadow o o o o o I m a g e G r a d i e n t ( a r b i t r a r y ) -10 -5 0 5 10Displacement (M) Black Hole Shadow o o o o o Figure 34.
The top row shows (left panel) a simulated 1.3 mm image of Sgr A ∗ , as calculated froma GRMHD simulation of accretion onto a Schwarzschild black hole with an inclination θ = 60 ◦ [394],and (right panel) the brightness profiles of the image along the three indicated cross sections at 0 ◦ ,45 ◦ , and 90 ◦ with respect to the equatorial plane shown on the left panel. In all cases, the rim of theblack-hole shadow corresponds to a sharp drop in the brightness which is consistent to within ∼ . r g .The bottom row shows (left panel) a map of the magnitude of the gradient of the image brightnessand (right panel) profiles of the magnitude of the gradient along the same cross sections as above. Thebright rim along the boundary of the black-hole shadow is clearly visible showing prominent peakswithin ∼ . r g of the location of the shadow. Taken from Ref. [456]. Reference [456] estimated the accuracy with which the size of the shadow can be determinedwith EHT observations at 1.3 mm employing an image of a Schwarzschild black hole from GRMHDsimulations of the accretion flow around Sgr A ∗ [394]. Figure 34 shows the simulated image together47 H o r i z o n t a l L o c a t i o n o f S h a do w C e n t e r ( G M B H / D c ) BH /Dc ) R a do n T r a n s f o r m BH /Dc ) Figure 35.
Left: two-dimensional cross section of the Radon transform of the top left panel inFig. 34 as a function of the opening angle of the shadow and the horizontal location of the center ofthe black-hole shadow when the vertical location of the center of the shadow is set to zero. Right:cross section of the Radon transform for black-hole shadows centered at the known location of theblack hole. The peak of the cross section is centered at the expected opening angle for the simulatedblack hole (vertical dashed line) and has a fractional HWHM of 9%. Taken from Ref. [456]. with the brightness profiles along three chords across the image which are sharply peaked near the rimof the shadow and consistent with the location of the shadow to within 0 . r g . Figure 34 also shows amap of the magnitude of the gradient of the image brightness together with profiles of the magnitudeof the gradient along the same cross sections as in the simulated image. The bright rim along theboundary of the black-hole shadow is clearly visible showing prominent peaks within ∼ . r g of thelocation of the shadow.Reference [456] then used an edge detection scheme for interferometric data and a pattern matchingalgorithm based on the Hough/Radon transform to demonstrate that the shadow of the black hole inthis image can be localized to within ∼ R anda potential offset ( x, y ) from the chosen center of the shadow in this image from Gaussian fits ofthe brightness profile along the chord sections labeled “1”, . . . ,“8” finding R = (26 . ± . µ as, x = ( − . ± . µ as, y = (1 . ± . µ as. This estimate of the angular radius is consistent withthe actual angular radius of the shadow R ≈ . µ as at the 1 σ level corresponding to the values ofthe mass M = 4 . × M (cid:12) , distance D = 8 kpc, and spin a = 0 used in the simulated image shown48 y 81 7 65432 ∆ RA ( µ as) ∆ D E C ( µ a s ) −50 −25 0 25 50−50−2502550 Figure 36.
The left panel shows a reconstructed image of Sgr A ∗ for a simulated EHT observationat 230 GHz with a seven-station array taken from Ref. [457]. The image shows seven chords for whichthe respective angular radii are determined from Gaussian fits of the brightness profile along the chordsections labeled “1”, . . . ,“8.” The right panel shows the resulting distributions of the angular radius R of the shadow and the offset ( x, y ) of the corresponding image center relative to the center of thechords using a Markov chain Monte Carlo sampling of a small region around the center of the chords.The inferred angular radius of ≈ . µ as corresponds to a precision of 6% and a length of ≈ . r g .Taken from Ref. [458]. in Fig. 36; there is no significant offset ( x, y ) of the image center. Figure 36 also shows a triangleplot of the 1 σ and 2 σ confidence contours of the resulting marginalized 2D probabilitiy densities andthe corresponding marginalized 1D probability densities with a Gaussian fit. Since the radius estimatewould be exact for a true image for which the specific intensity peaks at the shadow corresponding tothe longest optical path length of photons in the accretion flow, the method of Ref. [458] seems to haveno significant bias.Reference [458] combined the above simulated EHT measurement of the angular shadow radiusof Sgr A ∗ with existing measurements of its mass and distance assuming a nearly circular shape ofthe shadow and a Gaussian distribution P EHT (data | M, D, a, θ, α , β ) of the angular radius with anuncertainty σ = 1 . µ as and a mean corresponding to the maximum of the distribution P prior ( M, D )of the combined measurements of Refs. [23, 28, 29] assuming a Kerr black hole with spin a = 0 . r g andinclination θ = 60 ◦ . Then, they used Bayes’ theorem to express the likelihood of the mass, distance, anddeviation parameters given the data as P ( M, D, α , β | data) = C P
EHT (data | M, D, α , β ) P prior ( M, D ),where C is a normalization constant and where the likelihood was marginalized over the spin andinclination which only affect the size of the shadow marginally (see Figs. 24 and 25).Figure 37 shows the 1 σ and 2 σ confidence contours of the probability density of the mass anddistance and of the deviation parameters, respectively, for 10 EHT observations. Figure 37 also showsthe constraints on the deviation parameters for future measurements of the mass and distance of Sgr A ∗ obtainable with a 30m-class telescope with estimated uncertainties ∆ M, ∆ D ∼ .
1% [195] combinedwith 100 EHT observations. Here, all EHT measurements are assumed to be independent and identicalso that their uncertainty can be reduced by a factor of √ N .In this setup, the EHT alone can measure the mass-distance ratio (in units of 10 M (cid:12) / kpc)49 igure 37. Left panel: 1 σ and 2 σ confidence contours of the probability density of the massand distance of Sgr A ∗ for existing measurements (S-stars, “G09” [23]; masers, “R14” [29]; starcluster, “C15” [28]), a simulated measurement of the shadow size of Sgr A ∗ for N = 10 observationswith a seven-station EHT array (“EHT”), and several combinations thereof. The simulated EHTmeasurement improves the other constraints on the mass and distance significantly. Center and rightpanels: Simulated 1 σ and 2 σ confidence contours of the probability density of the deviation parameters α and β , respectively, corresponding to N = 10 and N = 100 EHT observations, each marginalizedover the mass and distance using the combination of all data sets (“all”) in the N = 10 case and ofsimulated stellar-orbit observations from a 30m-class telescope [195] in the N = 100 case. Taken fromRef. [458]. Data Mass (10 M (cid:12) ) Distance (kpc)EHT+G09 4 . +0 . . − . − . . +0 . . − . − . EHT+R14 4 . +0 . . − . − . . +0 . . − . − . EHT+C15 4 . +0 . . − . − . . +0 . . − . − . All 4 . +0 . . − . − . . +0 . . − . − . Table 3.
Simulated mass and distance measurements using existing data (G09 [23]; R14 [29]; C15 [28])as priors. Taken from Ref. [458].
M/R = 0 . +0 . . − . − . for N = 10 observations and M/R = 0 . +0 . . − . − . for N = 100observations, respectively. Table 3 lists constraints on the mass and distance corresponding to variouscombinations of the EHT measurements for 10 observations with existing data showing significantimprovements. In particular, combining the EHT result with the parallax measurement by Ref. [29] iscomparable to the mass and distance measurements from stellar orbits including the combined resultof Refs. [23, 28]. If all data sets are combined as shown in the left panel of Fig. 37, Ref. [458] obtainedthe constraints on the deviation parameters α = 0 . +0 . . − . − . , β = − . +0 . . − . − . in the N = 10 case,while, in the N = 100 case, they found α = − . +0 . . − . − . , β = 0 . +0 . . − . − . ; the uncertainties ofthe mass and distance remained at the ∼ .
1% level. Here, all results are quoted with 1 σ and 2 σ errorbars, respectively.The simulated constraints on the deviation parameters α and β also translate into specificconstraints on the parameters of known black-hole metrics in other theories of gravity (RS2, MOG,EdGB, Bardeen; see Sec. 5.1). These constraints are listed in Table 4. Note, however, that the couplingin quadratic gravity theories (i.e., theories that are quadratic in the Riemann tensor) such as EdGBhas units proportional to an inverse length squared (or inverse mass squared in gravitational units).Therefore, much stronger constraints on such couplings can be obtained from observations of stellar-50ass compact objects which have much lower masses and much stronger spacetime curvatures thansupermassive black holes [14, 106]. While the shadow size also depends on the parameter α , its effectis too weak to yield meaningful constraints in this scenario. Theory Constraints ( N = 10) Constraints ( N = 100)RS2 β tidal = − . +0 . . − . − . β tidal = 0 . +0 . . − . − . MOG α = − . +0 . . − . − . α = 0 . +0 . . − . − . EdGB ζ EdGB ≈ +0 . . − . − . ζ EdGB ≈ . +0 . . − . − . Bardeen g /r g ≈ − . +0 . . − . − . g /r g ≈ . +0 . . − . − . Table 4. σ and 2 σ constraints on the parameters of black holes in specific theories of modified gravity(RS2 [151]; MOG [152]; EdGB [153, 154, 156–159]; Bardeen [327, 329]) implied by the simulation ofRef. [458]. At least in the case when the metric of Ref. [66] is interpreted as a vacuum solution in f ( R )gravity, the constraint on the parameter β would imply a constraint on the quadrupole moment ofSgr A ∗ given by the expression M = − M √ − βa [in gravitational units; see Eq. (63) and Ref. [109]].Consequently, the above measurement of the shadow size would infer the quadrupole moment of Sgr A ∗ with a precision of ∼
9% and ∼
5% at the 1 σ level in the N = 10 and N = 100 cases, respectively.The analysis of Ref. [458] estimated the shadow radius from an image of Sgr A ∗ that is constant,thus neglecting small-scale variability in the image. This variability will originate first from the accretionflow itself with a characteristic timescale that is comparable to the period of the ISCO, which rangesfrom about half an hour for a Schwarzschild black hole to approximately four minutes for a maximallyrotating Kerr black hole. Second, electron scatter broadening of the image will blur the image, althoughthis effect is largely invertible as shown in Ref. [457]. Electron scattering will also introduce refractivesubstructure into the apparent image with a characteristic timescale of approximately one day, whichcan also cause image distortions that will vary stochastically from epoch to epoch [457, 459]. However,since Ref. [458] fit the brightness along the chords with Gaussians, their estimate of the shadow radiusis insensitive to remaining uncertainties in the interstellar scattering law. Therefore, in practice, oneimage of a quiescent accretion flow as the one shown in Fig. 36 likely corresponds to an average of severalEHT observations, over which time the source variability will average out [460] (but see Ref. [461]).Likewise, the effects of different realizations of refractive substructure on different observing days willaverage out.The results of Ref. [458] will also be affected moderately by uncertainties in the calibration of theEHT array and in the accretion flow model used for the image reconstruction. The former imposeda 5% uncertainty in early EHT observations with a three-station array, estimated from calibration fortheir visibility amplitudes [36]. For larger telescope arrays such as the seven-station array used in thissimulation, however, many more internal cross-checks will be available to improve the relative calibrationof stations (the absolute calibration is not important). In particular, the use of three individual phasedinterferometers (Hawaii, CARMA, ALMA) that simultaneously record conventional interferometric datawill permit scan-by-scan cross calibration of the amplitude scale of the array. In addition, measurementsof closure phases and closure amplitudes along different telescope triangles and quadrangles are immuneto calibration errors.In contrast to other accretion flow studies (see, e.g., Fig. 29), the uncertainties regarding theemployed accretion flow model likely only play a subdominant role in this simulation as long as a(nearly circular) shadow is clearly visible in the image, because the size and shape of the shadow arealmost entirely determined by the underlying spacetime alone (see Sec. 6.1). Although the method ofRef. [458] relies on the presence of an accretion flow which emits the radiation that comprises the brightring surrounding the shadow, the brightness profiles along the different chords in the image will have51ocal peaks near the location of the shadow corresponding to the longest optical photon path in theaccretion flow irrespective of the details of the accretion flow itself [458].Interpreting combined data sets as in the analysis of Ref. [458] must be done with great care, becauseit can be difficult to properly include their independent systematic uncertainties, which are likely todominate their error budgets. For example, the current orbit-based measurements of Refs. [23,25] nearlydisagree at a statistically-significant level and Ref. [29] neglects systematic errors arising from theirchoice of outlier removal and possible deviations from an axisymmetric velocity field. Before a 30m-classoptical telescope will be available, the uncertainties of mass and distance measurements based on stellarorbits will be further reduced by continued monitoring and the expected improvement in astrometrywith the second-generation instrument GRAVITY for the Very Large Telescope Interferometer [42] (seeSec. 3). Figure 38.
Sample crescent image (left panel), corresponding blurred crescent image (center panel),and visibility amplitudes of the blurred image (right panel) for a crescent model with radii 50 µ as and40 µ as of the outer and inner circles, resectively, where the inner circle is centered at the coordinates(8 µ as , µ as) in the image shown in the left panel. Taken from Ref. [462]. Figure 39.
Most likely image of Sgr A ∗ in the (left panel) image and (right panel) uv -planes inferredfrom a fit of the crescent model of Ref. [462] to early EHT data [35, 36]. Taken from Ref. [462]. References [456, 458] obtained estimates of the shadow radius from images in the image planeinstead of inferring the radius directly from the observed EHT data in the uv -plane, where u and v uv data during the image reconstruction, Ref. [462] constructed a analytical geometricfour-parameter crescent model of the shadow in the uv -plane, where the surface brightness across thetwo overlapping disks in the image is constant. Figure 38 shows an example of a crescent image togetherwith the corresponding scatter-broadened images in the image and uv -planes. Figure 39 shows the mostlikely image of Sgr A ∗ in the image and uv -planes, respectively, inferred from a fit of the crescent modelto the early EHT data [35, 36]. Identifying the inner circle as the shadow of the black hole, this modelcould provide another estimate of the shadow radius. More sophisticated models with a varying surfacebrightness across the disks have been explored in Ref. [463].
7. Variability
Variability in the emission of Sgr A ∗ has been observed at NIR/mm/sub-mm (e.g., [36,403,404,464–477])and x-ray [467, 470–475, 478–483] wavelengths. While the exact mechanism which causes the observedvariability remains unclear, several models for such flares have been proposed. These include the suddenheating of hot electrons in a jet [484], compact flaring structures (“hot spots”) on nearly circular orbitsin the accretion flow around Sgr A ∗ [420, 428, 485] (c.f., [486–488]), the ejection of a plasma blob out ofthe accretion flow [489,490], magnetohydrodynamic turbulence along with density fluctuations [491–493]and particle accelerations due to Rossby wave instabilities [494,495] (c.f., Ref. [496]), and red noise [497].Infalling material such as the gas cloud G2 [441,442], perhaps the product of a binary star merger [498],could also lead to a substantial flux increase over several months [499].For a Kerr black hole, a measurement of the orbital period of a hot spot can be used to infer thespin of Sgr A ∗ and several authors have argued that Sgr A ∗ must be rotating based on observed rapidperiodicities. Reference [500] found quasi-periodic variability at different periods ranging from ∼
100 sto ∼ M = 2 . +0 . − . × M (cid:12) and the spin a = 0 . +0 . − . r g forSgr A ∗ at 1 σ confidence. Reference [404] and Ref. [501] identified quasi-periodic variability with periodsof ∼
17 min and ∼
22 min, respectively, and inferred corresponding values of the spin of a ≈ . r g and a ≈ . r g assuming that the emission originates from the ISCO (see, also, Ref. [502]). Since theKeplerian frequency of a hot spot is highest at the ISCO, Ref. [503] argued that the spin of Sgr A ∗ hasa value a (cid:38) . r g based on a flare with a ∼
13 min period. Reference [504] analyzed the variabilitydetections of Ref. [470] in a two component hot spot/ring model, within which the hot spot travels ontop of a ring-like truncated disk, and found values of the spin 0 . r g ≤ a ≤ r g and inclination θ (cid:38) ◦ at 3 σ confidence. On the other hand, Rossby wave instabilities may naturally produce periodicities onthe order of tens of minutes even if Sgr A ∗ is not spinning [494, 495]. Therefore, these estimates ofthe parameters of Sgr A ∗ and, in particular, of the spin, remain uncertain and the underlying emissionmechanism of flares must be better understood.Deeper insight into the structure of flares is expected to be gained by observations with instrumentssuch as GRAVITY and with the EHT. References [505, 506] simulated GRAVITY observations of suchflares in different models and showed that moving and non-moving flares located at the ISCO radiuscan be distinghed even for faint flares with a K-band magintude of 15 and that flares originating from ablob ejected from the accretion flow can be distinguished from other flare models if the blob is ejectedat an inclination larger than ∼ ◦ and the flare has a duration of (cid:38) . µ as. Figure 40 also shows confidence contours of the directionaldispersion of the simulated flare locations for the three models at different inclinations.References [420,428] designed a 3D hot spot model with a Gaussian density profile of an overdensityof non-thermal electrons in the accretion flow with an extent of a few gravitational radii. The EHT is53
50 −40 −30 −20 −10−30−20−10 0 10 x ( m as) y ( m as ) −50 −40 −30 −20 −10−30−20−10 0 10 x ( m as) y ( m as ) −20 −10 0 10 20 50 100 150 x ( m as) y ( m as )
0 10 20 30 40 0 10 20 30 40
Dispersion x (µas)Inclination: 5 deg, max magnitude: 14 D i s p e r s i on y ( µ as ) Dispersion x (µas)Inclination: 5 deg, max magnitude: 14 D i s p e r s i on y ( µ as ) Dispersion x (µas)Inclination: 5 deg, max magnitude: 14 D i s p e r s i on y ( µ as )
0 10 20 30 40 0 10 20 30 40
Dispersion x (µas)Inclination: 45 deg, max magnitude: 14 D i s p e r s i on y ( µ as ) Dispersion x (µas)Inclination: 45 deg, max magnitude: 14 D i s p e r s i on y ( µ as ) Dispersion x (µas)Inclination: 45 deg, max magnitude: 14 D i s p e r s i on y ( µ as )
0 10 20 30 40 0 10 20 30 40
Dispersion x (µas)Inclination: 85 deg, max magnitude: 14 D i s p e r s i on y ( µ as ) Dispersion x (µas)Inclination: 85 deg, max magnitude: 14 D i s p e r s i on y ( µ as ) Dispersion x (µas)Inclination: 85 deg, max magnitude: 14 D i s p e r s i on y ( µ as ) Figure 40.
The panels in the top row show simulated one-night GRAVITY observations of a flarecaused by Rossby wave instability (left panel), red noise (center panel), or a plasma blob ejected fromthe accretion flow at an angle of 45 ◦ (right panel) assuming realistic astrometric performances of theinstrument with an integration time of 100 s. The dashed red line in each panel shows the theoreticalcentroid track of the flare. The panels in the bottom row show 68%, 95%, and 99% confidence contoursof the dispersions of the measured x and y locations for the flares observed with GRAVITY over 1000nights at inclinations of 5 ◦ , 45 ◦ , and 85 ◦ (left to right panels), corresponding to flares in the Rossbywave instability (blue contours), red noise (red contours), and ejected blob (black contours) models.The ejected blob model can easily be distinguished from the two other models at medium and highinclination, while the other two models cannot be distinguished regardless of the inclination. Takenfrom Ref. [506]. expected to be able to detect such flares and their orbital periods via closure phase/closure amplitudeanalysis [507] and via polarization measurements [508]. Reference [509] estimated that the EHT canmake such detections with a precision of ∼ µ as on timescales of minutes, which is comparable tothe anticipated precision of GRAVITY for similar observations [42, 505]. Reference [510] analyzed thelagged covariance between interferometric baselines of similar lengths but slightly different orientationsand demonstrated that the peak in the lagged covariance indicates the direction and angular velocityof the accretion flow, thus enabling the EHT to measure these quantities.Reference [104] pointed out that measurements of the orbital period of hot spots should be able tomeasure the spin of Sgr A ∗ in that model even if the no-hair theorem is violated, because the Keplerianfrequency of a given hot spot at a fixed radius depends only weakly on deviations from the Kerr metric(see Table 2). In addition to a measurement of the distance of the hot spot from the black hole withGRAVITY [506] or polarimetric VLBI observations [509], EHT observations could also determine that54istance either in combination with an analysis of the data similar to the ones of Refs. [16, 418, 419]or, perhaps, via observations of one hot spot and its tidal deformation [511]. Combined observations ofseveral hot spots at different radii with GRAVITY or the EHT could be used as tracers of the spacetime.The spin of Sgr A ∗ may also be measured by observations of infalling gas inside of the ISCO [512]. Thesetechniques could potentially also constrain deviations from the Kerr metric should they exist. M = N o r m a li ze d I n t e n s it y −3 0 5 10 ε T = N o r m a li ze d I n t e n s it y −3 0 5 10 ε Observer time (in units of
M = E ob s / E e m Figure 41.
The left and center panels show light curves of a compact hotspot centered at the ISCOradius orbiting around the compact object with values of the spin a = 0 . r g and inclination θ = 60 ◦ in the metric of Ref. [64] for different values of the deviation parameter (cid:15) , respectively measured ingravitational units and in units of the orbital period. The width of the light curve decreases and thefrequency of the hot spot increases for increasing values of the parameter (cid:15) , primarily because theISCO radius decreases. The right panel shows a spectrogram (observed to emitted energy ratio asa function of time) of a similar hot spot orbiting around a compact object with values of the spin a = 0 . r g and deviation parameter (cid:15) = − Reference [107] considered a 2D hotspot with a Gaussian density profile located in the equatorialplane of the compact object in the metric of Ref. [64] assuming monochromatic emission. Figure 41shows light curves and spectrograms of hotspots in this model for different values of the spin anddeviation parameter (cid:15) of the compact object. For increasing values of the parameter (cid:15) , the width ofthe light curve decreases and the frequency of the hot spot increases, which is caused primarily by thecorresponding decrease of the ISCO radius. For hot spots orbiting at the same ISCO radius aroundcompact objects with different sets of values of the spin and deviation parameter that correspond tothat radius, there is a slight phase shift between the primary and secondary curves in the spectrogrampotentially allowing these signals to be distinguished if the ISCO can be determined independently [107].Reference [108] considered a similar model, where the hotspot is located at a fixed (small) height aboveor below the equatorial plane and found slight changes of the brightness of the hot spot depending onits position. Reference [513] applied this model to wormholes.
8. Discussion
At present, general relativity remains the standard theory of gravity. Its validity has been confirmedby a number of experiments in the weak-field regime [1] and none of the few strong-field tests have, sofar, detected any deviation from it, neither with observations of neutron stars (see Ref. [4]; [2, 5–8]) norof black holes [9–17]; c.f., Ref. [2]. Likewise, there is no indication of a violation of general relativity oncosmological scales [21].However, general relativity is expected to break down at some level for several theoretical reasonssuch as its nonrenormalizability in a grand-unification scheme (see, e.g., Ref. [338]), the cosmological55onstant problem (see, e.g., Ref. [514]), and the hierarchy problem (see, e.g., Ref. [515]. Moreover, inΛCDM, the general-relativistic standard model of cosmology, dark matter and dark energy make upabout 26% and 69% of the total mass-energy content of the universe, respectively [516–522], but thenature of dark matter and, especially, of dark energy still remains largely mysterious.Thus far, we have barely begun to probe the strong-field regime of general relativity found aroundcompact objects (as well as the cosmological regime) and great progress is expected to be made inthe coming years and decades (c.f., Fig. 1). Tests of general relativity in both regimes require anappropriate underlying framework. For weak-field tests, a description of observables in terms of aparameterized post-Newtonian approach is sufficient [59], while for (model-independent) strong-fieldtests the spacetime itself has to be modelled carefully based on a Kerr-like metric (e.g., [60–68]).The nature of black holes as encapsulated by the (general-relativistic) no-hair theorem providesthe basis for unprecedented tests of general relativity with strong-field and weak-field probes. Sgr A ∗ is a prime target for such tests and three different experiments have high promise for a test of the no-hair theorem in the next few years and decades. NIR monitoring of stars orbiting around Sgr A ∗ has already led to precise measurements of the mass and distance of Sgr A ∗ [22–28]. Continuedmonitoring as well as the expected instrumental improvement with GRAVITY [42] and future 30m-classoptical telescopes (see Ref. [195]) will further improve upon these measurements. Such observationswill most likely detect orbital precessions and radial velocity corrections of stars induced by post-Newtonian effects including frame-dragging or even those caused by the quadrupole moment of Sgr A ∗ ,in particular during pericenter passages of the star S2 (which will take place in 2018) or of other S-stars(e.g., [41, 196, 202, 213]).Timing observations of radio pulsars on orbits around Sgr A ∗ could provide another precisemeasurement of the mass, spin, and quadrupole moment of Sgr A ∗ [53–55] or of stellar-mass blackholes in binaries with pulsars [523]. For Sgr A ∗ , such observations, carried out over about five yearswith an SKA-like telescope, could detect frame-dragging at the 10 − level and test the no-hair theoremat the 10 − level [55]. The recent discovery of a magnetar at a distance of only ∼ . ∗ [47–51] has spurred the hopes of finding a suitable pulsar that is close enough to the Galacticcenter. Although many pulsar searches have been conducted at high observing frequencies over severalyears [50, 261, 262, 265–268], the discovery of such a pulsar may require targeted surveys with theSKA [269–271]. Both of these methods (NIR and timing observations) track the orbits of stars orpulsars around Sgr A ∗ , which may be perturbed by surrounding stars [202], drag forces [203], stellarwinds and tidal disruptions [204, 209, 210], or other effects [208].The EHT is expected to probe Sgr A ∗ on event-horizon scales and to take the first direct imageof a black hole. The size and shape of the shadow of Sgr A ∗ (or any other black hole) dependsdirectly on the properties of the underlying spacetime, i.e., on the mass and spin for Kerr black hole(e.g., [363,368]) as well as on potential deviations from the Kerr metric if the no-hair theorem is violated(e.g., [367, 375, 377]). In addition, deviations from the Kerr metric can modify the properties of theaccretion flow surrounding the black hole (e.g., [12, 104, 340, 448]). Early EHT observations of Sgr A ∗ in 2007–2013 with a three-station array have resolved structure [35] and detected variability [36] aswell as polarized emission [37] on event horizon scales. Furthermore, Refs. [36, 353] detected a numberof closure phases along the initial CARMA–SMTO–Hawaii telescope array. As of 2015, the EHT iscomprised by eight different sites, and VLBI observations with telescope arrays that include more thanthree stations are scheduled to begin in spring 2016.Within the context of RIAF model images, the early EHT data favor small values of thespin [16,418,419,440], while constraints on deviations from the Kerr metric remain weak [16]. However,such constraints within a specific RIAF model will improve dramatically with EHT observations usinglarger telescope arrays [444]. Since the shadow itself is largely independent from the properties of theaccretion flow, measurements of its size and shape in combination with existing stellar-orbits data canbe used to improve upon current measurements of the mass and distance of Sgr A ∗ and to infer potential56eviations from the Kerr metric with high precision [456, 458].Both NIR observations with instruments such as GRAVITY and VLBI observations with the EHTshould also probe regions of quasi-periodic emission in the accretion flow of Sgr A ∗ with unprecedentedprecision [42, 505–508]. Such observations may also distinguish between different models [505, 506] andinfer other properties of Sgr A ∗ and its accretion flow (e.g., [506,509,511]). In particular, measurementsof the orbital period of flares can be used to constrain the spin (and perhaps even the quadrupolemoment) of Sgr A ∗ (e.g., [404, 470, 500, 503, 504]). Current spin estimates based on detections ofvariability, however, are uncertain and cover practically the entire range of spin values from ≈ ≈ f ( R )gravity [109]. One solution to this issue is perhaps the approach of Ref. [196] who analyzed stellarorbits in the Kerr spacetime using a ray-tracing algorithm. Performing such an analysis in a Kerr-like spacetime would directly link weak-field probes of stars orbiting around the Galactic center withstrong-field observables such as the shadow or the accretion flow of Sgr A ∗ .In addition to Sgr A ∗ , the supermassive black hole at the center of M87 is another prime targetof the EHT and early EHT observations at 230 GHz with three-station telescope arrays have alreadydetected structure on the order of ≈ . ∗ , EHT observations of this supermassiveblack hole do not face the same challenges with scattering or refractive time scales that require extraanalysis effort. Given its much greater mass ( ∼ − × M (cid:12) [526, 527]), the time scales for M87are also longer and the rotation of the Earth is less of a challenge. In addition, the spatial scales ofstrong-gravity signatures are approximately comparable to those in Sgr A ∗ , but the time scales forstrong-gravity effects such as the orbital period of matter particles near the ISCO are much longerand, therefore, tractable via time sequenced EHT observations that allow full imaging fidelity in eachepoch [38, 39].Reference [528] simulated images of the supermassive black hole at the center of M87 at 230 GHz and345 GHz based on a 7–8 station EHT array assuming realistic measurement conditions. Reference [528]showed that such an array would have a resolution of 20 − µ as (2 − × M (cid:12) for the supermassive black holein NGC 1277, which, therefore, has an angular shadow size of roughly 7 µ as in the sky (assuming adistance of 71 Mpc as in Ref. [530]). These black holes may be resolvable on horizon scales with futureVLBI stations in space [531–534]. In any case, the prospects for a test of the no-hair theorem and,57hereby, of general relativity within the coming years are great.I thank P. Cunha, S. Doeleman, C. Herdeiro, and P. Pani for useful comments. This work wassupported in part by Perimeter Institute for Theoretical Physics. Research at Perimeter Institute issupported through Industry Canada and by the Province of Ontario through the Ministry of Research& Innovation. Appendix A. Mappings to Black Holes in Alternative Gravity Theories
Here, I summarize the mappings of the metric of Ref. [66] [see Eq. (52)] to the analytically known blackhole solutions of EdGB and dCS gravity, as well as to other Kerr-like metrics. Some of these mappingswere derived in Ref. [66].
Appendix A.1. Einstein-dilaton-Gauss-Bonnet Gravity
Static and slowly-rotating black holes in gravity theories described by Lagrangians modified fromthe standard Einstein-Hilbert form by scalar fields coupled to quadratic curvature invariants wereinvestigated in Refs. [153, 154, 156–159]. In the limit of small deviations, the Kerr metric is modifiedby a perturbation h αβ , which, to linear order in the spin a and the parameter ζ EdGB , has the nonzerocomponents h EDBG tt = − ζ EdGB M r (cid:18) Mr + 665 M r + 965 M r − M r (cid:19) , (A.1) h EdGB rr = − ζ EdGB (cid:18) − Mr (cid:19) − M r × (cid:18) Mr + 523 M r + 2 M r + 165 M r − M r (cid:19) , (A.2) h EdGB tφ = 35 ζ EdBG aM sin θr × (cid:18) Mr + 10 M r + 16 M r − M r (cid:19) . (A.3)Here, the parameter ζ EdGB is defined by the equation ζ EdGB ≡ πα β EdGB M , (A.4)where α EdGB and β EdGB are the coupling constants of the theory; see, e.g., Ref. [158].The mapping is, then, given by the equations ∞ (cid:88) n =3 α n (cid:18) Mr (cid:19) n = − M ζ EdGB r − M ) r × (5 r + 130 M r + 66 M r + 96 M r − M ) , (A.5) ∞ (cid:88) n =2 α n (cid:18) Mr (cid:19) n = − M ζ EdGB r − M ) r (13 r + 134 M r + 74 M r + 96 M r − M ) , (A.6) ∞ (cid:88) n =2 α n (cid:18) Mr (cid:19) n = M ζ EdGB r − M ) r (15 r + 15 M r + 260 M r + 30 M r + 48 M r − M ) , (A.7)(A.8)58nd the lowest-order coefficients are: α = − ζ EdGB , α = − ζ EdGB , α = − ζ EdGB ,α = − ζ EdGB , α = − ζ EdGB , α = − ζ EdGB ,α = ζ EdGB , α = 3 ζ EdGB , α = 703 ζ EdGB . (A.9)Reference [66] only contains the mapping to the static black hole solution. In this case, A ( r ) = 1and all parameters α i , i ≥
2, vanish, leaving A ( r ) and A ( r ) as the remaining deviation functions.Note, however, that in Ref. [66] the mapping of the deviation function A ( r ) is missing and thatEqs. (133) and (134) contain a typo. As shown in Ref. [158], the black hole metric in EdGB gravity isnot integrable at O ( a ). Therefore, it can only be mapped to the metric of Ref. [66] up to O ( a ). Appendix A.2. Dynamical Chern-Simons Gravity
Slowly rotating black holes in dCS gravity were first analyzed in Ref. [161]. In these solutions, only the( t, φ ) component of the metric is modified, which is given by the expression h dCS tφ = 58 ζ dCS aM r sin θ (cid:18) M r + 27 M r (cid:19) . (A.10)In this case, the mapping is α = 58 ζ dCS , (A.11) α = 1514 ζ dCS , (A.12) α = 2716 ζ dCS . (A.13)All other deviation parameters vanish. Note that the metric to O ( a ) found in Ref. [162] is likewise notintegrable and, thus, cannot be mapped to the metric of Ref. [66]. Appendix A.3. The Modified Gravity Bumpy Kerr Metric
Reference [79] constructed an explicit form of the modified gravity bumpy Kerr metric [65], given bythe equation g MGBK µν = g K µν + h MGBK µν , (A.14)where the g K µν is the Kerr metric in Eq. (3). The correction h MGBK µν depends on three (nonzero) deviationfunctions given by the expansions γ A ≡ ∞ (cid:88) n =0 γ A,n (cid:18) Mr (cid:19) n , A = 1 , , (A.15) γ ≡ r ∞ (cid:88) n =0 γ ,n (cid:18) Mr (cid:19) n , (A.16)where γ , = γ , = γ , = γ , = γ , = γ , = 0 [79] as well as (preferentially) γ , = γ , = γ , =0 [66]. The fourth deviation function, Θ ( θ ), is set to zero [79].This metric can be mapped to the metric of Ref. [66] (expanded to linear order in the deviationparameters) via the relations (cid:15) n = 0 , n ≥ , (A.17) α n = γ ,n , n ≥ ∞ (cid:88) n =3 α n (cid:18) Mr (cid:19) n = 14( r + a )∆ { aM rh MGBK tφ + [2 r + a + a r (3 r + 4 M ) (A.19)+ a (∆ − M r ) cos 2 θ ] h MGBK tt } , (A.20) ∞ (cid:88) n =2 α n (cid:18) Mr (cid:19) n = − a ∆ [ a (Σ − M r ) h MGBK tt + 2(Σ − M r ) csc θh MGBK tφ ] . (A.21)At least up to order n = 5, the latter two equations can be written in the form α n − α n − γ ,n − + γ ,n ,α n − α n aM γ ,n − − Ma (2 γ ,n − γ ,n +1 ) , (A.22)where the first equation holds for n ≥
3, while the second equation holds for n ≥ Appendix A.4. The Metric of Ref. [68]
The metric of Ref. [68] has the following nonvanishing elements: g tt = − (cid:18) − m ( r ) r Σ (cid:19) ,g rr = Σ∆ ,g θθ = Σ ,g φφ = (cid:18) r + a + 2 a m ( r ) r sin θ Σ (cid:19) sin θ,g tφ = − am ( r ) r sin θ Σ , (A.23)where ∆ ≡ r − m ( r ) r + a (A.24)and where Σ is given by Eq. (5).There is no direct mapping between this metric and the Kerr-like metric of Ref. [66] except for thetrivial Kerr (or Kerr-Newman) case. Whether or not there exists a coordinate transformation that canrelate these metrics is unclear.The above metric depends on two deviation functions m ( r ) and m ( r ), where the former occurs inthe ( t, t ), ( t, φ ), and ( φ, φ ) elements and the latter in the ( r, r ) element; the ( θ, θ ) element is unmodified.It is straightforward to generalize this metric by introducing deviation functions m i ( r, θ ), i = 1 , . . . , ≡ Σ + f ( r ) as defined in Eq. (57) and writing the metric elements in the form g tt = − (cid:18) − m ( r ) r ˜Σ (cid:19) ,g rr = ˜Σ∆ ,g θθ = ˜Σ , φφ = (cid:18) r + a + 2 a m ( r ) r sin θ ˜Σ (cid:19) sin θ,g tφ = − am ( r ) r sin θ ˜Σ . (A.25)The resulting metric can be mapped to the metric of Ref. [66] by the choices m = ˜Σ2 r (cid:34) (cid:0) a A sin θ − ¯∆ (cid:1)(cid:2) A ( r + a ) − a A sin θ (cid:3) (cid:35) ,m = r + a − ¯∆ A r ,m = ˜Σ2 a r sin θ ˜Σ (cid:16) A (cid:0) r + a (cid:1) − a ¯∆ sin θ (cid:17)(cid:2) A ( r + a ) − a A sin θ (cid:3) − r − a ,m = ˜Σ (cid:2)(cid:0) r + a (cid:1) A A − ¯∆ (cid:3) r (cid:2) A ( r + a ) − a A sin θ (cid:3) (A.26)for the deviation functions m i ( r, θ ), i = 1 , . . . ,
4, as can be shown by equating the corresponding elementsof both metrics and solving for the deviation functions in the metric in Eq. (A.25). Here, ¯∆ is definedin Eq. (53). Expanding the RHS in powers of
M/r [including the functions A , A , A , and f as inEqs. (54)–(56) and (58)], the functions m i ( r, θ ), i = 1 , . . . ,
4, at the lowest two nonvanishing orders inthe deviation parameters are giving by the expressions m = M − βM r + (2 α − (cid:15) ) M r ,m = M − ( α + β ) M r + (2 α − α ) M r ,m = M + (cid:15) M a sin θ + a M (2 α − β ) + (cid:15) M sin θ a r + (cid:16) a α + a (cid:15) + M (cid:15) sin θ (cid:17) M a r ,m = M + ( α − β ) M r + ( α + α ) M r , (A.27)assuming a (cid:54) = 0 (as well as sin θ (cid:54) = 0) in the latter two equations. If a = 0 (or sin θ = 0), then the metricno longer depends on the functions m and m . Note that the function m can have a term of order M/r (which is not ruled out by weak-field constraints as claimed in Ref. [68]) and that m contains azeroth-order term. Appendix A.4.1. The Metric of Ref. [329]
In the rotating generalization of the static Bardeen metric [327] constructed in Ref. [329], the deviationfunctions m i ( r ), i = 1 , . . . ,
4, take the form m = . . . = m = r ( r + g ) / M. (A.28)Therefore, using the mapping in Eq. (A.27) up to O ( M /r ), the nonvanishing deviation parametersare given by the equations α = − g M , α = 2 α , α = 3 g (cid:0) a + 5 g − M (cid:1) M ,α = α , α = α , α = α ,α = − α , α = − α , α = − α . (A.29)61his seems to suggest a general mapping of the form A ( r ) = A ( r ) = − A ( r ), f ( r ) = 0 as well as β = 0. However, imposing this general mapping in Eq. (A.26) leads to the equation A ( r ) = − / which contradicts Eq. (54). Therefore, this general mapping cannot be correct. Appendix A.5. The Metric of Ref. [535]
Reference [535] propsed a modified Schwarzschild metric of the form g tt = − (cid:18) − MF (cid:19) ,g rr = F (cid:48) − MF ,g θθ = F r ,g φφ = F r sin θ, (A.30)where F is a function of radius fulfilling suitable boundary conditions in the limit r → ∞ and F (cid:48) is itsderivative.Setting a = 0 and β = 0 in the metric of Ref. [66] and comparing the respective metric elements, Iarrive at the equations F ( r ) = r (cid:2) r + f ( r ) (cid:3) ,F ( r ) = − (cid:104) M r ( r − M ) A ( r ) + (cid:112) r ( r − M ) [27 M r − M − r A ( r ) ] (cid:105) − × r − M ) (cid:20) r ( r − M ) A ( r ) + 3 (cid:20) M r ( r − M ) A ( r ) + (cid:112) r ( r − M ) [27 M r − M − r A ( r ) ] (cid:21) (cid:21) ,F (cid:48) ( r ) = ± (cid:115) F ( r )[ F ( r ) − M ] r ( r − M ) A ( r ) , (A.31)which can be rewritten as complicated relations between the deviation functions f , A , and A . Thedeviation function A does not occur in the metric elements due to the condition a = 0. Appendix B. Approximate Expressions of the Displacement, Diameter, and Asymmetryof the Shadow
The diameter, displacement, and asymmetry of a shadow around a black hole (or other compact object)can be determined approximately from fits of a large set of simulated shadows. If the metric possessesa third constant of motion, the locations and shapes of these shadows can be calculated analytically.Otherwise, they have to be calculated numerically.The diameter L K , displacement D K , and asymmetry A K of a Kerr black hole are given by theapproximate expressions [377] L K ( θ, a ) ≈ L K1 + L K2 cos (cid:0) n K1 θ + ϕ K (cid:1) , (B.1) D K ( θ, a ) ≈ D K1 sin (cid:0) n K2 θ (cid:1) , (B.2) A K ( θ, a ) ≈ A K1 θ n K3 + A K2 θ n K4 , (B.3)62here L K1 = 10 . − . a . − . a . ,L K2 = 0 . a . + 0 . a . − . a . + 0 . a . − . a . + 0 . a . − . a . ,n K1 = − . − . a . + 0 . a . + 0 . a . ,ϕ K = 3 . . a . + 0 . a . − . a . ,D K1 = 2 a + 0 . a . + 0 . a . + 0 . a . + 0 . a . ,n K2 = 1 − . a . + 0 . a . + 0 . a . + 0 . a . − . a . − . a . ,A K1 = − . − a ) . + 4 . a . + 1 . a . + 0 . a . + 0 . a . ,n K3 = − . . − a ) . + 0 . a . + 0 . a . − . a . − . a . ,A K2 = 0 . − a ) . + 1 . a . − . a . − . a . − . a . ,n K4 = 0 . . − a ) . + 1 . a . − . a . + 0 . a . + 0 . a . + 0 . a . . (B.4)These fits are valid for values of the spin 0 r g ≤ a ≤ . r g . The fit formula of the shadow diameteris accurate to < .
09% for spin values 0 r g ≤ a ≤ . r g and to < .
16% for spins 0 r g ≤ a ≤ . r g .The fit formula of the displacement is accurate to < .
1% for spin values 0 r g ≤ a ≤ . r g , to < . r g ≤ a ≤ . r + g , and to < .
7% for spins 0 r g ≤ a ≤ . r g . For values of the asymmetry A ≥ . r g , the asymmetry fit is accurate to <
1% for spin values 0 r g ≤ a ≤ . r g , to < .
8% for spins0 r g ≤ a ≤ . r g , and to < .
4% for spins 0 r g ≤ a ≤ . r g . The largest uncertainties in the fitoccur at low inclinations, and the fit is accurate at all spin values to < .
5% for inclinations θ ≥ . ◦ .Values of the ring asymmetry A < . r g , which occur only at spin values a ∼ θ ∼ A sin n θ as in Refs. [367,368]. Such a fit, however, introduces comparativelylarge errors at high spins, where the asymmetry deviates significantly from a sinosoidal form [377].For the shadows around the compact objects described by the quasi-Kerr metric [62], thedisplacement D (cid:15) and asymmetry A (cid:15) are given by the expressions [367] D (cid:15) ≈ a sin θ (1 − . (cid:15) sin θ ) , (B.5) A (cid:15) ≈ (cid:34) . (cid:15) + 0 . (cid:18) ar g (cid:19) (cid:35) sin / θ, (B.6)which are valid for 0 r g ≤ a ≤ . r g and 0 ≤ (cid:15) ≤ . L α , displacement D α , and asymmetry A α of a Kerr-like black hole described bythe metric of Ref. [66] as a function of the parameters α and α are given by the approximateexpressions [377] L α ( θ, a, α , α ) ≈ L K1 + L α α + L α α + L α α + L α α L K2 (1 + L α α )(1 + L α α ) cos (cid:0) n K1 θ + ϕ K (cid:1) , (B.7) D α ( θ, a, α , α ) ≈ D K1 (cid:0) D α α + D α α (cid:1) (cid:0) D α α + D α α (cid:1) × sin (cid:2) n K2 (1 + n α α + n α α )(1 + n α α + n α α ) θ (cid:3) , (B.8) A α ( θ, a, α , α ) ≈ (1 + A α α + A α α + A α α )(1 + A α α + A α α + A α α ) × (cid:104) A K1 (1 + A α α )(1 + A α α ) θ n α + A K2 (1 + A α α )(1 + A α α ) θ n α (cid:105) − , (B.9)where L α ≡ . . a . + 0 . a . ,L α ≡ − . − . a . − . a . ,L α ≡ − . a . − . a . ,L α ≡ − . a . − . a . ,L α ≡ − . . a . ,L α ≡ − . a . ,D α ≡ − . − . a . − . a . ,D α ≡ . . a . + 0 . a . ,D α ≡ . . a . − . a . ,D α ≡ . . a . − . a . ,n α ≡ − . − . a . + 2 . a . ,n α ≡ . − . a . + 1 . a . ,n α ≡ . − . a . + 0 . a . ,n α ≡ − . − . a . + 0 . a . ,A α ≡ . − . a . ,A α ≡ − . . a . ,A α ≡ . − . a . ,A α ≡ . . a . ,A α ≡ . − . a . ,A α ≡ . − . a . ,A α ≡ . . a . ,A α ≡ . − . a . ,A α ≡ . − . a . ,A α ≡ . . a . . (B.10)These fits are valid for values of the spin 0 r g ≤ a ≤ . r g and of the deviation parameters − ≤ α , α ≤
2. The fit of the diameter is accurate to < .
5% for spins 0 r g ≤ a ≤ . r g and to < .
5% for spins 0 r g ≤ a ≤ . r g . The fit of the displacement is accurate to < .
3% in spin range0 . r g ≤ a ≤ . r g with an average accuracy of 2%. The accuracy is significantly smaller at high andvery low spins and the error can exceed 100% in some cases. Finally, the fit of the asymmetry has anaverage accuracy of < . L β , displacement D β , and asymmetry A β of a Kerr-like black hole described by themetric of Ref. [66] as a function of the parameters β are given by the approximate expressions L β ( i, a, β ) ≈ L K1 + L β β + L β β + L K2 (1 + L β β ) cos (cid:0) n K1 i + ϕ K (cid:1) , (B.11) D β ( i, a, β ) ≈ D K1 (cid:16) D β β + D β β (cid:17) sin (cid:0) n K2 i (cid:1) , (B.12) A β ( i, a, β ) ≈ (1 + A β β + A β β + A β β ) × (cid:104) A K1 (1 + A β β ) i n K3 + A K2 (1 + A β β ) i n K4 (cid:105) − , (B.13)where L β ≡ − . − . a . − . a . ,L β ≡ − . . a . ,L β ≡ . − . a . ,D β ≡ . − . a . ,D β ≡ . − . a . ,A β ≡ . − . a . ,A β ≡ . − . a . ,A β ≡ . − . a . ,A β ≡ . − . a . ,A β ≡ − . − . a . . (B.14)These fits are valid for values of the spin 0 r g ≤ a ≤ . r g and of the deviation parameter − . ≤ β ≤ < . < < β close to the upper bound definedin Eq. (59). References [1] Will C M 2014, Living. Rev. Rel.
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