Can supermassive black hole shadows test the Kerr metric?
CCan supermassive black hole shadows test the Kerr metric?
Kostas Glampedakis
1, 2, ∗ and George Pappas † Departamento de F´ısica, Universidad de Murcia, Murcia, E-30100, Spain Theoretical Astrophysics, University of T¨ubingen,Auf der Morgenstelle 10, T¨ubingen, D-72076, Germany Department of Physics, Aristotle University of Thessaloniki, Thessaloniki 54124, Greece (Dated: March 1, 2021)The unprecedented image of the M87* supermassive black hole has sparked some controversy overits usefulness as a test of the general relativistic Kerr metric. The criticism is mainly related to theblack hole’s quasi-circular shadow and advocates that its radius depends not only on the black hole’strue spacetime properties but also on the poorly known physics of the illuminating accretion flow.In this paper we take a sober view of the problem and argue that our ability to probe gravity witha black hole shadow is only partially impaired by the matter degrees of freedom and the number ofnon-Kerr parameters used in the model. As we show here, a much more fatal complication arisesfrom the mass scaling of the dimensional coupling constants that typically appear in non-GR theoriesof gravity. Existing limits from gravitational wave observations imply that supermassive systemslike the M87* black hole would suffer a suppression of all non-GR deviation parameters in theirmetric, making the spacetime and the produced shadow virtually Kerr. Therefore, a supermassiveblack hole shadow is likely to probe only those extensions of General Relativity which are endowedwith dimensionless coupling constants.
Context.—
The centenary of the Eddington-Dyson1919 observation of light deflection by the sun [1] wasmarked by another important milestone in gravitationalphysics, the release of the direct image of the supermas-sive black hole in the nucleus of the M87 galaxy by theEvent Horizon Telescope (EHT) collaboration [2–4]. Thismillimeter-band radio image of unprecedented angularresolution, itself an example of extreme light deflection,has provided direct quantitative evidence of the presenceof supermassive black holes in galactic centers and hasshed some light in the inner workings of active galacticnuclei. A second image, that of our galactic SgrA* su-permassive black hole, is scheduled to be released by theEHT collaboration in the near future.From the point of view of fundamental physics, a keyelement of an image like that of the M87* black hole is thegeometric shape of the shadow seen by an asymptotic ob-server, as superimposed in the brighter background of theluminous matter surrounding the black hole. By its verynature, and in contrast to the observation of gravitationalwaves from compact binary systems, the generation andobservation of a black hole image is an ‘experiment’ onthe geodesic motion of photons emitted by the accretionflow, and therefore probes the geometry rather than thedynamical properties of the spacetime.One of the key motivations behind the conception ofEHT was to use the shadow as evidence for the exis-tence of black holes and as a probe of General Relativity(GR) [5]. This exciting possibility has spawn a signifi-cant amount of work over the last decade or so, mostlyfocused on the calculation of shadows of non-Kerr blackholes beyond GR [6–14] but also on improving our under- ∗ [email protected] † [email protected] standing of the image produced by garden-variety Kerrblack holes [15–17].Throughout this paper we use relativistic units G = c = 1. Probing gravity with black hole shadows.—
In a recentpaper, Psaltis et al. [18] used the physical shape of theM87* black hole shadow as a test of GR. A prerequisitefor this type of test is the independent knowledge of theblack hole’s mass, that is, the system’s intrinsic yard-stick. In the case of M87* the mass has been estimatedto be M = (6 . ± . × M (cid:12) by stellar kinematicswith the assumption of a distance of 17 . dimensionless parameters { α , α , α , ε } . TheJ-metric is Kerr-like in the sense that it is separable (thusadmitting a third constant of motion like the Carter con-stant), admits spherical photon orbits [21], and is en-dowed with a spherical event horizon with the same ra-dius r + = M + √ M − a as a Kerr black hole of the samemass M and spin a . Therefore, it is not too surprisingthat black holes in the J-metric cast Kerr-like shadows[7, 14], namely, shadows that have a nearly constant cir-cular radius unless the spin parameter a lies close to itsmaximum allowed value. Based on this property, one canfocus on a non-rotating system for which the shadow isexactly circular with its radius given by the impact pa-rameter b associated with the unstable circular photonorbit, i.e. the black hole’s photon ring. The photon ringradius r ph and the impact parameter depend on the sin-gle metric component g tt ( r ) = − (cid:18) − Mr (cid:19) (cid:0) ε M /r (cid:1) (1 + α M /r ) , (1) a r X i v : . [ g r- q c ] F e b and are given by b = r ph (cid:112) − g tt ( r ph ) , r ph dg tt dr ( r ph ) = 2 g tt ( r ph ) . (2)The aforementioned strategy was adopted in Ref. [18] andthe results are summarised in Fig. 1 where we show b fora non-rotating black hole in the J-metric as a function ofthe deformation parameters { a , ε } . The reported 17%uncertainty in the observed shadow radius [18] translatesinto an allowed range − . (cid:46) a (cid:46) − (cid:46) ε (cid:46)
12 for the deformation parameters. It can be noticedthat the two parameters are anti-correlated: a positive(negative) a ( ε ) gives rise to a bigger shadow relativeto the canonical GR radius b GR = 3 √ M (and vice versafor the opposite signs).This, however, is not the end of the story: the shadowradius is also a function of the geometry of the illuminat-ing accretion flow [15]. Assuming GR gravity, the radiusis given by b GR when the black hole is ‘backlit’ from adistant uniform source. The same is true for the more as-trophysically relevant scenario of a spherically symmetricflow in the vicinity of the black hole [15, 22]. In contrast,illumination by a thin accretion disk leads to a somewhatlarger shadow radius b ≈ . M [15]. A more realistic al-ternative possibility for a system like M87* is that of ageometrically thick/optically thin disk; in such a case theanalysis of Ref. [15] suggests a shadow radius of b ≈ . M that lies between the two previous values.The uncertainty in b caused by the unknown accretionphysics of M87* is shown in Fig. 1 as a grey band and hassome clear implications for the earlier constraints on thedeviation from GR. Most of the parameter space pre-viously associated with an enlarged black hole shadowas a result of deviations from GR is now occupied bythe more prosaic accretion physics ‘error’ [23]. Only thespacetime deviations for which b < b GR can be cleanlyprobed by the shadow measurement. Indeed, and giventhat it marks the peak of the geodesic potential, b GR isthe absolute minimum shadow radius irrespectively ofthe accretion physics details.The situation could deteriorate further if both defor-mation parameters are non-vanishing [24]. As a conse-quence of their anti-correlation, the shadow of a blackhole with α ∼ ε could lie significantly closer to b GR forthe same degree of deformation. This is exemplified bythe dashed curve in Fig. 1 which shows b for α = 1 . ε .It is clear that in such a case most of the deviation awayfrom b GR overlaps with the accretion physics error andthe constraints on α , ε are far less reliable. Of course,we would have drawn the exact opposite conclusion if α ∼ − ε .The upshot of this discussion is that the quality ofblack hole shadows as probes of GR gravity could bediluted by the system’s matter degrees of freedom andby possible (anti)correlations between the non-GR pa-rameters of the metric. An additional complication liesin the shadow’s shape itself; nearly circular shadows arerelatively ubiquitous in non-Kerr spacetimes, resembling - � � � ����������������� ����������� ��������� � / � α ε α = ε FIG. 1. The impact parameter/shadow radius b associatedwith the photon ring of a non-rotating black hole in the J-metric as a function of the deformation parameters α , ε .The middle dashed line indicates the GR value b GR = 3 √ M while the upper/lower dashed lines mark the ≈
17% observa-tional error in the image of M87*. The shaded region repre-sents the range in the GR value of b [as estimated by Grallaet al. (2019)] due to the variation in the geometry of the ac-cretion flow (spherical or thick disk accretion). The dashedcurve represents the case where the deformation parametersare correlated as α = 1 . ε (with the horizontal axis mea-suring ε ) , resulting in a shadow radius much closer to b GR . the shape of a Kerr shadow for most of the allowed spinrange. As a case in point, consider the parametrisedmetric of Carson & Yagi [25] which is an extension ofthe J-metric and represents the most general family ofseparable and asymptotically flat spacetimes. This is ametric that can be mapped on black hole solutions orig-inating from various modified theories of gravity; it toocan lead to Kerr-like quasi-circular shadows for a widerange of its deformation parameters, especially when thespin is not too high. The importance of non-GR coupling parameters.—
In asense, our discussion so far was just the tip of the prover-bial iceberg because even if we were to put aside the com-plications related to the physics of the accretion flow andthe commonness of quasi-circular shadows, we would stillhave to face a much more serious problem related to themass dependence of the non-Kerr deformation parame-ters. The key issue here is to understand to what extentthe constraints placed on these parameters by the M87*image are compatible with limits placed by gravitationalwave observations of merging black holes [26–28] or elec-tromagnetic observations of astrophysical black holes inX-ray binaries [29].The crucial importance of the mass scaling of a grav-ity theory’s non-GR parameters was recently emphasizedin Ref. [30], in the context of gravitational waves fromextreme mass ratio inspirals (EMRIs) in supermassiveblack holes. In order to understand the impact of themass, we follow the reasoning of this recent work and con-sider modified theories of gravity, as extensions of GR,described by an action of the following general form, S = S GR ( g µν , φ ) + α S c ( g µν , φ ) + S m ( g µν , φ, Ψ) , (3)where g µν is the metric, φ is the scalar field degree offreedom and Ψ stands for the matter fields. The firstterm represents the GR part of the action, S GR = 116 π (cid:90) dx √− g (cid:18) R − ∂ µ φ ∂ µ φ (cid:19) , (4)where R is the Ricci scalar and g is the metric determi-nant. The last term is the matter part of the action andcan be set to zero for the purposes of this paper. Thetheory’s non-GR physics is encapsulated in the term S c which describes non-minimal couplings between g µν and φ ; the factor in front of this term is the theory’s couplingconstant .First we need to distinguish between theories that ad-mit the Kerr metric as a black hole solution (e.g. [31]) andthose that do not. Given that a black hole shadow is es-sentially the result of photons moving along the geodesicsof the hole’s spacetime, it is clear that the shadow inthe former theories is identical to the GR Kerr shadow.The latter family of theories should generically lead tonon-Kerr shadows and hereafter we focus on them. Intheir vast majority, these theories are endowed with a dimensional coupling constant that scales as α ∼ M n with n ≥ M , a spin parameter a/M and a dimensionless coupling constant ζ ≡ αM n . (5)The hole’s scalar field is expanded in ζ around its con-stant asymptotic value and enters the metric through adimensionless function of order unity that multiplies ζ (e.g., see Ref. [33]). The appearance of ζ should notcome as a surprise since, as we have pointed out, themass is the system’s only available dimensional scale.Theory-agnostic deformed Kerr metrics can be typicallymapped onto specific ‘genuine’ theories with dimensionalconstants. Examples are provided by the general Carson-Yagi metric mentioned earlier [25], and the J-metric usedin this paper. As it turns out, in all cases the deformationparameters are simply related to the coupling constantof the corresponding gravity theory (e.g. [20, 25]). Forthe case of the J-metric we typically have α ∼ ζ k with k ≥
1, and similarly for the other parameters.At this point we may return to the analysis of theM87* shadow and examine what are the implications ofusing non-GR models with dimensional constants for itsdescription. As we have seen in the previous section theconstraint on α , when expressed in terms of the cou-pling constant ζ of a given theory, amounts to | ζ | (cid:46) s ⇒ | α | (cid:46) sM n , (6) The conclusions of this paper should be equally applicable toactions more general than (3), including additional scalar fieldsand coupling constants. where s ∼
10 approximately. We imagine that the samenon-GR model is also used in the study of the gravita-tional wave-driven inspiral and merger of a black holebinary system of typical mass M b , resulting in a similarconstraint | ζ b | (cid:46) s b , where ζ b = αM n b = ζ (cid:18) MM b (cid:19) n . (7)Existing limits from astrophysical observations (see e.g.[34, 35]) suggest s b (cid:46)
1. Using this as a fiducial limit forour J-metric model, we find | α | ∼ | ζ b | k (cid:18) M b M (cid:19) kn (cid:46) − kn (cid:18) M b10 M (cid:19) kn , (8)where the masses have been normalised to their typicalvalues, M = M/ M (cid:12) , M b10 = M b / M (cid:12) . Thus wehave shown that | α | (cid:28)
1; as a consequence of the massscaling of ζ , a similar result should hold for the rest ofthe parameters since all of them are comparable to ζ .The same argument can be turned around: a typi-cal deformation α ∼ O (1) coming from the shadow ofM87* would be stretched by a factor ∼ ( M/M b ) kn whenthe J-metric is used to model the celestial mechanics ofa merging binary system. This would cause an enormousdeviation from the GR black hole metric which wouldhave easily been seen in the system’s GW signal.The remarkable conclusion is that the black hole space-time of M87* is described by the Kerr metric to a veryhigh precision, with all non-Kerr deviations suppressedby the system’s enormous mass. Once the metric is ren-dered Kerr for all practical purposes, it follows that allgeodesic motion and the shadow itself must necessarilybe also Kerr [30]. Concluding remarks.—
The take home message of thispaper is rather clear: the shadow appearing in a blackhole image like that of M87* could be a viable probeof GR gravity (and more specifically of the Kerr space-time) but with some important caveats attached. Thisstandpoint lies somewhere in between the recent oppos-ing claims made in Refs. [15, 18] but at the same time itextends to a completely orthogonal direction.It is certainly true that the shadow radius is primar-ily a function of the black hole’s spacetime but also ofthe (largely unknown) accretion flow physics. However,if GR gravity is assumed, the radius cannot be pushedbelow b GR and therefore a b < b GR ought to be a cleanprobe of the black hole’s spacetime metric (provided itis observationally allowed in the first place). Moreover,the constraints placed on the deformation parameters ofthe non-Kerr model also depend on their actual number[24]. The quasi-circular shape of the M87* shadow is an-other complicating factor because similarly shaped shad-ows commonly emerge in non-GR gravity theories anddeformed black hole spacetimes alike. The very recentwork of [36] represents a detailed study of the degener-acy between Kerr and non-Kerr black holes in the strictsense of exactly matching shadows; of course a more em-pirical approach is also possible taking into considerationthat observational errors can easily accommodate a smallmismatch between shadows.Our discussion of non-GR theories with coupling con-stants has revealed a perhaps unexpected dichotomy: theshadow cast by a supermassive black hole is intrinsicallyinsensitive to deviations from Kerr when the underlyinggravity theory contains dimensional coupling constants(in which case Eq. (5) shows that the deviation param-eters are mass-suppressed). The majority of known ex-tensions of GR do indeed fall into this category. Never-theless, there exist notable exceptions like Einstein-æthertheory [37] where the non-GR parameters are dimension-less quantities. The non-Kerr character of black holesin such theories is equally prominent regardless of theirmass, and therefore their shadow could be used as a testof GR (see [38] for a calculation along these lines in the context of Einstein-æther theory) albeit subject to theinfluence of the factors discussed in this paper.The suppression of the non-GR coupling constants inthe metric of massive systems is likely to have much widerrepercussions than what discussed here. Apart from itsimpact on EMRIs [30], we would also expect that elec-tromagnetic radiation (such as the observed X-ray ironlines, continuum emission, or quasi-periodic oscillations)from accretion disks in active galactic nuclei, being emit-ted or reflected by matter moving on geodesics, to bealmost completely oblivious to deviations from GR grav-ity [39, 40]. 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