Can the EHT M87 results be used to test general relativity?
CCan the EHT M87 results be used to test general relativity?
Samuel E. Gralla ∗ Department of Physics, University of Arizona, Tucson, Arizona 85721, USA
No. All theoretical predictions for the observational appearance of an accreting supermassiveblack hole, as measured interferometrically by a sparse Earth-sized array at current observationfrequencies, are sensitive to many untested assumptions about accretion flow and emission physics.There is no way to distinguish a violation of general relativity (GR) from the much more likelyscenario that the relevant “gastrophysical” assumptions simply do not hold. Tests of GR willbecome possible with longer interferometric baselines (likely requiring a space mission) that reachthe resolution where astrophysics-independent predictions of the theory become observable.
I. INTRODUCTION
In April 2019, the Event Horizon Telescope (EHT) col-laboration released 1.3mm interferometric observationsof the core of the galaxy M87 (henceforth M87*) [1–6].These observations probe distance scales of order the sizeof the expected supermassive black hole, a remarkabletechnical achievement. As we move into this excitingnew era of horizon-scale electromagnetic astronomy, it isimportant to know what the data can—and cannot—tellus about the gravity and astrophysics of black holes.In their initial analysis, the collaboration argued thatthe data are consistent with expectations for an accret-ing black hole, bolstering their case with a new suite ofsimulations of accretion flow and associated 1.3mm emis-sion. Using these simulations as “ground truth”, theyestimated the mass of the black hole with a reported ∼
20% uncertainty. This result distinguished betweenconflicting prior mass measurements, favoring that basedon stellar dynamics [7] over that based on gas dynamics[8]. EHT did not claim any tests of GR or probes of ac-cretion physics; the idea was to assume the truth of GRand assume truth of the emission models, and to thereby measure the black hole mass.Recently, the collaboration reversed the logic [9].Without releasing new data or performing new analysisof existing data, they instead assumed that the stellar-dynamics mass [7] is correct and argued that their orig-inal analysis [6] is, in fact, a test of GR. This new ap-proach means that EHT now has more confidence in thestellar-dynamics mass measurement than it does in thecorrectness of GR. It also means that EHT has more con-fidence in its accretion and emission physics assumptionsthan it does in the correctness of GR. In this paper wewill compare the evidence for GR with the evidence forthe relevant features of the EHT emission models andconclude that GR is far better established. This impliesthat no analysis requiring these emission assumptions canbe used as a test of GR. ∗ [email protected] Not all present members of the collaboration are authors of Ref. [9],but the author list does state “the EHT collaboration”.
This difficulty is not unique to the modeling and anal-ysis choices made by EHT. The fundamental problem isthat current interferometric baselines (impressively longthough they are) are not sensitive to any unique GR pre-dictions. The (very interesting) “black hole shadow” ef-fect [10, 11] is unfortunately not generic, and does not oc-cur in the EHT models for M87*. The “photon ring” dueto photons that orbit the black hole [10–13] is a genericGR prediction, but it will be very challenging to observefrom Earth. In particular, analytical arguments [14] sup-ported by EHT simulations [15] establish that even inthe most favorable of circumstances, at most ∼
10% ofthe flux density is due to orbiting photons. These pho-tons occupy a narrow ring on the image plane (the pho-ton ring) that is well below the effective resolution of anEarth-sized array at currently envisaged observation fre-quencies. Photon ring tests of GR will almost certainlyrequire a space mission [15, 17].The contention of this paper is that while the originallogic of a mass estimate is reasonable, the new logic—aGR test—is not. There is, of course, a third possibility:Assume a mass, assume GR, and test the astrophysicalassumptions present in the underlying theoretical analy-sis. These assumptions touch on some of the most inter-esting questions in relativistic astrophysics: the natureof accretion flows, the behavior of strongly magnetizedplasma, and the mechanism(s) for powering and launch-ing relativistic jets. It may be that the greatest benefitsof continued ground-based interferometric measurementsof M87* will be for a better understanding of these andother fundamental processes at work in our universe. Ref. [15] considered a single simulation from the EHT suite and re-ported that ∼
10% of the flux comes from orbiting photons. (Thisnumber is given at the end of section 2 and can also be inferredfrom the ∼
20% figure quoted in Fig. 1 together with the split byorbit number shown in Fig. 3.) This simulation involves diffuseemission concentrated near the event horizon of a rapidly spinningblack hole, which are the conditions most favorable to having asignificant fraction of orbiting photons [16, 17]. Any departurefrom these conditions (a flatter emission profile, an emission profileending further away from the horizon, a more equatorial emissionprofile, or a less rapidly spinning black hole) will decrease the or-biting photon fraction. It would be interesting to know the orbitingphoton fraction for the entire suite of EHT simulations, on whichthe purported GR test [9] is based. a r X i v : . [ a s t r o - ph . H E ] O c t II. PREVIOUS EHT CLAIMS
The central claims of the M87* EHT papers [1–6] are:1. Observational Claim: The image of M87* is domi-nated by a ring of typical diameter ∼ µ as, whosethickness is estimated to be somewhere between10% and 50% of its diameter.2. Theoretical Claim: The diameter of the ring obeysan approximate scaling relation with the mass-to-distance ratio of the black hole: θ = (40 ± µ as M/ (6 . × M (cid:12) ) D/ (16 . . (1)Here M is the black hole mass, D is its distance, and θ is the angular diameter of the observed emission ring. A. Evidence for the observational claim(ring diameter)
Although I will not question the EHT observationalclaim in this paper, some readers may be interested in asummary of the evidence. This section may be skippedwithout loss of continuity.A radio interferometer measures the “complex visibil-ity”, which (under mild assumptions) is equal to theFourier transform of the sky brightness. Each pair oftelescopes is sensitive to the Fourier transform at vec-tor wavenumber equal to the telescope separation pro-jected along the line of sight, and divided by the obser-vation wavelength. An N -telescope array thus measures N ( N − / ∼
10 parameter freedom. Unfortunately, the models donot fit the data until an additional ∼
10 nuisance pa-rameters are included, corresponding to the addition of2–3 Gaussian blobs on the image. The preferred obser-vational appearance is a narrow ring of diameter 40 µ as(fractional width ∼ ∼ µ as, with fractional width varyingfrom ∼
30% to ∼ ∼ µ as ring suggests thatthis feature is indeed present in the data. B. Evidence for the theoretical claim(scaling relation)
I have distilled the EHT theoretical claims into thescaling relation (1) as follows. I begin with the statementin Ref. [6] that “the structure and extent of the emissionpreferentially from outside the photon ring leads to a (cid:46)
10% offset between the measured emission diameterin the model images and the size of the photon ring.”Here by “photon ring” EHT means the theoretical curveon the image plane defined by the arrival of photons thathave orbited the black hole arbitrarily many times beforearrival, which I discuss in Sec. III below as the “criticalcurve”. This curve has a typical diameter of 37 . µ aswith the mass and distance used in (1); I have increasedthis number by 7% to account for the offset describedby EHT, arriving at a central value of 40 µ as. I havecrudely estimated the claimed spread around 40 µ as bycomparing with the EHT reported posterior probabilityon the mass, folding in estimates of the uncertainty onthe distance D and the observed ring diameter θ .The precise values present in the scaling relation (1)are irrelevant to the main points of this paper. The es-sential point is that all EHT claims about a mass mea-surement or a GR test may be understood (at the order-of-magnitude level) with reference to a simple scaling re-lation of this form.Let us scrutinize the evidence for the scaling relation(1). The argument given by EHT involves a bank of gen-eral relativistic magnetohydrodynamic (GRMHD) simu-lations using the Kerr spacetime, together with a phe-nomenological prescription for the emission as a functionof the GRMHD variables. I will refer to this method ofgenerating images as GRMHD+, with the + standingfor the phenomenological emission prescription. EHTconsidered a large number of GRMHD+ models, vary-ing the black hole spin, the observer inclination, the netmagnetic flux on the black hole, and a parameter in theiremission model. They argued that the vast majority ofthese models are consistent with the data, but excludedsome fraction based on ancillary theoretical considera-tions. The scaling relation is derived from the remainingGRMHD+ models.Let us list the main assumptions of GRMHD+, in or-der from best established to least well established. By farthe best established assumption is the Kerr metric. GRhas been tested in a variety of ways in a variety of regimes[18–22], and the Kerr metric is an unambiguous predic-tion for the exterior of dark compact objects, supportedby an enormous body of mathematical proof, analyticalargument, and numerical evidence.The second-best-established assumption is the fluid de-scription of the plasma. Given that the collisional meanfree path is many orders of magnitude larger than theblack hole, the validity of a fluid description is far fromobvious. The EHT models further make the assumptionof ideal MHD (infinite conductivity), a very special casein the space of plausible fluid theories of plasma.The third-best-established assumption is the resolutionof the simulations. Global, self-consistent simulations op-erate at the limits of computational power, and conver-gence studies are challenging [23]. Local (shearing-box)simulations designed to study the magneto-rotational in-stability (MRI) are inconclusive regarding convergence,depending on the assumed magnetic environment [24,25]. This raises some doubt as to whether the globalGRMHD simulations adequately resolve the physics thatgives rise to the accretion they are designed to study.The fourth-best-established assumption is the initialconditions for the simulations. It is well known that thestructure of the MHD solution depends strongly on thechoice of initial conditions, most particularly with regardto the formation of Poynting-flux outflows (potentiallyable to power a relativistic jet) [26–28]. The more jet-promising initial conditions are naturally used to modelsystems with jets (and EHT specifically excluded someof their simulations based on their lack of Poynting flux),but the strong dependence of the simulation results onthe initial conditions suggests that other viable initialconditions may await discovery, especially as ideas forlaunching the jet continue to evolve [29].The fifth-best-established assumption is the EHT phe-nomenological prescription for the emission. GRMHD None of the approximately 60,000 GRMHD+ images provides aformally acceptable fit to the data (as judged by a reduced chi-squared). This is attributed to the high variability of the flow,and an alternative “average image scoring” (AIS) approach wasdeveloped to determine whether the variability in a GRMHD+simulation is statistically consistent with the observed data. TheAIS method eliminates high-magnetic-flux, retrograde flows aboutrapidly spinning black holes. The use of AIS means that the scalingrelation (1) does contain a modest amount of observational input.However, I will continue to regard it as a theoretical result, sincethe majority of input comes from theoretical considerations. In the models presented in table 2 of Ref. [5], more than halfwere eliminated based on ancillary theoretical concerns involvingjet power, X-ray luminosity, and radiative efficiency. follows the averaged properties of the ions, whereas itis the much-lighter electron component that is believedto give rise to the observed synchrotron emission. Evenassuming the correctness of the GRMHD results for theion properties, the synchrotron emissivity is unknown.EHT relied on a prescription that ties the synchrotronemissivity to the averaged properties of the ions, moti-vated by the idea that energy is dissipated exclusively byLandau damping. Needless to say, this phenomenologicalprescription has not been tested.The sixth-best-established assumption is the ad hoc zeroing of emission inside the jet region. GRMHD simu-lations fail inside a “jet region” that typically takes theshape of a paraboloid of revolution. In this region, thedensity becomes so small that the GRMHD code is un-able to continue the evolution. To avoid this difficulty,simulations impose an arbitrary floor on the density,which forces the dynamics to be approximately force-free.While perhaps reliable for determining the evolution ofthe large-scale electromagnetic field, this approach is in-capable of predicting the emission from the small-scaleparticle acceleration and pair-creation that is believed tooccur [30]. The EHT collaboration chose to make theemission zero in this region.
III. HEURISTIC INTERPRETATIONS
The EHT GRMHD+ models predict a relatively tightscaling (1) between black hole mass and observed ringdiameter. EHT suggests that this agreement in obser-vational appearance among their GRMHD+ models isa result of gravitational lensing, invoking the heuristicsof a “shadow” and a “photon ring”. In this section Iwill review these and other heuristics used to interpretblack hole images and argue that the agreement in ob-servational appearance cannot plausibly be attributed toa shadow or photon ring. Instead, the similar observa-tional appearance of the EHT GRMHD+ models likelymeans that all have very similar emission profiles, at leastwhen projected along the line of sight.
A. Backlit shadow
A black hole illuminated from behind by an isotropicsource of much larger angular radius will cast a darkshadow of size somewhat larger than the critical curve,inside of which will be a sequence of thin rings converg-ing to the critical curve [14]. If the illumination is in-stead from the entire celestial sphere (including behindthe observer), then the shadow shrinks down to the criti-cal curve. These scenarios are irrelevant for understand-ing the appearance of matter near black holes, and indeedwere not considered by EHT.
B. Doppler shadow
A black hole surrounded by optically thin, radially in-falling matter will appear as a dark hole whose outlineis the critical curve (Fig. 1(a) of Ref. [10]). This effectis caused by extreme Doppler deboosting [31] due to theassumption of radial infall. Since rays arriving within thecritical curve by definition have no turning point, all pho-tons in that area were emitted opposite to the radiallyinfalling flow and suffer strong Doppler deboosting.This effect was named the black hole “shadow”. Theterminology is somewhat unfortunate, since the phe-nomenon has nothing to do with backlighting, instead be-ing caused by special-relativistic Doppler deboosting [31].A more appropriate name would perhaps be “Dopplerdeficit”. Meeting halfway between these terms, I will usethe name “Doppler shadow” for the dimming inside thecritical curve due to radial infall [10, 31].
C. Discrete photon rings
The existence of unstable photon orbits gives rise tomultiple images of sources near black holes [32]. In prin-ciple there are infinitely many such images, indexed bythe total number of orbits and the direction of the orbit(clockwise or counter-clockwise). The images accumu-late near the critical curve and are increasingly demag-nified. When the source is in the shape of a disk, thehigh-order images are thin rings (“photon rings”) con-verging to the critical curve [12, 33, 34]. For opticallythin disks extending to near the horizon (as presumed tooccur in M87*), the photon rings comprise a distinctivemulti-peak structure [14] that is seen in (at least one of)the EHT GRMHD+ simulations when ray-traced at suf-ficiently high resolution [15]. The photon rings containat most ∼
10% of the total flux in such a simulation (seefootnote 2), consistent with theoretical expectations [14].
D. Continuous photon ring
When the source is diffuse (with no sharp featureslike point sources or disks) as well as optically thin, theheuristic of multiple images becomes less useful. Instead,we may consider the observed intensity to be proportionalto the optical path of a ray traced backward through thesource. The optical path diverges at the critical curve,leading to a smooth brightness enhancement [10, 13].However, the enhancement is only logarithmic [14–16]. Ref. [10] also considers matter orbiting on Keplerian shells[Fig. 1(d)]. This model produces a more gentle decrease not clearlyassociated with the critical curve.
E. Just add one
Although full ray-tracing in the Kerr metric is a com-putationally intensive process, some of its features canbe understood in simple terms. Consider, for example,the question of which point on the image corresponds towhich region of the source. If our source is a disk (thin orthick) centered on the spin-equator of the black hole, wemay ask for the relationship between the equatorial emis-sion radius r (using Boyer-Lindquist coordinates) and thearrival position on the image. In the face-on limit (suit-able as a first approximation for M87*), we may use ra-dial impact parameter b to represent radial distance onthe image. The answer for the direct photons (which donot orbit) is shockingly simple: b ≈ r + M. (2)That is, you “just add one”. This was noticed numeri-cally in Ref. [16] and derived as an analytic approxima-tion in Ref. [35]. We may interpret the formula as statingthat the primary image of the disk is essentially pristine,with no significant distortion due to lensing, and no in-fluence of the black hole spin. F. Interpretation of GRMHD+ images
Which heuristics are most useful for understanding theEHT GRMHD+ images? The backlit shadow is not rel-evant (and was not considered by EHT). The Dopplershadow is also not relevant, as the dark region can besignificantly displaced from the critical curve. The nar-row photon ring feature is present in all images, but itdoes not contain enough flux to significantly affect theobserved ring diameter (see footnote 2). Based on thestriking qualitative agreement with purely equatorial toymodels [14, 17] (for example, compare Fig. 1 of Ref. [14]to Fig. 3 of Ref. [15]), it appears that the most usefulheuristics for interpreting the GRMHD+ images are “justadd one” (Sec. III E) for the majority of the flux, togetherwith discrete photon rings (secondary images) appearingnear the critical curve (Sec. III C).In particular, gravitational lensing plays little role indetermining the observed ring diameter at EHT resolu-tion. Instead, the majority of the flux is in the relativelyundistorted “primary image” of the source, composedof the integrated emissivity along direct (non-orbiting)rays through the disk. That is, the structure of the ob-served ring is determined by the projected structure ofthe source disk. This lack of distortion persists in the modestly inclined case—see,for example, the second column of Fig. 6 of Ref. [16]. See, for example, the middle panel of Fig. 1 of Ref. [5], noting thecontrast between the thin ring (the photon ring, tracing the criticalcurve) and the dark region in the center.
IV. ALTERNATIVE SCENARIOS
I have argued that the scaling relation (1) is not due togravitational lensing, but rather a direct consequence ofthe assumed source structure. It is instructive to consideralternative source structures under which the scaling re-lation would be significantly altered. In fact, preciselysuch a scenario was given by the antecedents of EHT in2010 [36]. This paper presented the first evidence of a ∼ µ as 1.3mm structure in M87*. To interpret theirresults, the authors argued that the size of the emissionregion can be associated with the innermost stable cir-cular orbit (ISCO) of the black hole. Translated intoan effective scaling relation, the theoretical claims of [36]entail an effective allowed range for θ , θ min = 14 µ as M/ (6 . × M (cid:12) ) D/ (16 . θ max = 72 µ as M/ (6 . × M (cid:12) ) D/ (16 . . (4)As explained by the authors of [36], this “ISCO sce-nario” is natural because the mass density drops at theISCO (even if the disk is geometrically thick, with signif-icant stresses). The ISCO scenario predicts a wide range θ min < θ < θ max for the emission size, depending on blackhole spin. By constrast, the EHT GRMHD+ scenario (1)predicts a very narrow range of θ , independent of blackhole spin.Another alternative to the GRMHD+ scenario is theidea that the observed emission comes not from an accre-tion flow but from a near-horizon magnetosphere. Thisidea was discussed recently by Blandford, Meier, andReadhead [29]. After reviewing some issues with thestandard interpretation of the EHT observations, theauthors state that “a simpler hypothesis...is that EHTis observing dynamically insignificant relativistic plasmaorbiting with the angular velocity of an ordered magneticfield, confined by a much larger ejection disk.” Althoughthe emission from such a scenario has not yet been cal-culated, it would presumably come from very near thehorizon and therefore produce a scaling relation with asmaller prefactor than Eq. (1).Finally, even within the broad paradigm of an accre-tion flow extending down to the horizon [37, 38], somestudies have suggested that non-thermal emission fromnear the horizon (not included in GRMHD+) may playan important—or even dominant—role in the millimeter-wave observational appearance of M87* [39–43]. V. SUMMARY AND CONCLUSIONS
The scaling relation (1) claimed by EHT is derivedfrom a bank of GRMHD+ simulations (Sec. II B). Thescaling relation is very nearly identical to that whichwould arise by identifying the observed emission ring withthe critical curve (also sometimes called the “shadow” or“photon ring”). This is an accident of the particularassumptions in the EHT models of the source. Thereis no shadow effect in the EHT simulations used to de-rive the scaling relation (Sec. III F), and the photon ringcontributes minimally to the structure of these (or anyother) images on the effective resolution of Earth-sizedbaselines (Sec. III C). The scaling relation (1) is not auniversal prediction of GR, but a specific prediction of aspecific class of source models.On general grounds, this means that no analysis usingthe scaling relation can be considered a test of GR. Thegastrophysical assumptions underlying the scaling rela-tion (Sec. II B) are far less well-established than generalrelativity. Alternative scaling relations are conceivableand indeed have been considered in the past by someof the authors of [9] (Sec. IV). Given the overwhelmingevidence for GR and the much more limited evidencefor astrophysical assumptions used in deriving any givenscaling relation, null-hypothesis testing can only probeastrophysics, not gravity.Beyond the null-hypothesis idea, the recent paper [9]attempts to place constraints on alternative metrics. Theassumption is that, when these alternative metrics areconsidered, the new scaling relation will be determinedby identifying the observed ring with the critical curveof that new metric. This assumption is valid in GR onlyin special cases, and no justification for it holding in al-ternative theories is given in Ref. [9]. Even if an MHD+analysis were performed with an alternative metric to de-termine an alternative scaling relation, there would be noway to distinguish a failure of the gravity theory from afailure of the astrophysical assumptions.In conclusion, the diameter of the observed ring inM87*, as determined by the present EHT observationsand analysis, cannot be used to test GR.
ACKNOWLEDGEMENTS
This work was supported in part by NSF grant PHY-1752809 to the University of Arizona. For these estimates I use the range 1 M –9 M for the Boyer-Lindquistradius of the ISCO, together with the fact that the relevant lensingboils down to “just adding one” (Sec. III E). [1] Event Horizon Telescope Collaboration, K. Akiyama,A. Alberdi, et al. , “First M87 Event Horizon TelescopeResults. I. The Shadow of the Supermassive BlackHole,” ApJ (Apr., 2019) L1. [2] Event Horizon Telescope Collaboration, K. Akiyama,A. Alberdi, et al. , “First M87 Event Horizon TelescopeResults. II. Array and Instrumentation,” ApJ (Apr., 2019) L2.[3] Event Horizon Telescope Collaboration, K. Akiyama,A. Alberdi, et al. , “First M87 Event Horizon TelescopeResults. III. Data Processing and Calibration,” ApJ (Apr., 2019) L3.[4] Event Horizon Telescope Collaboration, K. Akiyama,A. Alberdi, et al. , “First M87 Event Horizon TelescopeResults. IV. Imaging the Central Supermassive BlackHole,” ApJ (Apr., 2019) L4.[5] Event Horizon Telescope Collaboration, K. Akiyama,A. Alberdi, et al. , “First M87 Event Horizon TelescopeResults. V. Physical Origin of the Asymmetric Ring,”ApJ (Apr., 2019) L5.[6] Event Horizon Telescope Collaboration, K. Akiyama,A. Alberdi, et al. , “First M87 Event Horizon TelescopeResults. VI. The Shadow and Mass of the Central BlackHole,” ApJ (Apr., 2019) L6.[7] K. Gebhardt, J. Adams, D. Richstone, et al. , “TheBlack Hole Mass in M87 from Gemini/NIFS AdaptiveOptics Observations,” ApJ (Mar., 2011) 119, arXiv:1101.1954 .[8] J. L. Walsh, A. J. Barth, L. C. Ho, and M. Sarzi, “TheM87 Black Hole Mass from Gas-dynamical Models ofSpace Telescope Imaging Spectrograph Observations,”ApJ (June, 2013) 86, arXiv:1304.7273 .[9] D. Psaltis, L. Medeiros, P. Christian, et al. ,“Gravitational Test Beyond the First Post-NewtonianOrder with the Shadow of the M87 Black Hole,” arXive-prints (Oct., 2020) arXiv:2010.01055, arXiv:2010.01055 [gr-qc] .[10] H. Falcke, F. Melia, and E. Agol, “Viewing the Shadowof the Black Hole at the Galactic Center,” ApJ (Jan., 2000) L13–L16, astro-ph/9912263 .[11] B. C. Bromley, F. Melia, and S. Liu, “PolarimetricImaging of the Massive Black Hole at the GalacticCenter,” ApJ no. 2, (July, 2001) L83–L86, arXiv:astro-ph/0106180 [astro-ph] .[12] J.-P. Luminet, “Image of a spherical black hole withthin accretion disk,” A&A (May, 1979) 228–235.[13] M. Jaroszynski and A. Kurpiewski, “Optics near Kerrblack holes: spectra of advection dominated accretionflows.,” A&A (Oct., 1997) 419–426, astro-ph/9705044 .[14] S. E. Gralla, D. E. Holz, and R. M. Wald, “Black holeshadows, photon rings, and lensing rings,” Phys. Rev. D no. 2, (July, 2019) 024018, arXiv:1906.00873[astro-ph.HE] .[15] M. D. Johnson, A. Lupsasca, A. Strominger, et al. ,“Universal interferometric signatures of a black hole’sphoton ring,” Science Advances no. 12, (2020) .[16] S. E. Gralla and A. Lupsasca, “Lensing by Kerr blackholes,” Phys. Rev. D no. 4, (Feb., 2020) 044031, arXiv:1910.12873 [gr-qc] .[17] S. E. Gralla, A. Lupsasca, and D. P. Marrone, “TheShape of the Black Hole Photon Ring: A Precise Test ofStrong-Field General Relativity,” arXiv e-prints (Aug.,2020) arXiv:2008.03879, arXiv:2008.03879 [gr-qc] .[18] M. Kramer, “Probing gravitation with pulsars,” in Neutron Stars and Pulsars: Challenges andOpportunities after 80 years , J. van Leeuwen, ed.,vol. 291 of
IAU Symposium , pp. 19–26. Mar., 2013. arXiv:1211.2457 [astro-ph.HE] .[19] C. M. Will, “The Confrontation between GeneralRelativity and Experiment,”
Living Reviews inRelativity no. 1, (Dec., 2014) 4, arXiv:1403.7377[gr-qc] .[20] E. Berti, E. Barausse, V. Cardoso, et al. , “Testinggeneral relativity with present and future astrophysicalobservations,” Classical and Quantum Gravity no. 24, (Dec., 2015) 243001, arXiv:1501.07274[gr-qc] .[21] B. P. Abbott, R. Abbott, T. D. Abbott, et al. , “Tests ofGeneral Relativity with GW150914,” Phys. Rev. Lett. no. 22, (June, 2016) 221101, arXiv:1602.03841[gr-qc] .[22] B. P. Abbott, R. Abbott, T. D. Abbott, et al. ,“Gravitational Waves and Gamma-Rays from a BinaryNeutron Star Merger: GW170817 and GRB 170817A,”ApJ no. 2, (Oct., 2017) L13, arXiv:1710.05834[astro-ph.HE] .[23] H. Shiokawa, J. C. Dolence, C. F. Gammie, and S. C.Noble, “Global General RelativisticMagnetohydrodynamic Simulations of Black HoleAccretion Flows: A Convergence Study,” ApJ no. 2, (Jan., 2012) 187, arXiv:1111.0396[astro-ph.HE] .[24] J. M. Stone, J. F. Hawley, C. F. Gammie, and S. A.Balbus, “Three-dimensional MagnetohydrodynamicalSimulations of Vertically Stratified Accretion Disks,”ApJ (June, 1996) 656.[25] S. W. Davis, J. M. Stone, and M. E. Pessah, “SustainedMagnetorotational Turbulence in Local Simulations ofStratified Disks with Zero Net Magnetic Flux,” ApJ no. 1, (Apr., 2010) 52–65, arXiv:0909.1570[astro-ph.HE] .[26] K. Beckwith, J. F. Hawley, and J. H. Krolik, “TheInfluence of Magnetic Field Geometry on the Evolutionof Black Hole Accretion Flows: Similar Disks,Drastically Different Jets,” ApJ no. 2, (May, 2008)1180–1199, arXiv:0709.3833 [astro-ph] .[27] R. F. Penna, A. Kulkarni, and R. Narayan, “A newequilibrium torus solution and GRMHD initialconditions,” A&A (Nov., 2013) A116, arXiv:1309.3680 [astro-ph.HE] .[28] R. F. Penna, J. C. McKinney, R. Narayan, et al. ,“Simulations of magnetized discs around black holes:effects of black hole spin, disc thickness and magneticfield geometry,” MNRAS no. 2, (Oct., 2010)752–782, arXiv:1003.0966 [astro-ph.HE] .[29] R. Blandford, D. Meier, and A. Readhead, “RelativisticJets from Active Galactic Nuclei,” ARA&A (Aug.,2019) 467–509, arXiv:1812.06025 [astro-ph.HE] .[30] R. D. Blandford and R. L. Znajek, “Electromagneticextraction of energy from Kerr black holes,” MNRAS (May, 1977) 433–456.[31] R. Narayan, M. D. Johnson, and C. F. Gammie, “TheShadow of a Spherically Accreting Black Hole,” arXive-prints (Oct, 2019) arXiv:1910.02957, arXiv:1910.02957 [astro-ph.HE] .[32] C. Darwin, “The Gravity Field of a Particle,” Proceedings of the Royal Society of London Series A (Jan., 1959) 180–194.[33] J. M. Hollywood and F. Melia, “General RelativisticEffects on the Infrared Spectrum of Thin AccretionDisks in Active Galactic Nuclei: Application to
Sagittarius A *,” ApJS no. 2, (Oct., 1997) 423–455.[34] K. Beckwith and C. Done, “Extreme gravitationallensing near rotating black holes,” MNRAS (June,2005) 1217–1228, astro-ph/0411339 .[35] D. E. A. Gates, S. Hadar, and A. Lupsasca, “MaximumObservable Blueshift from Circular Equatorial KerrOrbiters,” arXiv e-prints (Sept., 2020)arXiv:2009.03310, arXiv:2009.03310 [gr-qc] .[36] S. S. Doeleman, V. L. Fish, D. E. Schenck, et al. ,“Jet-Launching Structure Resolved Near theSupermassive Black Hole in M87,”
Science no. 6105, (Oct., 2012) 355, arXiv:1210.6132[astro-ph.HE] .[37] F. Melia, “An Accreting Black Hole Model forSagittarius A *,” ApJ (Mar., 1992) L25.[38] R. Narayan and I. Yi, “Advection-dominated Accretion:A Self-similar Solution,” ApJ (June, 1994) L13, arXiv:astro-ph/9403052 [astro-ph] . [39] A. E. Broderick and A. Loeb, “Imaging the Black HoleSilhouette of M87: Implications for Jet Formation andBlack Hole Spin,” ApJ no. 2, (June, 2009)1164–1179, arXiv:0812.0366 [astro-ph] .[40] A. Levinson and F. Rieger, “Variable TeV Emission asa Manifestation of Jet Formation in M87?,” ApJ no. 2, (Apr., 2011) 123, arXiv:1011.5319[astro-ph.HE] .[41] M. Mo´scibrodzka, C. F. Gammie, J. C. Dolence, andH. Shiokawa, “Pair Production in Low-luminosityGalactic Nuclei,” ApJ no. 1, (July, 2011) 9, arXiv:1104.2042 [astro-ph.HE] .[42] A. E. Broderick and A. Tchekhovskoy, “Horizon-scaleLepton Acceleration in Jets: Explaining the CompactRadio Emission in M87,” ApJ no. 1, (Aug., 2015)97, arXiv:1506.04754 [astro-ph.HE] .[43] K. Hirotani and H.-Y. Pu, “Energetic GammaRadiation from Rapidly Rotating Black Holes,” ApJ no. 1, (Feb., 2016) 50, arXiv:1512.05026[astro-ph.HE]arXiv:1512.05026[astro-ph.HE]