Testing General Relativity with supermassive black holes:M87*

John Moffat

Talk given at Miami 2019 Conference, Fort Lauderdale, Florida, December 13, 2019

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1. EHT observation of black hole M87*

• The aggregate of radio waves data was obtained by the VLBI over four days of observations at 1.33 mm wavelength corresponding to Gigahertz frequency. The data produced an amplitude and phases, that with a Fourier transform generates components of the brightness distribution.

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• Using CLEAN and RML (Regularized maximum likelihood) imaging algorithms the visibility distribution can be used to reconstruct an M87* image.

• Because the (u,v) coordinate plane is only sparsely sampled, the inverse Fourier transform problem is under-constrained (Astrophysical Journal Letters Paper IV). Second, the measured visibilities lack absolute phase calibration and can have large amplitude calibration uncertainties. The x-ray emissions from M87* undergo lensing and the construction of the image depends on the mass model of M87*. The imaging algorithms incorporate assumptions and constraints that produce a physically plausible smooth image.

• Theoretical modelling that can be compared with the reconstructed image uses the GRMHD (General Relativity Magneto-Hydrodynamic) simulation library, which describes a turbulent, hot, magnetized disk orbiting the Kerr black hole M87*. The GRMHD model depends on assuming that GR is correct.

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2. Testing General Relativity with M87*

• The black hole M87* observations by the EHT collaboration can test GR for strong gravitational fields.

• The formulas predicting the gravitational lensing of light by a black hole were published for the case of the Schwarzschild solution by J. L. Synge in 1966, and for the Kerr rotating black hole by J. Bardeen 1977.

• I will use my modified gravitational theory (MOG) to compare with the predictions of GR.

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• The MOG action and field equations are given by

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3. MOG field equations

• The equations of motion of a massive test particle in MOG:

• MOG satisfied the weak equivalence principle. The modified Newtonian acceleration is:

• MOG fits data for galaxies and clusters without dark matter. A fit to the RAR relation (McGough et al.) demonstrates that MOG provides a reasonable fit to 149 SPARC galaxy data (M. A. Green and JWM, Physics of the Dark Universe, 25, 100323 (2019)).

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• Photons and gravitons move along null geodesics:

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4. Matter-free MOG field equations and black hole solutions

• The matter-free MOG field equations are given by( )

• For the MOG black hole solution, we ignore the small mass μ corresponding to the mass mφ ~ 10-28 eV.

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• The Kerr-MOG solution derived from the field equations yields the metric:

• The event and ergosphere horizons are determined by

• When = 0, the metric reduces to the Schwarzschild-MOG metric:

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• The size of the black hole photon sphere for a = 0 and Schwarzschild-MOG black hole:

• The size of the shadow for a=0 is

( ,

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Size of black hole photon sphere (ring) in GR with a = 0 and α = 0:

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5. Mass determinations of M87* ( Jonelle Walsh, Texas A&M, M87 Workshop, ASIAA, May 23-27, 2016)

• Over a period of 20 years estimates of the mass of the black hole have been made by two methods. The stellar-dynamical method and the gas-dynamical method. To date, ~ 90 black hole estimates have been made. There remain open questions.

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• M87* photon ring angular diameter determined by EHT measurement

dγ = 42 ± 3 μas.

• Size of black hole photon sphere (ring) in GR with α = 0:

Two more M87* mass measurements: 1) MBH = 5.6 x 107 L. Titarchuk et al. A & A, to be published, arXiv:1911.03180. 2) MBH = 5.2 x 109 < M87* < 7.7 x 109 E . E. Nokhrina, et al. MNRAS, 489, 1197 (2019), arXiv: 1904.05665. For 1) we have = 154 and for 2) 0 < < 0.35.

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6. GR and MOG predictions for M87* ( JWM and V. T. Toth, arxiv:1904.04142,To be published in Phys. Rev. D)

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• For the EHT measured emission region of 42 ± 3 μas and for the distance of M87 adopted, D=16.8 ± 0.8 Mpc, the black hole mass MBH = (6.5 ± 0.7) x 109 is consistent with GR, and consistent with the stellar-dynamical mass model estimate.

• The Table shows different angular diameters of the emission region for MBH = 3.5 x 109 for different values of α and for a=0, obtained from the gas-dynamical model.

• The angular diameter of the photon sphere and shadow 42 ± 3 μas can be fitted by MOG with α = 1.13+0.30

-2.4 and MBH = 3.5 x 109 .

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7. Periastron (perihelion) shift

• The periastron (perihelion) shift in MOG is

• For Mercury GR predicts

Total Mercury perihelion precession is 5600’’/century.

• The difference between Newtonian gravity and GR is less than 1%!

• For S0-2 and Sagittarius A* black hole MBH = 4.3 x 106 , P = 15.92 year, a = 2.1 x 1016 cm and e = 0.89:

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330’’/yr = 33.3’’/yr

END

A. Hees et al. Phys. Rev. Lett. 118, 211101 (2019).

which gives α < 9.