Testing General Relativity with the Shadow Size of Sgr ${\mathrm{A}}^{*}$

In general relativity, the angular radius of the shadow of a black hole is primarily determined by its mass-to-distance ratio and depends only weakly on its spin and inclination. If general relativity is violated, however, the shadow size may also depend strongly on parametric deviations from the Kerr metric. Based on a reconstructed image of Sagittarius ${\mathrm{A}}^{*}$ (Sgr ${\mathrm{A}}^{*}$) from a simulated one-day observing run of a seven-station Event Horizon Telescope (EHT) array, we employ a Markov chain Monte Carlo algorithm to demonstrate that such an observation can measure the angular radius of the shadow of Sgr ${\mathrm{A}}^{*}$ with an uncertainty of $\sim 1.5\mu \mathrm{as}$ (6%). We show that existing mass and distance measurements can be improved significantly when combined with upcoming EHT measurements of the shadow size and that tight constraints on potential deviations from the Kerr metric can be obtained.

Figure 1

(Top panel) A reconstructed image of Sgr ${\mathrm{A}}^{*}$ for an EHT observation at 230 GHz with a seven-station array taken from Ref. [41]. The image shows seven chords for which we determined respective angular radii from Gaussian fits of the intensity profile along the chord sections labeled $1,\dots ,8$. (Bottom panel) 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 the chords using a Markov chain Monte Carlo sampling of a small region around the center of the chords.

Figure 2

(Left panel) $1\sigma$ and $2\sigma$ confidence contours of the probability density of the mass and distance of Sgr ${\mathrm{A}}^{*}$ for existing measurements ($S$ stars, G09 [2]; masers, R14 [6]; star cluster, C15 [5]), a simulated measurement of the shadow size of Sgr ${\mathrm{A}}^{*}$ for $N=10$ observations with a seven-station EHT array (EHT), and several combinations thereof. The simulated EHT measurement improves the other constraints on the mass and distance significantly. (Center and right panels) Simulated $1\sigma$ and $2\sigma$ confidence contours of the probability density of the deviation parameters ${\alpha }_{13}$ and $\beta$, respectively, corresponding to $N=10$ and $N=100$ EHT observations, each marginalized over the mass and distance using the combination of all data sets (all) in the $N=10$ case and of simulated stellar-orbit observations from a 30-m-class telescope [42] in the $N=100$ case.