Testing the Schwarzschild metric in a strong field region with the Event Horizon Telescope

Testing gravity theory in the strong field region becomes a reality due to the observations of gravitational waves and black hole shadows. In this paper, we discuss how to constrain the possible deviations of the classical general relativity with the image of M87* observed by the Event Horizon Telescope. More precisely, we want to know where is the event horizon for a nonrotating black hole. General relativity predicts the horizon is located at the Schwarzschild radius $r_{\mathrm{s}}$, while other gravity theories may give different predictions. We propose a parametrized Schwarzschild metric (PSM) in which the horizon is located at $r=n{r}_{\mathrm{s}}$, where $n$ is a real free parameter, and prove general relativity with nonlinear electrodynamics allows $n\ne 1$. In the weak field region, the PSM is equivalent to the Schwarzschild metric regardless of the value of $n$. In the strong field region, the difference between the PSM and Schwarzschild metric would leave an imprint on the shadow image. We present detailed calculations and discussions on the black hole shadows with large background light source and accretion disk in the PSM framework. More importantly, we point out that $n\approx 2$ can be used to explain why the black hole mass measured by the shadow is a factor of about two larger than the previous gas dynamics measurements. If this explanation is confirmed to be right, then this phenomenon, together with the late-time cosmological acceleration, will be very important to test gravity theories.

Figure 1

Representative plot shows the critical impact parameter. The value of $G(u)$ equals to the distance between the straight line and the curve line with constant $u$. Here we set $r_{\mathrm{s}}=2$.

Figure 2

The surface radiation flux with $n=0.75$, 1, 1.25. Here we set $r_{\mathrm{s}}=2$ and $\dot{M}=1$.

Figure 3

The coordinate system (see main text). The design of this figure is mainly taken from [11].

Figure 4

The projection factor $\cos i$ for $n=1$ and ${\theta }_0=80°(\mathrm{left}),45°(\mathrm{middle}),10°(\mathrm{right})$. Here we set $r_{\mathrm{s}}=2$.

Figure 5

The simulated image without considering the projection effect. The innermost dash-dotted loop corresponds to $r=r_{\mathrm{ISCO}}$, and the outer solid loops correspond to $r=10$, 15, 20, 30, respectively. The dashed line corresponds to $z=0$. The left side of this dashed line has $z<0$ (blueshift), and the right side has $z>0$ (redshift). $\cos i=1$ indicates this figure do not take into account the projection effect. Here, the horizontal and vertical directions correspond to the $x^{\prime \prime }$ and $y^{\prime \prime }$ axes in Fig. 3, respectively. We set $r_{\mathrm{s}}=2$.

Figure 6

The simulated image with considering the projection effect for ${\theta }_0=80°$ and $n=0.75(\mathrm{top}),1(\mathrm{middle}),1.25(\mathrm{bottom})$. The other marks are the same as in Fig. 5.

Figure 7

The simulated image with considering the projection effect for ${\theta }_0=45°$ and $n=0.75(\mathrm{left}),1(\mathrm{middle}),1.25(\mathrm{right})$. The other marks are the same as in Fig. 5.

Figure 8

The simulated image with considering the projection effect for ${\theta }_0=10°$ and $n=0.75(\mathrm{left}),1(\mathrm{middle}),1.25(\mathrm{right})$. The other marks are the same as in Fig. 5. Here no point corresponds to $z=0$ and all the region we are considering has $z>0$.

Figure 9

The Lagrangian density $\mathcal{L}(I)$ of nonlinear electrodynamics for $n=0.75$. Here we set $r_{\mathrm{s}}=2$.