Gravitational lensing of stars orbiting Sgr A* as a probe of the black hole metric in the Galactic center

We show that a possible astrophysical experiment, detection of lensed images of stars orbiting close to Sgr A*, can provide insight into the form of the metric around a black hole. We model Sgr A* as a black hole and add in a $\frac{1}{r^2}$ term to the Schwarzschild metric near the black hole. We then attempt to determine the effect of this extra term on the properties of the secondary images of the S stars in the Galactic center. When the $\frac{1}{r^2}$ term is positive, this represents a Reissner-Nordstrom metric. We show that there is little observational difference between a Schwarzschild black hole and a Reissner-Nordstrom black hole, leading to the conclusion that secondary images may not be a useful probe of electrical charge in black holes. A negative value for the $\frac{1}{r^2}$ term can enter through modified gravity scenarios. Although physically unlikely to apply in the case of a large black hole, the Randall-Sundrum II braneworld scenario admits a metric of this form, known as the tidal Reissner-Nordstrom metric. We use values of tidal charge ($Q$ in $\frac{Q}{r^2}$) ranging from $-6.4M^2$ to $1.6M^2$. A negative value of $Q$ enhances the brightness of images at all times and creates an increase in brightness of up to 0.4 magnitudes for the secondary image of the star S2 at periapse. We show that for other stars with brighter secondary images and positions more aligned with the optic axis, using the tidal Reissner-Nordstrom metric with negative $Q$ enhances the images as well. However, the effect is less pronounced. This effect is related to the increase in the size of the photon sphere in this spacetime and, therefore, should be noticeable in other metrics with a similar effect on the photon sphere. With the next generation of instruments and increased knowledge of radiation from Sgr A*, we may be able to use properties of secondary images to place constraints on the size of the $\frac{1}{r^2}$ term. This knowledge will be useful in constraining any modified gravity theory that adds a similar term into the strong field near a black hole.

Figure 1

The light curve for the secondary image of S2 is dependent on the choice of metric in the strong field. The top figure contains light curves for an entire period of the orbit of S2. The bottom image contains light curves in the year around periapse and peak brightness.

Figure 2

The light curve for the secondary image of S14 near its periapse and peak brightness (top) shows a small effect due to a $\frac{1}{r^2}$ term at peak. The effect for S6 around its peak brightness (bottom) is even smaller. In both cases, $q=-1.6$ for the TRN curve.

Figure 3

This figure displays the relationship between $q$ and image observables for S2. The relationship between $q$ and the apparent magnitude in the $K$ band (top) shows that brightness is inversely related to $q$. This is related to the fact that the angular position of the secondary image, $\theta$, also has an inverse relationship with $q$ (bottom).