Formalism for testing theories of gravity using lensing by compact objects: Static, spherically symmetric case

We are developing a general, unified, and rigorous analytical framework for using gravitational lensing by compact objects to test different theories of gravity beyond the weak-deflection limit. In this paper we present the formalism for computing corrections to lensing observables for static, spherically symmetric gravity theories in which the corrections to the weak-deflection limit can be expanded as a Taylor series in one parameter, namely, the gravitational radius of the lens object. We take care to derive coordinate-independent expressions and compute quantities that are directly observable. We compute series expansions for the observables that are accurate to second order in the ratio $\epsilon ={\vartheta }_{•}/{\vartheta }_E$ of the angle subtended by the lens’s gravitational radius to the weak-deflection Einstein radius, which scales with mass as $\epsilon \propto M_{•}^{1/2}$. The positions, magnifications, and time delays of the individual images have corrections at both first and second order in $\epsilon$, as does the differential time delay between the two images. Interestingly, we find that the first-order corrections to the total magnification and centroid position vanish in all gravity theories that agree with general relativity in the weak-deflection limit, but they can remain nonzero in modified theories that disagree with general relativity in the weak-deflection limit. For the Reissner-Nordström metric and a related metric from heterotic string theory, our formalism reveals an intriguing connection between lensing observables and the condition for having a naked singularity, which could provide an observational method for testing the existence of such objects. We apply our formalism to the galactic black hole and predict that the corrections to the image positions are at the level of $10\mu \mathrm{arc}\mathrm{s}$ (microarcseconds), while the correction to the time delay is a few hundredths of a second. These corrections would be measurable today if a pulsar were found to be lensed by the galactic black hole, and they should be readily detectable with planned missions like MAXIM.

Figure 1

Schematic diagram of the lensing geometry.

Figure 2

Terms in the series expansion (63) for the angular image position, as a function of the angular source position. Recall that $\beta >0$ corresponds to the positive-parity image while $\beta <0$ corresponds to the negative-parity image. Sources very close to the origin $|\beta |<0.1$ are not shown. (Top panel) The zeroth-order image position ${\theta }_0$, which is the same for all gravity theories. (Middle panel) The first-order term ${\theta }_1$; recall that the full correction term is $\epsilon {\theta }_1$. Solid curve: Schwarzschild metric in general relativity. Dotted and short-dashed curves: Reissner-Nordström metric in general relativity, with charge parameter $q=0.5$ and 2.5, respectively. Long-dashed curve: GMGHS metric in string theory (modified gravity), with charge parameter $q=2.5$. (The GMGHS case with $q=0.5$ is very similar to the Reissner-Nordström case with $q=0.5$.) (Bottom panel) The second-order term ${\theta }_2$. Again, the full correction term is ${\epsilon }^2{\theta }_2$. All angular lengths are in units of the angular Einstein radius ${\vartheta }_E$. For the galactic black hole, ${\vartheta }_E=0.068(d_{LS}/10\mathrm{pc}{)}^{1/2}$ arc s and the dimensionless expansion parameter is $\epsilon =1.3\times {10}^{-4}\times (d_{LS}/10\mathrm{pc}{)}^{-1/2}$.

Figure 3

Similar to Fig. 2, but showing the terms in the series expansion (76) of the individual image magnifications. Again recall that the full first- and second-order correction terms are $\epsilon {\mu }_1$ and ${\epsilon }^2{\mu }_2$, respectively.

Figure 4

Zeroth- and second-order terms in the series expansion (85) of the total magnification. Our examples involve gravity theories with $A_1=4$, so the first-order correction term vanishes identically and is not shown. For each $\beta >0$, we find the positive-parity image using the source position $\beta$, and the negative-parity image using the source position $-\beta$, and then combine them to obtain the total magnification. The line types are the same as in Fig. 2.

Figure 5

Similar to Fig. 4, but showing the zeroth- and second-order terms in the series expansion (88) of the magnification-weighted centroid position, in units of ${\vartheta }_E$. Again, for our sample gravity theories having $A_1=4$, the first-order correction term vanishes identically and is not shown.

Figure 6

Zeroth- and first-order terms in the series expansion (110) of the differential time delay between the two images. Note that we have defined $\Delta {\tau }_0$ and $\Delta {\tau }_1$ to have dimensions of time, so they are expressed in seconds. Recall that the full first-order correction is $\epsilon \Delta {\tau }_1$, where the expansion parameter is $\epsilon =1.3\times {10}^{-4}\times (d_{LS}/10\mathrm{pc}{)}^{-1/2}$ for the galactic black hole.