Gravitational Bending of Light Near Compact Objects

Abstract

A photon emitted near a compact object at an angle $\alpha$ with respect to the radial direction escapes to infinity at a different angle $\psi>\alpha$. This bending of light is caused by a strong gravitational field. We show that, in a Schwarzschild metric, the effect is described by $1-\cos\alpha=(1-\cos\psi)(1-r_g/R)$ where $R/r_g$ is the emission radius in Schwarzschild units. The formula is approximate and it applies at $R\geq 2r_g$ only, however at these radii it has amazing accuracy, fully sufficient in many applications. As one application we develop a new formulation for the light bending effects in pulsars. It reveals the simple character of these effects and gives their quantitative description with practically no losses of accuracy (for the typical radius of a neutron star $R=3r_g$ the error is 1%). The visible fraction of a star surface is shown to be $S_v/4\pi R^2=[2(1-r_g/R)]^{-1}$ which is 3/4 for $R=3r_g$. The instantaneous flux of a pulsar comes from one or two antipodal polar caps that rotate in the visible zone. The pulse produced by one blackbody cap is found to be sinusoidal (light bending impacts the pulse amplitude but not its shape). When both caps are visible, the pulse shows a plateau: the variable parts of the antipodal emissions precisely cancel each other. The pulsed fraction of blackbody emission with antipodal symmetry has an upper limit $A_{max}=(R-2r_g)/(R+2r_g)$. Pulsars with $A>A_{max}$ must be asymmetric.