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The bending angle of light is a central quantity in the theory ofgravitational lensing. We develop an analytical perturbation framework forcalculating the bending angle of light rays lensed by a Schwarzschild blackhole. Using a perturbation parameter given in terms of the gravitational radiusof the black hole and the light ray's impact parameter, we determine aninvariant series for the strong-deflection bending angle that extends beyondthe standard logarithmic deflection term used in the literature. In theprocess, we discovered an improvement to the standard logarithmic deflectionterm. Our perturbation framework is also used to derive as a consistency check,the recently found weak deflection bending angle series. We also reformulatethe latter series in terms of a more natural invariant perturbation parameter,one that smoothly transitions between the weak and strong deflection series. Wethen compare our invariant strong deflection bending-angle series with thenumerically integrated exact formal bending angle expression, and find lessthan 1% discrepancy for light rays as far out as twice the critical impactparameter. The paper concludes by showing that the strong and weak deflectionbending angle series together provide an approximation that is within 1% of theexact bending angle value for light rays traversing anywhere between the photonsphere and infinity.Comment: 22 pages, 5 figure

Light’s bending angle due to black holes: from the photon sphere to infinity