Gravitational lensing formalism

In general relativity, a point mass deflects a light ray with impact parameter by an angle approximately equal to where G is the gravitational constant, M the mass of the deflecting object and c the speed of light. A naive application of Newtonian gravity can yield exactly half this value, where the light ray is assumed as a massed particle and scattered by the gravitational potential well. This approximation is good when is small. In situations where general relativity can be approximated by linearized gravity, the deflection due to a spatially extended mass can be written simply as a vector sum over point masses. In the continuum limit, this becomes an integral over the density , and if the deflection is small we can approximate the gravitational potential along the deflected trajectory by the potential along the undeflected trajectory, as in the Born approximation in quantum mechanics. The deflection is then where is the line-of-sight coordinate, and is the vector impact parameter of the actual ray path from the infinitesimal mass located at the coordinates . In the limit of a "thin lens", where the distances between the source, lens, and observer are much larger than the size of the lens (this is almost always true for astronomical objects), we can define the projected mass density where is a vector in the plane of the sky. The deflection angle is then As shown in the diagram on the right, the difference between the unlensed angular position and the observed position is this deflection angle, reduced by a ratio of distances, described as the lens equation where is the distance from the lens to the source, is the distance from the observer to the source, and is the distance from the observer to the lens. For extragalactic lenses, these must be angular diameter distances. In strong gravitational lensing, this equation can have multiple solutions, because a single source at can be lensed into multiple images.