Gravitational lensing formalism

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Gravitational lensing
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In general relativity, a point mass deflects a light ray with impact parameter ${\displaystyle b~}$ by an angle approximately equal to

${\displaystyle {\hat {\alpha }}={\frac {4GM}{c^{2}b}}}$

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 ${\displaystyle 4GM/c^{2}b}$ 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 ${\displaystyle \rho ~}$, 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

${\displaystyle {\vec {\hat {\alpha }}}({\vec {\xi }})={\frac {4G}{c^{2}}}\int d^{2}\xi ^{\prime }\int dz\rho ({\vec {\xi }}^{\prime },z){\frac {\vec {b}}{|{\vec {b}}|^{2}}},~{\vec {b}}\equiv {\vec {\xi }}-{\vec {\xi ^{\prime }}}}$

where ${\displaystyle z}$ is the line-of-sight coordinate, and ${\displaystyle {\vec {b}}}$ is the vector impact parameter of the actual ray path from the infinitesimal mass ${\displaystyle d^{2}\xi ^{\prime }dz\rho ({\vec {\xi }}^{\prime },z)}$ located at the coordinates ${\displaystyle ({\vec {\xi }}^{\prime },z)}$.^[1]