A point mass at distance from an observer bends a light ray from a background source…

PHYSICS · Multi-part · Fermi University Physics · Relativity & Gravitation · Schwarzschild spacetime and the tests of general relativity · Schwarzschild spacetime and the tests of general relativity

The Geometry of Gravity, Ch Orbits and light rays in Schwarzschild spacetime

Problem

A point mass

M

at distance

D_{\rm L}

from an observer bends a light ray from a background source at distance

D_{\rm S}

(with

D_{\rm LS}

the distance from lens to source) by the angle

\Delta\phi=4GM/(bc^2)

, where

b

is the impact parameter of the ray at the lens. All angles are small, and

D_{\rm L}

D_{\rm S}

and

D_{\rm LS}

are treated as ordinary Euclidean distances. ((a)) Using the small-angle relation

b=D_{\rm L}\theta_{\rm I}

, where

\theta_{\rm I}

is the angle at which the image is seen, and the lens equation

\theta_{\rm S}=\theta_{\rm I}-(D_{\rm LS}/D_{\rm S})\Delta\phi

relating the source angle

\theta_{\rm S}

\theta_{\rm I}

, derive the Einstein ring radius

\theta_{\rm E}=\sqrt{\frac{4GM}{c^2}\,\frac{D_{\rm LS}}{D_{\rm L}D_{\rm S}}},

the value of

\theta_{\rm I}

when the source lies exactly behind the lens (

\theta_{\rm S}=0

). ((b)) A red dwarf of mass

0.30\,M_\odot

lies at distance

D_{\rm L}=4.0

kpc and lenses a background star at distance

D_{\rm S}=8.0

kpc, with

D_{\rm LS}\approx D_{\rm S}-D_{\rm L}=4.0

kpc. Using

GM_\odot/c^2=1.48

km and

1\ {\rm pc}=3.086\times10^{16}

m, find

\theta_{\rm E}

in milliarcseconds.

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