Schwarzschild Geodesics - Bending of Light By Gravity

Bending of Light By Gravity

See also: Gravitational lens and Shapiro delay

In the limit as the particle mass m goes to zero (or, equivalently, as the length-scale a goes to infinity), the equation for the orbit becomes

\varphi = \int \frac{dr}{r^{2} \sqrt{\frac{1}{b^{2}} - \left( 1 - \frac{r_{s}}{r} \right) \frac{1}{r^{2}}}}

Expanding in powers of rs/r, the leading order term in this formula gives the approximate angular deflection δφ for a massless particle coming in from infinity and going back out to infinity:

\delta \varphi \approx \frac{2r_{s}}{b} = \frac{4GM}{c^{2}b}.

Here, b can be interpreted as the distance of closest approach. Although this formula is approximate, it is accurate for most measurements of gravitational lensing, due to the smallness of the ratio rs/r. For light grazing the surface of the sun, the approximate angular deflection is roughly 1.75 arcseconds, roughly one millionth part of a circle.

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