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 r_s/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 r_s/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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