Monday, 11 May 2015

general relativity - Trajectory of a photon around a Schwarzschild black hole?


Consider a photon coming from the infinity in a unbounded orbit to a Schwarzschild black hole (Schwarzschild radius rs) (see this for illustration). Its impact parameter is b and its distance of closest approach is r0 with b2=r30r0rs

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Then its trajectory in polar coordinates is defined by :


dφdr=1r21b2(1rsr)1r2


Consequently : φ(r)=rr0dpp21b2(1rsp)1p2


and one can compute the total deviation using : Δφ=2×(limr+φ(r)π2)


But my question is : how can I plot/draw the trajectory using the integral expression of φ(r) ?





Because if I compute : f(r)=2×(rr0dpp21b2(1rsp)1p2π2)

I obtain f(r0)=π, and then f increases up to zero, crosses zero, and tends to its positive value at infinity Δφ. It does not make sense for me and I do not understand how to compute the trajectory from that.




If I compute : g(r)=rr0dpp21b2(1rsp)1p2

it starts from 0, and increase up to π2+Δφ2.




I would like to compute the trajectory in the (x,y) plane, so how to use the values of f(r) or g(r) to compute the function y(x) ?




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