Using similar reasoning to Section 5.4 of Relativity, Gravitation and Cosmology, it can be shown that photons passing ne

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Using Similar Reasoning To Section 5 4 Of Relativity Gravitation And Cosmology It Can Be Shown That Photons Passing Ne 1 (31.53 KiB) Viewed 35 times

Please help me with b and c, part b is needed to resolve c and
im stuck on both parts. I thought i had b but and i can't get c
from it. Really hope someone can help me.

Using Similar Reasoning To Section 5 4 Of Relativity Gravitation And Cosmology It Can Be Shown That Photons Passing Ne 2 (48.02 KiB) Viewed 35 times

Using similar reasoning to Section 5.4 of Relativity, Gravitation and Cosmology, it can be shown that photons passing near a Schwarzschild black hole, in the equatorial plane (@=1/2), have paths that satisfy 1 dr 1 2GM 1 = 1/62 (1) p2 do p2 rc2 where b is a constant. You are not required to prove this equation. The constant b happens to equal the ratio of photon angular momentum and orbital energy FCO ))+ (1- =

(b) The second term on the left hand side of Equation 1 can be interpreted as an effective potential V(r), following similar reasoning to that used for massive particles in Relativity, Gravitation and Cosmology. Show that b < 1/(V(r))1/2 (Hint: what is the requirement on dr/do for light to escape?) (c) Show that the function V(r) has a maximum value at r = (3/2)Rs, where Rs is the Schwarzschild radius, and that the maximum value is 1 (2) 27(R$/2)2 Hence show that the there is a critical impact parameter berit = 27Rs/2 such that when b > berit light rays are deflected rather than captured, but for b< berit light rays are absorbed by the black hole. Vmax = = =