- #1

.Scott

Science Advisor

Homework Helper

- 3,160

- 1,345

- TL;DR Summary
- Looking for 2 functions of the radius of a Schwarzschild Black Hole: the "vertical" escape velocity and the maximum angle for a photon to escape the photon sphere.

I've spent well over two hours searching the web for two functions of the radius of a Schwarzschild BH. One would give me the escape velocity of the BH assuming a perfectly vertical trajectory (so it isn't a normal escape velocity). The second relates to the trajectory of a photon that is outbound from inside the photon sphere. How far from vertical can its trajectory be and still escape?

For the first one, if I used the regular escape velocity formula - which happens to work at the event horizon, I get 0.8165 c (##(\sqrt{2/3})c##). But I have no faith in that result.

For the second one, clearly at the event horizon, only a vertical trajectory will allow the photon to reach the photon sphere - and even then, with no remaining energy ("fully" red-shifted). So if the radius is measured in Schwarzschild radii, then at the EH ##f(1)=0##. At the photon sphere ##f(1.5)=\pi/2##. But I would not hazard a guess as to what that function is. ... Or maybe I will: ##f(r)=arctan((2r-2)/(3-2r))## ?? Not likely.

BTW: I hope you don't mind layers because the reason I am trying to get these equations is to prepare another PF post.

For the first one, if I used the regular escape velocity formula - which happens to work at the event horizon, I get 0.8165 c (##(\sqrt{2/3})c##). But I have no faith in that result.

For the second one, clearly at the event horizon, only a vertical trajectory will allow the photon to reach the photon sphere - and even then, with no remaining energy ("fully" red-shifted). So if the radius is measured in Schwarzschild radii, then at the EH ##f(1)=0##. At the photon sphere ##f(1.5)=\pi/2##. But I would not hazard a guess as to what that function is. ... Or maybe I will: ##f(r)=arctan((2r-2)/(3-2r))## ?? Not likely.

BTW: I hope you don't mind layers because the reason I am trying to get these equations is to prepare another PF post.

Last edited: