Particle falling radially into a black hole

[Cythermax]

Homework Statement

Attached it below

Relevant Equations

Not sure which equations necessary here.

I've been stuck starting anywhere with this. I need to finish this class for graduation and i'd like a safety net of a passing grade with this.

 

[stevendaryl]

Well, do you know what the Schwarzschild metric is? In Schwarzschild (spherical) coordinates, if a particle starts at the point $r, \theta, \phi$ at time $t$ and travels a small distance to the point $r+\delta r, \theta + \delta \theta, \phi + \delta \phi$ by time $t+\delta t$, then the the change in proper time $\delta \tau$ satisfies:

$(\delta \tau)^2 = (1 - 2GM/(c^2 r))^{-1} (\delta t)^2 - (1 - 2GM/(c^2 r)) (\delta r)^2 - r^2 \delta \theta^2 - r^2 sin^2(\theta) (\delta \phi)^2$

You have to start with that as a "relevant equation".