1. Apr 24, 2013

### kikitard

1. The problem statement, all variables and given/known data

Consider Schwarzschild spacetime.
A) Show that the equation for ingoing/outgoing radial light rays is dt/dr = +-r/(r-2m) in t,r coordinates and dt*/dr = -1, dt*/dr=(r+2m)/(r-2m) in t*,r coordinates
B) Sketch the local light cones in t,r and t*,r coordinates
C) Explain which coordinates give a truer physical description of the local light cones.
D) Explain the motion of a radial light ray emitted near r=2m.

2. Relevant equations
schwarzschild metric 0=g$_{\mu\upsilon}$(x(t))$\dot{x}$$^{\mu}$(t)$\dot{x}$$^{\upsilon}$(t)= -$\frac{r-2m}{r}$ +$\frac{r}{r-2m}$$\dot{r}$$^{2}$+r$^{2}$sin$^{2}$$\theta\phi$$^{2}$+r$^{2}$$\dot{\theta}$$^{2}$

t$_{*}$=t+2mln($\frac{r}{2m}$-1)

3. The attempt at a solution
My main struggle here is with part A) ... C) I also am not 100% sure of.

A) I have managed to show dr/dt*=$\frac{r-2m}{r+2m}$ for the outgoing t*,r coordinates, by using null coordinates, but I am clearly missing something here.

B) *i have this one completed also*

C)I believe that the t coordinates give a truer physical description (for us), as these are introduced in order to adapt to the light rays. i.e. t* is the time coordinate adapted to the light rays, thus, this coordinate system should give the physical description for the light rays... but t should be physically truer for us (?)

D)I have said that all light rays near the r=2m will eventually hit the singularity, however the outward directed ray from outside the r=2m line will always remain outside r=2m, 'escaping' the blackhole.

Any hints/help are greatly appreciated!