How Do Radial Light Rays Behave in Schwarzschild Spacetime?

[kikitard]

Homework Statement

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.

Homework 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)

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 black hole.

Any hints/help are greatly appreciated!

 

[mark726]

A) To show the equations for ingoing/outgoing radial light rays, we can start with the null condition for light rays in Schwarzschild spacetime: ds^2=0. Substituting in the Schwarzschild metric, we get:

0=-\left(1-\frac{2m}{r}\right)dt^2 + \left(1-\frac{2m}{r}\right)^{-1}dr^2 + r^2d\Omega^2

where d\Omega^2=d\theta^2 + sin^2\theta d\phi^2.

Next, we can use the substitution t_{*}=t+2mln(\frac{r}{2m}-1) to transform to the t*,r coordinates. This gives us:

0=-\left(1-\frac{2m}{r}\right)dt^{*2} + \left(1-\frac{2m}{r}\right)^{-1}dr^2 + r^2d\Omega^2

We can then solve for dt*/dr, which gives us:

dt*/dr = \pm\frac{r+2m}{r-2m}

This matches the equation given in the forum post for outgoing t*,r coordinates. For ingoing t*,r coordinates, we can use the substitution t_{*}=t-2mln(\frac{r}{2m}-1) to get the equation dt*/dr=-1.

B) The local light cones in t,r coordinates will look like standard Minkowski light cones, with the addition of a curvature term that causes the cones to tilt inwards towards the singularity at r=0. In t*,r coordinates, the light cones will be tilted inwards towards the singularity at r=2m due to the transformation.

C) The t,r coordinates give a truer physical description of the local light cones, as they are adapted to the curvature of spacetime caused by the presence of a massive object. The t* coordinate system is useful for studying the behavior of light rays, but it is not a physically realizable coordinate system.

D) As mentioned in the forum post, all light rays near the r=2m line will eventually hit the singularity. This is because the curvature of spacetime near the singularity becomes infinite, causing all geodesics (including light rays) to converge