# General Relativity Energy Problem

1. Jan 31, 2014

### rrfergus

1. The problem statement, all variables and given/known data
An observer falls radially inward toward a black hole of mass M, which is equal to 3 solar masses, starting with zero kinetic energy at infinity. How much time is measured by this observer as he travels between radii r1 = 6GM/c2 and the event horizon?

2. Relevant equations
The only solution to Einstein's equations we have learned is the Schwarzschild radius, so the solution probably involves that (I'm guessing). The Schwarzschild solution is
ds2 = -c22(1 - 2GM/rc2)dt2 + dr2/(1 - 2GM/rc2) + r2(dθ2 + sin2θdσ2) in spherical coordinates.

3. The attempt at a solution
I'll be honest, I really have no idea where to even start with this problem

2. Jan 31, 2014

### WannabeNewton

What conserved quantities do we have at our disposal?

3. Jan 31, 2014

### rrfergus

Energy should be conserved. Momentum also, but I think energy conservation would be more relevant to the problem. I just don't know how to express the kinetic or potential energies in this context, or to calculate the time.

4. Jan 31, 2014

### strangerep

What do you know about the energy-momentum tensor, in the context of GR?

5. Jan 31, 2014

### WannabeNewton

Let's think about this from a top down point of view. What does the problem want and what does it tell us? Well it wants the proper time $\tau$ measured by the observer between two given radii $r = 6M$ and $r = 2M$ right? Well if we know how much proper time $d\tau$ is incremented when the observer travels an amount $dr$, we can integrate between the two radii and we're done. So we want to find an expression for $\frac{dr}{d\tau}$ somehow. Conserved quantities will be key here. In this case energy is all that matters since the observer falls in radially and hence has no angular momentum.

First, what's the conserved energy in Schwarzschild space-time?

Next, what do we know about the magnitude $u_{\mu}u^{\mu} = g_{\mu\nu}u^{\mu}u^{\mu}$ of the observer's 4-velocity $u^{\mu} = (\frac{dt}{d\tau}, \frac{dr}{d\tau}, \frac{d\theta}{d\tau}, \frac{d\phi}{d\tau})$? What is it always equal to?

Finally what can we say about $\frac{d\theta}{d\tau}$ and $\frac{d\phi}{d\tau}$ for this radially infalling observer?