How Much Proper Time Does an Observer Measure Falling Into a Black Hole?

[rrfergus]

Homework Statement

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?

Homework 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 = -c²dτ²(1 - 2GM/rc²)dt² + dr²/(1 - 2GM/rc²) + r²(dθ² + sin²θdσ²) in spherical coordinates.

The Attempt at a Solution

I'll be honest, I really have no idea where to even start with this problem

 

[WannabeNewton]

What conserved quantities do we have at our disposal?

 

[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.

 

[strangerep]

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.

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

 

[WannabeNewton]

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.

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?