some homework problems for Farsight, part 2
In an earlier post, I stated
the basic definitions that define locally Euclidean manifolds and differentiable manifolds. (The spacetime manifold is a
pseudo-Riemannian manifold, locally Minkowskian rather than locally Euclidean, but the basic definitions remain the same. The difference is that the metric tensor is not positive definite, merely nondegenerate.)
In
part 1, with
an important correction by ctamblyn, I gave
Farsight eight homework problems that would help him to understand the basic definition of a manifold. Those were the most difficult of the homework problems I will assign, and
Exercise 8 was by far the most difficult of the eight.
In this part 2, the homework exercises explore some properties of Schwarzschild coordinates and the Schwarzschild metric on the manifold M that surrounds (but does not include) the event horizon (aka gravitational radius) of a black hole.
In part 3, the homework exercises will explore Lemaître coordinates and the Lemaître metric on a larger manifold M' that includes M but also includes the event horizon and extends to (but does not include) the singularity at the heart of a black hole. Those exercises will prove that, contrary to
Farsight's repeated claims, the Schwarzschild metric's coordinate singularity at the gravitational radius is a mere artifact of the Schwarzschild coordinate system.
All of these homework exercises state well-known results, but I think it's useful to collect them within this thread and to give hints (in blue) that make it easy for anyone with an undergraduate education in math or the physical sciences to confirm some of the basic facts
Farsight's been denying.
With the (-,+,+,+) signature convention and natural units in which c=G=1, the Schwarzschild coordinates are (t, r, θ, φ), and the Schwarzschild metric is
[latex]
\[
ds^2 = - (1 - 2M/r)dt^2 + (1 - 2M/r)^{-1} dr^2 + r^2 d\Omega^2
\]
[/latex]
where
[latex]
\[
d \Omega^2 = d \theta^2 + \sin^2 \theta d \phi^2
\]
[/latex]
is the usual metric on a 2-sphere in spherical coordinates and
[latex]
\[
\begin{align*}
r &> 2M \\
0 &< \theta < \pi \\
- \pi &< \phi < \pi
\end{align*}
\]
[/latex]
(Note that the lower bound on θ was
incorrect when I stated that inequality in
part 1.)
We'll start out with some exercises that derive some properties of the metric dΩ
2 on the unit 2-sphere.
Exercise 9. Prove: When the azimuth φ is held constant,
[latex]
\[
\frac{d \Omega^2}{d \theta^2} = 1
\]
[/latex]
Hint: trivial algebra. φ is constant, so dφ=0.
Exercise 10. Prove: When the elevation θ is held constant,
[latex]
\[
\frac{d \Omega^2}{d \phi^2} = \sin^2 \theta
\]
[/latex]
Hint: trivial algebra.
Exercise 11. Prove:
[latex]
\[
\lim_{\theta \rightarrow 0} \frac{d \Omega^2}{d \phi^2} =
\lim_{\theta \rightarrow \pi} \frac{d \Omega^2}{d \phi^2} = 0
\]
[/latex]
Hint: trivial calculus.
Exercises 9 through 11 say the chart defined in exercise 2 involves spatial distortion away from the equator (where θ is pi/2, or 90 degrees) and coordinate singularities at the poles. Two of the Schwarzschild coordinates are the spherical coordinates of exercises 9 through 11, so every chart based on Schwarzschild coordinates will involve spatial distortion away from the equator and coordinate singularities at the poles.
Exercise 12. Define M as the idealized spacetime manifold that's covered by the complete set of Schwarzschild charts with coordinates restricted by the inequalities stated earlier. Prove that M cannot be covered by a single Schwarzschild chart.
Hint: Every Schwarzschild chart has coordinate singularities at the poles of its spherical coordinates.
Exercise 13. Prove that M can be covered by two Schwarzschild charts.
Hint: See exercise 7.
Exercise 14. Prove: If θ and φ are held constant, and ds=0, then
[latex]
\[
\frac{dr^2}{dt^2} = (1 - 2M/r)^2
\]
[/latex]
Hint: trivial algebra.
Exercise 15. Prove: If θ and φ are held constant, and ds=0, then
[latex]
\[
\lim_{r \rightarrow \infty} \frac{dr^2}{dt^2} = 1
\]
[/latex]
Hint: trivial calculus.
Exercise 16. Prove: If θ and φ are held constant, and ds=0, then
[latex]
\[
\lim_{r \rightarrow 2M} \frac{dr^2}{dt^2} = 0
\]
[/latex]
Hint: trivial calculus.
Light follows null geodesics, for which ds=0, so exercises 14 through 16 say something about the coordinate speed of light rays that are travelling radially. To an observer almost infinitely far away, the speed of light is essentially 1, which is the standard speed of light in the natural units we're using. That's why we say a Schwarzschild chart corresponds to an observer at infinity. Exercise 16 says that, as observed by an observer at infinity, the speed of light travelling directly toward or away from a black hole converges to zero as it approaches the event horizon (where r=2M).
Farsight has been misinterpreting the result of exercise 16 to mean light and/or time actually stops at that event horizon.
Compare exercise 16 with exercise 11. Does space stop at the poles of a 2-sphere? No. Exercise 11 therefore describes nothing more than a coordinate singularity. Might exercise 16 describe nothing more than a coordinate singularity?
Yes. In part 3, we'll prove that fact using Lemaître coordinates.
