Schwarzschild geodesics

Question

I've found on Wikipedia that energy $E$ and angular momentum $L$ of a particle are conserved quantities in Schwarzschild metric. It's written:

$$L=mr^2 \frac {d\phi} {d\tau},$$

$$E=mc^2\left(1-\frac{r_s}{r}\right)\frac{dt}{d\tau}.$$

And, from the metric, it finds these results:

$$\left(\frac{d\phi}{d\tau}\right)^2=\frac{L^2}{m^2r^4}$$

$$\left(\frac{dr}{d\tau}\right)^2=\frac{E^2}{m^2c^2}-\left(1-\frac{r_s}{r}\right)\left(c^2+\frac{L^2}{m^2r^2}\right).$$

$$\left(\frac{dt}{d\tau}\right)^2=\frac{E}{\left(1-\frac{r_s}{r}\right)mc^2}$$

That I need is to get the results including also the $\theta$ coordinate. So I tried this:

$$p_\phi=mr^2 \frac {d\phi} {d\tau}\qquad\qquad p_\theta=mr^2 \frac {d\theta} {d\tau}$$

$$L^2=p_\theta^2 + \sin^2\theta\ p_\phi^2.$$

$E$ is the same?

$$\left(\frac{d\phi}{d\tau}\right)^2=\frac{p_\phi^2}{m^2r^4}\qquad\qquad \left(\frac{d\theta}{d\tau}\right)^2=\frac{p_\theta^2}{m^2r^4}.$$

$\frac{dr}{d\tau}$ is the same?

$\frac{dt}{d\tau}$ is the same?

But I'm not sure. However, is it possible to get a similar result adding the $\theta$ coordinate?

PS: I am a beginner about GR, so I don't know many things about it.

Answer 1

Rotate the coordinates so that you reduce your problem to the 'standard' treatment with $\theta$ constant, solve it, and then rotate again the coordinates to get the solution in the original system.

The orbit is in the plane identified by the position ($\vec r$) and velocity ($\vec v$) vectors. This plane should be rotated so that it becomes the $xy$ plane. The vector normal to the orbit plane is $\hat n = (\vec r \times \vec v) / (rv)$, that should be rotated to become the $z$ axis - so the rotation axis is $\hat n \times \hat z$ and the rotation angle is $\arccos(\hat n \cdot \hat z)$

Answer 2

What you really want are two sections of wikipedia: https://en.wikipedia.org/wiki/Schwarzschild_geodesics#Mathematical_derivations_of_the_orbital_equation https://en.wikipedia.org/wiki/Schwarzschild_geodesics#Orbits_of_test_particles

It might be tough going for a while but understanding those will keep you from egregious errors. Such that I (and others) have made.

I think some expansion of the question is appropriate.
A more correct question would be:
1)Given a mathematical description of a physical situation can we find invariants for certain types of orbits/paths/histories.
i.e. In the case: What are the invariants of geodesic paths/orbits in the Schwarzschild metric? BTW a fixed point with "time" (time is not a fixed coordinate for different coordinate systems) as the only variable is not a geodesic.
Then:
2)Can we identify these invariants with energy and momentum conservation?

The first part is answered by a Theorem/construction/proof; inside of mathematical physics, it is not a given! Although in our limited physics experience the conservation of energy and angular momentum have proven true and useful; in mathematical physics they must be reproved for separate situations.
As far as I know the questions are unresolved for general situations in GR.