# General relativity question on mass conservation integral

• tcw

## Homework Statement

Starting off with a general axisymmetric metric:
$$ds^{2}=g_{tt}dt^{2}+2g_{t\phi }dtd\phi + g_{\phi \phi }d\phi^{2} +g_{rr}dr^2 + g_{\theta \theta }d\theta ^2...\left ( 1 \right )$$
where the metric components are functions of r and theta only.

I have deduced (using Euler-Lagrange equations) that:
$$E=g_{tt}\frac{dt}{d\tau}+g_{t\theta }\frac{d\phi}{d\tau}...\left ( 2 \right )\\ L=g_{t\phi }\frac{dt}{d\tau}+g_{\phi \phi}\frac{d\phi}{d\tau}...\left ( 3 \right )$$
where E and L are constants.

I am required to derive:
$$g_{rr}\left (\frac{dr}{d\tau} \right )^{2}+g_{\phi \phi}\left (\frac{d\theta }{d\tau} \right )^{2}=V_{eff}\left ( r,\theta ,E,L \right )$$
which is where I get stuck.

## The Attempt at a Solution

Dividing (1) by d(tau)^2 and substituting (2) and (3), and rearranging gives:
$$g_{rr}\left (\frac{dr}{d\tau} \right )^{2}+g_{\phi \phi}\left (\frac{d\theta }{d\tau} \right )^{2}=\left (\frac{ds}{d\tau} \right )^{2}-E\frac{dt}{d\tau}-L\frac{d\phi}{d\tau}$$
upon which I am unsure how to proceed.

I tried solving (2) and (3) simultaneously but to no avail.

Any help is appreciated. Thanks.

You only need to solve (2) and (3) algebraically for $$dt/d\tau$$ and $$d\phi/d\tau$$. You can deal with the $$(ds/d\tau)^2$$ term by recalling the relationship between the invariant interval and the proper time.