# Deriving the mass conservation integral in GR

tcw

## Homework Statement

Starting from a general axisymmetric metric
ds^2=g_tt dt^2 + 2g_tφ dtdφ +g_φφ dφ^2 + g_rr dr^2+g_θθ dθ^2 ...(0)
where the metric components are functions of the coordinates r and θ only.

I've managed to show (via Euler-Lagrange equations) that
g_tt dt/dτ + g_tφ dφ/dτ = E ...(1) and
g_tφ dt/dτ + g_φφ dφ/dτ = -L_z ...(2)
(where E and L_z are constants, and τ an affine parameter)

I am required to derive the mass conservation integral:
g_rr (dr/dτ)^2 + g_θθ (dθ/dτ)^2 = -V_eff (r,θ,E,L_z)
which I'm having trouble doing.

## The Attempt at a Solution

Dividing (0) by dτ^2 and substituting (1) and (2), and re-arranging gives:
g_rr (dr/dτ)^2 + g_θθ (dθ/dτ)^2 = (ds/dτ)^2 - E dt/dτ + L_z dφ/dτ ...(3)
but I'm not sure where to go from here.

I thought about trying to solve (1) and (2) simultaneously for dt/dτ and dφ/dτ to substitute into (3) but that doesn't seem to work.

I'd appreciate any help, thanks.

## Answers and Replies

tcw
I've spent a bit more time and still no luck. Any pointers?

tcw
Should I latex it up to get responses?

Homework Helper
Should I latex it up to get responses?
Just the fact that you didn't use LaTeX isn't itself a reason for people not to answer your question, but it would definitely help if you can put those formulas in LaTeX. I just tried reading through your question and it's kind of hard to parse with the formulas written out in text.

tcw
Just the fact that you didn't use LaTeX isn't itself a reason for people not to answer your question, but it would definitely help if you can put those formulas in LaTeX. I just tried reading through your question and it's kind of hard to parse with the formulas written out in text.

Thanks for your help, I'll do that then.