- #1

tcw

- 9

- 0

## 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.

## Homework Equations

## 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.