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.