ADM Mass for a diagonal metric

[praharmitra]

Given a metric of the form

$$ds^2 = A(r) dt^2 + B(r) dr^2 + C(r)^2 (d\theta^2+sin^2\theta d\phi^2 + sin^2\theta sin^2\phi d\psi^2)$$

I want to find the ADM mass of this black hole. Can anyone help me with the formula, or method to follow?

 

[bcrowell]

Is it a general metric of that form, or is it a black hole? Are we in 4+1 dimensions? Is it a vacuum solution? I don't know if Birkhoff's theorem applies in 4+1. Are you given the functions A, B, and C explicitly? I don't know if the definition of ADM mass generalizes trivially to 4+1 -- probably it does.

 

[praharmitra]

Is it a general metric of that form, or is it a black hole? Are we in 4+1 dimensions? Is it a vacuum solution? I don't know if Birkhoff's theorem applies in 4+1. Are you given the functions A, B, and C explicitly? I don't know if the definition of ADM mass generalizes trivially to 4+1 -- probably it does.

The metric is a black hole. The exact form of the functions A,B,C are

$$A(r) = -h(r)^{-2/3} f(r) \\ B(r) = h(r)^{1/3} f(r)^{-1} \\ C(r) = h(r)^{1/3} r^2 \\$$
where
$$h(r) = 1 + Q/r^2 \\ f(r) = 1 + r^2 + Q - M/r^2$$
for two constants Q and M.

We are working in 4+1 dimensional Asymptotically AdS Space.

Could you atleast give me the definition of ADM mass in 4 dimensions, I will try and figure out the generalisation.

 

[pervect]

There are some formulas in Wald, on pg 293, though he doesn't derive them, referring readers to the literature.

Unforutunately, they're presented in terms of asymptotically euclidean coordinates in an asymptotically flat space-time, so you'd have to read Wald's section in chapters 10 and 11 to make any sense of the formulas there - which still don't give any of the motivations, really.