# Total energy of matter of a star

1. Sep 23, 2009

### cosmogirl

1. The problem statement, all variables and given/known data

Suppose I have a spherically symmetric static star. What is the total energy of (baryonic) matter inside the star?

2. Relevant equations

The metric is (parametrizing as in Weinberg):

$$ds^2=-B(r)dt^2+A(r)dr^2+r^2d\Omega^2$$

I assume the energy-momentum tensor of a perfect fluid.
As it happens, the total energy (including the gravitational field) is given by:
$$M=\int{4\pi r^2\rho(r)dr}$$

However, I'm interested in the matter only.

3. The attempt at a solution

Weinberg (Gravitation... p. 302) gives the following expression, which seems correct:
$$M_m=\int{\sqrt{g}\rho(r)dr d\theta d\phi}$$

However, Shapiro (Black holes, white dwarfs and neutron stars, p. 125) gives another expression:
$$M_m=\int{\sqrt{A(r)}4\pi r^2 dr}$$

My questions are: how to reconcile between the two? Is Shapiro wrong? Is the quantity in question well defined?