Total energy of matter of a star

[cosmogirl]

Homework Statement

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

Homework 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.

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?

 

[Jhero]

Thank you for your question. The total energy of matter inside a spherically symmetric static star can be calculated using either of the two expressions you mentioned. Both expressions give the same result and are mathematically equivalent. The difference lies in the interpretation of the integral.

The first expression, given by Weinberg, calculates the total energy by integrating the energy density (ρ) over the volume of the star. This takes into account the curvature of space caused by the presence of matter.

The second expression, given by Shapiro, calculates the total energy by integrating the mass density (ρ) over the volume of the star, taking into account the metric (A(r)) which represents the curvature of space due to the presence of matter.

In summary, both expressions are correct and give the same result. It is a matter of preference which one to use, but both take into account the important factors of matter and space curvature. I hope this helps to clarify any confusion. Good luck with your research!