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Total energy of matter of a star

  1. Sep 23, 2009 #1
    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):

    [tex]ds^2=-B(r)dt^2+A(r)dr^2+r^2d\Omega^2 [/tex]

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

    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:
    [tex]M_m=\int{\sqrt{g}\rho(r)dr d\theta d\phi} [/tex]

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

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