How to understand the total proper mass?

Question

In Wald's General Relativity, page 126, he difined a function $m(r)$ as $$ m(r)=4\pi\int^r_0\rho(r')r'^2dr', $$ where $\rho(r)$ is just the $\rho$ appeared in a perfect fluid stress-energy tensor. From this, he indicated that the total mass in Newtonian gravity as $M=m(R)$. He also defined the total proper mass, which is $$ M_p=\int^R_0\rho(r)r^2\sqrt{^{(3)}\textrm{g}}\quad d^3x=4\pi\int^R_0\rho(r)r^2\big[1-\frac{2m(r)}{r}\big]^{-1/2}dr. $$ Then he gave the gravitational binding energy $E_B$ as the difference between $M$ and $M_p$: $$ E_B=M_p-M. $$ I am having a difficulty in recognizing the physical meaning of these three quantities. Is the "proper" here the same meaning as the proper time? Could someone give an explanation?

Answer 1

The "proper volume" of a spherical shell is $$dV = 4\pi r^2\,\left(1 - \frac{2m(r)}{r}\right)^{-1/2}dr\ ,$$ where $m(r)$ is the mass that a distant observer would say is responsible for the gravitational effect exterior to $r$ and $G=c=1$. This expression takes account of the fact that the proper distance between $r_2$ and $r_1$ is not $r_2-r_1$.

Whilst the surface area of a sphere is $4\pi r^2$ for the Schwarzschild metric (outside a spherically symmetric mass, e.g., see here), the proper distance between two points separated on a radial line (i.e. $ds$ when $dt=d\theta=d\phi=0$) is $$ ds = \left(1 - \frac{2m(r)}{r}\right)^{-1/2}dr\ .$$ Hence the expression for "proper volume". The proper mass is just the proper volume multiplied by the density.

$M(R)$ is the observable mass/energy of the star - the mass that you would deduce for example by studying the motion of a binary companion.

However, $M_p$, the total proper mass/energy is the mass that the consituents of the star would have if they were dispersed to infinity by giving them exactly the gravitational binding energy.

The binding energy is a negative energy term and the gravitational effects of the mass of the star are reduced by the binding energy. i.e. $M < M_p$. I therefore assume that Wald is defining $E_B$ as a positive quantity?