# How to understand the total proper mass?

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?

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?