Local mass function in spherically symmetric spacetime

Question

I am studying the paper `Inflation and de Sitter Thermodynamics' at https://arxiv.org/abs/hep-th/0212327 . I have problems with the way they define a local mass function in a general spherically symmetric spacetime. I am quoting from Appendix B of the paper: "A general spherical spacetime is described by the metric \begin{align} ds^2 = g_{ab}dx^a dx^b + \rho^2 d\Omega^2 \qquad \end{align} where $g_{ab}$ is the metric on a two manifold with coordinates $(t,r)$ and $\rho$ is the physical radius of spherical slices."

Firstly, I do not understand the difference between $r$ and $\rho$.

Then, before eq. (35) they say, "In spherical symmetry, it is possible to define a local mass function $M(x^a)$ by \begin{align} 1 - \frac{2GM}{\rho} = (\nabla \rho)^2~." \end{align} It is not at all obvious to me how the above relation comes about. Any insight into this will be very helpful.

Answer 1

1) You can choose whichever coordinates you like in GR, in order to aid physical intuition and/or to ease calculations, etc. In this case, the manifold is sliced into 2-spheres which is done by choosing $r=$ constant and $t=$ constant, to get the metric

$$ds^2 = \rho^2 d \Omega^2$$

which is exactly the metric for a 2-sphere of radius $\rho$. What they mean is that, in general, $r$ may be different from $\rho$, but it is $\rho$ that defines the sizes of the spherical slices.

2) That's a definition, as stated in the paper, so they're relating $\rho$ to a new function $M$

$$\frac{2GM}{\rho} = 2 \kappa \rho$$

$$\Rightarrow M = \frac{\kappa \rho^2}{G}$$