# Local mass function in spherically symmetric spacetime

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.

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 $$M$$

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

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

• Have you written $2GM/\rho = 2\kappa \rho$ because, in the paper ,the surface gravity $\kappa$ has been defined as $\kappa = [1- (\nabla \rho)^2]/2\rho$? I understand that they are defining a local mass function $M(x^a)$ but I do not get the logic behind the definition. – damaihati Feb 13 '19 at 10:08