Beyond Schwarzschild: new pulsating coordinates for spherically symmetric metrics

Abstract

Starting from a general transformation for spherically symmetric metrics where g_11=-1/g_00, we analyze coordinates with the common property of conformal flatness at constant solid angle element. Three general possibilities arise: one where tortoise coordinate appears as the unique solution, other that includes Kruskal-Szekeres coordinates as a very specific case, but that also allows other similar transformations, and finally a new set of coordinates with very different properties than the other two. In particular, it represents any causal patch of the spherically symmetric metrics in a compactified form. We analyze general properties of the new proposed “pulsating coordinates”, and then proceed to apply the transformation for the Schwarzschild spacetime, as well as for several cosmological solutions, contrasting properties with the Kruskal case. In particular, Anti-de-Sitter solution presents interesting features.

Appendix. Existence of conformal flat transformation in (1+1) dimensions.

Any (1+1) metric $g_{ab}dx^{a}dx^{b}$ can be put in a conformal flat way $\Omega \eta _{ab}dx^{a\prime }dx^{b\prime }$, where $\eta _{ab}=diag(-1,1)$. The plainer way to see it is to consider null coordinates v and u. Here ${\textbf{g}}(\partial _{v},\partial _{v})=0$, where $\partial _{v}$ is the basis for any null vector in the v-direction. Since the dual version in terms of the inverse metric is ${\textbf{g}}^{-1}(dv,dv)=0$, and given the one-form $dv=(\partial v/\partial x^{a})dx^{a}$, this is the same as $g^{ab}\partial v/\partial x^{a}\partial v/\partial x^{b}=0$, that can be seen as the definition of a null coordinate, and analog definitions for the other null coordinate u [22]. Defining $x^{0^{\prime }}=v$ and $x^{1^{\prime }}=u$, those relations can be casted as the transformation of the metric to new coordinates, in the form

$$\begin{aligned} g^{\alpha \beta }\frac{\partial v}{\partial x^{\alpha }}\frac{\partial v}{ \partial x^{\beta }}=g^{0^{\prime }0^{\prime }}=0 \end{aligned}$$

(A.1)

and

$$\begin{aligned} g^{\alpha \beta }\frac{\partial u}{\partial x^{\alpha }}\frac{\partial u}{ \partial x^{\beta }}=g^{1^{\prime }1^{\prime }}=0. \end{aligned}$$

(A.2)

Now, if $g^{ab}$ has diagonal elements equal to zero, so does $g_{ab}$, since in this case $g^{00}=(g^{01})^{2}g_{11}$ and $g^{01}\ne 0$ for the inverse to exist. It follows that $g_{11}=0$ and similar argument yields $g_{00}=0$. Then, in terms of null coordinates the system simplifies to

$$\begin{aligned} ds^{2}=g_{\alpha ^{\prime }\beta ^{\prime }}dx^{\alpha ^{\prime }}dx^{\beta ^{\prime }}=g_{0^{\prime }1^{\prime }}(dvdu+dudv). \end{aligned}$$

(A.3)

Rotating coordinates by means of $T=v+u$ and $X=v-u$, the metric takes the assumed conformal form

$$\begin{aligned} ds^{2}=\Omega (-dT^{2}+dX^{2}), \end{aligned}$$

(A.4)

where $\Omega =-2g_{0^{\prime }1^{\prime }}$.