Life cycle of a black hole


One of the difficult part of talking about black holes is the wide array of metrics which fall under that label. For the most part, the static black holes (such as the Schwarzschild or Kerr metric) will be used for that purpose, but more realistic black holes will not be static, making any statement from a static black hole hard to check for a dynamical one.

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There is already another article on this site on the topic of black holes, if you need some basics on the topic. Our static black hole here will simply be the maximally extended Schwarzschild metric, with interior and exterior metric

\begin{equation} ds^2 = -(1 - \frac{r_s}{r}) dt^2 + (1 - \frac{r_s}{r})^{-1} dr^2 + r^2 d\Omega^2 \end{equation}

with the usual well-known properties of a black hole.

The problem is the following : we need to see how a black hole is born, how it dies, and, joining the two together, what an observer will see.

The Oppenheimer-Snyder collapse

There are quite a lot of ways to make a black hole thanks to the singularity theorem, including a few pathological cases of naked singularities, but the most simple and standard way of doing so is the Oppenheimer-Snyder collapse, which is the collapse of a spherical ball of uncharged dust.

Since this is a spherically symmetric spacetime, the exterior portion will simply be the Schwarzschild metric. If our ball is of radius $R(t)$, then the metric is, for $r > R(t)$,

\begin{equation} ds^2 = -(1 - \frac{r_s}{r}) dt^2 + (1 - \frac{r_s}{r})^{-1} dr^2 + r^2 d\Omega^2 \end{equation}

Keeping the Schwarzschild coordinates, the interior metric is of the form

\begin{equation} ds^2 = - e^{2\alpha} dt^2 + e^{2\beta} dr^2 + r^2 d\Omega^2 \end{equation} the stress-energy tensor of a spherically symmetric metric is

\begin{eqnarray} R_{tt} &=& e^{2(\alpha - \beta)} \left[ \alpha'' + (\alpha')^2 - \alpha'\beta' + \frac{2}{r} \alpha' \right]\\ R_{rr} &=& - \left[ \alpha'' + (\alpha')^2 - \alpha' \beta' - \frac{2}{r} \beta' \right]\\ R_{\theta\theta} &=& e^{-2\beta} \left[ r (\beta' - \alpha') - 1 \right] + 1\\ R_{\phi\phi} &=& R_{\theta\theta} \sin^2 \theta \end{eqnarray}

The Oppenheimer-Snyder collapse is the case of a zero pressure, uniform density varying in time, so that

\begin{eqnarray} e^{2(\alpha - \beta)} \left[ \alpha'' + (\alpha')^2 - \alpha'\beta' + \frac{2}{r} \alpha' \right] &=& \rho(t)\\ - \left[ \alpha'' + (\alpha')^2 - \alpha' \beta' - \frac{2}{r} \beta' \right] &=& 0\\ e^{-2\beta} \left[ r (\beta' - \alpha') - 1 \right] + 1 &=& 0\\ \end{eqnarray}
Last updated : 2019-09-17 11:19:37
Tags : physics , general-relativity , black-hole