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.
...
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 (via Birkhoff's theorem). 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}Black hole thermodynamics
Hawking radiation
The process of Hawking radiation relates to the behaviour of quantum fields in a black hole background. For instance, let's consider the Klein-Gordon field in curved spacetime :
\begin{equation} S[\phi] = \int_M \left[ g(\nabla \phi, \nabla \phi) + m^2 \phi^2 + \xi R \phi \right] d\mu[g] \end{equation}with the classical equation of motion
\begin{equation} \Box \phi - m^2 \phi - \xi R = 0 \end{equation}Since we are dealing with the Schwarzschild solution, we have $R = 0$ and the Laplace-Beltrami operator is
\begin{eqnarray} \Box \phi &=& \frac{1}{\sqrt{-g}} \partial_{\mu} (g^{\mu\nu} \sqrt{-g} \partial_\nu \phi) \end{eqnarray} \begin{eqnarray} \sqrt{-g} &=& r^2 \sin(\theta) \end{eqnarray} \begin{eqnarray} \Box \phi &=& g^{\mu\nu} \partial_{\mu} \partial_\nu \phi + g^{\mu\nu} \frac{\partial_{\mu}\sqrt{-g}}{\sqrt{-g}} \partial_\nu \phi \\ &=& -\frac{1}{(1 - \frac{r_s}{r})} \partial^2_t \phi + (1 - \frac{r_s}{r}) \partial^2_r \phi + \frac{1}{r^2} \partial^2_\theta \phi + \frac{1}{r^2 \sin^2 (\theta)} \partial^2_\varphi \phi + 2 \frac{(1 - \frac{r_s}{r})}{r} \partial_r \phi + \cot(\theta) \partial_\varphi \phi \end{eqnarray}The angular part is simply the spherical Laplacian, which has as its solution the spherical harmonics $Y^m_l(\theta, \varphi)$
...
The associated quantum theory for this will be given by the Schwinger-Dyson equation for it
\begin{equation} \Box \hat{\phi} - m^2 \hat{\phi} = 0 \end{equation}If we consider the proper definition of field operators as operator-valued distributions,
The Vaidya metric
If we stop considering our Hawking radiations as a test field, what happens to our black hole? The black hole thermodynamics we've seen relies on the assumption
Bibliography
Last updated : 2023-04-20 11:08:41