# Schwarzschild black hole

If law of conservation of angular momentum holds good then how is it possible for a Schwarzschild black hole to exist? Say the momentum of the star initially before the supernova was 1000 kg m/s. Now when this star collapses into a black hole it should spin fast. But a Schwarzschild black hole does not spin. So is conservation of angular momentum violated?

• ...who claims that a true idealized Schwarzschild black hole exists? Also note that a Schwarzschild black hole is by definition eternal and did not arise from the collapse of a star. – ACuriousMind Feb 10 '17 at 14:27
• If we talk about black holes we often talk about the Kerr-black-hole solution. Theoretically we can extrcat energy from the rotating black hole (energy of its rotation) and by doing so we reduce its spin. For this see: en.wikipedia.org/wiki/Penrose_process – Alpha001 Feb 10 '17 at 14:31

The Schwarzschild black hole does not spin, true, but as ACuriousMind says, it is an idealized model version of a Black Hole, which probably does not exist in nature, due to the conservation of angular momentum reason you stated above.

The Schwarzschild black hole was the first solution of a Black Hole using the then brand new General Relativity theory, in 1918. It is used as a first step in teaching students at least 4 different concepts in the derviation of physical results from the Einstein equation and is simplified in very many ways.

The Schwarzschild metric may be written in the form:

$$ds^{2}=\left(1-{\frac {2Gm}{c^{2}r}}\right)^{-1}dr^{2}+r^{2}(d\theta ^{2}+\sin ^{2}\theta d\phi ^{2})-c^{2}\left(1-{\frac {2Gm}{c^{2}r}}\right)dt^{2}}$$

Note that:

$${\frac {2Gm}{c^{2}}}=r_{s}}$$

If you compare the above to the line element of real rotating black holes, you will see that the absence of cross terms and the ability to concentrate on the $r$ and $t$ variables, whilst holding $\theta$ and $\phi$ constant, allows the basic manipulations involved, as well as the physical significance of the various terms, to be learnt more easily.

A Schwarzschild black hole is indeed unrealistic, due to the conservation of angular momentum as you mentioned. However, the Kerr metric can be used to describe a rotation black hole, wherein,

$$ds^2 = \left(1-\frac{r_sr}{\rho^2} \right) dt^2 - \frac{\rho^2}{\Delta}dr^2 - \rho^2 d\theta^2 - \left(r^2+\alpha^2+\frac{r_s r\alpha^2}{\rho^2} \right)\sin^2\theta\, d\phi^2 +\frac{2r_sr\alpha \sin^2 \theta}{\rho^2}dt d\phi$$

where $r_s$ is the Schwarzschild radius, $\rho^2 = r^2 + \alpha^2 \cos^2 \theta$ and $\Delta = r^2-r_sr+\alpha^2$, with the parameter $\alpha = J/M$ dependent upon the angular momentum, $J$ of the black hole. Nevertheless, it is still unrealistic to expect to model any black hole observed in Nature using this exact solution.

Realistically, one would consider perturbations of the metric, perhaps away from this idealised form, to model a black hole, and with perturbation theory one can solve for corrections. However, it is in general a highly non-trivial problem, even numerically, as there are various constraints to consider and gauge issues, as well as the construction of the appropriate initial data.