# What equation (/solution) predicts the existence of black holes?

Where does our theoretical prediction of the existence of black holes come from? If it is (as I am guessing) from the Einstein Field Equations, which solution predicts it and why?

• Sep 11, 2015 at 9:25
• That would be the Schwarzschild solution, or more generally the Kerr-Newman solution, for the static case. Sep 11, 2015 at 9:31

Where does our theoretical prediction of the existence of black holes come from?

It comes from the singularity theorems of Hawking and Penrose. Before then, people were aware of solutions to Einstein's Field Equations that had singularities but they required absolutely perfect symmetries, such as perfect radial symmetry for Schwarzschild, or perfect axial symmetry for Kerr.

Lifshitz and Khalatnikov (1963) should have an account of the work people did trying to show that perfect and unachievable exactness would be required to make a black hole. And at that point people really thought that singularities might be impossible to actually make. They thought if you you tried to make one then the absolutely slightest mistake could instead make one not form. Which means there wouldn't really be any in nature, just some mathematical solutions that never apply to the real world

Then everything changed in 1965 when Penrose showed (roughly) that if you pushed something inside the event horizon (roughly) it would have to form a singularity (with some other assumptions). This is in contrast to say, Newtonian physics where small enough particles can swirl around the common center of mass and never collide.

There is a big caveat in that you have to assume it gets compressed enough and then the singularity forms. Observers on the outside don't see it get compressed enough, so there is still no evidence of a singularity. Hawking then made some singularity theorems too, for instance which implied that if you look at earlier and earlier times the big bang was itself a singularity. And then by the 1980's Hawking said that really he thinks quantum effects come up before a singularity is formed so we don't really know.

But the singularity theorems starting in the mid 1960's are when we first thought that singularities were a real world prediction of General Relativity. I.e. that General Relativity predicts that singularities would exist in the real world if General Relativity is true of the real world at all length and time scales. But the caveat still applies. And the caveat has been updated to the Cosmic Censorship Conjecture, where people hypothesize that we will never have evidence of a singularity formation because we will always avoid seeing too much matter or energy or stress or pressure or momentum get trapped by a surface of too small a surface area.

From a historical perspective black holes weren't predicted. In 1916 Karl Schwarzschild found a solution to Einstein's equations for a spherically symmetric mass. It was only subsequently realised that the Schwarzschild metric is a vacuum solution with an event horizon and a curvature singularity at its centre, and that the metric describes a static uncharged black hole. It took until the end of the fifties (over 40 years!) before the black hole nature of Schwarzschild's solution was fully understood.

Following Schwarzschild's solution, three more solutions were found describing charged, rotating and charged-rotating black holes. These are the Reisner-Nordström, Kerr and Kerr-Newman metrics. These four metrics are the only known black hole solutions.

• Can the table on the wiki page for Kerr be interpreted as saying that: The Schwarzschild solution predicts black holes for $J=0$ and $Q=0$, Kerr for $J \ne 0$ and $Q=0$, Reissner-Nordtrom for $J =0$ and $Q \ne 0$ and Kerr-Newman for $J \ne 0$ and $Q \ne 0$. Or is predicts to strong a word? Sep 11, 2015 at 10:59
• @Joseph: all four metrics predict that a horizon can form for sufficiently high density. The RN, Kerr and KN metrics predict that the horizon will disappear for sufficiently high values of $Q$ and $J$, but these are thought to be unphysical and in practice a horizon will always exist if and only if the density is high enough. So it's really only the density of the spherically symmetric body that matters. Low density (relatively low density that is) objects like the Earth, Sun or even neutron stars won't form a horizon but denser objects will. Sep 11, 2015 at 11:45
• @JohnRennie I think the focus on density can be misleading because people can think it is energy (or mass) per volume that matters instead of energy per area of enclosing surface. You can have a surface density of energy and not form a black hole even though the volume density is infinite. An explanation can be as simple as possible, but don't make it simpler than that. Sep 11, 2015 at 15:13
• @Timaeus: I take your point, and can say only that it's a judgement call how far to simplify an answer for non-GR heads. Sep 11, 2015 at 15:16
• @JohnRennie I think you can write a good post, and have comments clearly labelled "for experts" that aren't really for experts in GR but are for people more experienced in physics in general. Because the idea that it is volume density develops early and some people never get rid of it. So if people pick it up here before they formally study GR then it could be permanently crippling to them if they never gets remediated properly. Because they can think they know it right and not listen to people that know better. When a respected person like you can be read as agreeing with them it's a barrier Sep 11, 2015 at 15:22