So, the canned explanation that I always hear about why the white hole solution of the extended Schwarzschild solution is non-physical is that "The matter distribution cuts off the white hole solution."

If I look at Oppenheimer-Snyder collapse, however, the cosmological solution that I paste to in the dust's interior is just the closed pressureless model of the Robertson-Walker metric, which is perfectly time-symmetric about the point of maximal expansion. If I look at this naïvely, then, the fluid in the Oppenheimer-Snyder solution originated in a singularity, which then disintegrates, to form an expanded fluid, which reaches some maximal extent, and then recollapses to form a singularity in the future, after some finite time. That past singularity sure seems like a white hole to me.

What am I misinterpreting here? Or is it just a pathology of the Oppenheimer-Snyder construction that it is over-simple, and more physically realistic matter models WOULD cut off the past singularity?


4 Answers 4


The point is that the configuration that you are describing as the time-reversed Oppenheimer-Snyder collapse would need very specific initial conditions to occur. A collapse process on the other hand is very generic. You can have a collapse with any number of initial configurations. In that sense a collapse process is more physical than the reverse.

It is like dropping a needle and having it land on its tip. There are many ways to drop it and have it lying down on its side, but there is only one that you can drop it and have it land and stay balanced on its tip.

It is a mathematical possibility but a physical improbability.

  • $\begingroup$ OK. Take any singularity free spacetime with a moment of time symmetry whose future development has a black hole in it. Why does that spacetime not have a white hole in its past? Alternately, why would small perturbations of the Oppenheimer-Snyder spacetime cause the past singularity to dissapear, but the future singularity to remain? $\endgroup$ Commented Nov 25, 2010 at 1:29
  • $\begingroup$ For the same reason that one can break an egg but not reconstitute a whole egg from a broken one. It's a question of entropy. $\endgroup$
    – Matt Reece
    Commented Nov 25, 2010 at 5:21
  • $\begingroup$ @Jerry Schirmer: I think that you are looking at it backwards. It is not a matter of whether there should be in the past of every black hole a white hole, but whether a dust configuration you could have now has originated from the inside of a white hole. Snyder collapse is not the only way to get a black hole. Christodoulou for example has shown that you can have a convergence of gravitational waves from null infinity with general initial conditions that produces a trapped surface and ends up forming a black hole. There is no white hole in the past of that. $\endgroup$
    – Vagelford
    Commented Nov 25, 2010 at 8:27
  • $\begingroup$ Sure. the whole argument collapses if there is no moment of time symmetry in the solution, but that doesn't seem to be a radical requirement to make. And one would think that most black holes came from particulate matter, and not things like geons and Brill waves. But if you can find physically realistic matter distributions that seem to have white hole solutions in their past, $\endgroup$ Commented Nov 25, 2010 at 11:48
  • $\begingroup$ The basic argument is that white holes have very low entropy, just like the big bang. $\endgroup$
    – Sklivvz
    Commented Nov 25, 2010 at 19:58

So, I brought this up at my research group meeting this week. Turns out my initial guess was right--the past development of the Oppenheimer-Snyder spacetime does contain a white hole. When people say that the matter distribution cuts off the white hole, what they generally assume is that the spacetime does not contain a moment of time symmetry, and that the matter distribution is something like a star, that goes into the distant past.

  • $\begingroup$ Yes, Oppenheimer Snyder time reversed contains a white hole. So what. You have to set up the white hole, and with very specific conditions on the singularity that it is emitting the Oppenheimer Snyder dust shells to exit. This is considered unphysical. A better question would be for charged dust-shells collapsing to form a near-extremal Reissner Nordstrom black hole (or white hole, whichever you like, they're the same thing). $\endgroup$
    – Ron Maimon
    Commented Jul 28, 2012 at 7:57

I can't make sense of your quoted phrase "The matter distribution cuts off the white hole solution.", but I'll try to answer the rest of the question

The Oppenheimer-Synder model for black hole collapse is a solution of the equations of classical GR where a uniform speherically symmetrical dust cloud with no pressure or rotation collapses to form a black hole. The time reversal is also a valid solution of the field equations but it would represent a white hole and is therefore normally considered unphysical.

Inside the sphere of dust the solution matches the well known cosmological Freidman, Robertson Walker solutions for the expanding universe. Outiside the collapsing sphere it looks like the Schwartzchild static black hole solutions.

In the cosmological solution, there are different cases where the universe can expand forever or recollapse. The same thing happens in the Oppenheimer-Synder solutuon. So there are cases where it starts with a white-hole and the dust expands outwards, but the dust does not have enough energy to escap so it recollapses to form a black hole. However, there are other cases where the dust does escape from the whitehole. The time reversal of this is also the formation of a blackhole without the whitehole in the past

By the way, these are all special cases of a more general class of spherically symmetric solutions called the Lamaitre-Tolman solutions.


A white hole is in a sense a time reversed black hole. If you take an Eddington-Finkelstein diagram and turn it upside down you have a white hole. A physical sense of things suggests these do not exist. The argument for “cutting off” is usually interpreted as how the Penrose conformal diagram with two square patches for time like regions and two triangular spacelike regions is reduced so the bottom spacelike region for the white hole is removed. Pretty clearly there is an asymmetry between the two halves of the pure or “eternal” solution.

White holes may have played a role in the early universe though. The Hubble constant is ${\dot a}/a~=~H$, ${\dot a}~=~da/dt$, and $a$ the scale factor. The FLRW differential equation of motion for a constant vacuum for the de Sitter spacetime is $$ ({\dot a}/a)^2~=~8πG\Lambda/3, $$ which has an exponential solution. So we have the scale factor evolving as $$ a~\simeq~\sqrt{3/8πG\Lambda}exp(\sqrt{8πG\Lambda/3}t). $$ This exponential expansion is what smoothed out anisotropies in the universe.

So we consider this with some anisotropy which is rewound backwards in time. So the small anisotropies rapidly clump back together during the inflationary period, which in a time reversed setting can mean the collapse of matter into black holes. Now this is the time reversal of inflation, which means the inflationary period may have had white holes. The white holes might have been some perturbation on the inflationary cosmology and their disappearance in the exponentially expanding space some aspect of inflation.


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