Why are white holes the same thing as black holes in quantum gravity?

Question

Why are white holes the same thing as black holes in quantum gravity? Their Penrose diagrams in semiclassical gravity are utterly different.

Answer 1

The reason is that black holes are thermal equilibrium states, they emit Hawking radiation. So if you put a black hole in a box with thermal radiation, it will reach equilibrium, sometimes it will fluctuate out of existence, increasing the temperature of the radiation slightly, sometimes it will fluctuate back. The whole thing is CPT invariant. But if you CPT reverse the thing, it will be a white hole fluctuating back and forth. So black holes and white holes must be the same. This was hawking's argument in 1978 (maybe '77).

Notice that this is a strange result--- the classical picture is totally different for the two. But the difference is an artifact of Penrose descriptions, and shows you that they are suspect in a quantum theory.

A white hole that disappears a long time in the future is exactly the same in the exterior solution as a black hole which appeared a long time ago. The two attract matter (the time reverse of a particle accelerating toward a black hole is a particle accelerating toward a white hole), they both aggregate matter on the horizon from an exterior point of view.

The only difference is the structure of the horizon. For a black hole, the infalling matter crosses the horizon, and outgoing modes are smooshed really close to the initial formation point of the horizon. For a white hole, it's the reverse--- the infalling matter is smooshed really close to the final point, while the outgoing matter is past-extendible.

The resolution of the two pictures is to view things in Susskind's way, as a complementary description. All the regions which are inaccessible, both the white hole region and the black hole region, are reconstructions from the exterior description, they are related to the oscillations of the black hole horizon itself. If you do one map, you extend to the future interior, and the past interior is not accessible in these variables. If you do another map, you get the past region variables, but then the future region variables are not accessible. This picture is clearly correct, since it produces AdS/CFT and all that, but it is very hard to make it precise for neutral Schwarzschild.

The end result is that you just think of the regions in continuation past a horizon as reconstructions, and the same horizon variables are encoding both regions. These horizon variables also encode all near-horizon matter, to the extent that something near the horizon is surrounded by horizon (meaning that if you follow light rays out from a given point, if nearly all of these end on the horizon, the point should be described by horizon degrees of freedom). This preceding principle was explicitly described by Susskind in the mid-90s.

That the Penrose picture is unreliable, and should be considered a reconstruction, can already be seen in classical solutions. Consider the deSitter Schwartschild solution, describing a black hole in deSitter space.

$$ ds^2 = f(r) dt^2 - {dr^2\over f(r)} - r^2 d\Omega^2 $$

$$ f(r) = 1- {2a\over r} - b r^2 $$

For small a, the coefficient a is the mass of the black hole, and b is the deSitter horizon radius. As you increase a keeping b fixed, you reach a limit where the two positive zeros of f collide. This is the Nariai solution. At this point, the space between the black hole horizon and the cosmological horizon is regular, and the singularity recedes infinitely far to the point where it is no longer accessible to the oberver sandwiched between the two horizons. The black hole and cosmological horizon are symmetric, and the space is $dS_2 \times S_2$.

Further, if you start near a Nariai solution, the black hole and cosmological horizon both emit Hawking radiation, and the net flow is according to statistics. So if you get lucky, you can swap the two horizons, and the universe turns inside out like a sock. What you called the black hole is now the cosmological horizon, and what you called the cosmological horizon is now the black hole.

If you imagine that there is a singularity in the black hole, this singularity is disappearing through the improbable Hawking radiation, and appearing on the other side! This is clearly ridiculous if you imagine the spacetime beyond the horizon is not a reconstruction of the horizon degrees of freedom.

While this picture is not proved, nor is it embedded in a consistent theory of deSitter space string theory, it is internally consistent, and consistent with the known holographic dualities that work. So one should just not take the Penrose picture too seriously--- always work in one causal patch, and consider the Penrose extensions reconstructions of spacetime from exterior available data.

If one followed logical positivism assiduously, this would not have come as a surprise. The regions outside the cosmological or black hole horizons are not directly experimentally accessible, and it is only an extrapolation that gives them meaning.