Black Holes are for Real: Horizons, Black Hole Interiors, and Singularities

Andrew J. S. Hamilton (JILA, U. Colorado, Boulder)

Solution of the firewall problem

A Penrose diagram consistent with general relativity, quantum field theory, and causality, that evades any firewall paradox. (Why?)

AMPS (Almheiri, Marolf, Polchinski, Sully arXiv:1207.3123, updated arXiv:1304.6483)

Title:
Black Holes: Complementarity or Firewalls?

Abstract:
We argue that the following three statements cannot all be true:

  1. Hawking radiation is in a pure state,
  2. the information carried by the radiation is emitted from the region near the horizon, with low energy effective field theory valid beyond some microscopic distance from the horizon, and
  3. the infalling observer encounters nothing unusual at the horizon.
Perhaps the most conservative resolution is that the infalling observer burns up at the horizon.
Alternatives would seem to require novel dynamics that nevertheless cause notable violations of semiclassical physics at macroscopic distances from the horizon.

The paper caused a mini-storm of positive and negative reaction on the arXiv.

No-no words All these are observer-dependent, as they are in cosmology.

AMPS: “To restate our paradox in brief, the purity of the Hawking radiation implies that the late radiation is fully entangled with the early radiation, and the absence of drama for the infalling observer implies that it is fully entangled with the modes behind the horizon. This is tantamount to cloning.”

Oppenheimer-Snyder (1939) collapse of a uniform pressureless star to a black hole


Hawking effect redux

Hawking emission results whenever an observer watches an accelerating emitter.

Waves that are pure positive frequency (positive energy) in the emitter's frame translate to a mix of positive and negative frequencies in the observer's frame. The vacuum in any frame is defined to be the absence of any positive frequency waves. The emitter's vacuum (“in”-vacuum) is therefore not the same as the observer's vacuum (“out”-vacuum).

A classic calculation shows that if an observer watches an emitter who is accelerating away at constant acceleration for several acceleration times, the observer will see the emitter's vacuum as a thermal state at temperature T = κ/(2π).

Outside observers see black hole horizons emit Hawking radiation because matter that collapsed to form the black hole long ago remains frozen at the horizon, accelerating and redshifting into the indefinite future.


The Information Paradox (Hawking 1976)

In the presence of black hole horizons, one of two sacred tenets of QFT must break down:
Unitarity:  Quantum mechanics is fundamentally deterministic and lossless (information is conserved)
Locality:  Spacelike (FTL) separated field operators commute, enforcing causality (information cannot travel FTL)

Information inside the black hole can reach an outside observer only along a spacelike-separated path.

So if locality holds, evaporation cannot be unitary.

Conversely if evaporation is unitary, then locality cannot hold.

Why locality must break down inside black holes

Locality (commutation of spacelike-separated field operators) implies that entropy is additive over any spacelike surface.

Feed a (charged, spherical) black hole with matter. Allow (ohmic) dissipation inside. The entropy accumulated along a spacelike surface near the singularity can exceed the Bekenstein-Hawking (BH) entropy by many orders of magnitude, 1010 times in the model of Wallace, Hamilton & Polhemus arXiv:0801.4415. This overproduction of entropy (a) violates the notion that the entropy of a black hole is the BH entropy, (b) violates the second law if the black hole subsequently evaporates radiating only the BH entropy.

Solution? Impose locality only between points whose future lightcones intersect before hitting the singularity.

Locality must hold where causality requires it

Locality must hold between two spacelike-separated observers whose future lightcones intersect, to enforce causality in accordance with standard quantum field theory.

Therefore, if locality breaks down as holography postulates, then that breakdown must occur between spacelike-separated points whose future lightcones fail to intersect.

Therefore, the quantum “interior” of a black hole, the region that is holographically connected to the outside, must be observer-dependent. The quantum interior must lie in that region whose future hits the singularity before intersecting the future of the observer.

Contradiction

Observers observing the black hole along the same past lightcone access different sets of interior states of the black hole in the Hawking radiation that they see. The more distant observer accesses more states.

This contradicts postulate (ii) of AMPS “low energy effective field theory valid beyond some microscopic distance from the horizon.”

Hawking radiation becomes real only when an observer observes it.

Note that the vast majority of the interior states are concentrated to the leftmost point of the Penrose diagram.

The Singularity (of a Schwarzschild black hole) is a Wall, not a Point

Conclusions
 1.  Arguments in favor of firewalls implicitly assume that the quantum interior of a black hole is the region inside the true (future event) horizon. The assumption is perhaps so obvious that it has not been considered worth stating. But the assumption is wrong.
 2.  The illusory horizon (the redshifting surface of the collapsed star), not the true horizon, is the observational boundary of an observer's spacetime, is the origin of Hawking radiation, and is the hologram of the black hole, for insiders as well as outsiders.
 3.  The singularity is where general relativity fails, and must be replaced by new physics. General relativity suggests that the singularity is a wall (a brane?), not a point.
 4.  If Hawking evaporation is to be unitary, then there must be some non-local (FTL) “holographic” connection between the singularity and the illusory horizon. The nature of that connection remains an unsolved problem of quantum gravity.