The standard setting for the black hole information paradox is semiclassical gravity, where quantum fields backreact on the metric via the expectation value of their stress-energy tensor. Then the paradox is as follows:

  1. By solving the Heisenberg equations of motion (as Hawking did), we find that the late-time state of the exterior of the black hole is thermal.

  2. In the semiclassical approximation, since Hawking radiation carries away energy, the black hole gets smaller over time.

  3. Once the black hole is small enough, the dimension of its Hilbert space is too small to purify the state of the exterior. Therefore the late-time state of the exterior is thermal, violating unitarity.

Step 1 is known to be dubious: small corrections to Hawking's calculation are enough to destroy the conclusion that the exterior state is thermal at late times. In the well-known recent Page curve calculations, the saddle-point approximation of the gravitational path integral was shown to give a pure exterior state at late times, as expected, provided the correct saddle is chosen. This at least partially resolves Hawking's information paradox.

In this resolution, it was crucial that gravity was turned on. The whole argument was done using the gravitational path integral, albeit only in a saddle-point approximation.

But isn't there an information paradox even when gravity is turned off?

That is, suppose we fix a background geometry corresponding to a slowly shrinking black hole. I don't care whether it solves Einstein's equations or not: right now, we're just doing QFT on a fixed curved spacetime. There's no reason we can't just choose an arbitrary spacetime by hand. Sure, it's not physical, but this is just a thought experiment anyway.

Now we solve the Heisenberg equations of motion, and just as before we find that the late-time state is thermal. Again, this presents a paradox once the black hole is too small to purify the external state. But now, the clever gravitational path integral resolution is not available to us, since there's no gravity: the background is fixed. So unitarity is violated, which is worrying since QFT on curved spacetime is essentially just a quantum system with time-dependent Hamiltonian.

What have I missed?


1 Answer 1


… when gravity is turned off …

… we fix a background geometry corresponding to a slowly shrinking black hole …

… this presents a paradox once the black hole is too small to purify the external state …

The error in OP's reasoning lies in that last sentence. Once the gravity is turned off, the Bekenstein bound no longer provides the measure of too small black hole since now you can “cram” arbitrary large Hilbert spaces into a fixed volume.

Consider a box with impenetrable walls. If there is no gravity we can place arbitrary large number of quanta into it with arbitrarily high energy. So the total entropy contained in such a box could be arbitrary large. It is only gravitational backreaction that puts a cap on the number of microstates in such a box, because large enough energy in a box leads to a black hole formation. A fixed background geometry of blackhole interior provides precisely the impenetrable box for quantum fields. If quantum fields do not leave an imprint on geometry, there is no bound on information such a geometry can contain.

So, no, there is no paradox without gravity.

  • $\begingroup$ Thank you. This is a great answer. $\endgroup$ Nov 15, 2022 at 18:16

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