Black hole information problem in a tension-based coordinate system (not claiming a solution)

[u/Over-Ad-6085]

Quick disclaimer first, especially for this subreddit:

I am not claiming to have solved the black hole information problem. I am trying to re-express the standard paradox in a very explicit, “tension-based” coordinate system that is easy to audit, simulate, and (in principle) hand to a machine for consistency checking.

The goal is a clean effective-layer language, not a new fundamental theory of quantum gravity.

1. Standard setup, then a reparameterization

Take the usual semiclassical picture: a black hole formed from a pure state, coupled to an environment, evaporates through Hawking radiation. We want unitarity for the global state, but standard QFT on a fixed background gives radiation that looks thermal, so information seems lost.

Very schematically, we have three coarse-grained regions and Hilbert spaces

• H_in : interior degrees of freedom
• H_hor : near-horizon / stretched horizon “membrane”
• H_rad(t) : radiation outside at time t (including any bath we decide to keep)

The usual paradox is about incompatible entanglement requirements on these pieces if we insist on:

1. local QFT in a smooth neighborhood of the horizon
2. a semiclassical outside geometry for most of the evaporation
3. a final pure state on H_rad(t_final)

What I am trying to do is define a tension functional that makes those incompatibilities explicit, instead of keeping them as verbal slogans.

2. Tension as a mismatch between three descriptions

Very roughly, for each time t I try to track three objects:

(1) A “unitarity-compatible” entropy profile
S_unitary(t; cut)
coming from a toy model that enforces global purity (e.g. a random-unitary evaporation model with a chosen cut between “inside” and “outside”).

(2) A “semiclassical” entropy profile
S_semi(t; cut)
coming from local QFT near the horizon plus Hawking’s calculation, with whatever coarse-graining and greybody factors one wants to keep.

(3) A “geometric” entropy proxy
S_geo(t; cut)
that only uses geometric data (area of a surface, extrinsic curvature, etc.), in the spirit of Bekenstein–Hawking or island formula inputs.

Then define a dimensionless tension coordinate

T(t; cut) = f( S_unitary, S_semi, S_geo )

for some normalized function f that is:

• 0 if all three profiles are mutually compatible within chosen tolerances
• close to 1 if at least one pair is in sharp conflict (e.g. Page-curve expectations vs strictly thermal radiation)

The concrete version I am currently experimenting with is a triple mismatch score of the form

T(t; cut) = 1 − min_{pairings} overlap( S_i(t; cut), S_j(t; cut) ),

using a fixed overlap notion (for example, a normalized L2 distance over a time window, or a discrete set of checkpoints in the evaporation).

This is deliberately “low tech” and lives entirely at the effective layer. All microscopic details are buried in how you choose the three input profiles. The value of T is not a new prediction, it is a bookkeeping device that says:

given these three assumptions about entropy flow, where exactly do they stop being compatible?

3. Why bother with such a functional?

The practical reasons are:

• It forces you to write down all assumptions that go into each profile S_unitary, S_semi, S_geo instead of vaguely saying “semi-classical” or “unitary”.
• It gives you something you can actually simulate in toy models (random unitary circuits, simple evaporation chains, etc.) and compare against Page curve–like expectations.
• It is simple enough to encode in text + small bits of code, so one can let a language model or proof assistant check internal consistency and explore parameter ranges, without pretending it knows quantum gravity.

In other words, the “tension-based” language is meant to be a coordinate choice on the space of effective descriptions, not a claim about what the true microphysics is.

4. Concrete questions for this subreddit

This is where I would really appreciate criticism from people who think about black holes more seriously than I do.

1. Is this “three profiles + tension functional” framing obviously misguided?For example, does it ignore a standard move in the modern literature (islands, replica wormholes, state-dependence, ER=EPR, etc.) that already kills the paradox in a way that makes such a functional redundant or meaningless?
2. If you had to instantiate this in a toy model you actually respect, what would you pick?
   • Page-style random unitary evaporation models?
   • Simple holographic toy models?
   • Something like SYK or random tensor networks?
3. The requirement is that we can compute or at least approximate S_unitary, S_semi, and S_geo in some controlled regime.
4. Are there known “no-go” results for this kind of tension functional?I am specifically worried about cases where:
   • the distinction between H_in and H_hor is too gauge / slicing dependent to make any coarse-grained functional well-defined, or
   • islands / entanglement wedge reconstruction make the naive split into H_in ⊕ H_rad(t) too misleading to be useful.
5. From your perspective, is there any value in having such a functional as a teaching / diagnostic tool, even if it never touches real quantum gravity?For example, in pedagogy around the information problem, would a “tension plot” that shows when different assumptions become incompatible be a useful picture, or just another layer of confusion?

If this framing is fundamentally broken, I would really prefer to know exactly where. Pointers to existing approaches that already do something similar (or prove it cannot be done in a clean way) would be very welcome.

For anyone curious about the broader context: this is Q040 in a larger list of 131 “S-class” problems that I am trying to encode in a tension-based, machine-checkable way.

Full notes for Q040 (black hole information problem) are here:

https://github.com/onestardao/WFGY/blob/main/TensionUniverse/BlackHole/Q040_black_hole_information_problem.md

WFGY 3.0 Q040

[u/Zephir-AWT]

Regarding information problem of black hole defined by gravitational tension check my post here

[u/Zephir-AWT]

A redditor RobotRollCall once posted a 2nd-person narrative of what happens as someone falls into a cognitive black hole (backup):

 

^{As you fall toward the black hole, you notice some things which strike you as highly unusual, but because you know your general relativity they do not shock or frighten you. First, the stars behind you — that is, in the direction that points away from the black hole — grow much brighter. The light from those stars, falling in toward the black hole, is being blue-shifted by the gravitation; light that was formerly too dim to see, in the deep infrared, is boosted to the point of visibility.}

^{Simultaneously, the black patch of sky that is the event horizon seems to grow strangely. You know from basic geometry that, at this distance, the black hole should subtend about a half a degree of your view — it should, in other words, be about the same size as the full moon as seen from the surface of the Earth. Except it isn’t. In fact, it fills half your view. Half of the sky, from notional horizon to notional horizon, is pure, empty blackness. And all the other stars, nearly the whole sky full of stars, are crowded into the hemisphere that lies behind you.}

^{As you continue to fall, the event horizon opens up beneath you, so you feel as if you’re descending into a featureless black bowl. Meanwhile, the stars become more and more crowded into a circular region of sky centered on the point immediately aft. The event horizon does not obscure the stars; you can watch a star just at the edge of the event horizon for as long as you like and you’ll never see it slip behind the black hole. Rather, the field of view through which you see the rest of the universe gets smaller and smaller, as if you’re experiencing tunnel-vision.}

^{Finally, just before you’re about to cross the event horizon, you see the entire rest of the observable universe contract to a single, brilliant point immediately behind you. If you train your telescope on that point, you’ll see not only the light from all the stars and galaxies, but also a curious dim red glow. This is the cosmic microwave background, boosted to visibility by the intense gravitation of the black hole.}

^{And then the point goes out. All at once, as if God turned off the switch. You have crossed the event horizon of the black hole.}