Entropy of a black hole after evaporation

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Discussion Overview

The discussion revolves around the entropy of black holes, particularly focusing on the changes in entropy during and after the evaporation process. Participants explore various theoretical frameworks and models, including semiclassical descriptions, quantum gravity theories, and the implications of the Planck scale on black hole entropy.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant suggests using a semiclassical description involving Hawking radiation and classical spacetime to calculate entropy, noting challenges with adapting standard formalism and the non-constancy of geometry and temperature.
  • Another participant raises concerns about the breakdown of current theories for black holes smaller than the Planck mass, questioning the possibility of obtaining a quantitative value for entropy after complete evaporation.
  • A similar point is reiterated, emphasizing the limitations of semiclassical calculations and suggesting alternative approaches such as microscopic state counting in Loop Quantum Gravity or string theory.
  • One participant humorously proposes that the entropy change for a Schwarzschild black hole evaporating could be quantitatively zero, linking it to a ground state energy related to the Planck mass.
  • Another participant expresses skepticism about the assumption that black hole entropy resides solely at the event horizon.

Areas of Agreement / Disagreement

Participants express multiple competing views regarding the calculation of black hole entropy during evaporation, with no consensus reached on the validity of different theoretical approaches or the implications of the Planck scale.

Contextual Notes

Participants highlight limitations in current theories, particularly regarding the behavior of black holes at the Planck scale and the assumptions made in traditional entropy calculations.

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Black holes have an entropy, but they evaporate. At the end of the evaporation, the entropy is greater than the entropy at the beginning of the evaporation. I am looking for an example of a quantitative result for the entropy of the black hole after evaporation (or the entropy difference between the beginning and the end of the evaporation). You can use your favorite theory (General relativity, f(R), String theory, Loop Quantum Gravity, etc..), you can use your favorite kind of black hole (Schwarzschild, rotating, charged, extremal, BTZ, etc... ), and you can use your favorite dimension (from 4 to 11...), but I am looking for a quantitative result.
 
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I would try a semiclassical description (Hawking radiation + classical spacetime; after evaporation this is Hawking radiation + flat spacetime). For Hawking radiation the entropy calculation is done using Bose-Einstein statistics for non-interacting photons of given temperature. There are two problems: i) one has to adapt the standard plane wave formalism to (distorted) spherical waves (have a look at Hawking's paper); ii) geometry and therefore temperature are not constant.
 
Don't all our current theories break down for a black hole smaller than a Planck mass (or a ratio thereof, such as √∏), thus we can't arrive at a quantitative (integrative) value for the entropy of complete evaporation?
 
ryan albery said:
Don't all our current theories break down for a black hole smaller than a Planck mass (or a ratio thereof, such as √∏), thus we can't arrive at a quantitative (integrative) value for the entropy of complete evaporation?
The semiclassical calculation (Hawking) is certainly invalid; there are proposals for microscopic state counting in LQG or string theory (but I think that evaporation with thermal radiations + corrections is not understood); anyway - why not try to calculate the BH entropy using Bekensteins formula, and then calculate the entropy for the thermal radiation?
 
Does the Planck scale represent the realm where the horizon of a black hole is basically the same as the singularity itself, at least in regards to entropy?

With a sense of humor I hope others can appreciate, the theory I choose to answer this question in is that the universe has a ground state energy/entorpy (or related frequency) that's closely proportional to the Planck mass. Taking that same mass/energy/frequency for my Schwarzschild black hole, then the entropy change for such a black hole evaporating is quantitatively zero. Funny, that same ground state also gives (in a space domain) a value for dark energy that's pretty close to what we observe.

I obviously think I might need some help with my thinking.
 
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Typically, black hole entropy calculations assume it all resides at the event horizon. I'm unconvinced that is correct.
 

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