We present a semi-rigorous justification of Bekenstein’s Generalized Second

Law of Thermodynamics applicable to a universe with black holes present, based

on a generic quantum gravity formulation of a black hole spacetime, where the

bulk Hamiltonian constraint plays a central role. Specializing to Loop Quantum

Gravity, and considering the inspiral and post-ringdown stages of binary black

hole merger into a remnant black hole, we show that the Generalized Second Law

implies a lower bound on the non-perturbative LQG correction to the

Bekenstein-Hawking area law for black hole entropy. This lower bound itself is

expressed as a function of the Bekenstein-Hawking area formula for entropy.

Results of the analyses of LIGO-VIRGO-KAGRA data recently performed to verify

the Hawking Area Theorem for binary black hole merger, are shown to be entirely

consistent with this Loop Quantum Gravity-induced inequality. However, the

consistency is independent of the magnitude of the Loop Quantum Gravity

corrections to black hole entropy, depending only on the negative algebraic

sign of the quantum correction. We argue that results of alternative quantum

gravity computations of quantum black hole entropy, where the quantum entropy

exceeds the Bekenstein-Hawking value, may not share this consistency.

## The Future of Quantum Gravity and Black Hole Entropy

In this article, we have presented a justification of Bekenstein’s Generalized Second Law of Thermodynamics as it applies to a universe with black holes, using a quantum gravity formulation of black hole spacetime. Specifically, we have focused on the Loop Quantum Gravity (LQG) approach and examined the inspiral and post-ringdown stages of binary black hole merger into a remnant black hole.

One of the main conclusions we have drawn is that the Generalized Second Law implies a lower bound on the non-perturbative LQG correction to the Bekenstein-Hawking area law for black hole entropy. This means that the entropy of a black hole, as described by LQG, must be at least a certain value determined by the Bekenstein-Hawking formula.

Furthermore, we have shown that recent analyses of data from LIGO-VIRGO-KAGRA are consistent with this LQG-induced lower bound on black hole entropy. This consistency is based on the negative algebraic sign of the quantum correction in LQG and not on the magnitude of the correction itself.

However, it is important to note that alternative quantum gravity computations of black hole entropy, where the quantum entropy exceeds the Bekenstein-Hawking value, may not share this consistency. This raises the possibility of different approaches within quantum gravity leading to different predictions about black hole entropy.

### Roadmap for the Future

The research presented in this article opens up several avenues for future exploration in the field of quantum gravity and black hole entropy. Here is a roadmap for readers interested in this topic:

- Further Investigation of Loop Quantum Gravity: Continued research into the LQG approach is necessary to better understand the nature of black hole entropy and its relationship to the Bekenstein-Hawking formula. This could involve refining the calculations of LQG corrections or exploring different scenarios for black hole mergers.
- Alternative Approaches to Quantum Gravity: The article highlights the potential inconsistencies between LQG and other quantum gravity theories regarding black hole entropy. Future studies should focus on these alternative approaches to determine if they provide a more complete and unified description of black hole thermodynamics.
- Experimental Verification: While the consistency between LQG and LIGO-VIRGO-KAGRA data is promising, further experimental verification is crucial. Ongoing observations and advancements in gravitational wave detection technology can offer valuable insights into the nature of black holes and the validity of different quantum gravity theories.
- Philosophical Implications: The discrepancies between quantum gravity theories may have philosophical implications, questioning the nature of entropy and the fundamental laws of thermodynamics. Exploring these philosophical aspects can deepen our understanding and guide future research directions.

Overall, the future of quantum gravity and black hole entropy research is full of challenges and opportunities. By further investigating different quantum gravity theories, conducting experimental tests, and contemplating the philosophical implications, we can uncover deeper truths about the nature of the universe and its fundamental laws.