Do Black Holes Really Have High Entropy?

1. Introduction

The idea that black holes are highly entropic objects originates in the Bekenstein-Hawking entropy formula:

S = (k_B * c^3 * A) / (4 * G * ħ)

where A is the area of the event horizon. This formula has been deeply influential, leading to the development of black hole thermodynamics. However, it relies on an analogy between Hawking temperature and classical temperature that is difficult, if not impossible, to test empirically. In this post, I question the assumption that black hole entropy must be as high as the Bekenstein-Hawking formula suggests.

2. Gravitational Collapse and the Asymptotic Horizon

From the frame of a distant observer, a collapsing star never crosses its Schwarzschild radius in finite time. Due to extreme gravitational time dilation, the surface of the star appears to freeze ever closer to the would-be event horizon, asymptotically approaching it.

There seems to be no reason that the entropy of the object should suddenly jump, or, as was assumed before Bekenstein, suddenly drop. Instead, the entropy of the object should evolve continuously as more matter collapses or falls in. It would seem that the entropy of the object should be traceable to its matter content and configuration, much as in any other thermodynamic system.

3. Rethinking Bekenstein's Argument

Bekenstein originally introduced black hole entropy to resolve apparent violations of the second law of thermodynamics. His concern was that objects with entropy could fall into a black hole, causing the total entropy of the universe to decrease unless the black hole itself was assigned an entropy. However, Bekenstein was largely concerned with avoiding the conclusion that black holes had zero entropy.

From the perspective of a distant observer, collapsing matter retains its identity outside the horizon for all time. Assigning it its original entropy—without invoking an area law—is sufficient to satisfy the second law in this frame.

4. Hawking Radiation and Thermodynamic Analogies

Hawking's conclusion that black hole entropy is extraordinarily large -- proportional to the area of the event horizon -- relies on the formal analogy between the surface gravity of a black hole and the temperature of a classical thermodynamic system. This analogy, while elegant, is still a conjecture. It is not derived from a statistical mechanics account of microstates, but from a semi-classical field theory calculation. The identification of entropy as proportional to the area is not a necessity of the theory but an interpretive leap.

5. Relevance to the Information Paradox

The question of whether black holes are highly entropic is often linked to the black hole information paradox. However, this link may be overstated. It is generally accepted that even asymptotically collapsing black holes emit Hawking radiation and can evaporate, regardless of whether a true event horizon forms. The existence of Hawking radiation and black hole evaporation depends on quantum field behavior in curved spacetime, not on entropy accounting.

Therefore, the claim that black holes must be highly entropic is not essential for understanding or resolving the information paradox. Even if the entropy remains equivalent to that of the infalling matter in the frame of a distant observer, the mechanisms for evaporation and information retention (or loss) remain intact. The paradox should be addressed on its own terms, without assuming a horizon-based entropy law.

6. Observational Considerations

It remains unclear whether any empirical observation could definitively distinguish between the standard claim that black holes have an enormous entropy proportional to horizon area and the alternative proposal that their entropy corresponds to the infalling matter. Both perspectives make the same predictions for external observers in terms of Hawking radiation and gravitational dynamics. As a result, the entropy-area law may be more a theoretical convention than an empirically testable fact.

7. Conclusion

I have presented a conceptual argument for skepticism about the claim that black holes are intrinsically highly entropic. From the frame of a distant observer, where an event horizon is never fully realized, there is no compelling reason to assume that black hole entropy must obey the area law. Instead, it may be more natural to assign entropy based on the matter content that is collapsing, without invoking an enormous entropy jump.