Probing the quantum nature of black holes through entropy
In a study published in Physical Review Letters, physicists have demonstrated that black holes satisfy the third law of thermodynamics, which states that entropy remains positive and vanishes at extremely low temperatures, just like ordinary quantum systems. The finding provides strong evidence that black holes possess isolated ground states, a hallmark of quantum mechanical behavior.
Understanding gravity’s quantum behavior is among the biggest open questions facing modern physics. Black holes are used as laboratories for investigating quantum gravity, particularly at low temperatures where quantum effects become visible.
Prior calculations showed that black hole entropy might become negative at low temperatures, a result that appeared physically puzzling. In this work, researchers addressed the paradox by incorporating wormhole effects in the two-dimensional Jackiw-Teitelboim (JT) gravity model.
Phys.org spoke to the authors of the study, Stefano Antonini, Prof. Luca Victor Iliesiu, Pratik Rath, and Patrick Duy Tran, to gain insight into their work.
“By describing black holes at extremely low temperatures, and understanding whether they have an isolated ground state just like most conventional quantum systems, we hope to unveil quantum properties of gravity,” the researchers explain.
The entropy problem
In quantum systems, entropy measures the number of possible microscopic configurations. If a system has an isolated ground state—a unique lowest energy configuration—its entropy should vanish as temperature approaches absolute zero.
However, entropy calculations in gravitational theories always involve an average over an ensemble of possible configurations, making them tricky.
Two different averaging procedures, called annealed and quenched entropy, can give different answers. Annealed entropy calculates the average first and then the entropy, while quenched entropy calculates the entropy first for each configuration and then averages.
“The necessity boils down to an order of operations issue,” the researchers explain. “Suppose you are given an assortment of quantum systems and tasked to calculate the average entropy. Ideally, you would calculate the entropy of each system and then average over these entropies. This is called the quenched entropy.”
“Instead, it is often easier for physicists to calculate the annealed entropy, which takes averages first and then calculates entropy—a wrong order of operations.”
At high temperatures, these two methods agree. But at low temperatures, they diverge dramatically: the quenched entropy approaches zero, reflecting an isolated ground state, while the annealed entropy goes negative. This result is nonsensical because the third law of thermodynamics dictates that entropy must be non-negative and vanish as temperature approaches absolute zero.
Introducing a new quantity
While the quenched entropy offers the correct conceptual way to calculate entropy, it is often very difficult to compute precisely in gravitational systems. This difficulty arises because it requires detailed knowledge of the full distribution of quantum states and fluctuations within the ensemble, which is mathematically and numerically challenging.
To address this, the researchers introduced a new intermediate quantity called semiquenched entropy.
“We had to introduce semiquenched entropy, which is simpler to compute than quenched entropy,” the team said. “Nevertheless, this quantity still captures similar properties to the quenched entropy: for instance, proving that either quantity is positive at low temperatures implies that the ground states of the assortment of quantum systems are all isolated.”
The key advantage is that proving the semiquenched entropy remains positive across all temperatures is sufficient to show that black holes have isolated ground states—and by extension, that the quenched entropy also stays positive.
This is because semiquenched entropy, despite being easier to calculate, is similar to the quenched entropy in the sense that it shares the same qualitative behavior and probes the same physical properties of the ground state. Positivity and vanishing of semiquenched entropy at zero temperature therefore confirm that black holes behave like conventional quantum systems with unique lowest energy states.
Airy tail and wormholes
The Airy edge is a mathematical concept from random matrix theory describing a universal pattern in how eigenvalues are distributed near the boundary of their spectrum. This pattern shows up in many complex systems across physics and mathematics.
In the context of JT gravity, the black hole energy spectrum is mathematically equivalent to the spectrum of eigenvalues of an ensemble of random matrices. This equivalence allows physicists to apply the Airy edge statistics to understand subtle quantum behaviors of black holes at very low temperatures.
“By going to low temperatures, we begin to probe the edge statistics in the black hole spectrum and see that it shares the same universal statistics as in matrix integrals,” the researchers explain.
The team performed their calculations using two complementary approaches. The first involved summing over wormhole contributions—geometric structures that connect different regions of spacetime—in the gravitational path integral.
The second used random matrix theory techniques to show that the dual matrix integral is dominated by a new configuration, a one-eigenvalue instanton. Remarkably, both approaches agreed on their common regime of validity, providing a powerful consistency check.
“This agreement seems to tell us a strange and surprising result: that these one-eigenvalue instantons correspond to not just a single wormhole, but a resummation of an infinite number of wormholes,” the team noted.
“If we were to instead sum over a finite number of wormhole corrections, we would not see that the semiquenched entropy is positive. This means that accounting for all wormholes is critical to understand the quantum nature of black holes and get results consistent with a conventional quantum system.”
Implications and next steps
Demonstrating that black holes have isolated ground states carries implications for our understanding of quantum gravity.
“By proving an isolated ground state, we show that black holes in JT gravity behave like quantum mechanical systems. In other words, their lowest energy states are quantized,” the researchers explain.
“This provides evidence in favor of the microstate interpretation of black hole entropy and advances theoretical probes of the quantum nature of gravity.”
The results also highlight the role of wormholes in gravitational physics. Without summing over the full infinite series of wormhole contributions, the calculations would not yield physically sensible, positive entropy.
Looking ahead, the researchers identify intriguing open questions: What is the gravitational interpretation of the one-eigenvalue instantons? Can these methods extend to higher-dimensional black holes? Is semiquenched entropy useful beyond gravity, like in condensed matter or quantum computing?
The team has already taken steps toward answering these questions. They generalize their results in a follow-up paper, released on the preprint server arXiv, to a broader class of black holes with matter excitations, strengthening the case that black holes behave as generic, chaotic quantum systems.
Source: phys.org
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