Thermodynamics of frozen stars (2310.11572v1)

Abstract: The frozen star is a recent proposal for a non-singular solution of Einstein's equations that describes an ultracompact object which closely resembles a black hole from an external perspective. The frozen star is also meant to be an alternative, classical description of an earlier proposal, the highly quantum polymer model. Here, we show that the thermodynamic properties of frozen stars closely resemble those of black holes: Frozen stars radiate thermally, with a temperature and an entropy that are perturbatively close to those of black holes of the same mass. Their entropy is calculated using the Euclidean-action method of Gibbons and Hawking. We then discuss their dynamical formation by estimating the probability for a collapsing shell of "normal" matter to transition, quantum mechanically, into a frozen star. This calculation followed from a reinterpretation of a transitional region between the Euclidean frozen star and its Schwarzschild exterior as a Euclidean instanton that mediates a phase transition from the Minkowski interior of an incipient Schwarzschild black hole to a microstate of the frozen star interior. It is shown that, up to negligible corrections, the probability of this transition is $e^{-A/4}$, with $A$ being the star's surface area. Taking into account that the dimension of the phase space is $e^{+A/4}$, we conclude that the total probability for the formation of the frozen star is of order unity. The duration of this transition is estimated, which we then use to argue, relying on an analogy to previous results, about the scaling of the magnitude of the off-diagonal corrections to the number operator for the Hawking-like particles. Such scaling was shown to imply that the corresponding Page curve indeed starts to go down at about the Page time, as required by unitarity.

• The paper demonstrates that frozen stars replicate black hole thermodynamics by emitting thermal radiation and conforming to the area-entropy law.
• It employs Euclidean-action methods and quantum tunneling estimates to show that the transition probability for their formation approaches unity.
• The findings suggest that frozen stars could resolve information paradox issues and bridge gaps between general relativity and quantum gravity.

Essay on "Thermodynamics of Frozen Stars"

The paper "Thermodynamics of Frozen Stars" by Ram Brustein, A.J.M. Medved, and Tamar Simhon explores an innovative model for ultracompact objects, known as frozen stars, proposed as non-singular and horizonless alternatives to traditional black holes (BHs). The authors provide a detailed analysis of the thermodynamic and formation characteristics of these objects, aligning them closely with the familiar properties of black holes. This essay offers a systematic examination of the paper’s findings, emphasizing its implications for theories of quantum gravity and black hole thermodynamics.

Thermodynamic Similarity to Black Holes

Central to the paper is the demonstration that frozen stars mimic the thermodynamic behavior of black holes. The paper evidences that frozen stars emit thermal radiation at temperatures and entropies perturbatively close to those of BHs of equivalent mass. This emission aligns with the Hawking radiation paradigm, suggesting that frozen stars share key thermodynamic aspects with black holes despite their non-singularity and absence of event horizons.

The calculation of entropy via the Euclidean-action method parallels the approach of Gibbons and Hawking, reinforcing that the entropy of frozen stars conforms to the Bekenstein-Hawking area law. Such conformity is noteworthy because replicating this law has been challenging for alternative models of compact objects, often failing to capture the area/entropy proportionality, including its prefactor.

Dynamical Formation and Quantum Transition

The paper ambitiously extends its scope to discuss the dynamic formation of frozen stars. Here, it applies concepts from quantum mechanics to estimate the probability of a collapsing shell of matter transitioning into a frozen star. By interpreting the transitional region as a Euclidean instanton, the authors argue that the quantum mechanical phase transition entails that the transition probability is proportional to $e^{-A/4}$, where $A$ represents the star’s surface area. Remarkably, when considering the degeneracy of microstates (quantified by $e^{+A/4}$), the total probability approaches unity. This implies that the emergence of frozen stars is not only possible but probable under specific conditions.

Implications and Prospective Developments

The implications of these findings are substantial, both theoretically and practically. Theoretically, the concept of frozen stars challenges the conventional dualities of space-time and singularity, suggesting a paradigm where classical geometries can substantially encapsulate quantum properties. The model addresses long-standing issues such as information paradoxes by providing a horizonless structure that maintains entropy-area law, preserving unitarity over Hawking evaporation processes.

Furthermore, the research proposes that frozen stars can potentially present a unitary Page curve during evaporation, addressing unsolved problems in traditional black hole thermodynamics. Notably, the investigation proposes a substructure to GR (General Relativity) solutions that may align closer with quantum gravity theories including string theory, hinting at stringy behaviors akin to Hagedorn states.

Future developments in observational astrophysics could test the observational footprints and theoretical assumptions of frozen stars. The theoretical implications signal pertinent advances, particularly in understating the underlying quantum structure of space-time, potentially opening avenues for reconciling general relativity with quantum mechanics.

Conclusion

The paper presents a compelling narrative that ultracompact, horizonless frozen stars could viably serve as alternatives to conventional black holes while maintaining thermodynamic characteristics. These findings pave the way for more in-depth analyses of non-singular BH analogues and could significantly impact the approach towards comprehension of quantum gravity and cosmic censorship conjectures. Future research should endeavor to test these theoretical predictions through simulations or observable phenomena, thereby cementing the role of frozen stars within the broader specter of astrophysical objects.