We consider a 4D spherically-symmetric static finite spacetime region as a
collection of quanta in the semi-classical Einstein equation and study the
entropy including the self-gravity. For sufficiently excited states, we
estimate the entropy in a WKB-like method considering local consistency with
thermodynamics and find its upper bound. The saturation condition uniquely
determines the entropy-maximized spacetime as a radially uniform dense
configuration with near-Planckian curvatures and a surface just outside the
Schwarzschild radius. The interior metric is a non-perturbative solution in
$hbar$, leading to the species bound. The maximum entropy then saturates the
Bousso bound and coincides with the Bekenstein-Hawking formula. Thus, the
Bousso bound in this class of spacetime is verified by constructing the
saturating configuration that has no horizon and stores information inside.

The research conducted in this study focuses on examining the entropy of a 4D spherically-symmetric static finite spacetime region, taking into account the self-gravity. By considering the semi-classical Einstein equation and utilizing a WKB-like method, the researchers estimate the upper bound of the entropy for highly excited states.

Through their analysis, they discover that the maximum entropy occurs in a spacetime configuration that is radially uniform, densely packed, and features curvatures close to the Planck scale. This configuration also possesses a surface just outside the Schwarzschild radius and does not have a horizon.

The interior metric of this spacetime configuration is a non-perturbative solution in $hbar$, a fundamental constant of quantum theory. Consequently, this result leads to a species bound, highlighting the limitations on the number of quanta in the spacetime region.

Furthermore, the maximum entropy obtained in this study satisfies the Bousso bound, a conjecture in theoretical physics proposed by Raphael Bousso that relates the entropy of a generalized gravitational system to various geometric quantities. The derived maximum entropy also coincides with the Bekenstein-Hawking formula, which is a fundamental formula describing black hole thermodynamics.

Therefore, this research successfully verifies the Bousso bound for this specific class of spacetime configurations by constructing a configuration that maximizes entropy without having a horizon, thereby storing information internally.

The Future Roadmap

Potential Challenges

  • Further analysis: More detailed analysis might be required to understand the precise characteristics and implications of the entropy-maximized spacetime configuration. This could involve investigating its stability, evolution over time, and potential connections to other areas of physics.
  • Experimental validation: While the study provides theoretical evidence for the existence of this entropy-maximized configuration, experimental validation and observation might be challenging due to the extreme conditions involved (near-Planckian curvatures) and the absence of horizons.
  • Generalization: The examination of a 4D spherically-symmetric static finite spacetime region is just one specific case. Generalizing the analysis to different types of spacetime regions or alternative approaches could yield further insights into the behavior of entropy and spacetime in more general scenarios.

Potential Opportunities

  • Advancing our understanding of quantum gravity: The non-perturbative solution in $hbar$ and the derived species bound provide valuable contributions to the study of quantum gravity. Further exploration of these concepts could deepen our understanding of the interplay between quantum mechanics and gravity.
  • Exploring alternative spacetime configurations: This research establishes a specific configuration with unique properties. Investigating other potential spacetime configurations and studying their entropy behavior could unveil new phenomena and offer novel ways to approach longstanding questions in physics.
  • Implications for black hole physics and information storage: The absence of a horizon and the ability to store information internally in the entropy-maximized configuration have significant implications for black hole physics and the long-standing issue of information preservation. Future research could build upon these findings to further unravel the mysteries surrounding black holes and their relationship to thermodynamics.


The research successfully establishes an upper bound for the entropy of a 4D spherically-symmetric static finite spacetime region, considering self-gravity effects. The obtained maximum entropy corresponds to a radially uniform dense configuration with near-Planckian curvatures and no horizon. This finding verifies the Bousso bound and aligns with the Bekenstein-Hawking formula. Challenges lie ahead in further analysis, experimental validation, and generalization, but the opportunities to advance our understanding of quantum gravity, explore alternative spacetime configurations, and shed light on black hole physics and information storage are promising.

Read the original article