Abstract: The 3d volume inside a spherical black hole can be defined by extending an intrinsic flat-spacetime characterization of the volume inside a 2-sphere. For a collapsed object, the volume grows with time since the collapse, reaching a simple asymptotic form, which has a compelling geometrical interpretation. Perhaps surprising, it is large. The result may have relevance for the discussion on the information paradox.
The paper presents a novel method for defining black hole volume using maximal spacelike surfaces bounded by the event horizon.
It demonstrates analytically and numerically that the interior volume grows linearly with asymptotic time, with a form of 3√3π m² v.
The findings offer new insights into the black hole information paradox and quantum gravity, suggesting further exploration in these areas.
How Big is a Black Hole? An Exploration of Interior Volume
The paper "How Big is a Black Hole?" by Marios Christodoulou and Carlo Rovelli provides a novel perspective on the volumetric analysis of black holes, challenging preconceived ideas about spatial configurations within these enigmatic cosmic phenomena. By analyzing the volume within spherical black holes, the authors address the intriguing question: just how much space resides inside a black hole?
Conceptual Approach and Formal Definition
The paper introduces an innovative method to assess the volume inside a black hole by extending flat-space-time concepts traditionally applied to 2-spheres into the curved geometry context of black holes. This novel approach acknowledges the intrinsic time-dependent nature of the interior of a black hole post-collapse, diverging from the static perception of the external Schwarzschild metric. The authors underscore the need for a time-varying definition of black hole volume and argue against using coordinate-dependent measurements which often complicate the estimations of interior volumes in curved spaces.
Focusing on spherically symmetric sections of the Schwarzschild black hole, the authors redefine black hole volume as the largest spacelike surface, denoted as Σv, bounded by the event horizon, or more specifically, by a specific two-sphere Sv. This characterization enables a meaningful realization of "interior volume," robustly aligning with considerations in both flat and curved spaces alike.
Analytical and Numerical Results
The theoretical framework culminates in a striking revelation: the derived volume V(v) inside the black hole grows linearly with asymptotic time v, and assumes a form of 33πm2v for a Schwarzschild black hole. This formula emphasizes a significant, although somewhat unexpected, spatial largeness even within relatively old black holes like the Sagittarius A*.
The paper proceeds with numerical simulations to concretely visualize and compute the geodesics that define these maximal-volume surfaces inside the black hole. These simulations reveal how, under large v, the spacelike hyper-surfaces primarily converge onto a region characterized by a constant radial coordinate, forming what resembles a long, three-dimensional cylinder radiating from the black hole core. This spatial arrangement accounts for the voluminous property calculated theoretically.
Theoretical and Practical Implications
The assertion of substantial interior volumes bears relevance in the discourse surrounding the black hole information paradox. If the interior of a black hole is substantially voluminous, it prompts a reconsideration of how information is retained or lost within such confines, potentially offering alternative pathways to resolve this paradox beyond current Hawking radiation-based understanding.
Extending their findings, the authors offer a framework applicable to more complex geometries beyond the canonical Schwarzschild solution, such as Reissner-Nordström black holes. They note the potential for higher-order extensions and deeper insights into non-singular metrics and collapse scenarios.
Speculation on Future Developments
The implications of these findings for black hole thermodynamics and quantum gravity are profound and call for continued exploration of how volume and entropy interplay within these extreme cosmic environments. It invites a re-examination of entropy's classical assumptions against the backdrop of spacelike volumes in quantum settings, a promising avenue for gaining insights into black hole quantum mechanics.
In conclusion, the paper provides an articulate and detailed investigation into the spatial characteristics of black holes, unraveling new layers in our understanding of this cosmic phenomenon. The potential theoretical ramifications extend into discussions on entropy, information paradoxes, and quantum gravitational theories, highlighting the importance of ongoing research in these domains.
