Black hole volume
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Understanding Black Hole Volume: Insights from Recent Research
Black Hole Volume Definition and Growth
Intrinsic Volume Characterization
The volume inside a black hole can be defined by extending the intrinsic flat-spacetime characterization of the volume inside a two-sphere. This approach reveals that the volume grows with time since the collapse of the black hole, reaching a large asymptotic form . This growth is significant despite the shrinking exterior area due to Hawking radiation, indicating that the interior volume of black holes remains large and continues to grow over time 14.
Maximal Surfaces and Geodesic Problems
A generally covariant definition of the volume inside a black hole using maximal surfaces has been proposed. This method shows that finding the maximal surfaces in an arbitrary spherically symmetric spacetime is equivalent to solving a 1+1 geodesic problem. The volume of these maximal surfaces inside the apparent horizon of an evaporating black hole grows monotonically with advanced time, even as the horizon shrinks to Planckian dimensions .
Time Dependence and Geometric Interpretation
Time-Dependent Volume
The three-dimensional spatial volume inside a black hole depends explicitly on the definition of time and can vary or even be zero depending on the chosen time-coordinate definitions. This variability highlights the complex nature of black hole interiors and provides a new pedagogical resource for understanding these volumes .
Geometric and Holographic Interpretations
The volume inside black holes, particularly in Kerr-AdS spacetimes, exhibits surprising features such as non-monotonic growth relative to the black hole's area. This volume is useful in the context of holography, suggesting a dual relationship with the complexity of the boundary field theory . Additionally, the concept of vector volume, defined through divergence-free vector fields, offers a useful tool for describing black hole volumes and has several interesting properties, including linearity with respect to the choice of vector .
Implications for Black Hole Thermodynamics
Thermodynamic Volume and Pressure
In black hole thermodynamics, the mass of a black hole is interpreted as enthalpy, with the pressure given by the cosmological constant. The volume is then defined as the Legendre transform of the pressure. This relationship is explored in the context of the first law of black hole thermodynamics, which includes a PdV term that modifies the maximal efficiency of processes like the Penrose process . The thermodynamic volume also plays a role in the extended thermodynamics of AdS black holes, interacting with proposals such as the complexity = action (CA) duality .
Gravitational Tension and Spacetime Pressure
Variations in surrounding gravitational fields can be incorporated into the first law of black hole thermodynamics by defining gravitational tension or binding energy. This approach shows that spacetime volume and gravitational tension encode similar physics to spacetime pressure and black hole volume, providing a unified framework for understanding these concepts .
Conclusion
The study of black hole volume reveals a complex and dynamic interior that grows over time, even as the exterior shrinks. This growth is intricately linked to the definitions of time and space within the black hole, and has significant implications for black hole thermodynamics and the information paradox. The geometric and holographic interpretations of black hole volume further enrich our understanding of these enigmatic objects, offering new insights into their fundamental nature.
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