Boundary of a black hole
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Understanding the Boundary of a Black Hole
Introduction to Black Hole Boundaries
The boundary of a black hole, often referred to as the event horizon, is a critical concept in understanding black hole physics. This boundary marks the point beyond which nothing, not even light, can escape the gravitational pull of the black hole. Recent research has provided new insights and methods for characterizing and understanding these boundaries.
Alternative Characterization of Black Hole Boundaries
Traditional definitions of black hole boundaries have faced several shortcomings. A new method has been proposed to address these issues, particularly in general spherically symmetric spacetimes. This method suggests that an observer falling into a black hole would encounter infinite energy density, pressure, and flux at the boundary, which is interpreted as a firewall preventing the growth of classical black holes. However, the possibility of black hole growth through quantum mechanical tunneling remains open .
Boundary Conditions and Coordinate Conditions
Boundary conditions at black hole horizons are essential for solving the Einstein equations in (3+1)D spacetime. The apparent horizon is often used as the inner boundary on a space slice. To ensure the system is well-posed, "prescribed curvature conditions" are proposed, leading to a system of 2D elliptic differential equations on the inner boundary surface. This system coexists with the 3D equation for maximal slicing, ensuring the overall system is globally well-behaved .
Excision Boundary Conditions for Initial Data
A set of boundary conditions has been defined for black-hole excision surfaces within the conformal thin-sandwich formalism. These conditions are designed to result in black holes in quasiequilibrium and can be applied with any conformal three-geometry and slicing condition. Tests on single black holes and binary systems of equal-mass black holes show that these boundary conditions work effectively, allowing for the specification of arbitrary spin on each black hole .
Five-Dimensional Stationary Rotating Black Holes
In the context of five-dimensional vacuum Einstein equations, the boundary value problem for stationary rotating black holes has been studied. Assuming the existence of two additional commuting rotational Killing vector fields and sphericity of the horizon topology, it is shown that a black hole with a regular event horizon is uniquely characterized by its mass and a pair of angular momenta .
Holographic Duality and Black Hole Interiors
Holographic duality provides a framework for understanding the boundary descriptions of event horizons, black hole interiors, and singularities. In this context, an eternal AdS black hole is described by two copies of the boundary CFT in the thermal field double state. This framework allows for the construction of an evolution operator for a bulk in-falling observer, making the boundary emergence of black hole horizons and interiors manifest .
Thermodynamics and Action of de Sitter Black Holes
The thermodynamics of rotating black holes in the presence of a positive cosmological constant are analyzed by considering either of the two horizons as a boundary. Depending on whether the cosmological horizon or the black hole horizon is taken as the boundary, the action and thermodynamics of the system differ. The energy and angular momentum are obtained as surface terms on the boundary, and the temperature and rotational chemical potential are expressed in terms of boundary magnitudes .
Definitions Without Boundary at Infinity
Two definitions of black holes that do not reference the boundary at infinity have been proposed. One definition does not require any global causality condition, while the other is applicable to a closed Friedmann-like spacetime. The area theorem holds in both cases, providing a robust framework for understanding black holes without relying on asymptotic boundaries .
Extended Symmetries at the Horizon
Non-extremal black holes in four-dimensional general relativity exhibit an infinite-dimensional symmetry in their near-horizon region. By prescribing a set of boundary conditions at the horizon, an algebra of asymptotic Killing vectors is derived, which includes two sets of supertranslations and two mutually commuting copies of the Witt algebra. These symmetries are crucial for understanding the surface charges and the entropy of black holes .
Gauge Theories and Nontrivial Boundary Conditions
The partition function and entropy of U(1) gauge theories with multiple boundaries on black holes have been studied. Nontrivial boundary conditions allow for residual zero longitudinal momentum modes and Wilson lines stretched between boundaries. These conditions lead to different area contributions and logarithmic corrections in the entropy, providing insights into the behavior of gauge fluctuations on black hole backgrounds .
Conclusion
The boundary of a black hole is a complex and multifaceted concept that has been explored through various theoretical frameworks and boundary conditions. From new characterizations and boundary conditions to holographic duality and extended symmetries, recent research continues to deepen our understanding of black hole boundaries and their implications in general relativity and quantum mechanics.
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