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| Boulware vacuum | |
|---|---|
| Name | Boulware vacuum |
| Type | Quantum state |
| Field | Quantum field theory in curved spacetime |
| Introduced | 1975 |
| Introduced by | David G. Boulware |
| Related | Unruh vacuum, Hartle–Hawking vacuum, Schwarzschild spacetime, Hawking radiation |
Boulware vacuum
The Boulware vacuum is a quantum state for fields on static, spherically symmetric spacetimes associated with the absence of particles for static observers at spatial infinity, originally defined by David G. Boulware. It plays a central role in analyses of quantum fields in Schwarzschild, Reissner–Nordström, and Kerr backgrounds and is contrasted with other vacuum choices such as the Unruh vacuum and the Hartle–Hawking vacuum. Important in discussions of vacuum polarization, singular behavior on horizons, and semiclassical backreaction, it informs studies linking quantum field theory with general relativity and black hole thermodynamics.
The Boulware vacuum is defined as the state annihilated by mode operators associated with positive-frequency modes with respect to the static Killing vector of a background such as Schwarzschild metric, Reissner–Nordström metric, or Kerr metric. For an observer following a static worldline at spatial infinity in Minkowski space limits, this state appears empty, whereas near the event horizon of a black hole like those in Event horizon studies it exhibits divergent stress-energy similar to the divergences discussed in analyses by Stephen Hawking and James B. Hartle. Boulware's construction contrasts with particle definitions tied to collapsing geometries as in the work leading to Hawking radiation. It is frequently used when considering static stars modeled by Tolman–Oppenheimer–Volkoff equation solutions or in exteriors matched to Oppenheimer–Snyder collapse interior solutions.
Construction proceeds by solving mode equations for a linear field (e.g., a scalar field obeying the Klein–Gordon equation) on a static background such as Schwarzschild solution and imposing positive-frequency boundary conditions relative to the timelike Killing vector field generating time translations in spacetimes like Stationary spacetime examples. One expands the field operator in a complete set of mode functions labeled by quantum numbers analogous to those in analyses by Paul Dirac and Lev Landau, and defines annihilation operators a_lmω that annihilate the Boulware state, following canonical quantization procedures developed in the tradition of Petr Kapitsa-era quantum field theory texts and the canonical formalism used by Bryce DeWitt. Mathematical tools include the use of tortoise coordinate r* as in treatments by Regge–Wheeler and scattering theory methods employed in studies by V. Frolov and A. Zelnikov. Mode matching at spatial infinity uses asymptotics similar to plane waves in Minkowski space and spherical harmonics from the theory developed by Simon Newcomb and Georg Friedrich Bernhard Riemann.
The renormalized expectation value of the stress–energy tensor ⟨T_ab⟩ in the Boulware vacuum vanishes at spatial infinity for free fields and exhibits vacuum polarization effects near compact objects studied in works by Paul Davies, Stephen Fulling, and William Unruh. For a Schwarzschild exterior, ⟨T_ab⟩ diverges on the event horizon in a manner analyzed by B. Kay and R. Wald, producing negative energy densities that have been compared with Casimir-type energies explored by Hendrik Casimir and semiclassical backreaction computations associated with the semiclassical Einstein equation favored by John Wheeler. Numerical and analytic evaluations often use point-splitting renormalization techniques developed by DeWitt and stress-tensor renormalization schemes from the literature of Christensen and Brown and Ottewill. For fields with spin, the behavior parallels results obtained in studies involving Dirac fields inspired by work of P. A. M. Dirac and electromagnetic fields traced to analyses by James Clerk Maxwell.
Unlike the Boulware vacuum, the Unruh vacuum models the late-time vacuum appropriate to a collapsing star and yields outgoing Hawking flux at infinity as in Hawking’s seminal papers, while the Hartle–Hawking vacuum represents a thermal equilibrium state at the Hawking temperature for eternal black holes, as formalized by Hartle and Hawking. The Unruh state is regular on the future horizon as studied in treatments by Paul Davies and Stephen Hawking, whereas the Hartle–Hawking state is regular on both future and past horizons and linked to path-integral derivations associated with work by Gary Gibbons and Stephen Hawking on black hole thermodynamics. Comparisons often invoke Bogoliubov transformations as in the formalism advanced by Niels Bohr-era quantum theorists and illustrate differences in particle content measured by static observers at infinity versus freely falling observers connected to analyses by John Wheeler and Roger Penrose.
The Boulware vacuum is used in modeling vacuum polarization around static stars and in exterior regions of compact objects for matching conditions in semiclassical gravity calculations by researchers like Paul Anderson and P. R. Johnson. It provides a baseline for studying quantum corrections to classical black hole metrics in frameworks influenced by Stephen Hawking and Jacob Bekenstein on entropy and thermodynamics, and in analyses of quantum energy inequalities related to work by Lawrence Ford and Thomas Roman. Applications include numerical evaluation of backreaction employing methods from C. O. Lousto and L. C. B. Crispino, studies of vacuum-induced forces in analog gravity setups inspired by experiments at institutions like CERN and Max Planck Institute for Gravitational Physics, and comparative studies of detector response functions building on the Unruh–DeWitt detector model developed by Bill Unruh and Bryce DeWitt.
The primary criticism of the Boulware vacuum is its singular behavior at event horizons, making it unsuitable for modeling states that are regular across horizons as required in descriptions of evaporating black holes, highlighted in critiques by Robert Wald and S. A. Fulling. Its physical relevance is thus limited to exteriors of static, horizonless compact objects or to approximations where horizon divergences are ignored or regulated, a limitation emphasized in semiclassical stability analyses by P. Candelas and studies of quantum backreaction by Ya. B. Zel'dovich. Additional limitations include dependence on global staticity assumptions similar to those questioned in debates involving Kip Thorne and Stephen Hawking about the end state of collapse, and technical issues in renormalization and mode-sum convergence discussed in the literature by Paul Anderson and Eric Poisson.
Category:Quantum field theory in curved spacetime Category:Black hole physics