The tetralogy of Birkhoff theorems

Schmidt, Hans‐Jürgen

doi:10.1007/s10714-012-1478-5

Cited by 21 publications

(28 citation statements)

References 98 publications

(123 reference statements)

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“…Generalization to higher dimensions was achieved by K. A. Bronnikov and V. N. Melnikov [33] and a thorough discussion on the relationship between manifold dimensionality and the existence of Birkhoff-like theorems was made by H.-J. Schmidt [34]. R. Goswami and G.F.R.…”

Section: General Formulations Of the Birkhoff Theoremmentioning

confidence: 99%

Revisiting the Birkhoff theorem from a dual null point of view

2018

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The Birkhoff theorem is a well-known result in general relativity and it is used in many applications. However, its most general version, due to Bona, is almost unknown and presented in a form less accessible to the relativist and cosmologist community. Moreover, many wield it mistakenly as a simple transposition of Newton's iron sphere theorem. In the present work, we propose a modern, dual null, presentation -useful in many explorations, including black holes -of the theorem that renders accessible most of the results of Bona's version. In addition, we discuss the fluid contents admissible for the application of the theorem, beyond a vacuum, and we demonstrate how the formalism greatly simplifies solving the dynamical equations and allows one to express the solution as a power expansion in r. We present a family of solutions that share the properties predicted by the Birkhoff theorem and discuss the existence of trapped and antitrapped regions. The formalism manifestly shows how the type of region -trapped or untrapped -determines the character of the Killing vector. *

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Section: General Formulations Of the Birkhoff Theoremmentioning

confidence: 99%

Revisiting the Birkhoff theorem from a dual null point of view

2018

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“…As a closing note let us briefly mention that quite recently, several papers appeared which are related to different other aspects of the study of (2 + 1)-dimensional gravity: In [25] and [26], massive gravity and supergravity are discussed; in [27], higher spin in topologically massive gravity is discussed; in [28] and [29] wormholes and star models are constructed in 2 + 1 dimensions; in [30], Birkhoff's theorem is generalized; in [31][32][33][34][35][36][37][38] various properties of the BTZ geometry are discussed; in [39], massive particles with spin in 2 + 1 dimension are constructed; in [40][41][42][43][44] further aspects of (2 + 1)-dimensional gravity are deduced and discussed; and finally in [45], the observability of strong gravitational sources (black hole, naked singularities) via lensing is discussed.…”

Section: Conclusion and Summarymentioning

confidence: 99%

Exact radial solution in 2+1 gravity with a real scalar field

Schmidt

Singleton

2013

Physics Letters B

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Exact solution BTZ black hole Self-interacting scalar field In this Letter we give some general considerations about circularly symmetric, static space-times in 2 + 1 dimensions, focusing first on the surprising (at the time) existence of the BTZ black hole solution. We show that BTZ black holes and Schwarzschild black holes in 3 + 1 dimensions originate from different definitions of a black hole. There are two by-products of this general discussion: (i) we give a new and simple derivation of (2 + 1)-dimensional Anti-de Sitter (AdS) space-time; (ii) we present an exact solution to (2 + 1)-dimensional gravity coupled to a self-interacting real scalar field. The spatial part of the metric of this solution is flat but the temporal part behaves asymptotically like AdS space-time. The scalar field has logarithmic behavior as one would expect for a massless scalar field in flat space-time. The solution can be compared to gravitating scalar field solutions in 3 + 1 dimensions but with certain oddities connected with the (2 + 1)-dimensional character of the space-time. The solution is unique to 2 + 1 dimensions; it does not carry over to 3 + 1 dimensions.

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“…Likewise, anti-de Sitter space plays a prominent role in string theories and in the AdS/CFT correspondence [12] which have been the subject of a large literature (see [13] for recent reviews). It is surprising, therefore, that modern relativity * vfaraoni@ubishops.ca † acardini15@ubishops.ca ‡ wchung13@ubishops.ca textbooks do not mention the Jebsen-Birkhoff theorem in the presence of a cosmological constant, although occasionally one finds in the literature an explicit statement about the uniqueness of the Schwarzschild-(anti-)de Sitter space (e.g., [5,14,15]). A proof of the Jebsen-Birkhoff theorem extended to include a non-vanishing Λ is available in Synge's 1960 textbook 1 on general relativity [16].…”

Section: Introductionmentioning

confidence: 99%