Birkhoff-like theorem for rotating stars in (2+1) dimensions
Jozef Skakala, Matt Visser
备注：
4 pages; uses revtex4; V2: added 1 very important reference; significantly expanded discussion; many small changes in the text but no significant change in physics conclusions; V3: minor typos fixed
Consider a rotating and possibly pulsating "star" in (2+1) dimensions. If the star is axially symmetric, then in the vacuum region surrounding the star, (a region that we assume at most contains a cosmological constant), the Einstein equations imply that under physically plausible conditions the geometry is in fact stationary. Furthermore, the geometry external to the star is then uniquely guaranteed to be the (2+1) dimensional analogue of the Kerr-de Sitter spacetime, the BTZ geometry. This Birkhoff-like theorem is very special to (2+1) dimensions, and fails in (3+1) dimensions. Effectively, this is a "no hair" theorem for (2+1) dimensional axially symmetric stars: the exterior geometry is completely specified by the mass, angular momentum, and cosmological constant.
Spherically symmetric shock waves are shown to exist in Lovelock gravity. They amount to a change of branch of the spherically symmetric solutions across a null hypersurface. The implications of their existence for the status of Birkhoff's theorem in the theory is discussed.
Birkhoff's theorem is one of the most important statements of Einstein's general relativity, which generally can not be extended to modified theories of gravity. Here we study the validity of the theorem in scalar-tensor theories using a perturbative approach, and compare the results in the so-called Einstein and Jordan frames. The implications of the results question the physical equivalence between both frames, at least in perturbations.
The analysis of the validity of Birkhoff's theorem about the uniqueness of the spherically symmetric solution of the gravitational field equations is extended to the framework of the Poincaré gauge gravity theory. The class of models with the most general Lagrangians of the Yang-Mills type constructed from all possible quadratic invariants of the curvature and the torsion is considered, including both parity-even and parity-odd sectors. We find the models in which the weak and strong versions of the generalized Birkhoff theorem are valid, by making use of the double duality technique.
We study the local causality issue via the Shapiro time-delay computations in the on-shell consistent exotic massive gravity in three dimensions. The theory shows time-delay as opposed to time-advance despite having a ghost at the linearized level both for asymptotically flat and anti-de Sitter spacetimes. We also prove a Birkhoff-like theorem: any solution with a hypersurface orthogonal non-null Killing vector field is conformally flat; and find some exact solutions.
General two-dimensional pure dilaton-gravity can be discussed in a unitary way by introducing suitable field redefinitions. The new fields are directly related to the original spacetime geometry and in the canonical picture they generalize the well-known geometrodynamical variables used in the discussion of the vacuum dilatonic black hole. So the model can be quantized using the techniques developed for the latter case. The resulting quantum theory coincides with the quantum theory obtained imposing the Birkhoff theorem at the classical level.
Birkhoff's theorem is discussed in the frame of f(R) gravity by using its scalar-tensor representation. Modified gravity has become very popular at recent times as it is able to reproduce the unification of inflation and late-time acceleration with no need of a dark energy component or an inflaton field. Here, another aspect of modified f(R) gravity is studied, specifically the range of validity of Birkhoff's theorem, compared with another alternative to General Relativity, the well known Brans-Dicke theory. As a novelty, here both theories are studied by using a conformal transformation and w…
We develop a simple scheme of quantization for the dilaton CGHS model without scalar fields, that uses the Gupta-Bleuler approach for the string fields. This is possible because the constraints can be linearized classically, due to positivity conditions that are present in the model (and not in the general string case). There is no ambiguity nor anomalies in the quantization. The expectation values of the metric and dilaton fields obey the classical requirements, thus exhibiting at the quantum level the Birkhoff theorem.
In this paper we present a class of higher derivative theories of gravity which admit Birkhoff's theorem. In particular, we explicitly show that in this class of theories, although generically the field equations are of fourth order, under spherical (plane or hyperbolic) symmetry, all the field equations reduce to second order and have exactly the same or similar structure to those of Lovelock theories, depending on the spacetime dimensions and the order of the Lagrangian.
All three-dimensional matter-free spacetimes with negative cosmological constant, compatible with cyclic symmetry are identified. The only cyclic solutions are the 2+1 (BTZ) black hole with SO(2) x R isometry, and the self-dual Coussaert-Henneaux spacetimes, with isometry groups SO(2) x SO(2,1) or SO(2) x SO(2).
We extend the Birkhoff's theorem in Lovelock gravity for arbitrary base manifolds using an elementary method. In particular, it is shown that any solution of the form of a warped product of a two-dimensional transverse space and an arbitrary base manifold must be static. Moreover, the field equations restrict the base manifold such that all the non-trivial intrinsic Lovelock tensors of the base manifold are constants, which can be chosen arbitrarily, and the metric in the transverse space is determined by a single function of a spacelike coordinate which satisfies an algebraic equation involvi…
We show that the generic solutions of the Lovelock equations with spherical, planar or hyperbolic symmetry are locally isometric to the corresponding static Lovelock black hole. As a consequence, these solutions are locally static: they admit an additional Killing vector that can either be space-like or time-like, depending on the position. This result also holds in the presence of an abelian gauge field, in which case the solutions are locally isometric to a charged static black hole.
We consider the Hassan-Rosen bimetric field equations in vacuum when the two metrics share a single common null direction in a spherically symmetric configuration. By solving these equations, we obtain a class of exact solutions of the generalized Vaidya type parametrized by an arbitrary function. Besides not being asymptotically flat, the found solutions are nonstationary admitting only three global spacelike Killing vector fields which are the generators of spatial rotations. Hence, these are spherically symmetric bimetric vacuum solutions with the minimal number of isometries. The absence o…
In classical two-dimensional pure dilaton gravity, and in particular in spherically symmetric pure gravity in d dimensions, the generalized Birkhoff theorem states that, for a suitable choice of coordinates, the metric coefficients are only functions of a single coordinate. It is interesting to see how this result is recovered in quantum theory by the explicit construction of the Hilbert space. We examine the CGHS model, enforce the set of auxiliary conditions that select physical states a` la Gupta-Bleuler, and prove that the matrix elements of the metric and of the dilaton field obey the cla…
It is known that the Jebsen-Birkhoff theorem is valid for vacuum solutions to Einstein's equation, as well as some of its generalizations. Using symmetry inheritance properties we investigate in detail the additional constraints that fields have to satisfy in order to allow the Jebsen-Birkhoff theorem in the non-vacuum cases of a wide class of gravitational field equations.
We discuss the validity, or lack thereof, of the Jebsen-Birkhoff theorem in scalar-tensor theories by generalizing it and regarding the Brans-Dicke-like scalar as effective matter. Both the Jordan and Einstein frames are discussed and an apparent contradiction between static spherical solutions of scalar-tensor gravity and Hawking's theorem on Brans-Dicke black holes is clarified. The results are applied to metric and Palatini f(R) gravity.
The validity conditions for the extended Birkhoff theorem in multidimensional gravity with $n$ internal spaces are formulated, with no restriction on space-time dimensionality and signature. Examples of matter sources and geometries for which the theorem is valid are given. Further generalization of the theorem is discussed.
We prove a Jebsen-Birkhoff like theorem for f(R) theories of gravity in order to to find the necessary conditions required for the existence of the Schwarzschild solution in these theories and demonstrate that the rigidity of such solutions of f(R) gravity is valid even in the perturbed scenario.
The 5-dimensional (5d) Birkhoff theorem gives the class of 5d vacuum space-times containing spatial hypersurfaces with cosmological symmetries. This theorem is violated by the 5d vacuum Gergely-Maartens (GM) space-time, which is not a representant of the above class, but contains the static Einstein brane as embedded hypersurface. We prove that the 5d Birkhoff theorem is still satisfied in a weaker sense: the GM space-time is related to the degenerated horizon metric of certain black-hole space-times of the allowed class. This result resembles the connection between the Bertotti-Robinson space…
We provide a simple, unified proof of Birkhoff's theorem for the vacuum and cosmological constant case, emphasizing its local nature. We discuss its implications for the maximal analytic extensions of Schwarzschild, Schwarzschild(-anti)-de Sitter and Nariai spacetimes. In particular, we note that the maximal analytic extensions of extremal and over-extremal Schwarzschild-de Sitter spacetimes exhibit no static region. Hence the common belief that Birkhoff's theorem implies staticity is false for the case of positive cosmological constant. Instead, the correct point of view is that generalized B…
We extend Birkhoff's theorem for almost LRS-II vacuum spacetimes to show that the rigidity of spherical vacuum solutions of Einstein's field equations continues even in the perturbed scenario.
We study the space-times admitting a null conformal Killing-Yano tensor whose divergence defines a Killing vector. We analyze the similitudes and differences with the recently studied non null case (Gen. Relativ. Grav. (2015) {\bf 47} 1911). The results by Barnes concerning the Birkhoff theorem for the case of null orbits are analyzed and generalized.
We analyze the main geometric conditions imposed by the hypothesis of the Jebsen-Birkhoff theorem. We show that the result (existence of an additional Killing vector) does not necessarily require a three-dimensional isometry group on two-dimensional orbits but only the existence of a conformal Killing-Yano tensor. In this approach the (additional) isometry appears as the known invariant Killing vector that the ${\cal D}$-metrics admit.
We classify the existent Birkhoff-type theorems into four classes: First, in field theory, the theorem states the absence of helicity 0- and spin 0-parts of the gravitational field. Second, in relativistic astrophysics, it is the statement that the gravitational far-field of a spherically symmetric star carries, apart from its mass, no information about the star; therefore, a radially oscillating star has a static gravitational far-field. Third, in mathematical physics, Birkhoff's theorem reads: up to singular exceptions of measure zero, the spherically symmetric solutions of Einstein's vacuum…
The relation between the expanding universe and local vacuum solutions, such as that for the Solar System, is crucially mediated by Birkhoff's theorem. Here we consider how that relation works, and give generalizations of Birkhoff's theorem when there are geometric and matter and perturbations. The issue of to what degree dark matter might influence the solar system emerges as a significant question.
We present a novel approach to establish the Birkhoff's theorem validity in the so-called quadratic Poincaré Gauge theories of gravity. By obtaining the field equations via the Palatini formalism, we find paradigmatic scenarios where the theorem applies neatly. For more general and physically relevant situations, a suitable decomposition of the torsion tensor also allows us to establish the validity of the theorem. Our analysis shows rigorously how for all stable cases under consideration, the only solution of the vacuum field equations is a torsionless Schwarzschild spacetime, although it is …
We derive the general $Σ_2\times S$ solution of topologically massive gravity in vacuum and in presence of a cosmological constant. The field equations reduce to three-dimensional Einstein equations and the solution has constant Ricci tensor. We briefly discuss the emergence of non-Ricci flat solutions when spin is introduced.
As a continuation of a previous work, here we examine the admittance of Birkhoff's theorem in a class of higher derivative theories of gravity. This class is contained in a larger class of theories which are characterized by the property that the trace of the field equations are of second order in the metric. The action representing these theories are given by a sum of higher curvature terms. Moreover the terms of a fixed order k in the curvature are constructed by taking a complete contraction of k conformal tensors. The general spherically (hyperbolic or plane) symmetric solution is then giv…
Assuming SO(3)-spherical symmetry, the 4-dimensional Einstein equation reduces to an equation conformally related to the field equation for 2-dimensional gravity following from the Lagrangian L = R^(1/3). Solutions for 2-dimensional gravity always possess a local isometry because the traceless part of its Ricci tensor identically vanishes. Combining both facts, we get a new proof of Birkhoff's theorem; contrary to other proofs, no coordinates must be introduced. The SO(m)-spherically symmetric solutions of the (m+1)-dimensional Einstein equation can be found by considering L = R^(1/m) in two d…
Birkhoff's theorem for spherically symmetric vacuum spacetimes is a key theorem in studying local systems in general relativity theory. However realistic local systems are only approximately spherically symmetric and only approximately vacuum. In a previous paper, we showed the theorem remains approximately true in an approximately spherically symmetric vacuum space time. In this paper we prove the converse case: the theorem remains approximately true in a spherically symmetric, approximately vacuum space time.
Birkhoff's theorem is a classic result that characterizes locally spherically symmetric solutions of the Einstein equations. In this paper, we illustrate the consequences of its local nature for the cases of vacuum and positive cosmological constant. We construct several examples of initial data for spherically symmetric spacetimes on Cauchy surfaces of different topology than that of the maximal analytic extension of Schwarzschild and Schwarzschild-de Sitter spacetimes. The spacetimes formed from the evolution of these initial data sets also have very different physical properties; in particu…
We consider spherically symmetric space-times in GR under the unconventional assumptions that the spherical radius $r$ is either a constant or has a null gradient in the $(t,x)$ subspace orthogonal to the symmetry spheres (i.e., $(\partial r)^2 = 0$). It is shown that solutions to the Einstein equations with $r = \rm const$ contain an extra (fourth) spatial or temporal Killing vector and thus satisfy the Birkhoff theorem under an additional physically motivated condition that the lateral pressure is functionally related to the energy density. This leads to solutions that directly generalize th…
Birkhoff showed in 1923 that the Schwarzschild solution for the metric from a point particle was also valid in the a priori non-static case as long as the spherical symmetry was maintained. This theorem was actually discovered and published two years earlier by an unknown Norwegian physicist, J.T. Jebsen. His life and scientific career is briefly chronicled.
We present a new cubic theory of gravity in five dimensions which has second order traced field equations, analogous to BHT new massive gravity in three dimensions. Moreover, for static spherically symmetric spacetimes all the field equations are of second order, and the theory admits a new asymptotically locally flat black hole. Furthermore, we prove the uniqueness of this solution, study its thermodynamical properties, and show the existence of a C-function for the theory following the arguments of Anber and Kastor (arXiv:0802.1290 [hep-th]) in pure Lovelock theories. Finally, we include the…
A recent precise formulation of the hoop conjecture in four spacetime dimensions is that the Birkhoff invariant $β$ (the least maximal length of any sweepout or foliation by circles) of an apparent horizon of energy $E$ and area $A$ should satisfy $β\le 4 πE$. This conjecture together with the Cosmic Censorship or Isoperimetric inequality implies that the length $\ell$ of the shortest non-trivial closed geodesic satisfies $\ell^2 \le πA$. We have tested these conjectures on the horizons of all four-charged rotating black hole solutions of ungauged supergravity theories and find that they alway…
The f(T) theory, a generally modified teleparallel gravity, has been proposed as an alternative gravity model to account for the dark energy phenomena. Following our previous work [Xin-he Meng and Ying-bin Wang, EPJC(2011), arXiv:1107.0629v1], we prove that the Birkhoff's theorem holds in a more general context, specifically with the off diagonal tetrad case, in this communication letter. Then, we discuss respectively the results of the external vacuum and internal gravitational field in the f(T) gravity framework, as well as the extended meaning of this theorem. We also investigate the validi…
There is undetermined potential function $V(ϕ)$ in the action of mimetic gravity which should be resolved through physical means. In general relativity(GR), the static spherically symmetric(SSS) solution to the Einstein equation is a benchmark and its deformation also plays a crucial role in mimetic gravity. The equation of motion is provided with high nonlinearity, but we can reduce primal nonlinearity to a frequent Riccati form in the SSS case of mimetic gravity. In other words, we obtain an expression of solution to the functional differential equation of motion with any potential function.…
Generalized from the so-called teleparallel gravity which is exactly equivalent to general relativity, the $f(T)$ gravity has been proposed as an alternative gravity model to account for the dark energy phenomena. In this letter we prove that the external vacuum gravitational field for a spherically symmetric distribution of source matter in the $f(T)$ gravity framework must be static and the conclusion is independent of the radial distribution and spherically symmetric motion of the source matter that is, whether it is in motion or static. As a consequence, the Birkhoff's theorem is valid in …
We report the explicit form of the general static, spherically symmetric, and asymptotically flat solution of vacuum Brans-Dicke gravity in the Jordan frame, assuming that the Brans-Dicke scalar field has no singularities or zeros (except possibly for a central singularity). This general solution is conformal to the Fisher-Wyman geometry of Einstein theory and its nature depends on a scalar charge parameter. Apart from the Schwarzschild black hole, only wormhole throats and central naked singularities are possible
Spherically symmetric, asymptotically flat solutions of Shape Dynamics were previously studied assuming standard falloff conditions for the metric and the momenta. These ensure that the spacetime is asymptotically Minkowski, and that the falloff conditions are Poincaré-invariant. These requirements however are not legitimate in Shape Dynamics, which does not make assumptions on the structure or regularity of spacetime. Analyzing the same problem in full generality, I find that the system is underdetermined, as there is one function of time that is not fixed by any condition and appears to have…
Shape Dynamics is a theory of gravity that replaces refoliation invariance for spatial Weyl invariance. Those solutions of the Einstein equations that have global, constant mean curvature slicings, are mirrored by solutions in Shape Dynamics. However, there are solutions of Shape Dynamics that have no counterpart in General relativity, just as there are solutions of GR that are not completely foliable by global constant mean curvature slicings (such as the Schwarzschild spacetime). It is therefore interesting to analyze directly the equations of motion of Shape Dynamics in order to find its ow…
f(T) gravity, a generally modified teleparallel gravity, has become very popular in recent times as it is able to reproduce the unification of inflation and late-time acceleration without the need of a dark energy component or an inflation field. In this present work, we investigate specifically the range of validity of Birkhoff's theorem with the general tetrad field via perturbative approach. At zero order, Birkhoff's theorem is valid and the solution is the well known Schwarzschild-(A)dS metric. Then considering the special case of the diagonal tetrad field, we present a new spherically sym…
In Newtonian gravity (NG) it is known that the gravitational field anywhere inside a spherically symmetric distribution of mass is determined only by the enclosed mass. This is also widely believed to be true in general relativity (GR), and the Birkhoff theorem is often invoked to support this analogy between NG and GR. Here we show that such an understanding of the Birkhoff theorem is incorrect and leads to erroneous calculations of light deflection and delay time through matter. The correct metric, matching continuously to the location of an external observer, is determined both by the enclo…
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 admiss…
As a consequence of Birkhoff's theorem, the exterior gravitational field of a spherically symmetric star or black hole is always given by the Schwarzschild metric. In contrast, the exterior gravitational field of a rotating (axisymmetric) star differs, in general, from the Kerr metric, which describes a stationary, rotating black hole. In this paper, I discuss the possibility of a quasi-stationary transition from rotating equilibrium configurations of normal matter to rotating black holes.
It is fair to say that our current mathematical understanding of the dynamics of gravitational collapse to a black hole is limited to the spherically symmetric situation and, in fact, even in this case much remains to be learned. The reason is that Einstein's equations become tractable only if they are reduced to a 1+1 dimensional system of partial differential equations. Due to this technical obstacle, very little is known about the collapse of pure gravitational waves because by Birkhoff's theorem there is no spherical collapse in vacuum. In this essay we describe a new cohomogeneity-two sym…
Space-time is spherically symmetric if it admits the group of SO(3) as a group of isometries,with the group orbits spacelike two-surfaces. These orbits are necessarily two-surface of constant positive curavture. One commonly chooses coordinate ${t,r,θ,ϕ}$ so that the group orbits become surfaces ${t,r = const.}$ and the radial coordinate r is defined by the requirment that $4πr^2$ is the area of these spacelike two-surfaces with the range of zero to infinity. According to the Birkhoff's theorem upon the above assumptions, the Schwarzschild metric is the only soltion of the vacuum Einstein fiel…
We generalize Birkhoff's Theorem in the following fashion. We find necessary and sufficient conditions for any spherically symmetric space-time to be static in terms of the eigenvalues of the stress-energy tensor. In particular, we generalize the Tolman-Oppenheimer-Volkoff equation and prove that Birkhoff's theorem holds under the weaker hypothesis of no pressure (with respect to an appropriate frame.) We provide equations that show how the coefficients of the metric relate to the eigenvalues of the stress-energy tensor. These involve integrals that are simple functions of those eigenvalues. W…
Oppenheimer and Snyder found in 1939 that gravitational collapse in vacuum produces a "frozen star", i.e., the collapsing matter only asymptotically approaches the gravitational radius (event horizon) of the mass, but never crosses it within a finite time for an external observer. Based upon our recent publication on the problem of gravitational collapse in the physical universe for an external observer, the following results are reported here: (1) Matter can indeed fall across the event horizon within a finite time and thus BHs, rather than "frozen stars", are formed in gravitational collapse…
