Jiri Bicak, Martin Scholtz, Paul Tod
By an argument similar to that of Gibbons and Stewart, but in a different coordinate system and less restrictive gauge, we show that any weakly-asymptotically-simple, analytic vacuum or electrovacuum solutions of the Einstein equations which are periodic in time are necessarily stationary.
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It is proven that a solution to the Einstein-Maxwell equations whose gravitational and electromagnetic radiation fields vanish is in fact stationary in a neighbourhood of spatial infinity. That is, if the Weyl and Faraday tensors decay suitably fast, then there exists a time-like Killing vector field in the region outside the bifurcate horizon of a sphere of sufficiently large radius. In particular, truly dynamical time-periodic electrovacuum spacetimes do not exist. This is an extension of earlier work by Alexakis and Schlue and Bičák, Sholtz and Tod to include electromagnetism.
A theorem related to the Newman-Penrose constants is proven. The theorem states that all the Newman-Penrose constants of asymptotically flat, stationary, asymptotically algebraically special electrovacuum spacetimes are zero. Straightforward application of this theorem shows that all the Newman-Penrose constants of the Kerr-Newman spacetime must vanish.
Symmetries compatible with asymptotic flatness and admitting gravitational and electromagnetic radiation are studied by using the Bondi-Sachs-van der Burg formalism. It is shown that in axially symmetric electrovacuum spacetimes in which at least locally a smooth null infinity in the sense of Penrose exists, the only second allowable symmetry is either the translational symmetry or the boost symmetry. Translationally invariant spacetimes with in general a straight "cosmic string" along the axis of symmetry are non-radiative although they can have a non-vanishing news function. The boost-rotati…
The multipole moments of stationary axially symmetric vacuum or electrovacuum spacetimes can be expressed in terms of the power series expansion coefficients of the Ernst potential on the axis. In this paper we present a simpler, more efficient calculation of the multipole moments, applying methods introduced by Bäckdahl and Herberthson. For the non-vacuum electromagnetic case, our results for the octupole and higher moments differ from the results already published in the literature. The reason for this difference is that we correct an earlier unnoticed mistake in the power series solution of…
We show a uniqueness result for the n-dimensional spatial Reissner-Nordström manifold: a static, electrovacuum, asymptotically flat system which is asymptotically Reissner-Nordström is a subextremal Reissner-Nordström manifold with positive mass, provided that its inner boundary is a (possibly disconnected) photon sphere that fulfils a suitably defined quasilocal subextremality condition. Our result implies a number of earlier uniqueness results for the Schwarzschild and the Reissner-Nordström manifolds in the static, (electro-)vacuum, asymptotically flat context, both for photon sphere and bl…
"Smooth Gowdy-symmetric generalized Taub-NUT solutions" are a class of inhomogeneous cosmological vacuum models with a past and a future Cauchy horizon. In this proceedings contribution, we present families of exact solutions within that class, which contain the Taub solution as a special case, and discuss their properties. In particular, we show that, for a special choice of the parameters, the solutions have a curvature singularity with directional behaviour. For other parameter choices, the maximal globally hyperbolic region is singularity-free. We also construct extensions through the Cauc…
Vacuum, ma non troppo: hidden matter distribution in symmetry-transformed electrovacuum spacetimes
本文研究了通过不同对称变换生成的两个静态时空，展示了在Weyl坐标下，这两个解由半无限环状质量分布支持，揭示了隐藏的物质源。
We construct a 4-parameter family of inhomogeneous cosmological models, which contains two recently derived 3-parameter families as special cases. The corresponding exact vacuum solution to Einstein's field equations is obtained with methods from soliton theory. We also study properties of these models and find that they combine all interesting features of both earlier solution families: general regularity within the maximal globally hyperbolic region, particular singular cases in which a curvature singularity with a directional behaviour forms, a highly non-trivial causal structure, and Cauch…
We obtain a purely local characterisation that singles out the Majumdar-Papapetrou class, the near-horizon Bertotti-Robinson geometry and the Reissner-Nordström exterior solution, together with its plane and hyperbolic counterparts, among the static electrovacuum spacetimes. These five classes are found to form the whole set of static Einstein-Maxwell fields without sources and conformally flat space of orbits, this is, the conformastat electrovacuum spacetimes. The main part of the proof consists in showing that a functional relationship between the gravitational and electromagnetic potential…
We show that all Majumdar--Papapetrou electrovacuum space--times with a non--empty black hole region and with a non--singular domain of outer communications are the standard Majumdar--Papapetrou space--times.
Stable photon orbits in stationary axisymmetric spacetimes with an electromagnetic field and a cosmological constant
本研究将稳定光环的存在性结论推广至含宇宙学常数的时空，证明其可存在于Λ-电真空时空中，却被Λ-真空时空排除，拓展了稳态轴对称时空不稳定性相关研究。
It is shown that there are three vacuum and one electrovacuum solutions of diagonal plane waves with M=0 and constant Maxwell scalars. Namely, these are the single wave, Stoyanov, Babala and Bell-Szekeres solutions. A comparison is made with the planar solutions of Taub.
We determine all infinitesimal transverse deformations of extreme horizons in Einstein-Maxwell theory that preserve axisymmetry. In particular, we show that the general static transverse deformation of the AdS(2) X S(2) near-horizon geometry is a two-parameter family, which contains the known extreme charged, accelerating, static black hole solution held in equilibrium by an external electric or magnetic field (Ernst solution) and a special case of the extreme Kerr-Newman-Melvin solution. More generally, we find a three-parameter family of deformations of the extreme Kerr-Newman horizon, which…
We show that the hypothesis of analyticity in the uniqueness theory of vacuum, or electrovacuum, static black holes is not needed. More generally, we show that prehorizons covering a closed set cannot occur in well-behaved domains of outer communications.
The general relativity vacuum black holes (BHs) can be scalarised in models where a scalar field non-minimally couples to the Gauss-Bonnet (GB) invariant. Such GB scalarisation comes in two flavours, depending on the GB sign that triggers the phenomenon. Hereafter these two cases are termed GB$^\pm$ scalarisation. For vacuum BHs, only GB$^+$ scalarisation is possible in the static case, while GB$^-$ scalarisation is $spin$ induced. But for electrovacuum BHs, GB$^-$ is also $charged$ induced. We discuss the GB$^-$ scalarisation of Reissner-Nordström and Kerr-Newman BHs, discussing zero modes an…
From Bertotti--Robinson to Vacuum: New Exact Solutions in General Relativity via Harrison and Inversion Symmetries
本文通过Harrison和反演对称性，从Bertotti-Robinson黑洞出发，构造了新的真空解，展示了电磁嵌套如何产生非平凡的真空度规。
We explicitly construct static black hole solutions to the fully non-linear, D=4, Einstein-Maxwell-AdS equations that have no continuous spatial symmetries. These black holes have a smooth, topologically spherical horizon (section), but without isometries, and approach, asymptotically, global AdS spacetime. They are interpreted as bound states of a horizon with the Einstein-Maxwell-AdS solitons recently discovered, for appropriate boundary data. In sharp contrast with the uniqueness results for Minkowski electrovacuum, the existence of these black holes shows that single, equilibrium, BH solut…
In this paper we continue earlier investigations of evolutionary formulations of the Einstein vacuum constraint equations originally introduced by Rácz. Motivated by the strong evidence from these works that the resulting vacuum initial data sets are generically not asymptotically flat we analyse the asymptotics of the solutions of a modified formulation by a combination of analytical and numerical techniques. We conclude that the vacuum initial data sets generated with this new formulation are generically asymptotically flat.
We show that time-reflection symmetric, asymptotically flat, static vacuum data which admit a non-trivial conformal rescaling which leads again to such data must be axi-symmetric and admit a conformal Killing field. Moreover, it is shown that there exists a 3-parameter family of such data.
Extending the results of an earlier article we give a complete description of the asymptotically flat, conformally non-flat, static vacuum data which admit non-trivial, asymptotically smooth conformal mappings onto other such data. These data form a 3-parameter family which decomposes into 1-parameter families of data which are conformal to each other. The data and the associated static vacuum solutions are given explicitly in terms of elliptic and, in a special case, elementary functions.
By an argument similar to that of Gibbons and Stewart (1984), we show that there are no weakly-asymptotically-simple, analytic vacuum or electrovac solutions of the Einstein equations which are periodic in time.
We introduce a new class of inhomogeneous cosmological models as solutions to the Einstein-Maxwell equations in electrovacuum. The new models can be considered to be nonlinear perturbations, through an electromagnetic field, of the previously studied `smooth Gowdy-symmetric generalised Taub-NUT solutions' in vacuum. Utilising methods from soliton theory, we analyse the effects of the Maxwell field on global properties of the solutions. In particular, we show existence of regular Cauchy horizons, and we investigate special singular cases in which curvature singularities form.
We show that static electro--vacuum black hole space--times containing an asymptotically flat spacelike hypersurface with compact interior and with both degenerate and non--degenerate components of the event horizon do not exist. This is done by a careful study of the near-horizon geometry of degenerate horizons, which allows us to eliminate the last restriction of the static electro-vacuum no-hair theory.
In this paper, we continue our investigations of Rácz's parabolic-hyperbolic formulation of the Einstein vacuum constraints. Our previous studies of the asymptotically flat setting provided strong evidence for unstable asymptotics which we were able to resolve by introducing a certain modification of Rácz's parabolic-hyperbolic formulation. The primary focus of the present paper here is the asymptotically hyperboloidal setting. We provide evidence through a mixture of numerical and analytical methods that the asymptotics of the solutions of Rácz's parabolic-hyperbolic formulation is stable, an…
We present an axially symmetric, asymptotically flat empty space solution of the Einstein field equations containing a naked singularity. The spacetime is regular everywhere except on the symmetry axis where it possess a true curvature singularity. The spacetime is of type D in the Petrov classification scheme and is locally isometric to the metrics of case IV in the Kinnersley classification of type D vacuum metrics. Additionally, the spacetime also shows the evolution of closed timelike curves (CTCs) from an initial hypersurface free from CTCs.
We construct non-trivial vacuum space-times with a global Scri. The construction proceeds by proving extension results across compact boundaries for initial data sets, adapting the gluing arguments of Corvino and Schoen. Another application of the extension results is existence of initial data which are exactly Schwarzschild both near infinity and near each of the connected component of the apparent horizon.
Recently, Bañados, Teitelboim and Zanelli obtained spherically symmetric black hole solutions in a particular class of Einstein--Lovelock gravity. We derive the propagator in an exact form for a conformal scalar field in the asymptotically anti-de Sitter black hole spacetime so as to study the quantum effects of the scalar fields. We treat the cases in odd dimensions in this paper. We calculate the vacuum expectation value of $\langleφ^2\rangle$ and show its dependence on the radial coordinate for the five-dimensional case as an example.
We prove non-existence of static, vacuum, appropriately regular, asymptotically flat black hole space-times with degenerate (not necessarily connected) components of the event horizon. This finishes the classification of static, vacuum, asymptotically flat domains of outer communication in an appropriate class of space-times, showing that the domains of outer communication of the Schwarzschild black holes exhaust the space of appropriately regular black hole exteriors.
We present a new asymptotically-flat time-machine model made solely of vacuum and dust. The spacetime evolves from a regular spacelike initial hypersurface S and subsequently develops closed timelike curves. The initial hypersurface S is asymptotically flat and topologically trivial. The chronology violation occurs in a compact manner; namely the first closed causal curves form at the boundary of the future domain of dependence of a compact region in S (the core). This central core is empty, and so is the external asymptotically flat region. The intermediate region surrounding the core (the en…
Modulo conventional scale factors, the Simon and Simon-Mars tensors are defined for stationary vacuum spacetimes so that their equality follows from the Bianchi identities of the second kind. In the nonvacuum case one can absorb additional source terms into a redefinition of the Simon tensor so that this equality is maintained. Among the electrovacuum class of solutions of the Einstein-Maxwell equations, the expression for the Simon tensor in the Kerr-Newman-Taub-NUT spacetime in terms of the Ernst potential is formally the same as in the vacuum case (modulo a scale factor), and its vanishing …
We show that static electro-vacuum black hole space-times containing an asymptotically flat spacelike hypersurface with compact interior and with both degenerate and non-degenerate components of the event horizon do not exist, under the supplementary hypothesis that all degenerate components of the event horizon have charges of the same sign. This extends previous uniqueness theorems of Simon and Masood-ul-Alam (where only non-degenerate horizons were allowed) and Heusler (where only degenerate horizons were allowed).
A global model of a slowly rotating perfect fluid ball in general relativity is presented. To second order in the rotation parameter, the junction surface is an ellipsoidal cylinder. The interior is given by a limiting case of the Wahlquist solution, and the vacuum region is not asymptotically flat. The impossibility of joining an asymptotically flat vacuum region has been shown in a preceding work.
The BKL conjecture, stated in the 60s and early 70s by Belinski, Khalatnikov and Lifshitz, proposes a detailed description of the generic asymptotic dynamics of spacetimes as they approach a spacelike singularity. It predicts complicated chaotic behaviour in the generic case, but simpler non-chaotic one in cases with symmetry assumptions or certain kinds of matter fields. Here we construct a new class of four-dimensional vacuum spacetimes containing spacelike singularities which show non-chaotic behaviour. In contrast with previous constructions, no symmetry assumptions are made. Rather, the m…
We give a complete classification of all connected isometry groups, together with their actions in the asymptotic region, in asymptotically flat, asymptotically vacuum space--times with timelike ADM four--momentum.
We study formal expansions of asymptotically flat solutions to the static vacuum field equations which are determined by minimal sets of freely specifyable data referred to as `null data'. These are given by sequences of symmetric trace free tensors at space-like infinity of increasing order. They are 1:1 related to the sequences of Geroch multipoles. Necessary and sufficient growth estimates on the null data are obtained for the formal expansions to be absolutely convergent. This provides a complete characterization of all asymptotically flat solutions to the static vacuum field equations.
We compute the vacuum energy of three-dimensional asymptotically flat space based on a Chern-Simons formulation for the Poincare group. The equivalent action is nothing but the Einstein-Hilbert term in the bulk plus half of the Gibbons-Hawking term at the boundary. The derivation is based on the evaluation of the Noether charges in the vacuum. We obtain that the vacuum energy of this space has the same value as the one of the asymptotically flat limit of three-dimensional anti-de Sitter space.
On vacuum and charged asymptotically (A)dS black holes in quadratic gravity
本文研究了二次引力中带电和非带电渐近(A)dS黑洞的性质，发现其渐近行为与Bach参数和电荷相关，并提出了两种不同的黑洞家族。
We present a characterization of the asymptotics of all asymptotically flat stationary vacuum solutions of Einstein's field equations. This characterization is given in terms of two sequences of symmetric trace free tensors (we call them the `null data'), which determine a formal expansion of the solution, and which are in a one to one correspondence to Hansen's multipoles. We obtain necessary and sufficient growth estimates on the null data to define an absolutely convergent series in a neighbourhood of spatial infinity. This provides a complete characterization of all asymptotically flat sta…
We study the geometry of a general class of vacuum asymptotically Anti-de Sitter spacetimes near the conformal boundary. In particular, the spacetime is only assumed to have finite regularity, and it is allowed to have arbitrary boundary topology and geometry. For the main results, we derive limits at the conformal boundary of various geometric quantities, and we use these limits to construct partial Fefferman--Graham expansions from the boundary. The results of this article will be applied, in upcoming papers, toward proving symmetry extension and gravity--boundary correspondence theorems for…
We present a family of four-dimensional vacuum space-times with asymptotically velocity dominated singularities and without symmetries.
Adapting Israel's proof of static black hole uniqueness, we show that the Schwarzschild spacetime is the only static vacuum asymptotically flat spacetime that possesses a suitably defined photon sphere.
In this paper we investigate the parabolic-hyperbolic formulation of the vacuum constraint equations introduced by R{á}cz with a view to constructing multiple black hole initial data sets without spin. In order to respect the natural properties of this configuration, we foliate the spatial domain with 2-spheres. It is then a consequence of these equations that they must be solved as an initial value problem evolving outwards towards spacelike infinity. Choosing the free data and the "strong field boundary conditions" for these equations in a way which mimics asymptotically flat and asymptotica…
In the present article we solve the Dirac-Pauli and Klein Gordon equations in an asymptotically uniformly accelerated frame when a constant magnetic field is present. We compute, via the Bogoliubov coefficients, the density of scalar and spin 1/2 particles created. We discuss the role played by the magnetic field and the thermal character of the spectrum.
There has been considerable interest in applying the gauge/gravity duality to condensed matter theories with particular attention being devoted to gravity duals (Lifshitz spacetimes) of theories that exhibit anisotropic scaling. In this context, black hole solutions with Lifshitz asymptotics have also been constructed aiming at incorporating finite temperature effects. The goal here is to look at quantum polarization effects in these spacetimes, and to this aim, we develop a way to compute the coincidence limit of the Green's function for massive, non-minimally coupled scalar fields, adapting …
Uniqueness of the static vacuum asymptotically flat spacetimes with massive particle spheres
本研究拓展了静态真空渐近平直时空的唯一性定理，证明含单个大质量粒子球的四维该类时空与史瓦西时空等距，且存在覆盖不同能量的大质量粒子球构成的完整时空叶状结构。
We show that a set of conformally invariant equations derived from the Fefferman-Graham tensor can be used to construct global solutions of the vacuum Einstein equations, in all even dimensions. This gives, in particular, a new, simple proof of Friedrich's result on the future hyperboloidal stability of Minkowski space-time, and extends its validity to even dimensions.
We consider the problem of matching two spacetimes, the previous and present aeons, in the Conformal Cyclic Cosmology model. The common boundary between them inherits two sets of constraints -- one for each solution of the Einstein field equations extended to the conformal boundaries. The previous aeon is assumed to be an asymptotically de Sitter spacetime, so the standard conformal formulation of the Einstein field equations suffice to derive the constraints on the future null infinity. For the future aeon, which is supposed to evolve from an initial singularity, they are obtained with the us…
The assumption of asymptotic flatness for isolated astrophysical bodies may be considered an approximation when one considers a cosmological context where a cosmological constant or vacuum energy is present. In this framework we study the motion of spinning particles in static, spherically symmetric and asymptotically non-flat spacetimes with repulsive cosmological vacuum energy and quintessential field. Due to the combined effects of gravitational attraction and cosmological repulsion, the region where stable circular orbits are allowed is restricted by an innermost and an outermost stable ci…
Various works have suggested that the Bondi--Sachs--Penrose decay conditions on the gravitational field at null infinity are not generally representative of asymptotically flat space--times. We have made a detailed analysis of the constraint equations for ``asymptotically hyperboloidal'' initial data and find that log terms arise generically in asymptotic expansions. These terms are absent in the corresponding Bondi--Sachs--Penrose expansions, and can be related to explicit geometric quantities. We have nevertheless shown that there exists a large class of ``non--generic'' solutions of the con…
We show that stationary, asymptotically flat solutions of the electro-vacuum Einstein equations are analytic at $i^0$, for a large family of gauges, in odd space-time dimensions higher than seven. The same is true in space-time dimension five for static vacuum solutions with non-vanishing mass.
We reformulate the standard local equations of general relativity for asymptotically flat spacetimes in terms of two non-local quantities, the holonomy $H$ around certain closed null loops on characteristic surfaces and the light cone cut function $Z$, which describes the intersection of the future null cones from arbitrary spacetime points, with future null infinity. We obtain a set of differential equations for $H$ and $Z$ equivalent to the vacuum Einstein equations. By finding an algebraic relation between $H$ and $Z$ this set of equations is reduced to just two coupled equations: an integr…
Perturbations of the linearized vacuum Einstein equations on a null cone in the Bondi-Sachs formulation of General Relativity can be derived from a single master function with spin weight two, which is related to the Weyl scalar Ψ_0, and which is determined by a simple wave equation. Utilizing a standard spin representation of the tensors on a sphere and two different approaches to solve the master equation, we are able to determine two simple and explicitly time-dependent solutions. Both solutions, of which one is asymptotically flat, comply with the regularity conditions at the vertex of the…
Static asymptotically Lifshitz wormholes and black holes in vacuum are shown to exist for a class of Lovelock theories in d=2n+1>7 dimensions, selected by requiring that all but one of their n maximally symmetric vacua are AdS of radius l and degenerate. The wormhole geometry is regular everywhere and connects two Lifshitz spacetimes with a nontrivial geometry at the boundary. The dynamical exponent z is determined by the quotient of the curvature radii of the maximally symmetric vacua according to n(z^2-1)+1=(l/L)^2, where L corresponds to the curvature radius of the nondegenerate vacuum. Lig…
It is shown that the class of asymptotically flat solutions to the axisymmetric and stationary vacuum Einstein equations with reflection symmetry of the metric is uniquely characterized by a simple relation for the Ernst potential on the upper part of the symmetry axis. This result generalizes a well-known fact from potential theory.
We obtain a characterization of the Kerr metric among stationary, asymptotically flat, vacuum spacetimes, which extends the characterization in terms of the Simon tensor (defined only in the manifold of trajectories) to the whole spacetime. More precisely, we define a three index tensor on any spacetime with a Killing field, which vanishes identically for Kerr and which coincides in the strictly stationary region with the Simon tensor when projected down into the manifold of trajectories. We prove that a stationary asymptotically flat vacuum spacetime with vanishing spacetime Simon tensor is l…
We discuss the issue of radiation extraction in asymptotically flat space-times within the framework of conformal methods for numerical relativity. Our aim is to show that there exists a well defined and accurate extraction procedure which mimics the physical measurement process. It operates entirely intrisically within $\scri^+$ so that there is no further approximation necessary apart from the basic assumption that the arena be an asymptotically flat space-time. We define the notion of a detector at infinity by idealising local observers in Minkowski space. A detailed discussion is presented…
We present a class of curved-spacetime vacuum solutions which develope closed timelike curves at some particular moment. We then use these vacuum solutions to construct a time-machine model. The causality violation occurs inside an empty torus, which constitutes the time-machine core. The matter field surrounding this empty torus satisfies the weak, dominant, and strong energy conditions. The model is regular, asymptotically-flat, and topologically-trivial. Stability remains the main open question.
New asymptotically flat static vacuum metrics with near Euclidean boundary data
针对Bartnik近欧几里得边界数据的静态真空延拓猜想，研究改进了此前的静态正则超曲面结果，证明一类光滑单侧超曲面族的开稠子族中任意超曲面均为静态正则，采用了受相关猜想研究启发的新论证。
We present a new five-parameter class of Ricci-flat solutions in four dimensions with Euclidean signature. The solution is asymptotically locally flat (ALF), and contains a finite asymptotic NUT charge. When this charge is sent to infinity, the solution becomes asymptotically locally Euclidean (ALE), and one in fact obtains the Ricci-flat Plebanski-Demianski solution. The solution we have found can thus be regarded as an ALF generalisation of the latter solution. We also show that it can be interpreted as a system consisting of two touching Kerr-NUTs: the south pole of one Kerr-NUT touches the…
Black Hole and Equipotential Photon Surface Uniqueness in 4-dimensional Asymptotically Flat Electrostatic Electro-Vacuum Spacetimes
本文研究四维渐近平坦静电电真空时空中黑洞与光子面的唯一性，采用位势理论方法，统一证明并扩展了Reissner-Nordström时空的唯一性定理，特别填补了超极端情形下连通光子面唯一性结果的空白。
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