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On space-times with U(1) × U(1) symmetric compact Cauchy surfaces
Abstract
General space-times evolving from U(1) × U(1) symmetric Cauchy data prescribed on compact Cauchy surfaces are studied. Existence and properties of solutions of the constraint equations are analyzed. Some “canonical” forms of the metric are derived. When the spatial topology is S3 or S2 × S1 or L(p, q) we show that no singularities form before “the spacelike boundary of Gowdy's square” is reached.
... We quickly introduce important and relevant facts about Gowdy symmetry on S 1 × S 2 . General aspects of U(1) × U(1)-symmetric solutions of Einstein's field equations were discussed in [17] for the first time and later reconsidered in [9]. ...
... All functions involved here only depend on θ. For the smoothness of h, it is sufficient and necessary [9] that there are functionR,P andλ so that R =R sin θ, P =P + ln sin θ, λ = (P + lnR)/2 +λ, (2.2a) ...
... Gowdy symmetric spacetimes with spatial S 1 × S 2 -topology Now let us consider a globally hyperbolic spacetime (M, g) foliated with U(1) × U(1)-invariant Cauchy surfaces of topology S 1 × S 2 , in the sense that the first and second fundamental form of each surface are invariant under the U(1) × U(1)-action before. It has been shown before [9,10] that this action is orthogonally transitive, i.e. the twist constants ...
This is the second of two papers where we study the asymptotics of the generalized Nariai solutions and its relation to the cosmic no-hair conjecture. In the first paper, the author suggested that according to the cosmic no-hair conjecture, the Nariai solutions are non-generic among general solutions of Einstein's field equations in vacuum with a positive cosmological constant. We checked that this is true within the class of spatially homogeneous solutions. In this paper now, we continue these investigations within the spatially inhomogeneous Gowdy case. On the one hand, we are motivated to understand the fundamental question of cosmic no-hair and its dynamical realization in more general classes than the spatially homogeneous case. On the other hand, the results of the first paper suggest that the instability of the Nariai solutions can be exploited to construct and analyze physically interesting cosmological black hole solutions in the Gowdy class, consistent with certain claims by Bousso in the spherically symmetric case. However, in contrast to that, we find that it is not possible to construct cosmological black hole solutions by means of small Gowdy symmetric perturbations of the Nariai solutions and that the dynamics shows a certain new critical behavior. For our investigations, we use the numerical techniques based on spectral methods which we introduced in a previous publication.
... In this work, we show that the Fuchsian algorithm is an effective tool for proving that AVD behavior occurs in a wider class of spacetimes: those which possess, like the Gowdy spacetimes, a T 2 isometry group with spacelike generators, but in which, unlike the Gowdy case, the Killing vectors have a non-vanishing twist. The main new difficulty is that this non-vanishing twist prevents the constraint equations from decoupling from the evolution equations, resulting in a considerably more complicated PDE system than what obtains in the case of Gowdy spacetimes [16,17,18]. This difficulty is overcome by abandoning the separation of constraint and evolution equations. ...
... While the Gowdy T 3 spacetimes [22] have been extensively studied over the years [5,16,6,12,7], and are relatively well-understood, the more general T 2symmetric spacetimes have only recently begun to be considered [16,17,18]. The technical condition which distinguishes the Gowdy sub-family is the requirement that the Killing fields X and Y which generate the isometry group have vanishing twist constants κ ...
... While the Gowdy T 3 spacetimes [22] have been extensively studied over the years [5,16,6,12,7], and are relatively well-understood, the more general T 2symmetric spacetimes have only recently begun to be considered [16,17,18]. The technical condition which distinguishes the Gowdy sub-family is the requirement that the Killing fields X and Y which generate the isometry group have vanishing twist constants κ ...
We use Fuchsian Reduction to study the behavior near the singularity of a class of solutions of Einstein's vacuum equations. These solutions admit two commuting spacelike Killing fields like the Gowdy spacetimes, but their twist does not vanish. The spacetimes are also polarized in the sense that one of the `gravitational degrees of freedom' is turned off. Examining an analytic family of solutions with the maximum number of arbitrary functions, we find that they are all asymptotically velocity-term dominated as one approaches the singularity.
... A T 2 -symmetric cosmological space-time (M, g) is by definition globally hyperbolic and characterized by the existence of an effective smooth action of the isometry group U(1) × U(1) ∼ = T 2 [50], which preserves the Cauchy hypersurfaces of the space-time. The corresponding Lie algebra of Killing vector fields is spanned by two commuting spacelike Killing vector fields X and Y with closed orbits. ...
... We recall that a T 2 -symmetric metric g in areal coordinates (t, θ, x, y) [50,51] on (0, ∞) × T 3 takes the form g = e 2(ν−u) (−αdt 2 + dθ 2 ) + e 2u (dx + Qdy + (G + QH)dθ) 2 + e −2u t 2 (dy + Hdθ) 2 , ...
... are defined as the twist constants of g. These names are motivated by the fact [50] that J X and J Y are constant for T 2 -symmetric solutions of the vacuum Einstein equations (1.1) with arbitrary cosmological constant. It can also be shown that the twist one-forms are given by ...
We prove the nonlinear stability of the asymptotic behaviour of perturbations of subfamilies of Kasner solutions in the contracting time direction within the class of polarized T 2 -symmetric solutions of the vacuum Einstein equations with arbitrary cosmological constant Λ . This stability result generalizes the results proven in Ames E et al. (2022 Stability of AVTD Behavior within the Polarized T 2 -symmetric vacuum spacetimes. Ann. Henri Poincaré . ( doi:10.1007/s00023-021-01142-0 )), which focus on the Λ = 0 case, and as in that article, the proof relies on an areal time foliation and Fuchsian techniques. Even for Λ = 0 , the results established here apply to a wider class of perturbations of Kasner solutions within the family of polarized T 2 -symmetric vacuum solutions than those considered in Ames E et al. (2022 Stability of AVTD Behavior within the Polarized T 2 -symmetric vacuum spacetimes. Ann. Henri Poincaré . ( doi:10.1007/s00023-021-01142-0 )) and Fournodavlos G et al. (2020 Stable Big Bang formation for Einstein’s equations: the complete sub-critical regime . Preprint. ( http://arxiv.org/abs/2012.05888 )). Our results establish that the areal time coordinate takes all values in ( 0 , T 0 ] for some T 0 > 0 , for certain families of polarized T 2 -symmetric solutions with cosmological constant.
This article is part of the theme issue ‘The future of mathematical cosmology, Volume 1’.
... T 2 -symmetry and areal gauge. A T 2 -symmetric cosmological spacetime (M, g) is by definition globally hyperbolic and characterised by the existence of an effective smooth action of the isometry group U (1) × U (1) ∼ = T 2 [17], which preserves the Cauchy hypersurfaces of the spacetime. The corresponding Lie algebra of Killing vector fields is spanned by two commuting spacelike Killing vector fields X and Y with closed orbits. ...
... We recall that a T 2 -symmetric metric g in areal coordinates (t, θ, x, y) [7,17] on (0, ∞) × T 3 takes the form ...
... are defined as the twist constants of g. These names are motivated by the fact [17] that J X and J Y are constant for T 2 -symmetric solutions of the vacuum Einstein equations (1.1) with arbitrary cosmological constant. It can also be shown that the twist one-forms are given by ...
We prove the nonlinear stability of the asymptotic behavior of perturbations of subfamilies of Kasner solutions in the contracting time direction within the class of polarised -symmetric solutions of the vacuum Einstein equations with arbitrary cosmological constant . This stability result generalizes the results proven in [3], which focus on the case, and as in that article, the proof relies on an areal time foliation and Fuchsian techniques. Even for , the results established here apply to a wider class of perturbations of Kasner solutions within the family of polarised -symmetric vacuum solutions than those considered in [3] and [26]. Our results establish that the areal time coordinate takes all values in for some , for certain families of polarised -symmetric solutions with cosmological constant.
... This is shown for Gowdy intial data with the time t taking values in (0, ∞) in the case of T 3 spatial topology, and (0, π) in the remaining cases [26,62]. A similar result is proved for the T 2 -symmetric spacetimes with non-vanishing twist. ...
... Beyond the fact that the areal coordinates cover the T 2 -symmetric spacetimes, they are useful in studying T 2 -symmetric solutions for two additional reasons. The first is that due to the results [12,26,46,62] mentioned above, in these coordinates one approaches the cosmological singularity precisely as the time coordinate approaches t = 0. The second is that in these coordinates the Einstein equations can be brought into hyperbolic form. ...
... where 26. ...
Characterizing the long-time behavior of solutions to the Einstein field equations remains an active area of research today. In certain types of coordinates the Einstein equations form a coupled system of quasilinear wave equations. The investigation of the nature and properties of solutions to these equations lies in the field of geometric analysis. We make several contributions to the study of solution dynamics near singularities. While singularities are known to occur quite generally in solutions to the Einstein equations, the singular behavior of solutions is not well-understood. A valuable tool in this program has been to prove the existence of families of solutions which are so-called asymptotically velocity term dominated (AVTD). It turns out that a method, known as the Fuchsian method, is well-suited to proving the existence of families of such solutions. We formulate and prove a Fuchsian-type theorem for a class of quasilinear hyperbolic partial differential equations and show that the Einstein equations can be formulated as such a Fuchsian system in certain gauges, notably wave gauges. This formulation of Einstein equations provides a convenient general framework with which to study solutions within particular symmetry classes. The theorem mentioned above is applied to the class of solutions with two spatial symmetries -- both in the polarized and in the Gowdy cases -- in order to prove the existence of families of AVTD solutions. In the polarized case we find families of solutions in the smooth and Sobolev regularity classes in the areal gauge. In the Gowdy case we find a family of wave gauges, which contain the areal gauge, such that there exists a family of smooth AVTD solutions in each gauge.
... In the following, we assume that 0 ≤ K ≤ 1 so that the speed of sound is less than or equal to the speed of light. As discussed in the introduction, we restrict our attention to solutions of the Einstein-Euler equations with a Gowdy symmetry [13,24] by considering Gowdy metrics in areal coordinates on R >0 × T 3 of the form ...
... As the solution becomes discontinuous, the ratio (4.16) should blow up. In [17], a solution was classified as shock-forming if this ratio was 13 Note, however, that one can employ so-called well-balanced schemes which preserve certain steady states to much higher accuracy, see for example [33]. larger than 10 6 . ...
We numerically study, under a Gowdy symmetry assumption, nonlinear perturbations of the decelerated FLRW fluid solutions to the Einstein-Euler system toward the future for linear equations of state with . This article builds on the work of Fajman et al. \cite{Fajman_et_al:2024} in which perturbations of the homogeneous fluid solution on a fixed, decelerating FLRW background were studied. Our numerical results show that for all values of K, perturbations of the FLRW solution develop shocks in finite time. This behaviour contrasts known results for spacetimes with accelerated expansion in which shock formation is suppressed.
... One of the first examples of global existence for arbitrary initial data (in a symmetry class of the type of interest here), is the one concerning Gowdy spacetimes obtained in [13]. However, it was followed by a large collection of results, proving global existence of geometrically natural foliations for several different matter fields; see, e.g., [116][117][118][119][120][121][122][123][124][125][126][127]. The symmetry classes studied are Gowdy symmetry, T 2 -symmetry and surface symmetry. ...
... Due to [128,129], this means that the group has to be T 2 . Moreover, the initial hypersurface must be T 3 , S 2 × S 1 , S 3 or one of the Lens spaces (which means that a covering space of it is S 3 , so that we ignore Lens spaces from now on); see [117]. The 2-torus action on the initial hypersurface gives rise to two Killing vector fields of the corresponding development, say X i , i = 1, 2. Then T i := ϵ αβγδ X α 1 X β 2 ∇ γ X δ i are the associated twist constants, where ϵ αβγδ are the components of the volume form and ∇ is the Levi-Civita connection of the development. ...
... The advantage of considering Gowdy spacetimes is that the presence of two Killing fields allows us to reduce the Einstein-Euler-scalar field equations to a 1 + 1-dimensional problem with periodic boundary conditions. This type of simplification has been exploited both analytically and numerically many times in previous studies of the Einstein equations [1,2,8,9,10,11,15,16,28,29,30,40,43]. ...
... As discussed in the introduction, we restrict our attention to solutions of the Einstein-Euler-scalar field equations with a Gowdy symmetry [16,23] by considering Gowdy metrics in areal coordinates on R >0 × T 3 of the form ...
Using numerical methods, we examine, under a Gowdy symmetry assumption, the dynamics of nonlinearly perturbed FLRW fluid solutions of the Einstein-Euler-scalar field equations in the contracting direction for linear equations of state and sound speeds . This article builds upon the numerical work from \cite{BMO:2023} in which perturbations of FLRW solutions to the Einstein-Euler equations with positive cosmological constant in the expanding time direction were studied. The numerical results presented here confirm that the instabilities observed in \cite{BMO:2023,MarshallOliynyk:2022} for , first conjectured to occur in the expanding direction by Rendall in \cite{Rendall:2004}, are also present in the contracting direction over the complementary parameter range . Our numerical solutions show that the fractional density gradient of the nonlinear perturbations develop steep gradients near a finite number of spatial points and become unbounded towards the big bang. This behaviour, and in particular the characteristic profile of the fractional density gradient near the big bang, is strikingly similar to what was observed in the expanding direction near timelike infinity in the article \cite{BMO:2023}.
... The Gowdy spacetimes we consider in this article are especially well-suited to both analytical and numerical treatments (e.g. [1,2,3,4,5,6,7,8,16,17,18,29,32]) due to the presence of two Killing fields, which reduces the Einstein-Euler equations to a 1 + 1-dimensional problem with periodic boundary conditions. ...
... As discussed in the introduction, we restrict our attention to solutions of the Einstein-Euler equations with a Gowdy symmetry [8,13]. We do this by considering Gowdy metrics in areal coordinates on R >0 × T 3 that are of the form ...
Using numerical methods, we examine the dynamics of nonlinear perturbations in the expanding time direction, under a Gowdy symmetry assumption, of FLRW fluid solutions to the Einstein-Euler equations with a positive cosmological constant and a linear equation of state for the parameter values . This paper builds upon the numerical work in \cite{Marshalloliynyk:2022} in which the simpler case of a fluid on a fixed FLRW background spacetime was studied. The numerical results presented here confirm that the instabilities observed in \cite{Marshalloliynyk:2022} are also present when coupling to gravity is included as was previously conjectured in \cite{Rendall:2004,Speck:2013}. In particular, for the full parameter range , we find that the density contrast of the nonlinear perturbations develop steep gradients near a finite number of spatial points and becomes unbounded there at future timelike infinity. This instability is of particular interest since it is not consistent with the standard picture for late time expansion in cosmology.
... In the last 20-30 years, a substantial number of results concerning quiescent big bang singularities have appeared; cf., e.g., [8,9,16,14,18,5,19,32,10,15,22,23,1,17,2,29,30,31,11,12,3,4,7]. In some of the results, solutions are constructed given initial data on the singularity. ...
... where λ, P and Q only depend on t and ϑ; we refer the interested reader to [13,8] for a description of the origin of this class of spacetimes. The underlying manifold is M := T 3 × (0, ∞). ...
In a recent article, we propose a general geometric notion of initial data on big bang singularities. This notion is of interest in its own right. However, it also serves the purpose of giving a unified perspective on many of the results in the literature. In the present article, we give a partial justification of this statement by rephrasing the results concerning Bianchi class A orthogonal stiff solutions and solutions in the -Gowdy symmetric vacuum setting in terms of our general geometric notion of initial data on the big bang singularity.
... Strong Cosmic Censorship in vacuum under Gowdy-symmetry. Strong Cosmic Censorship was studied for a class of spacetimes with a compact spatial topology T 3 , S 3 or S 2 × S 1 admitting a two-dimensional isometry group U(1) × U(1) with spacelike orbits, under an additional two-surface orthogonality condition (Gowdy-symmetry, see [164,165,166]). The Gowdy polarized subcase is defined as the instance in which the two independent Killing vector fields generated by the isometry group are orthogonal. In view of Theorem 7.1, one may expect these spacetimes to feature a spacelike singularity at {t = 0}. ...
In the wake of major breakthroughs in General Relativity during the 1960s, Roger Penrose introduced Strong Cosmic Censorship, a profound conjecture regarding the deterministic nature of the theory. Penrose's proposal has since opened far-reaching new mathematical avenues, revealing connections to fundamental questions about black holes and the nature of gravitational singularities. We review recent advances arising from modern techniques in the theory of partial differential equations as applied to Strong Cosmic Censorship, maintaining a focus on the context of gravitational collapse that gave birth to the conjecture.
... This equation is similar to one which arises in a similar context for Gowdy spacetimes [5] and can be treated in exactly the same way. In fact if l is an arc length parameter which is zero at x 0 then the solution of (2.20) is ...
It is shown that the initial singularities in spatially compact spacetimes with spherical, plane or hyperbolic symmetry admitting a compact constant mean curvature hypersurface are crushing singularities when the matter content of spacetime is described by the Vlasov equation (collisionless matter) or the wave equation (massless scalar field). In the spherically symmetric case it is further shown that if the spacetime admits a maximal slice then there are crushing singularities both in the past and in the future. The essential properties of the matter models chosen are that their energy-momentum tensors satisfy certain inequalities and that they do not develop singularities in a given regular background spacetime.
... The first rigorous work in [3] on this topic has recently been extended in [15]. The numerical studies in [4] of Gowdy-symmetric [14,25] (see Section 2.1 for more details on Gowdy symmetry) inhomogeneous fully nonlinear perturbations of the Nariai solution have revealed evidence that the analogous critical phenomenon also exists in much larger classes of spacetimes. In particular, it was found that all solutions, which are obtained from initial data not too far away from the Nariai solutions, always either globally collapse or expand in the same manner as in the spatially homogeneous case -with the exception of critical solutions which are exactly at the borderline between these two cases. ...
In this paper, we construct and study solutions of Einstein's equations in vacuum with a positive cosmological constant which can be considered as inhomogeneous generalizations of the Nariai cosmological model. Similar to this Nariai spacetime, our solutions are at the borderline between gravitational collapse and de-Sitter-like exponential expansion. Our studies focus in particular on the intriguing oscillatory dynamics which we discover. Our investigations are carried out both analytically (using heuristic mode analysis arguments) and numerically (using the numerical infrastructure recently introduced by us).
... In the present paper, we are interested in T 2 -symmetric solutions to Einstein's equations. There are various geometric ways of imposing this type of symmetry (cf., e.g., [7,33]), but for the purposes of the present paper, we simply assume the topology to be of the form I × T 3 , where I is an open interval contained in (0, ∞). If θ, x and y are 'coordinates' on T 3 and t is the coordinate on I, we also assume the metric to be of the form ...
The currently preferred models of the universe undergo accelerated expansion induced by dark energy. One model for dark energy is a positive cosmological constant. It is consequently of interest to study Einstein's equations with a positive cosmological constant coupled to matter satisfying the ordinary energy conditions; the dominant energy condition etc. Due to the difficulty of analysing the behaviour of solutions to Einstein's equations in general, it is common to either study situations with symmetry, or to prove stability results. In the present paper, we do both. In fact, we analyse, in detail, the future asymptotic behaviour of T^3-Gowdy symmetric solutions to the Einstein-Vlasov equations with a positive cosmological constant. In particular, we prove the cosmic no-hair conjecture in this setting. However, we also prove that the solutions are future stable (in the class of all solutions). Some of the results hold in a more general setting. In fact, we obtain conclusions concerning the causal structure of T^2-symmetric solutions, assuming only the presence of a positive cosmological constant, matter satisfying various energy conditions and future global existence. Adding the assumption of T^3-Gowdy symmetry to this list of requirements, we obtain C^0-estimates for all but one of the metric components. There is consequently reason to expect that many of the results presented in this paper can be generalised to other types of matter.
... Here H is the trace of the second fundamental form k ab (i.e. the mean curvature), K is the Gaussian curvature of F ,κ AB andλ AB are the trace-free parts of κ AB and λ AB respectively and J is the contraction of the unit normal vector to F in S with j a . This way of writing the constraints generalizes an approach used by Malec andÓ Murchadha [15], for spherically symmetric asymptotically flat spacetimes, by the author [17] for spatially compact surface symmetric spacetimes and by Chruściel [5] for vacuum spacetimes with U (1) × U (1) symmetry. In the following these equations are only used in the case of local U (1)×U (1) symmetry. ...
It is shown that in a class of maximal globally hyperbolic spacetimes admitting two local Killing vectors, the past (defined with respect to an appropriate time orientation) of any compact constant mean curvature hypersurface can be covered by a foliation of compact constant mean curvature hypersurfaces. Moreover, the mean curvature of the leaves of this foliation takes on arbitrarily negative values and so the initial singularity in these spacetimes is a crushing singularity. The simplest examples occur when the spatial topology is that of a torus, with the standard global Killing vectors, but more exotic topologies are also covered. In the course of the proof it is shown that in this class of spacetimes a kind of positive mass theorem holds. The symmetry singles out a compact surface passing through any given point of spacetime and the Hawking mass of any such surface is non-negative. If the Hawking mass of any one of these surfaces is zero then the entire spacetime is flat.
... There are few cases in which this has been done successfully. Notable examples are Gowdy spacetimes [58,97,61] and solutions of the Einstein-Vlasov system with spherical and plane symmetry [130]. Progress in constructing spacetimes with prescribed singularities will be described in section 6. ...
This article is a guide to the literature on existence theorems for the Einstein equations which also draws attention to open problems in the field. The local in time Cauchy problem, which is relatively well understood, is treated first. Next global results for solutions with symmetry are discussed. A selection of results from Newtonian theory and special relativity which offer useful comparisons is presented. This is followed by a survey of global results in the case of small data and results on constructing spacetimes with given singularity structure. The article ends with some miscellaneous topics connected with the main theme.
... There are few cases in which this has been done successfully. Notable examples are Gowdy spacetimes [84,139,87] and solutions of the Einstein-Vlasov system with spherical and plane symmetry [189]. Progress in constructing spacetimes with prescribed singularities will be described in section 6. ...
This article is a guide to theorems on existence and global dynamics of solutions of the Einstein equations. It draws attention to open questions in the field. The local in time Cauchy problem, which is relatively well understood, is surveyed. Global results for solutions with various types of symmetry are discussed. A selection of results from Newtonian theory and special relativity which offer useful comparisons is presented. Treatments of global results in the case of small data and results on constructing spacetimes with prescribed singularity structure are given. A conjectural picture of the asymptotic behaviour of general cosmological solutions of the Einstein equations is built up. Some miscellaneous topics connected with the main theme are collected in a separate section.
... See [Wald84] Theorem 7.1.1. and [Chr90]. ...
Variables for constraint free null canonical vacuum general relativity are presented which have simple Poisson brackets that facilitate quantization. Free initial data for vacuum general relativity on a pair of intersecting null hypersurfaces has been known since the 1960s. These consist of the "main" data which are set on the bulk of the two null hypersurfaces, and additional "surface" data set only on their intersection 2-surface. More recently the complete set of Poisson brackets of such data has been obtained. However the complexity of these brackets is an obstacle to their quantization. Part of this difficulty may be overcome using methods from the treatment of cylindrically symmetric gravity. Specializing from general to cylindrically symmetric solutions changes the Poisson algebra of the null initial data surprisingly little, but cylindrically symmetric vacuum general relativity is an integrable system, making powerful tools available. Here a transformation is constructed at the cylindrically symmetric level which maps the main initial data to new data forming a Poisson algebra for which an exact deformation quantization is known. (Although an auxiliary condition on the data has been quantized only in the asymptotically flat case, and a suitable representation of the algebra of quantum data by operators on a Hilbert space has not yet been found.) The definition of the new main data generalizes naturally to arbitrary, symmetryless gravitational fields, with the Poisson brackets retaining their simplicity. The corresponding generalization of the quantization is however ambiguous and requires further analysis.
... In D = 5 the only possible closed oriented 3-manifolds which admit an effective U(1) 2 action are: T 3 , S 1 × S 2 and S 3 (as well as the Lens spaces L(p, q) which occur as it's quotients by discrete isometry subgroups), see e.g. [49] and references therein. Note that only in the T 3 case is the action free -in the other cases there are fixed points. ...
We consider the near-horizon geometries of extremal, rotating black hole solutions of the vacuum Einstein equations, including a negative cosmological constant, in four and five dimensions. We assume the existence of one rotational symmetry in 4d, two commuting rotational symmetries in 5d and in both cases non-toroidal horizon topology. In 4d we determine the most general near-horizon geometry of such a black hole, and prove it is the same as the near-horizon limit of the extremal Kerr-AdS(4) black hole. In 5d, without a cosmological constant, we determine all possible near-horizon geometries of such black holes. We prove that the only possibilities are one family with a topologically S^1 X S^2 horizon and two distinct families with topologically S^3 horizons. The S^1 X S^2 family contains the near-horizon limit of the boosted extremal Kerr string and the extremal vacuum black ring. The first topologically spherical case is identical to the near-horizon limit of two different black hole solutions: the extremal Myers-Perry black hole and the slowly rotating extremal Kaluza-Klein (KK) black hole. The second topologically spherical case contains the near-horizon limit of the fast rotating extremal KK black hole. Finally, in 5d with a negative cosmological constant, we reduce the problem to solving a sixth-order non-linear ODE of one function. This allows us to recover the near-horizon limit of the known, topologically S^3, extremal rotating AdS(5) black hole. Further, we construct an approximate solution corresponding to the near-horizon geometry of a small, extremal AdS(5) black ring.
... A simple characterization of T 3 Gowdy symmetry 1 is a spacetime 1 A more geometric characterization is a spacetime with T 3 spatial topology and containing two spacetlike Killing vector fields, here given by ∂ ∂σ and ∂ ∂δ , and whose associated twist constants vanish. See the original paper of Gowdy [13], or [7,28]. ...
We study the phenomenon of bounces, as predicted by Belinski, Khalatnikov and Lifshitz (BKL), as an instability mechanism within the setting of the Einstein vacuum equations in Gowdy symmetry. In particular, for a wide class of inhomogeneous initial data we prove that the dynamics near the singularity are well-described by ODEs reminiscent of Kasner bounces. Unlike previous works regarding bounces, our spacetimes are not necessarily spatially homogeneous, and a crucial step is proving so-called asymptotically velocity term dominated (AVTD) behaviour, even in the presence of nonlinear BKL bounces and other phenomena such as spikes. (A similar phenomenon involving bounces and AVTD behaviour, though not spikes, can also be seen in our companion paper "BKL bounces outside homogeneity: Einstein-Maxwell-scalar field in surface symmetry", albeit in the context of the Einstein-Maxwell-scalar field model in surface symmetry.) One particular application is the study of (past) instability of certain polarized Gowdy spacetimes, including some Kasner spacetimes. Perturbations of such spacetimes are such that the singularity persists, but the intermediate dynamics -- between initial data and the singularity -- feature BKL bounces.
... Looking for an appropriate gauge field such that the gauge covariant derivative is / D A → / D + iq / A with the 1 It is noteworthy that the phenomenology of the gauge group U (1)⊗U (1) also manifests in other contexts beyond brane physics [54][55][56][57][58][59][60][61][62][63]. ...
A model of baryogenesis is introduced where our usual visible Universe is a 3-brane coevolving with a hidden 3-brane in a multidimensional bulk. The visible matter and antimatter sectors are naturally coupled with the hidden matter and antimatter sectors, breaking the C/CP invariance and leading to baryogenesis occurring after the quark-gluon era. The issue of leptogenesis is also discussed. The symmetry breaking spontaneously occurs due to the presence of an extra scalar field supported by the U ( 1 ) ⊗ U ( 1 ) gauge group, which extends the conventional electromagnetic gauge field in the two-brane universe. Observational consequences are discussed.
Published by the American Physical Society 2024
... It is noteworthy that the phenomenology of the gauge group U (1)⊗U (1) also manifests in other contexts beyond brane physics[26][27][28][29][30][31][32][33][34][35]. ...
For the past decade, there has been significant interest in the experimental search for neutron-hidden neutron transitions as predicted by various theoretical models, such as braneworld scenarios where the dark sector resides on a hidden brane. In a recent study, it was demonstrated that a C/CP asymmetry between and transitions can explain baryogenesis. However, the origins of this asymmetry and its required magnitude were only suggested. In this paper, we demonstrate that both aspects naturally occur due to the presence of an extra scalar field supported by the gauge group, which extends the conventional electromagnetic gauge field in the two-brane universe.
... It is worth remarking how non-uniqueness theorems become quite easier to prove if one allows for matter sources, as was indeed done in the earlier contributions Baumgarte et al. (2007) and Walsh (2007) by a clever design of non-scaling sources; see also Pfeiffer and York (2005) for non-uniqueness results for the extended conformal thin-sandwich method. We also stress that this study should be regarded as a test for the conformal method in its full generality rather than an attempt to smartly parametrising data on symmetric background manifolds, like Uð1Þ Â Uð1Þ symmetric initial data sets on the 3-torus, for indeed that case had already been handled, through a different and better adapted methodology, in Chruściel (1990) (among other things, the author gives Ernst and Gowdy-like parameterisations for the full class of Uð1Þ Â Uð1Þ symmetric spacetimes with compact Cauchy hypersurfaces). ...
We present the state-of-the-art concerning the relativistic constraints, which describe the geometry of hypersurfaces in a spacetime subject to the Einstein field equations. We review a variety of solvability results, the construction of several classes of solutions of special relevance and place results in the broader context of mathematical general relativity. Apart from providing an overview of the subject, this paper includes a selection of open questions, as well as a few complements to some significant contributions in the literature.
... We assume a Lorentzian manifold with topology M = I × T 3 for some interval I ⊂ (0, ∞). The class of T 2 -symmetric spacetimes are characterized by a T 2 isometry group acting on T 3 effectively [13]. The time-dependent areas of the symmetry orbits provide a useful time coordinate, which along with a coordinatization (x, y, θ) of T 3 are known as the areal coordinates. ...
We prove stability of the family of Kasner solutions within the class of polarized \Tbb^2-symmetric solutions of the vacuum Einstein equations in the contracting time direction with respect to an areal time foliation. All Kasner solutions for which the asymptotic velocity parameter K satisfies are non-linearly stable, and all sufficiently small perturbations exhibit asymptotically velocity term dominated (AVTD) behavior.
... It is not as stringent a condition as it might appear, since it is actually enforced by the vacuum field equations provided only two numbers, the so called twist constants, vanish. See [Wald84], theorem 7.1.1 and [Chr90]. ...
The Geroch group is an infinite dimensional transitive group of symmetries of classical cylindrically symmetric gravitational waves which acts by non-canonical transformations on the phase space of these waves. Here this symmetry is re derived and the unique Poisson bracket on the Geroch group which makes its action on the gravitational phase space Lie–Poisson is obtained. Two possible notions of asymptotic flatness are proposed that are compatible with the Poisson bracket on the phase space, and corresponding asymptotic flatness preserving subgroups of the Geroch group are defined which turn out to be compatible with the Poisson bracket on the group. A quantization of the Geroch group is proposed that is similar to, but distinct from, the Yangian, and a certain action of this quantum Geroch group on gravitational observables is shown to preserve the commutation relations of Korotkin and Samtleben’s quantization of asymptotically flat cylindrically symmetric gravitational waves. The action also preserves three of the additional conditions that define their quantization. It is conjectured that the action preserves the remaining two conditions (asymptotic flatness and a unit determinant condition on a certain basic field) as well and is, in fact, a symmetry of their model. Our results on the quantum theory are formal, but a fairly detailed outline of how these results might be formulated rigorously within the framework of algebraic quantum theory is provided.
... See [Wald84] Theorem 7.1.1. and [Chr90]. ...
The Geroch group is an infinite dimensional transitive group of symmetries of cylindrically symmetric gravitational waves which acts by non-canonical transformations on the phase space of these waves. The unique Poisson bracket on the Geroch group which makes this action Lie-Poisson is obtained. A quantization of the Geroch group is proposed, at a formal level, that is very similar to an Yangian, and a certain action of this quantum Geroch group on gravitational observables is shown to preserve the commutation relations of Korotkin and Samtleben's quantization of cylindrically symmetric gravitational waves. The action also preserves three of the four additional conditions that define their quantization. It is conjectured that the action preserves the remaining condition as well and is, in fact, a symmetry of their model.
... conjecture [28]-mainly by focusing on gravitational dust collapse processes [29][30][31][32][33][34][35][36][37][38]. None came with conclusive definitive proof of whether naked singularities could or could not physically exist and a question is raised if it is necessary to change the methodology which was used to deal with the cosmic censorship conjecture [39]. ...
Motivated by the endeavors of Li Xiang and You-Gen Shen on naked singularities, we investigate the validity of the cosmic censorship conjecture in the context of the generalized uncertainty principle. In particular, upon considering both linear and quadratic terms of momentum in the uncertainty principle, we first compute the entropy of a massless charged black hole in de Sitter spacetime at a given modified temperature. Then, we compute the corresponding modified cosmological radius and express the black hole electric charge in terms of this modified cosmological radius and, thus, in terms of the generalized uncertainty principle parameter. Finally, we examine whether such a system will end up being a naked singularity or might be protected by the cosmic censorship conjecture and how that might be related to the possible existence of massless charged particles.
... Let us discuss first vacuum solutions [5,10]. We are interested in spacetimes (M, g) with T 2 symmetry on T 3 , that is, (3+1)-dimensional Lorentzian manifolds with topology I × T 3 (where I is an interval) admitting the Lie group T 2 as an isometry group acting on the spatial leaves T 3 . ...
We are interested in the evolution of a compressible fluid under its self-generated gravitational field. Assuming here Gowdy symmetry, we investigate the algebraic structure of the Euler equations satisfied by the mass density and velocity field. We exhibit several interaction functionals that provide us with a uniform control on weak solutions in suitable Sobolev norms or in bounded variation. These functionals allow us to study the local regularity and nonlinear stability properties of weakly regular fluid flows governed by the Euler–Gowdy system. In particular, for the Gowdy equations, we prove that a spurious matter field arises under weak convergence, and we establish the nonlinear stability of weak solutions.
... Once local existence is ensured (due to an appropriate choice of data), global existence in the entire Gowdy square -with possible exception of the future boundary t = π (y = −1) -follows immediately. This was shown by Chruściel for the vacuum case [7], and the argument carries over to electrovacuum [8]. However, it is not immediately clear how the solution behaves at t = π, which could be the location of a curvature singularity, a future Cauchy horizon, or something else. ...
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 refer to [1,7], or [8] for details on the notion of Gowdy symmetry. There are several choices of spacetime manifolds compatible with Gowdy symmetry. ...
We prove in the case of cosmological models for the Einstein-Vlasov-scalar field system with Gowdy symmetry, that the solutions exist globally in the past. The sources of the equations are generated by a distribution function and a scalar field, subject to the Vlasov and the wave equations respectively. The result is generalized for the case of T ² symmetry. Using previous results, we deduce geodesic completeness. © 2018, Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Nature Singapore Pte Ltd.
... See [Wald84] theorem 7.1.1. and [Chr90]. ...
Variables for constraint free null canonical vacuum general relativity are presented which have simple Poisson brackets that facilitate quantization. Free initial data for vacuum general relativity on a pair of intersecting null hypersurfaces has been known since the 1960s. These consist of the "main" data which are set on the bulk of the two null hypersurfaces, and additional "surface" data set only on their intersection 2-surface. More recently the complete set of Poisson brackets of such data has been obtained. However the complexity of these brackets is an obstacle to their quantization. Part of this difficulty may be overcome using methods from the treatment of cylindrically symmetric gravity. Specializing from general to cylindrically symmetric solutions changes the Poisson algebra of the null initial data surprisingly little, but cylindrically symmetric vacuum general relativity is an integrable system, making powerful tools available. Here a transformation is constructed at the cylindrically symmetric level which maps the main initial data to new data forming a Poisson algebra for which an exact deformation quantization is known. (Although an auxiliary condition on the data has been quantized only in the asymptotically flat case, and a suitable representation of the algebra of quantum data by operators on a Hilbert space has not yet been found.) The definition of the new main data generalizes naturally to arbitrary, symmetryless gravitational fields, with the Poisson brackets retaining their simplicity. The corresponding generalization of the quantization is however ambiguous and requires further analysis.
... There are few cases in which this has been done successfully. Notable examples are Gowdy spacetimes [32,50,35] and solutions of the Einstein-Vlasov system with spherical and plane symmetry [63]. ...
This article is a guide to the literature on existence theorems for the Einstein equations which also draws attention to open problems in the field. The local in time Cauchy problem, which is relatively well understood, is treated first. Next global results for solutions with symmetry are discussed. This is followed by a presentation of global results in the case of small data, and some miscellaneous topics connected with the main theme.
... The first rigorous work in [3] on this topic has recently been extended in [15]. The numerical studies in [4] of Gowdy-symmetric [14,25] (see Section 2.1 for more details on Gowdy symmetry) inhomogeneous fully nonlinear perturbations of the Nariai solution have revealed evidence that the analogous critical phenomenon also exists in much larger classes of spacetimes. In particular, it was found that all solutions, which are obtained from initial data not too far away from the Nariai solutions, always either globally collapse or expand in the same manner as in the spatially homogeneous case -with the exception of critical solutions which are exactly at the borderline between these two cases. ...
In this paper, we construct and study solutions of Einstein's equations in vacuum with a positive cosmological constant which can be considered as inhomogeneous generalizations of the Nariai cosmological model. Similar to this Nariai spacetime, our solutions are at the borderline between gravitational collapse and de-Sitter-like exponential expansion. Our studies focus in particular on the intriguing oscillatory dynamics which we discover. Our investigations are carried out both analytically (using heuristic mode analysis arguments) and numerically (using the numerical infrastructure recently introduced by us).
... There are few cases in which this has been done successfully. Notable examples are Gowdy spacetimes [84,139,87] and solutions of the Einstein-Vlasov system with spherical and plane symmetry [189]. Progress in constructing spacetimes with prescribed singularities will be described in section 6. ...
This article is a guide to theorems on existence and global dynamics of solutions of the Einstein equations. It draws attention to open questions in the field. The local in time Cauchy problem, which is relatively well understood, is surveyed. Global results for solutions with various types of symmetry are discussed. A selection of results from Newtonian theory and special relativity which offer useful comparisons is presented. Treatments of global results in the case of small data and results on constructing spacetimes with prescribed singularity structure are given. A conjectural picture of the asymptotic behaviour of general cosmological solutions of the Einstein equations is built up. Some miscellaneous topics connected with the main theme are collected in a separate section.
We study global existence problems and asymptotic behavior of higher-dimensional inhomogeneous spacetimes with a compact Cauchy surface in the Einstein-Maxwell-dilaton (EMD) system. Spacelike -symmetry is assumed, where is spacetime dimension. The system of the evolution equations of the EMD equations in the areal time coordinate is reduced to a wave map system, and a global existence theorem for the system is shown. As a corollary of this theorem, a global existence theorem in the constant mean curvature time coordinate is obtained. Finally, for vacuum Einstein gravity in arbitrary dimension, we show existence theorems of asymptotically velocity-terms dominated singularities in the both cases which free functions are analytic and smooth.
"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 Cauchy horizons and analyse the causal structure of the solutions. Finally, we discuss the generalization from vacuum to electrovacuum and present an exact family of solutions for that case.
We construct a solution satisfying initial conditions for accelerating cosmologies from string/M-theory. Gowdy symmetric spacetimes with a positive potential are considered. Also, a global existence theorem for the spacetimes is shown.
Using numerical methods, we examine, under a Gowdy symmetry assumption, the dynamics of nonlinearly perturbed Friedmann-Lemaître-Robertson-Walker (FLRW) fluid solutions of the Einstein-Euler-scalar field equations in the contracting direction for linear equations of state p=Kρ and sound speeds 0≤K<1/3. This article builds upon the numerical work from Beyer et al. [Phys. Rev. D 107, 104030 (2023)] in which perturbations of FLRW solutions to the Einstein-Euler equations with positive cosmological constant in the expanding time direction were studied. The numerical results presented here confirm that the instabilities observed in previous work [Phys. Rev. D 107, 104030 (2023), Lett. Math. Phys. 113, 102 (2023).] for 1/3<K<1, first conjectured to occur in the expanding direction by Rendall [Ann. Henri Poincaré 5, 1041 (2004)], are also present in the contracting direction over the complementary parameter range 0≤K<1/3. Our numerical solutions show that the fractional density gradient of the nonlinear perturbations develop steep gradients near a finite number of spatial points and become unbounded toward the big bang. This behavior, and in particular the characteristic profile of the fractional density gradient near the big bang, is strikingly similar to what was observed in the expanding direction near timelike infinity by Beyer et al. [Phys. Rev. D 107, 104030 (2023)].
Fitting the thermal continuum emission of accreting black holes observed across X-ray bands represents one of the principle means of constraining the properties (mass and spin) of astrophysical black holes. Recent “continuum fitting” studies of Galactic X-ray binaries in the soft state have found best fitting dimensionless spin values which run into the prior bounds placed on traditional models (a⋆ = 0.9999). It is of critical importance that these results are robust, and not a result solely of the presence of these prior bounds and deficiencies in conventional models of accretion. Motivated by these results we derive and present superkerr, an XSPEC model comprising of a thin accretion disc solution valid in the Kerr geometry for arbitrary spin parameter a⋆, extending previous models valid only for black holes (|a⋆| < 1). This extension into “superextremal” spacetimes with |a⋆| > 1 includes solutions which describe discs evolving around naked singularities, not black holes. While being valid solutions of Einstein’s field equations these naked singularities are not expected to be present in nature. We discuss how the “measurement” of a Kerr spin parameter 1 < a⋆ < 5/3 would present compelling evidence for the requirement of a rethink in either standard accretion theory, or our theories of gravity.
Using numerical methods, we examine the dynamics of nonlinear perturbations in the expanding time direction, under a Gowdy symmetry assumption, of Friedmann-Lemaître-Robertson-Walker (FLRW) fluid solutions to the Einstein-Euler equations with a positive cosmological constant Λ>0 and a linear equation of state p=Kρ for the parameter values 1/3<K<1. This paper builds upon the numerical work in [arXiv:2209.06982] in which the simpler case of a fluid on a fixed FLRW background spacetime was studied. The numerical results presented here confirm that the instabilities observed in [arXiv:2209.06982] are also present when coupling to gravity is included as was previously conjectured in [A. D. Rendall, Asymptotics of solutions of the Einstein equations with positive cosmological constant, Ann. Henri Poincaré 5, 1041 (2004); J. Speck, The stabilizing effect of spacetime expansion on relativistic fluids with sharp results for the radiation equation of state, Arch. Ration. Mech. Anal. 210, 535 (2013)]. In particular, for the full parameter range 1/3<K<1, we find that the fractional density gradient of the nonlinear perturbations develop steep gradients near a finite number of spatial points and becomes unbounded there at future timelike infinity.
We prove stability of the family of Kasner solutions within the class of polarized T2-symmetric solutions of the vacuum Einstein equations in the contracting time direction with respect to an areal time foliation. All Kasner solutions for which the asymptotic velocity parameter K satisfies |K-1|>2 are non-linearly stable, and all sufficiently small perturbations exhibit asymptotically velocity term dominated (AVTD) behavior and blow-up of the Kretschmann scalar.
We present a number of open problems within general relativity. After a brief introduction to some technical mathematical issues and the famous singularity theorems, we discuss the cosmic censorship hypothesis and the Penrose inequality, the uniqueness of black hole solutions and the stability of Kerr spacetime and the final state conjecture, critical phenomena and the Einstein–Yang–Mills equations, and a number of other problems in classical general relativity. We then broaden the scope and discuss some mathematical problems motivated by quantum gravity, including AdS/CFT correspondence and problems in higher dimensions and, in particular, the instability of anti-de Sitter spacetime, and in cosmology, including the cosmological constant problem and dark energy, the stability of de Sitter spacetime and cosmological singularities and spikes. Finally, we briefly discuss some problems in numerical relativity and relativistic astrophysics.
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 consider self-gravitating fluids in cosmological spacetimes with Gowdy symmetry on the torus T³ and, in this class, we solve the singular initial value problem for the Einstein–Euler system of general relativity, when an initial data set is prescribed on the hypersurface of singularity. We specify initial conditions for the geometric and matter variables and identify the asymptotic behavior of these variables near the cosmological singularity. Our analysis of this class of nonlinear and singular partial differential equations exhibits a condition on the sound speed, which leads us to the notion of sub-critical, critical, and super-critical regimes. Solutions to the Einstein–Euler systems when the fluid is governed by a linear equation of state are constructed in the first two regimes, while additional difficulties arise in the latter one. All previous studies on inhomogeneous spacetimes concerned vacuum cosmological spacetimes only.
We present a list of open questions in mathematical physics. After a historical introduction, a number of problems in a variety of different fields are discussed, with the intention of giving an overall impression of the current status of mathematical physics, particularly in the topical fields of classical general relativity, cosmology and the quantum realm. This list is motivated by the recent article proposing 42 fundamental questions (in physics) which must be answered on the road to full enlightenment (Allen and Lidstrom 2017 Phys. Scr. 92 012501). But paraphrasing a famous quote by the British football manager Bill Shankly, in response to the question of whether mathematics can answer the Ultimate Question of Life, the Universe, and Everything, mathematics is, of course, much more important than that.
We use qualitative arguments combined with numerical simulations to argue that, in the approach to the singularity in a vacuum solution of Einstein's equations with T2 isometry, the evolution at a generic point in space is an endless succession of Kasner epochs, punctuated by bounces in which either a curvature term or a twist term becomes important in the evolution equations for a brief time. Both curvature bounces and twist bounces may be understood within the context of local mixmaster dynamics although the latter have never been seen before in spatially inhomogeneous cosmological spacetimes.
Existence, uniqueness and well-posedness for a general class of quasi-linear
evolution equations on a short time interval are established. These results,
generalizing those of [29], are applied to second-order quasi-linear hyperbolic
systems on IR" whose solutions (u(t),il(t)) lie in the Sobolev space H^(s+1) x H^5.
Our results improve existing theorems by lowering the required value of s to
s > (n/2) + 1, or s > n/2 in case the coefficients of the highest order terms do not
involve derivatives of the unknown, and by establishing continuous dependence
on the initial data for these values. As consequences we obtain well-posedness of the
equations of elastodynamics if s>2.5 and of general relativity if s>l.5; s>3
was the best known previous value for systems of the type occuring in general
relativity ([12], [16], [23]).
A scheme is introduced which yields, beginning with any source‐free solution of Einstein's equation with two commuting Killing fields for which a certain pair of constants vanish (e.g., the exterior field of a rotating star), a family of new exact solutions. To obtain a new solution, one must specify an arbitrary curve (up to parametrization) in a certain three‐dimensional vector space. Thus, a single solution of Einstein's equationgenerates a family of new solutions involving two arbitrary functions of one variable. These transformations on exact solutions form a non‐Abelian group. The extensive algebraic structure thereby induced on such solutions is studied.
A method is described for constructing, from any source‐free solution of Einstein's equations which possesses a Killing vector, a one‐parameter family of new solutions. The group properties of this transformation are discussed. A new formalism is given for treating space‐times having a Killing vector.
We show that for a wide class of field equations the orbits of the isometry group defining axial symmetry and stationarity admit orthogonal 2-surfaces. The field equations covered by this result include those of a perfect fluid.
We consider vacuum space-times with spacelike U(1) isometry groups which are defined on manifolds of the form R × Bn, where Bn is an arbitrary S1-bundle over the two-sphere. We reduce the Einstein equations for this problem to a system of “harmonic map” equations defined over the base manifold R × S2 equipped with a Lorentzian metric determined uniquely by the solution of an associated nonlinear elliptic system. The harmonic map fields (which have a range space diffeomorphic to the hyperbolic two-plane) represent the unconstrained, dynamical degrees of freedom of the gravitational field. We give a complete discussion of the existence and uniqueness of solutions of the associated elliptic system and also display a Geroch-type solution generating technique for globally transforming the space of solutions associated with any one non-trivial bundle, R × Bn → R × S2, to that of any other such bundle. The basic techniques could readily be generalized to treat S1-bundles over R × M where M is a compact two-manifold of arbitrary genus. In the higher genus cases the Teichmüller space of conformally diffeomorphic Riemannian metrics over M arises as an additional component of the configuration manifold. For M ≈ S2 this space collapses to a point, which slightly simplifies the analysis.
Spacetimes with closed spacelike hypersurfaces and spacelike two-parameter isometry groups are investigated to determine their possible global structures. It is shown that the two spacelike Killing vectors always commute with each other. Connected group-invariant spacelike hypersurfaces must be homeomorphic to S1 ⊗ S1 ⊗ S1 (three-torus), S1 ⊗ S2 (three-handle), S3 (three-sphere), or to a manifold which is covered by one of these. The spacetime metric and Einstein equations are simplified in the absence of nongravitational sources and used to establish the impossibility of topology change as well as other limitations on global structure. Regularity conditions for spacetimes of this type are derived and shown to be compatible with Einstein's equations.
We study the global properties of the Gowdy metrics generated by Cauchy data on the 3-torus. We show that the boundaries of the maximal Cauchy developments of Gowdy initial data sets are always “crushing singularities” in the sense of Eardley and Smarr. This means that each solution admits a slicing in which tr K(t) (the trace of the second fundamental form induced on the surface Σt of the slicing) uniformly blows up as t approaches its limiting value. A theorem of Hawking shows that the maximal Cauchy development cannot extend beyond the boundary at which tr K blows up and our result shows that no singularities arise to halt the evolution until this boundary is reached. Thus each maximal Cauchy development is always as large as it can be, consistent with Hawking's theorem. We discuss the relevance of this result to the strong cosmic censorship conjecture and the question of when the crushing singularities are in fact curvature singularities.
It is shown that, given any set of initial data for Einstein's equations which satisfy the constraint conditions, there exists a development of that data which is maximal in the sense that it is an extension of every other development. These maximal developments form a well-defined class of solutions of Einstein's equations. Any solution of Einstein's equations which has a Cauchy surface may be embedded in exactly one such maximal development.
Proceedings of the Tulane Conference on Transformation Groups
- Neumann
General Relativity and Gravitation
- Choquet-Bruhat
Proceedings of the Relativity Conference in the Midwest
- Fischer