Asymptotic Behavior of the T3×RT^3 \times R Gowdy Spacetimes

We present new evidence in support of the Penrose's strong cosmic censorship conjecture in the class of Gowdy spacetimes with $T^3$ spatial topology. Solving Einstein's equations perturbatively to all orders we show that asymptotically close to the boundary of the maximal Cauchy development the dominant term in the expansion gives rise to curvature singularity for almost all initial data. The dominant term, which we call the ``geodesic loop solution'', is a solution of the Einstein's equations with all space derivatives dropped. We also describe the extent to which our perturbative results can be rigorously justified.Comment: 30 page

Blow-Up of Test Fields Near Cauchy Horizons

The behaviour of test fields near a compact Cauchy horizon is investigated. It is shown that solutions of nonlinear wave equations on Taub spacetime with generic initial data cannot be continued smoothly to both extensions of the spacetime through the Cauchy horizon. This is proved using an energy method. Similar results are obtained for the spacetimes of Moncrief containing a compact Cauchy horizon and for more general matter models.Comment: 10 pages, Plain TeX, MPA-AR-92-

Global constants in (2+1)--dimensional gravity

The extended conformal algebra (so)(2,3) of global, quantum, constants of motion in 2+1 dimensional gravity with topology R x T^2 and negative cosmological constant is reviewed. It is shown that the 10 global constants form a complete set by expressing them in terms of two commuting spinors and the Dirac gamma matrices. The spinor components are the globally constant holonomy parameters, and their respective spinor norms are their quantum commutators.Comment: 14 pages, to appear in Classical and Quantum Gravity, Spacetime Safari: Essays in Honor of Vincent Moncrief on the Classical Physics of Strong Gravitational Field

Dipole Perturbations of the Reissner-Nordstrom Solution: The Polar Case

The formalism developed by Chandrasekhar for the linear polar perturbations of the Reissner-Nordstrom solution is generalized to include the case of dipole (l=1) perturbations. Then, the perturbed metric coefficients and components of the Maxwell tensor are computed.Comment: 16 pages, LaTeX, no figures. Submitted for publication in Physical Review

The Mixmaster Spacetime, Geroch's Transformation and Constants of Motion

We show that for $U(1)$-symmetric spacetimes on $S^3 \times R$ a constant of motion associated with the well known Geroch transformation, a functional $K[h_{ij},\pi^{ij}]$, quadratic in gravitational momenta, is strictly positive in an open subset of the set of all $U(1)$-symmetric initial data, and therefore not weakly zero. The Mixmaster initial data appear to be on the boundary of that set. We calculate the constant of motion perturbatively for the Mixmaster spacetime and find it to be proportional to the minisuperspace Hamiltonian to the first order in the Misner anisotropy variables, i.e. weakly zero. Assuming that $K$ is exactly zero for the Mixmaster spacetime, we show that Geroch's transformation, when applied to the Mixmaster spacetime, gives a new \mbox{$U(1)$-symmetric} solution of the vacuum Einstein equations, globally defined on \mbox{$S^2 \times S^1 \times R$},which is non-homogeneous and presumably exhibits Mixmaster-like complicated dynamical behavior.Comment: 25 pages, preprint YCTP-20-93, Revte

Gravitational waves in general relativity: XIV. Bondi expansions and the ``polyhomogeneity'' of \Scri

The structure of polyhomogeneous space-times (i.e., space-times with metrics which admit an expansion in terms of $r^{-j}\log^i r$) constructed by a Bondi--Sachs type method is analysed. The occurrence of some log terms in an asymptotic expansion of the metric is related to the non--vanishing of the Weyl tensor at Scri. Various quantities of interest, including the Bondi mass loss formula, the peeling--off of the Riemann tensor and the Newman--Penrose constants of motion are re-examined in this context.Comment: LaTeX, 28pp, CMA-MR14-9

Static Black Hole Solutions without Rotational Symmetry

We construct static black hole solutions that have no rotational symmetry. These arise in theories, including the standard electroweak model, that include charged vector mesons with mass $m\ne 0$. In such theories, a magnetically charged Reissner-Nordstrom black hole with horizon radius less than a critical value of the order of $m^{-1}$ is classically unstable against the development of a nonzero vector meson field just outside the horizon, indicating the existence of static black hole solutions with vector meson hair. For the case of unit magnetic charge, spherically symmetric solutions of this type have previously been studied. For other values of the magnetic charge, general arguments show that any new solution with hair cannot be spherically symmetric. In this paper we develop and apply a perturbative scheme (which may have applicability in other contexts) for constructing such solutions in the case where the Reissner-Nordstrom solution is just barely unstable. For a few low values of the magnetic charge the black holes retain a rotational symmetry about a single axis, but this axial symmetry disappears for higher charges. While the vector meson fields vanish exponentially fast at distances greater than $O(m^{-1})$, the magnetic field and the metric have higher multipole components that decrease only as powers of the distance from the black hole.Comment: 42 pages, phyzzx. 4 figures (PostScript, 1.7 MB when uncompressed) available by email from the Authors on reques

On Further Generalization of the Rigidity Theorem for Spacetimes with a Stationary Event Horizon or a Compact Cauchy Horizon

A rigidity theorem that applies to smooth electrovac spacetimes which represent either (A) an asymptotically flat stationary black hole or (B) a cosmological spacetime with a compact Cauchy horizon ruled by closed null geodesics was given in a recent work \cite{frw}. Here we enlarge the framework of the corresponding investigations by allowing the presence of other type of matter fields. In the first part the matter fields are involved merely implicitly via the assumption that the dominant energy condition is satisfied. In the second part Einstein-Klein-Gordon (EKG), Einstein-[non-Abelian] Higgs (E[nA]H), Einstein-[Maxwell]-Yang-Mills-dilaton (E[M]YMd) and Einstein-Yang-Mills-Higgs (EYMH) systems are studied. The black hole event horizon or, respectively, the compact Cauchy horizon of the considered spacetimes is assumed to be a smooth non-degenerate null hypersurface. It is proven that there exists a Killing vector field in a one-sided neighborhood of the horizon in EKG, E[nA]H, E[M]YMd and EYMH spacetimes. This Killing vector field is normal to the horizon, moreover, the associated matter fields are also shown to be invariant with respect to it. The presented results provide generalizations of the rigidity theorems of Hawking (for case A) and of Moncrief and Isenberg (for case B) and, in turn, they strengthen the validity of both the black hole rigidity scenario and the strong cosmic censor conjecture of classical general relativity.Comment: 25 pages, LaTex, a shortened version, including a new proof for lemma 5.1, the additional case of Einstein-Yang-Mills-Higgs systems is also covered, to appear in Class. Quant. Gra

The Quantum Modular Group in (2+1)-Dimensional Gravity

The role of the modular group in the holonomy representation of (2+1)-dimensional quantum gravity is studied. This representation can be viewed as a "Heisenberg picture", and for simple topologies, the transformation to the ADM "Schr{\"o}dinger picture" may be found. For spacetimes with the spatial topology of a torus, this transformation and an explicit operator representation of the mapping class group are constructed. It is shown that the quantum modular group splits the holonomy representation Hilbert space into physically equivalent orthogonal ``fundamental regions'' that are interchanged by modular transformations.Comment: 23 pages, LaTeX, no figures; minor changes and clarifications in response to referee (basic argument and conclusions unaffected

Strong Cosmic Censorship and Causality Violation

We investigate the instability of the Cauchy horizon caused by causality violation in the compact vacuum universe with the topology $B\times {\bf S}^{1}\times {\bf R}$, which Moncrief and Isenberg considered. We show that if the occurrence of curvature singularities are restricted to the boundary of causality violating region, the whole segments of the boundary become curvature singularities. This implies that the strong cosmic censorship holds in the spatially compact vacuum space-time in the case of the causality violation. This also suggests that causality violation cannot occur for a compact universe.Comment: corrected version, 8 pages, one eps figure is include