arXiv subjects

Lars Andersson

Publications and source records attributed to Lars Andersson.

105 records · Page 6Linked to original sources

arXiv2003

dS/CFT and spacetime topology

Motivated by recent proposals for a de Sitter version of the AdS/CFT correspondence, we give some topological restrictions on spacetimes of de Sitter type, i.e., spacetimes with $Λ>0$, which admit a regular past and/or future conformal boundary. For example we show that if $M^{n+1}$, $n \ge 2$, is a globally hyperbolic spacetime obeying suitable energy conditions, which is of de Sitter type, with a conformal boundary to both the past and future, then if one of these boundaries is compact, it must have finite fundamental group and its conformal class must contain a metric of positive scalar curvature. Our results are closely related to theorems of Witten and Yau hep-th/9910245 pertaining to the Euclidean formulation of the AdS/CFT correspondence.

hep-th↗

arXiv2003

Gowdy phenomenology in scale-invariant variables

The dynamics of Gowdy vacuum spacetimes is considered in terms of Hubble-normalized scale-invariant variables, using the timelike area temporal gauge. The resulting state space formulation provides for a simple mechanism for the formation of ``false'' and ``true spikes'' in the approach to the singularity, and a geometrical formulation for the local attractor.

gr-qc↗

arXiv2003

Constant mean curvature foliations of simplicial flat spacetimes

Benedetti and Guadagnini have conjectured that the marked lenght spectrum of the constant mean curvature foliation $M_τ$ in a 2+1 dimensional flat spacetime $V$ with compact hyperbolic Cauchy surfaces converges, in the direction of the singularity, to that of the marked measure spectrum of the R-tree dual to the measured foliation corresponding to the translational part of the holonomy of $V$. We prove that this is the case for $n+1$ dimensional, $n \geq 2$, {\em simplicial} flat spacetimes with compact hyperbolic Cauchy surface. A simplicial spacetime is obtained from the Lorentz cone over a hyperbolic manifold by deformations corresponding to a simple measured foliation.

math.DG↗

arXiv2003

Bel--Robinson energy and constant mean curvature foliations

An energy estimate is proved for the Bel--Robinson energy along a constant mean curvature foliation in a spatially compact vacuum spacetime, assuming an $L^{\infty}$ bound on the second fundamental form, and a bound on a spacetime version of Bel--Robinson energy.

math.DG↗

arXiv2003

Future complete vacuum spacetimes

In this paper we prove a global existence theorem, in the direction of cosmological expansion, for sufficiently small perturbations of a family of spatially compact variants of the $k=-1$ Friedmann--Robertson--Walker vacuum spacetime. We use a special gauge defined by constant mean curvature slicing and a spatial harmonic coordinate condition, and develop energy estimates through the use of the Bel-Robinson energy and its higher order generalizations. In addition to the smallness condition on the data, we need a topological constraint on the spatial manifold to exclude the possibility of a non--trivial moduli space of flat spacetime perturbations, since the latter could not be controlled by curvature--based energies such as those of Bel--Robinson type. Our results also demonstrate causal geodesic completeness of the perturbed spacetimes (in the expanding direction) and establish precise rates of decay towards the background solution which serves as an attractor asymptotically.

gr-qc↗

arXiv2002

Construction of hyperboloidal initial data

I describe the conformal method for constructing solutions of the hyperboloidal constraint equations as well as the conditions needed on the free data in order to have regularity up to boundary for the solutions to the constraint equations. A brief discussion of the Einstein evolution equations in the unphysical setting is given.

gr-qc↗

arXiv2002

Report on GR16, Session A3: Mathematical Studies of the Field Equations

In this report, which is an extended version of that appearing in the Proceedings of GR16, I will give a summary of the main topics covered in Session A.3. on mathematical relativity at GR16, Durban. The summary is mainly based on extended abstracts submitted by the speakers. I would like to thank all participants for their contributions and help with this summary.

gr-qc↗

arXiv2002

Elliptic-hyperbolic systems and the Einstein equations

The Einstein evolution equations are studied in a gauge given by a combination of the constant mean curvature and spatial harmonic coordinate conditions. This leads to a coupled quasilinear elliptic--hyperbolic system of evolution equations. We prove that the Cauchy problem is locally strongly well--posed and that a continuation principle holds. For initial data satisfying the Einstein constraint and gauge conditions, the solutions to the elliptic-hyperbolic system defined by the gauge fixed Einstein evolution equations are shown to give vacuum spacetimes.

gr-qc↗

arXiv2001

Constant mean curvature foliations of flat space--times

Let $V$ be a maximal globally hyperbolic flat $n+1$--dimensional space--time with compact Cauchy surface of hyperbolic type. We prove that $V$ is globally foliated by constant mean curvature hypersurfaces $M_τ$, with mean curvature $τ$ taking all values in $(-\infty, 0)$. For $n \geq 3$, define the rescaled volume of $M_τ$ by $\Ham = |τ|^n \Vol(M,g)$, where $g$ is the induced metric. Then $\Ham \geq n^n \Vol(M,g_0)$ where $g_0$ is the hyperbolic metric on $M$ with sectional curvature -1. Equality holds if and only if $(M,g)$ is isometric to $(M,g_0)$.

math.DG↗

arXiv2000

Quiescent cosmological singularities

I discuss recent work (gr-qc/0001047) on non-oscillatory singularities in four dimensional space-times with scalar field or stiff fluid matter, in the context of the BKL proposal.

gr-qc↗

arXiv2000

Quiescent cosmological singularities

The most detailed existing proposal for the structure of spacetime singularities originates in the work of Belinskii, Khalatnikov and Lifshitz. We show rigorously the correctness of this proposal in the case of analytic solutions of the Einstein equations coupled to a scalar field or stiff fluid. More specifically, we prove the existence of a family of spacetimes depending on the same number of free functions as the general solution which have the asymptotics suggested by the Belinskii-Khalatnikov-Lifshitz proposal near their singularities. In these spacetimes a neighbourhood of the singularity can be covered by a Gaussian coordinate system in which the singularity is simultaneous and the evolution at different spatial points decouples.

gr-qc↗

arXiv1998

Finite dimensional approximations to Wiener measure and path integral formulas on manifolds

Certain natural geometric approximation schemes are developed for Wiener measure on a compact Riemannian manifold. These approximations closely mimic the informal path integral formulas used in the physics literature for representing the heat semi-group on Riemannian manifolds. The path space is approximated by finite dimensional manifolds consisting of piecewise geodesic paths adapted to partitions $P$ of $[0,1]$. The finite dimensional manifolds of piecewise geodesics carry both an $H^{1}$ and a $L^{2}$ type Riemannian structures $G^i_P$. It is proved that as the mesh of the partition tends to $0$, $$ 1/Z_P^i e^{- 1/2 E(σ)} Vol_{G^i_P}(σ) \to ρ_i(σ)ν(σ) $$ where $E(σ)$ is the energy of the piecewise geodesic path $σ$, and for $i=0$ and $1$, $Z_P^i$ is a ``normalization'' constant, $Vol_{G^i_P}$ is the Riemannian volume form relative $G^i_P$, and $ν$ is Wiener measure on paths on $M$. Here $ρ_1 = 1$ and $$ ρ_0 (σ) = \exp( -1/6 \int_0^1 Scal(σ(s))ds ) $$ where $Scal$ is the scalar curvature of $M$. These results are also shown to imply the well know integration by parts formula for the Wiener measure.

math.DG↗

arXiv1997

Existence of Constant Mean Curvature Hypersurfaces in Asymptotically Flat Spacetimes

The problem of existence of spacelike hypersurfaces with constant mean curvature in asymptotically flat spacetimes is considered for a class of asymptotically Schwarzschild spacetimes satisfying an interior condition. Using a barrier construction, a proof is given of the existence of complete hypersurfaces with constant mean cuvature which intersect null infinity in a regular cut.

gr-qc↗

arXiv1996

On the global evolution problem in 2+1 gravity

Existence of global CMC foliations of constant curvature 3-dimensional maximal globally hyperbolic Lorentzian manifolds, containing a constant mean curvature hypersurface with $\genus(Σ) > 1$ is proved. Constant curvature 3-dimensional Lorentzian manifolds can be viewed as solutions to the 2+1 vacuum Einstein equations with a cosmological constant. The proof is based on the reduction of the corresponding Hamiltonian system in constant mean curvature gauge to a time dependent Hamiltonian system on the cotangent bundle of Teichmüller space. Estimates of the Dirichlet energy of the induced metric play an essential role in the proof.

gr-qc↗

arXiv1993

On ``hyperboloidal'' Cauchy data for vacuum Einstein equations and obstructions to smoothness of ``null infinity''

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 constraint equations, the evolution of which leads to space--times satisfying the Bondi--Sachs--Penrose smoothness conditions.

gr-qc↗