Large data decay of Yang-Mills-Higgs fields on Minkowski and de Sitter spacetimes

We extend Eardley and Moncrief's $L^\infty$ estimates for the conformally invariant Yang-Mills-Higgs equations to the Einstein cylinder. Our method is to first work on Minkowski space and localise their estimates, and then carry them to the Einstein cylinder by a conformal transformation. By patching local estimates together, we deduce global $L^\infty$ estimates on the cylinder, and extend Choquet-Bruhat and Christodoulou's small data well-posedness result to large data. Finally, by employing another conformal transformation, we deduce exponential decay rates for Yang-Mills-Higgs fields on de Sitter space, and inverse polynomial decay rates on Minkowski space.Comment: 20 page

Multipole structure of current vectors in curved spacetime

A method is presented which allows the exact construction of conserved (i.e. divergence-free) current vectors from appropriate sets of multipole moments. Physically, such objects may be taken to represent the flux of particles or electric charge inside some classical extended body. Several applications are discussed. In particular, it is shown how to easily write down the class of all smooth and spatially-bounded currents with a given total charge. This implicitly provides restrictions on the moments arising from the smoothness of physically reasonable vector fields. We also show that requiring all of the moments to be constant in an appropriate sense is often impossible; likely limiting the applicability of the Ehlers-Rudolph-Dixon notion of quasirigid motion. A simple condition is also derived that allows currents to exist in two different spacetimes with identical sets of multipole moments (in a natural sense).Comment: 13 pages, minor changes, accepted to J. Math. Phy

Bose-Einstein condensate and Spontaneous Breaking of Conformal Symmetry on Killing Horizons

Local scalar QFT (in Weyl algebraic approach) is constructed on degenerate semi-Riemannian manifolds corresponding to Killing horizons in spacetime. Covariance properties of the $C^*$-algebra of observables with respect to the conformal group PSL(2,\bR) are studied.It is shown that, in addition to the state studied by Guido, Longo, Roberts and Verch for bifurcated Killing horizons, which is conformally invariant and KMS at Hawking temperature with respect to the Killing flow and defines a conformal net of von Neumann algebras, there is a further wide class of algebraic (coherent) states representing spontaneous breaking of PSL(2,\bR) symmetry. This class is labeled by functions in a suitable Hilbert space and their GNS representations enjoy remarkable properties. The states are non equivalent extremal KMS states at Hawking temperature with respect to the residual one-parameter subgroup of PSL(2,\bR) associated with the Killing flow. The KMS property is valid for the two local sub algebras of observables uniquely determined by covariance and invariance under the residual symmetry unitarily represented. These algebras rely on the physical region of the manifold corresponding to a Killing horizon cleaned up by removing the unphysical points at infinity (necessary to describe the whole PSL(2,\bR) action).Each of the found states can be interpreted as a different thermodynamic phase, containing Bose-Einstein condensate,for the considered quantum field. It is finally suggested that the found states could describe different black holes.Comment: 36 pages, 1 figure. Formula of condensate energy density modified. Accepted for pubblication in Journal of Mathematical Physic

The self-consistent gravitational self-force

I review the problem of motion for small bodies in General Relativity, with an emphasis on developing a self-consistent treatment of the gravitational self-force. An analysis of the various derivations extant in the literature leads me to formulate an asymptotic expansion in which the metric is expanded while a representative worldline is held fixed; I discuss the utility of this expansion for both exact point particles and asymptotically small bodies, contrasting it with a regular expansion in which both the metric and the worldline are expanded. Based on these preliminary analyses, I present a general method of deriving self-consistent equations of motion for arbitrarily structured (sufficiently compact) small bodies. My method utilizes two expansions: an inner expansion that keeps the size of the body fixed, and an outer expansion that lets the body shrink while holding its worldline fixed. By imposing the Lorenz gauge, I express the global solution to the Einstein equation in the outer expansion in terms of an integral over a worldtube of small radius surrounding the body. Appropriate boundary data on the tube are determined from a local-in-space expansion in a buffer region where both the inner and outer expansions are valid. This buffer-region expansion also results in an expression for the self-force in terms of irreducible pieces of the metric perturbation on the worldline. Based on the global solution, these pieces of the perturbation can be written in terms of a tail integral over the body's past history. This approach can be applied at any order to obtain a self-consistent approximation that is valid on long timescales, both near and far from the small body. I conclude by discussing possible extensions of my method and comparing it to alternative approaches.Comment: 44 pages, 4 figure

"Peeling property" for linearized gravity in null coordinates

A complete description of the linearized gravitational field on a flat background is given in terms of gauge-independent quasilocal quantities. This is an extension of the results from gr-qc/9801068. Asymptotic spherical quasilocal parameterization of the Weyl field and its relation with Einstein equations is presented. The field equations are equivalent to the wave equation. A generalization for Schwarzschild background is developed and the axial part of gravitational field is fully analyzed. In the case of axial degree of freedom for linearized gravitational field the corresponding generalization of the d'Alembert operator is a Regge-Wheeler equation. Finally, the asymptotics at null infinity is investigated and strong peeling property for axial waves is proved.Comment: 27 page

Second-order gravitational self-force

We derive an expression for the second-order gravitational self-force that acts on a self-gravitating compact-object moving in a curved background spacetime. First we develop a new method of derivation and apply it to the derivation of the first-order gravitational self-force. Here we find that our result conforms with the previously derived expression. Next we generalize our method and derive a new expression for the second-order gravitational self-force. This study also has a practical motivation: The data analysis for the planned gravitational wave detector LISA requires construction of waveforms templates for the expected gravitational waves. Calculation of the two leading orders of the gravitational self-force will enable one to construct highly accurate waveform templates, which are needed for the data analysis of gravitational-waves that are emitted from extreme mass-ratio binaries.Comment: 35 page

Algebraic approach to quantum field theory on non-globally-hyperbolic spacetimes

The mathematical formalism for linear quantum field theory on curved spacetime depends in an essential way on the assumption of global hyperbolicity. Physically, what lie at the foundation of any formalism for quantization in curved spacetime are the canonical commutation relations, imposed on the field operators evaluated at a global Cauchy surface. In the algebraic formulation of linear quantum field theory, the canonical commutation relations are restated in terms of a well-defined symplectic structure on the space of smooth solutions, and the local field algebra is constructed as the Weyl algebra associated to this symplectic vector space. When spacetime is not globally hyperbolic, e.g. when it contains naked singularities or closed timelike curves, a global Cauchy surface does not exist, and there is no obvious way to formulate the canonical commutation relations, hence no obvious way to construct the field algebra. In a paper submitted elsewhere, we report on a generalization of the algebraic framework for quantum field theory to arbitrary topological spaces which do not necessarily have a spacetime metric defined on them at the outset. Taking this generalization as a starting point, in this paper we give a prescription for constructing the field algebra of a (massless or massive) Klein-Gordon field on an arbitrary background spacetime. When spacetime is globally hyperbolic, the theory defined by our construction coincides with the ordinary Klein-Gordon field theory on aComment: 21 pages, UCSBTH-92-4

Rigorous steps towards holography in asymptotically flat spacetimes

Scalar QFT on the boundary $\Im^+$ at null infinity of a general asymptotically flat 4D spacetime is constructed using the algebraic approach based on Weyl algebra associated to a BMS-invariant symplectic form. The constructed theory is invariant under a suitable unitary representation of the BMS group with manifest meaning when the fields are interpreted as suitable extensions to $\Im^+$ of massless minimally coupled fields propagating in the bulk. The analysis of the found unitary BMS representation proves that such a field on $\Im^+$ coincides with the natural wave function constructed out of the unitary BMS irreducible representation induced from the little group $\Delta$, the semidirect product between SO(2) and the two dimensional translational group. The result proposes a natural criterion to solve the long standing problem of the topology of BMS group. Indeed the found natural correspondence of quantum field theories holds only if the BMS group is equipped with the nuclear topology rejecting instead the Hilbert one. Eventually some theorems towards a holographic description on $\Im^+$ of QFT in the bulk are established at level of $C^*$ algebras of fields for strongly asymptotically predictable spacetimes. It is proved that preservation of a certain symplectic form implies the existence of an injective $*$-homomorphism from the Weyl algebra of fields of the bulk into that associated with the boundary $\Im^+$. Those results are, in particular, applied to 4D Minkowski spacetime where a nice interplay between Poincar\'e invariance in the bulk and BMS invariance on the boundary at $\Im^+$ is established at level of QFT. It arises that the $*$-homomorphism admits unitary implementation and Minkowski vacuum is mapped into the BMS invariant vacuum on $\Im^+$.Comment: 62 pages, amslatex, xy package; revised section 2 and the conclusions; corrected some typos; added some references; accepted for pubblication on Rev. Math. Phy

Gravitational waves about curved backgrounds: a consistency analysis in de Sitter spacetime

Gravitational waves are considered as metric perturbations about a curved background metric, rather than the flat Minkowski metric since several situations of physical interest can be discussed by this generalization. In this case, when the de Donder gauge is imposed, its preservation under infinitesimal spacetime diffeomorphisms is guaranteed if and only if the associated covector is ruled by a second-order hyperbolic operator which is the classical counterpart of the ghost operator in quantum gravity. In such a wave equation, the Ricci term has opposite sign with respect to the wave equation for Maxwell theory in the Lorenz gauge. We are, nevertheless, able to relate the solutions of the two problems, and the algorithm is applied to the case when the curved background geometry is the de Sitter spacetime. Such vector wave equations are studied in two different ways: i) an integral representation, ii) through a solution by factorization of the hyperbolic equation. The latter method is extended to the wave equation of metric perturbations in the de Sitter spacetime. This approach is a step towards a general discussion of gravitational waves in the de Sitter spacetime and might assume relevance in cosmology in order to study the stochastic background emerging from inflation.Comment: 17 pages. Misprints amended in Eqs. 50, 54, 55, 75, 7

Kinetic Anomalies in Addition-Aggregation Processes

We investigate irreversible aggregation in which monomer-monomer, monomer-cluster, and cluster-cluster reactions occur with constant but distinct rates K_{MM}, K_{MC}, and K_{CC}, respectively. The dynamics crucially depends on the ratio gamma=K_{CC}/K_{MC} and secondarily on epsilon=K_{MM}/K_{MC}. For epsilon=0 and gamma<2, there is conventional scaling in the long-time limit, with a single mass scale that grows linearly in time. For gamma >= 2, there is unusual behavior in which the concentration of clusters of mass k, c_k decays as a stretched exponential in time within a boundary layer k<k* propto t^{1-2/gamma} (k* propto ln t for gamma=2), while c_k propto t^{-2} in the bulk region k>k*. When epsilon>0, analogous behaviors emerge for gamma<2 and gamma >= 2.Comment: 6 pages, 2 column revtex4 format, for submission to J. Phys.