Vector Fields In Space-time Research Articles

The Homotopy Momentum Map of General Relativity

AbstractWe show that the action of spacetime vector fields on the variational bicomplex of general relativity has a homotopy momentum map that extends the map from vector fields to conserved currents given by Noether’s 1st theorem to a morphism of $L_\infty $-algebras.

Maxwell\u2019s Equations are Universal for Locally Conserved Quantities

A fundamental result of classical electromagnetism is that Maxwell’s equations imply that electric charge is locally conserved. Here we show the converse: Local charge conservation implies the local existence of fields satisfying Maxwell’s equations. This holds true for any conserved quantity satisfying a continuity equation. It is obtained by means of a strong form of the Poincaré lemma presented here that states: Divergence-free multivector fields locally possess curl-free antiderivatives on flat manifolds. The above converse is an application of this lemma in the case of divergence-free vector fields in spacetime. We also provide conditions under which the result generalizes to curved manifolds.

On the Local Extension of Killing Vector Fields in Electrovacuum Spacetimes

We revisit the problem of extension of a Killing vector field in a spacetime which is solution to the Einstein-Maxwell equation. This extension has been proved to be unique in the case of a Killing vector field which is normal to a bifurcate horizon by Yu. Here we generalize the extension of the vector field to a strong null convex domain in an electrovacuum spacetime, inspired by the same technique used by Ionescu-Klainerman in the setting of Ricci flat manifolds. We also prove a result concerning non-extendibility: we show that one can find local, stationary electrovacuum extension of a Kerr-Newman solution in a full neighborhood of a point of the horizon (that is not on the bifurcation sphere) which admits no extension of the Hawking vector field. This generalizes the construction by Ionescu-Klainerman to the electrovacuum case.

On homogeneous and isotropic universe

We give a simple example of spacetime metric, illustrating that homogeneity and isotropy of space slices at all moments of time is not obligatory lifted to a full system of six Killing vector fields in spacetime, thus it cannot be interpreted as a symmetry of a four-dimensional metric. The metric depends on two arbitrary and independent functions of time. One of these functions is the usual scale factor. The second function cannot be removed by coordinate transformations. We prove that it must be equal to zero, if the metric satisfies Einstein’s equations and the matter energy–momentum tensor is homogeneous and isotropic. A new, equivalent, definition of homogeneous and isotropic spacetime is given.

On the theory of Killing orbits in spacetime

This paper gives a theoretical discussion of the orbits and isotropies which arise in a spacetime which admits a Lie algebra of Killing vector fields. The submanifold structure of the orbits is explored together with their induced Killing vector structure. A general decomposition of a spacetime in terms of the nature and dimension of its orbits is given and the concept of stability and instability for orbits introduced. A general relation is shown linking the dimensions of the Killing algebra, the orbits and the isotropies. The well-behaved nature of ‘stable’ orbits and the possible misbehaviour of the ‘unstable’ ones is pointed out and, in particular, the fact that independent Killing vector fields in spacetime may not induce independent Killing vector fields on unstable orbits. Several examples are presented to exhibit these features. Finally, an appendix is given which revisits and attempts to clarify the well-known theorem of Fubini on the dimension of Killing orbits.

Covariant charges in Chern–Simons AdS3 gravity

We try to give hereafter an answer to some open questions about the definition of conserved quantities in Chern–Simons theory, with particular reference to Chern–Simons AdS3 gravity. Our attention is focused on the problem of global covariance and gauge invariance of the variation of Noether charges. A theory which satisfies the principle of covariance on each step of its construction is developed, starting from a gauge invariant Chern–Simons Lagrangian and using a recipe developed by Allemandi et al (2002 Class. Quantum Grav. 19 2633–55, 237–58) to calculate the variation of conserved quantities. The problem of giving a mathematical well-defined expression for the infinitesimal generators of symmetries is pointed out and it is shown that the generalized Kosmann lift of spacetime vector fields leads to the expected numerical values for the conserved quantities when the solution corresponds to the BTZ black hole. The fist law of black-hole mechanics for the BTZ solution is then proved and the transition between the variation of conserved quantities in Chern–Simons AdS3 gravity theory and the variation of conserved quantities in general relativity is analysed in detail.

Conformal Vector Fields and Conformal–Type Collineations in Space-Times

Some restrictions on the existence of homothetic and conformal vector fields in space-times which already admit some Killing symmetry are established. In particular, the behaviour of Weyl invariants and the nature of the Petrov type of the Weyl tensor along the integral curves of conformal vector fields are studied. This results in important restrictions between conformal vector fields and Killing orbits. A brief remark is made on Weyl collineations.

Covariant conservation laws from the palatini formalism

Covariant conservation laws in the Palatini formalism are derived. The result indicates that the gravitational part of conserved charges in general relativity should be calculated from a combination of Komar's strongly conserved current and the Einstein tensor. This implies that the set of complete diffeomorphism charges of a gravitating system consisting of scalar matter is described by Komar's vector density, and that the identification of gravitational energy and momentum reduces to two choices: a choice of relative weights of the contributions resulting from Komar's current and from the Einstein tensor, and a choice of preferred vector fields in space-time. A proposal is made which yields energy and momentum as scalars under diffeomorphisms and as a Lorentz vector in tangent space. Furthermore, the result can be used to identify covariant conservation laws holding separately for the matter contributions to diffeomorphism charges.

Conformal vector fields in general relativity

A general discussion of conformal vector fields in space-times is given. Amongst the topics considered are the maximum dimension of the conformal algebra for space-times that are not conformally flat, the nature of conformal isotropies and a new approach to the theorem of Bilyalov and Defrise-Carter concerning the reduction of the conformal algebra to a Killing or homothetic algebra. Some deficiencies in the original statements of this theorem are discussed (with reference to a general class of counterexamples) and corrected. The proof offered is geometrical in nature and has the advantage of displaying some of the more general features and properties of conformal vector fields and the ways in which they can differ from Killing vector fields.

Homothety groups in space-time

This paper investigates space-times which admit anr-dimensional Lie algebra of homothetic vector fields at least one of which is proer homothetic. The situation is resolved completely ifr ≥ 6 and some general results are given whenr ≤ 5. Some applications to the study of affine vector fields in space-times are also given and the maximum dimension of this latter algebra is computed and corrects an earlier error.

Special conformal symmetries in general relativity

A geometrical discussion of special conformal vector fields in space-time is given. In particular, it is shown that if such a vector fieldξ is admitted, it is unique up to a constant scaling and the addition of a homothetic or a Killing vector field. In the case when the gradient of the conformal scalar associated withξ is non-null it is shown that other homothetic and affine symmetries are necessarily admitted by the space-time, that an intrinsic family of 2-dimensional flat submanifolds is determined in the space-time, thatξ is, in general, hypersurface orthogonal and that the space-time, if non-flat, is necessarily (geodesically) incomplete. Other geometrical features of such space-times are also considered.

Killing tensor conservation laws and their generators

The generators of Killing vector and tensor geodesic conservation laws are derived. It is shown that the generator of a Killing vector conservation law coincides with the Killing field itself. For Killing tensors the generators are not space-time vector fields but rather depend on the geodesic tangent vector and therefore lie in a jet space of the geodesic equations. By regarding the metric as a field on the one-jet space of the geodesic equations, the action of the Killing tensor generators on the metric can be defined in a natural way. It is found that the metric is not invariant under Killing tensor symmetries. This happens because the Killing tensor symmetries, unlike the Killing vector symmetries, are divergence symmetries.

Dirac constraint quantization of a parametrized field theory by anomaly-free operator representations of spacetime diffeomorphisms.

We construct a consistent Dirac constraint quantization of a parametrized massless scalar field propagating on a two-dimensional cylindrical Minkowskian background. The constraints are taken in the form of ``diffeomorphism Hamiltonians'' whose Poisson-brackets algebra is homomorphic to the Lie algebra of spacetime diffeomorphisms. The fundamental canonical variables are represented by operators acting on an embedding-dependent Fock space H which is based on the Heisenberg modes that are geometrically specified with respect to the Killing vector structure of the background. In the Heisenberg picture, the constraints become the Heisenberg embedding momenta and their Abelian Poisson algebra is homomorphically mapped into the operator commutator algebra without any anomaly. The algebra of normal-ordered Heisenberg evolution generators (which propagate the field operators) develops a covariantly defined anomaly. This anomaly is an exact two-form on the space of embeddings Emb(\ensuremath{\Sigma},M) and can thus be written as a functional curl of an anomaly potential on Emb(\ensuremath{\Sigma},M). By subtracting this potential from the normal-ordered Heisenberg generators (which amounts to their embedding-dependent reordering) we arrive at a commuting set of operators which we identify with the Schr\"odinger embedding momenta. By smearing the Heisenberg and the Schr\"odinger embedding momenta by spacetime vector fields we obtain a pair of anomaly-free operator representations of L DiffM. The diffeomorphism Hamiltonians annihilate the physical states and the smeared reordered Heisenberg evolution generators propagate the fields. We present the operator transformation from the Schr\"odinger to the Heisenberg picture. The two operator representations of L DiffM, by diffeomorphism Hamiltonians and by smeared Heisenberg evolution generators, guarantee that the Dirac constraint quantization is consistent, covariant, and leads to foliation-independent dynamics both in the Heisenberg and in the Schr\"odinger pictures.The appropriate factor ordering of the Hamiltonian flux operator and of the constraints is rewritten in terms of the fundamental Schr\"odinger variables with help of a normal-ordering kernel which is reconstructed from the intrinsic metric and the extrinsic curvature on a given embedding. All operators are defined and dynamics takes place on a single function space which is then restricted by the constraints to the space of physical states with a Hilbert-space structure.

Representations of spacetime diffeomorphisms. I. Canonical parametrized field theories

The super-Hamiltonian and supermomentum in canonical geometrodynamics or in a parametried field theory on a given Riemannian background have Poisson brackets which obey the Dirac relations. By smearing the supermomentum with vector fields V →∈ L Diff Σ on the space manifold Σ, the Lie algebra L Diff Σ of the spatial diffeomorphism group Diff Σ can be mapped antihomomorphically into the Poisson bracket algebra on the phase space of the system. The explicit dependence of the Poisson brackets between two super-Hamiltonians on canonical coordinates (spatial metrics in geometrodynamics and embedding variables in parametrized theories) is usually regarded as an indication that the Dirac relations cannot be connected with a representation of the complete Lie algebra L Diff M of spacetime diffeomorphisms. We show how this difficulty may be overcome and construct a homomorphic mapping of spacetime vector fields V ∈ L Diff M into the Poisson bracket algebra on the phase space of the system. In the present paper, I, we explain how the technique works in the case of a parametrized field theory and in the following paper, II, we generalize it to canonical geometrodynamics. In a parametrized theory, the phase space of the system is the ordinary phase space of the field augmented by the embedding variables X: Σ →Mand their conjugate momenta. The dynamical variable H(V) which represents V ∈ L Diff M generates a deformation of the embedding along the flow lines of V accompanied by the correct dynamical evolution of the field data and preserves the constraints in the extended phase space of the system. We also establish the relation between the representations of Diff Σ and DiffM.