Quantities In General Relativity Research Articles

Calculation of asymptotic charges at the critical sets of null infinity.

The asymptotic structure of null and spatial infinities of asymptotically flat spacetimes plays an essential role in discussing gravitational radiation, gravitational memory effect, and conserved quantities in General Relativity (GR). Bondi, Metzner and Sachs (BMS) established that the asymptotic symmetry group for asymptotically simple spacetimes is the infinite-dimensional BMS group. Given that null infinity is divided into two sets: past null infinity [Formula: see text] and future null infinity [Formula: see text], one can identify two independent symmetry groups: [Formula: see text] at [Formula: see text] and [Formula: see text] at [Formula: see text]. Associated with these symmetries are the so-called BMS charges. A recent conjecture by Strominger suggests that the generators of [Formula: see text] and [Formula: see text] and their associated charges are related via an antipodal reflection map near spatial infinity. To verify this matching, an analysis of the gravitational field near spatial infinity is required. This task is complicated due to the singular nature of spatial infinity for spacetimes with non-vanishing ADM mass. Different frameworks have been introduced in the literature to address this singularity, e.g. Friedrich's cylinder, Ashtekar-Hansen's hyperboloid and Ashtekar-Romano's asymptote at spatial infinity. This paper reviews the role of Friedrich's formulation of spatial infinity in the investigation of the matching of the spin-2 charges on Minkowski spacetime and in the full GR setting. This article is part of a discussion meeting issue 'At the interface of asymptotics, conformal methods and analysis in general relativity'.

From conformal infinity to equations of motion: conserved quantities in general relativity.

This paper describes conservation laws in general relativity (GR) dating back to the mass-energy conservation of Bondi and Sachs in the early 1960s but using 2-spinor techniques. The notion of conformal infinity is employed, and the highly original ideas of E. T. Newman are discussed in relation to twistor theory. The controversial NP constants are introduced, and their meaning is considered in a new light related to the problem of equations of motion in GR. This article is part of a discussion meeting issue 'At the interface of asymptotics, conformal methods and analysis in general relativity'.

Counterfactuals in the Initial Value Formulation of General Relativity

How precisely to understand and evaluate counterfactuals can be an intricate issue. The aim of this article is to examine a new set of difficulties for evaluating counterfactuals that arise in the context of the dynamical spacetimes described by the theory of general relativity (GR). The initial value formulation provides us with a methodology to pin down the specific combination of features of the theory at the origin of the difficulties, namely, non-linearity and certain non-local aspects (typically captured by ellipticity at the analytical level), in particular when combined with the global and/or quasi-local character of physical quantities in GR. Finally, we connect the philosophical question about counterfactuals with concrete applied physical issues in GR about extracting meaningful predictions by constructing appropriate initial data sets, leading us to question the very suitability of the Cauchy approach to fully account for the explanatory power of the theory.

Slowly rotating black holes in Einstein-æther theory

We study slowly rotating, asymptotically flat black holes in Einstein-aether theory and show that solutions that are free from naked finite area singularities form a two-parameter family. These parameters can be thought of as the mass and angular momentum of the black hole, while there are no independent aether charges. We also show that the aether has non-vanishing vorticity throughout the spacetime, as a result of which there is no hypersurface that resembles the universal horizon found in static, spherically symmetric solutions. Moreover, for experimentally viable choices of the coupling constants, the frame-dragging potential of our solutions only shows percent-level deviations from the corresponding quantities in General Relativity and Horava gravity. Finally, we uncover and discuss several subtleties in the correspondence between Einstein-aether theory and Horava gravity solutions in the $c_\omega\to\infty$ limit.

Quasilocal angular momentum and center of mass in general relativity

For a spacelike 2-surface in spacetime, we propose a new definition of quasi-local angular momentum and quasi-local center of mass, as an element in the dual space of the Lie algebra of the Lorentz group. Together with previous defined quasi-local energy-momentum, this completes the definition of conserved quantities in general relativity at the quasi-local level. We justify this definition by showing the consistency with the theory of special relativity and expectations on an axially symmetric spacetime. The limits at spatial infinity provide new definitions for total conserved quantities of an isolated system, which do not depend on any asymptotically flat coordinate system or asymptotic Killing field. The new proposal is free of ambiguities found in existing definitions and presents the first definition that precisely describes the dynamics of the Einstein equation.

Conserved Quantities in General Relativity: From the Quasi-Local Level to Spatial Infinity

We define quasi-local conserved quantities in general relativity by using the optimal isometric embedding in [26] to transplant Killing fields in the Minkowski spacetime back to the 2-surface of interest in a physical spacetime. To each optimal isometric embedding, a dual element of the Lie algebra of the Lorentz group is assigned. Quasi-local angular momentum and quasi-local center of mass correspond to pairing this element with rotation Killing fields and boost Killing fields, respectively. They obey classical transformation laws under the action of the Poincar\'e group. We further justify these definitions by considering their limits as the total angular momentum and the total center of mass of an isolated system. These expressions were derived from the Hamilton-Jacobi analysis of gravitation action and thus satisfy conservation laws. As a result, we obtained an invariant total angular momentum theorem in the Kerr spacetime. For a vacuum asymptotically flat initial data set of order $1$, it is shown that the limits are always finite without any extra assumptions. We also study these total conserved quantities on a family of asymptotically flat initial data sets evolving by the vacuum Einstein evolution equation. It is shown that the total angular momentum is conserved under the evolution. For the total center of mass, the classical dynamical formula relating the center of mass, energy, and linear momentum is recovered, in the nonlinear context of initial data sets evolving by the vacuum Einstein evolution equation. The definition of quasi-local angular momentum provides an answer to the second problem in classical general relativity on Penrose's list [18].

THE RELATIVE ENERGY OF HOMOGENEOUS AND ISOTROPIC UNIVERSES FROM VARIATIONAL PRINCIPLES

We calculate the relative conserved currents, superpotentials and conserved quantities between two homogeneous and isotropic universes. In particular, we prove that their relative "energy" (defined as the conserved quantity associated to cosmic time coordinate translations for a comoving observer) is vanishing and so are the other conserved quantities related to a Lie subalgebra of vector fields isomorphic to the Poincaré algebra. These quantities are also conserved in time. We also find a relative conserved quantity for such a kind of solution which is conserved in time though non-vanishing. This example provides at least two insights in the theory of conserved quantities in General Relativity. First, the contribution of the cosmological matter fluid to the conserved quantities is carefully studied and proved to be vanishing. Second, we explicitly show that our superpotential (that happens to coincide with the so-called KBL potential although it is generated differently) provides strong conservation laws under much weaker hypotheses than the ones usually required. In particular, the symmetry generator is not needed to be Killing (nor Killing of the background, nor asymptotically Killing), the prescription is quasi-local and it works fine in a finite region too and no matching condition on the boundary is required.

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.

On the dynamics of characteristic surfaces

We formulate the vacuum Einstein equations as differential equations for two functions, one complex and one real on a six-dimensional manifold, ℳ×S2, with ℳ eventually becoming the space–time and the S2 becoming the sphere of null directions over ℳ. At the start there is no other further structure available: the structure arising from the two functions. The complex function, referred to as Λ[ℳ×S2], encodes information about a sphere’s worth of surfaces through each point of ℳ. From knowledge of Λ one can define a second rank tensor on ℳ which can be interpreted as a conformal metric, so that the ‘‘surfaces’’ are automatically null or characteristics of this conformal metric. The real function, Ω, plays the role of a conformal factor: it converts the conformal metric into a vacuum Einstein metric. Locally, all Einstein metrics can be obtained in this manner. In this work, we fully develop this ‘‘null surface version of general relativity (GR):’’ we display, discuss and analyze the equations, we show that many of the usual geometric quantities of GR (e.g., the Weyl and Ricci tensors, the optical parameters, etc.) can be easily expressed in terms of the Λ and Ω, we study the gauge freedom and develop a perturbation theory. To conclude, we speculate on the significance and possible classical and quantum uses of this formulation.

On conserved quantities in general relativity

Recently, definitions of total 4-momentum and angular momentum of isolated gravitating systems have been introduced in terms of the asymptotic behavior of the Weyl curvature (of the underlying space–time) at spatial infinity. Given a space–time equipped with isometries, on the other hand, one can also construct conserved quantities using the presence of the Killing fields. Thus, for example, for stationary space–times, the Komar integral can be used to define the total mass, and, the asymptotic value of the twist of the Killing field, to introduce the dipole angular momentum moment. Similarly, for axisymmetric space–times, one can obtain the (’’z-component’’ of the) total angular momentum in terms of the Komar integral. It is shown that, in spite of their apparently distinct origin, in the presence of isometries, quantities defined at spatial infinity reduce to the ones constructed from Killing fields. This agreement reflects one of the many subtle aspects of Einstein’s (vacuum) equation.

Group-Theoretic Approach to the New Conserved Quantities in General Relativity

The Newman-Penrose formalism for obtaining the recent conserved quantities in general relativity is discussed and a group-theoretic interpretation is given to it. This is done by relating each triad of the orthonormal vectors on the sphere to an orthogonal matrix g. As a result, the spin-weighted quantities η become functions on the group of three-dimensional rotation, η = η(g), where g ∈ O3. An explicit form for the matrix g is given and a prescription for rewriting η(g) as functions of the spherical coordinates is also given. We show that a quantity of spin weight s can be expanded as a series in the matrix elements Tsmj of the irreducible representation of O3, where s is fixed. Infinite- and finite-dimensional representations of the group SU2 are then realized in the spaces of η's and Tsmj. It is shown that the infinite-dimensional representation is not irreducible; its decomposition into irreducible parts leads to the expansion of η in the Tsmj, the latter providing invariant subspaces in which irreducible representations act.

Construction of a Complete Set of Independent Observables in the General Theory of Relativity

The construction of a complete set of quantities in general relativity, whose functional form is invariant under coordinate transformations, is indicated. The set obtained is highly redundant. The Cauchy problem for obtaining an independent complete set of such quantities (observables) is therefore discussed. It is also pointed out that the observables obtained may alternatively be viewed as the metric tensor in a special gauge (i.e., with a special coordinate condition). This latter viewpoint may facilitate the quantization of general relativity.