We study the covariant phase space of vacuum general relativity at the null boundary of causal diamonds. The past and future components of such a null boundary each have an infinite-dimensional symmetry algebra consisting of diffeomorphisms of the $2$-sphere and boost supertranslations corresponding to angle-dependent rescalings of affine parameter along the null generators. Associated to these symmetries are charges and fluxes obtained from the covariant phase space formalism using the prescription of Wald and Zoupas. By analyzing the behavior of the spacetime metric near the corners of the causal diamond, we show that the fluxes are also Hamiltonian generators of the symmetries on the phase space. In particular, the supertranslation fluxes yield an infinite family of boost Hamiltonians acting on the gravitational data of causal diamonds. We show that the smoothness of the vector fields representing such symmetries at the bifurcation edge of the causal diamond implies suitable matching conditions between the symmetries on the past and future components of the null boundary. Similarly, the smoothness of the spacetime metric implies that the fluxes of all such symmetries is conserved between the past and future components of the null boundary. This establishes an infinite set of conservation laws for finite subregions in gravity analogous to those at null infinity. We also show that the symmetry algebra at the causal diamond has a non-trivial center corresponding to constant boosts. The central charges associated to these constant boosts are proportional to the area of the bifurcation edge, for any causal diamond, in analogy with the Wald entropy formula.
We explain the necessity of application of semi-metric in general relativity. A detailed discussion on the energy-momentum conservation in the general relativity is presented using the mathematical tool of semi-metric. By means of the general covariant spacetime translation transformation, the most general covariant conservation law of energy-momentum is obtained, which is valid for any coordinates and overcomes the flaws of the expressions of Einstain, Landau and Moller.
A covariant formula for conserved currents of energy, momentum and angular-momentum is derived from a general form of Noethers theorem applied directly to the Einstein-Hilbert action of classical general relativity. Energy conservation in a closed big-bang cosmology is discussed as a special case. Special care is taken to distinguish between kinematic and dynamic expressions.
The generalized Stokes theorem (connecting integrals of dimensions 3 and 4) is formulated in a curved space-time in terms of paths in Minkowski space (forming Path Group). A covariant integral form of the conservation law for the energy-momentum of matter is then derived in General Relativity. It extends Einstein's equivalence principle on the energy conservation, since it formulates the conservation law for the energy-momentum of matter without explicit including the gravitational field in the formulation.
We prove that the field equations of general relativity and other metric theories can be derived from the conservation of energy-momentum without using the assumption of least action principle. We show a new procedure for perturbative derivation of symmetric pseudo-energy momentum for the gravity field.
This work refers to the new formula for the superpotential Uikl in conservation laws in general relativity satisfying the integral and differential conservation laws within the Schwarzschild metric. The new superpotential is composed of two terms. The first term is based on Mollers concept and its a function of the metric gik and its first derivative only. The second term is the antisymmetric tensor density of weight plus one and it consists of higher derivatives of the metric gik. Although the new superpotential consists of higher derivatives of the metric gik it might bring a new evaluation …
We consider the collision of self-gravitating n-branes in a (n+2)-dimensional spacetime. We show that there is a geometrical constraint which can be expressed as a simple sum rule for angles characterizing Lorentz boosts between branes and the intervening spacetime regions. This constraint can then be re-interpreted as either energy or momentum conservation at the collision.
We discuss general properties of the conservation law associated with a local symmetry. Using Noether's theorem and a generalized Belinfante symmetrization procedure in 3+1 dimensions, a symmetric energy-momentum (pseudo) tensor for the gravitational Einstein-Hilbert action is derived and discussed in detail. In 2+1 dimensions, expressions are obtained for energy and angular momentum arising in the $ISO(2,1)$ gauge theoretical formulation of Einstein gravity. In addition, an expression for energy in a gauge theoretical formulation of the string-inspired 1+1 dimensional gravity is derived and c…
A set of exact quasi-local conservation equations is derived from the Einstein's equations using the first-order Kaluza-Klein formalism of general relativity in the (2,2)-splitting of 4-dimensional spacetime. These equations are interpreted as quasi-local energy, momentum, and angular momentum conservation equations. In the asymptotic region of asymptotically flat spacetimes, it is shown that the quasi-local energy and energy-flux integral reduce to the Bondi energy and energy-flux, respectively. In spherically symmetric spacetimes, the quasi-local energy becomes the Misner-Sharp energy. Moreo…
A new quasigroup approach to conservation laws in general relativity is applied to study asymptotically flat at future null infinity spacetime. The infinite-parametric Newman-Unti group of asymptotic symmetries is reduced to the Poincaré quasigroup and the Noether charge associated with any element of the Poincaré quasialgebra is defined. The integral conserved quantities of energy-momentum and angular momentum are linear on generators of Poincaré quasigroup, free of the supertranslation ambiguity, posess the flux and identically equal to zero in Minkowski spacetime.
All the Doubly Special Relativity (DSR) models studied in literature so far involve a deformation of the energy conservation rule that forces us to release the hypothesis of the additivity of the energy for composite systems. In view of the importance of the issue for a consistent formulation of a DSR statistical mechanics and a DSR thermodynamics, we show that DSR models preserving the usual (i.e. additive) energy conservation rule can be found. These models allow the construction of a DSR-covariant extensive energy. The implications of the analysis for the dynamics of DSR-covariant multipart…
We show that the singularity in the General Theory of Relativity (GTR) is the expression of a non-Machian feature. It can be avoided with a scale-invariant dynamical theory, a property lacking in GTR. It is further argued that the global non-conservation of energy in GTR also results from the lack of scale-invariance and the field formulation presented by several authors can only resolve the problem in part. Assuming the global energy conservation, we propose a negative energy density component with positive equation of state that can drive the late-time acceleration in the universe, while the…
During the First World War, the status of energy conservation in general relativity was one of the most hotly debated questions surrounding Einstein's new theory of gravitation. His approach to this aspect of general relativity differed sharply from another set forth by Hilbert, even though the latter conjectured in 1916 that both theories were probably equivalent. Rather than pursue this question himself, Hilbert chose to charge Emmy Noether with the task of probing the mathematical foundations of these two theories. Indirect references to her results came out two years later when Klein began…
We show that the BMS-supertranslations and their associated supermomenta on past null infinity can be related to those on future null infinity, proving the conjecture of Strominger for a class of spacetimes which are asymptotically-flat in the sense of Ashtekar and Hansen. Using a cylindrical 3-manifold of both null and spatial directions of approach towards spatial infinity, we impose appropriate regularity conditions on the Weyl tensor near spatial infinity along null directions. The asymptotic Einstein equations on this 3-manifold and the regularity conditions imply that the relevant Weyl t…
In this work we revisit the study of the gravitational interaction in the context of the Special Theory of Relativity. It is found that, as long as the equivalence principle is respected, a relativistic non-linear energy conservation theorem arises in a natural way. We interpret that this non-linear conservation law stresses the non-linear character of the gravitational interaction.The theorem reproduces the energy conservation theorem of Newtonian mechanics in the corresponding low energy limit, but also allows to derive some standard results of post-Newtonian gravity, such as the formula of …
The ADM formalism is reviewed and techniques for decomposing generic components of metric, connection and curvature are obtained. These techniques will turn out to be enough to decompose not only Einstein equations but also covariant conservation laws. Then a number of independent sets of hypotheses that are sufficient (though non-necessary) to obtain standard ADM quantities (and Hamiltonian) from covariant conservation laws are considered. This determines explicitely the range in which standard techniques are equivalent to covariant conserved quantities. The Schwarzschild metric in different …
We present a practical framework for ideal hyperelasticity in numerical relativity. For this purpose, we recast the formalism of Carter and Quintana as a set of Eulerian conservation laws in an arbitrary 3+1 split of spacetime. The resulting equations are presented as an extension of the standard Valencia formalism for a perfect fluid, with additional terms in the stress-energy tensor, plus a set of kinematic conservation laws that evolve a configuration gradient. We prove that the equations can be made symmetric hyperbolic by suitable constraint additions, at least in a neighbourhood of the u…
The geometry of small causal diamonds is systematically studied, based on three distinct constructions that are common in the literature, namely the geodesic ball, the Alexandrov interval and the lightcone cut. The causal diamond geometry is calculated perturbatively using Riemann normal coordinate expansion up to the leading order in both vacuum and non-vacuum. We provide a collection of results including the area of the codimension-two edge, the maximal hypersurface volume and their isoperimetric ratio for each construction, which will be useful for any applications involving the quantitativ…
In the present work, the spin-coefficient formalism of Newman and Penrose is used to formulate geometric constraints for the existence of Kerr-Schild groups, i.e. continuous groups of generalized Kerr-Schild transformations. In addition, by characterizing the geometric structure of the deformed Einstein tensor of the generalized Kerr-Schild class, restrictions are imposed on the existence of apparent conservation laws in generic spacetimes, which are defined via considering special Kerr-Schild currents whose associated Kerr-Schild vector fields coincide with timelike Killing vector fields of p…
In de Sitter (dS) special relativity (SR), two kinds of conserved currents are derived. The first kind is a 5-dimensional (5d) dS-covariant angular momentum (AM) current, which unites the energy-momentum (EM) and 4d AM current in an inertial-type coordinate system. The second kind is a dS-invariant AM current, which can be generalized to a conserved current for the coupling system of the matter field and gravitational field in dS gravity. Moreover, an inherent EM tensor is predicted, which comes from the spin part of the dS-covariant current. All the above results are compared to the ordinary …
We show that the asymptotic charges associated with Lorentz symmetries on past and future null infinity match in the limit to spatial infinity in a class of asymptotically-flat spacetimes. These are spacetimes that obey the Ashtekar-Hansen definition of asymptotic flatness at null and spatial infinity and satisfy an additional set of conditions which we lay out explicitly. Combined with earlier results on the matching of supertranslation charges, this shows that all Bondi-Metzner-Sachs (BMS) charges on past and future null infinity match in the limit to spatial infinity in this class of spacet…
As Harvey Brown emphasizes in his book Physical Relativity, inertial motion in general relativity is best understood as a theorem, and not a postulate. Here I discuss the status of the "conservation condition", which states that the energy-momentum tensor associated with non-interacting matter is covariantly divergence-free, in connection with such theorems. I argue that the conservation condition is best understood as a consequence of the differential equations governing the evolution of matter in general relativity and many other theories. I conclude by discussing what it means to posit a ce…
In this article, we will review Noether's Theorems and their application in General Relativity. We will present Noether's Theorems in their original form and restate them as they are usually applied to physics. Some basic equations of Special Relativity will be reviewed and contrasted with the equations in General Relativity. We will be most interested in the role of Noether's Theorems in conservation laws. Several applications in flat spaces are examined. A formulation of Noether's Theorems in curved space is presented. Of particular interest is the conservation of energy in General Relativit…
We review the energy concept in the case of a continuum or a system of fields. First, we analyze the emergence of a true local conservation equation for the energy of a continuous medium, taking the example of an isentropic continuum in Newtonian gravity. Next, we consider a continuum or a system of fields in special relativity: we recall that the conservation of the energy-momentum tensor contains two local conservation equations of the same kind as before. We show that both of these equations depend on the reference frame, and that, however, they can be given a rigorous meaning. Then we revi…
We reformulate the standard Lagrangian formalism to a reparameterisation invariant Lagrangian formalism by means of Finsler and Kawaguchi geometry. In our formalism, various types of symmetries that appears in theories of physics are expressed geometrically by symmetries of Finsler (Kawaguchi) metric, and the conservation law of energy-momentum is a part of Euler-Lagrange equations. The application to scalar field, Dirac field, electromagnetic field and general relativity coupled to perfect fluid (added: ver.3) are discussed. By this formalism, we try to propose an alternative definition of en…
General relativity predicts that energy and momentum conservation laws hold and that preferred frames do not exist. The parametrised post-Newtonian formalism (PPN) phenomenologically quantifies possible deviations from general relativity. The PPN parameter alpha_3 (which identically vanishes in general relativity) plays a dual role in that it is associated both with a violation of the momentum conservation law, and with the existence of a preferred frame. By considering the effects of alpha_3 neq 0 in certain binary pulsar systems, it is shown that alpha_3 < 2.2 x 10^-20 (90% CL). This limit i…
Motivated by various theoretical arguments that the Planck energy (Ep - 10^19 GeV) - should herald departures from Lorentz invariance, and the possibility of testing these expectations in the not too distant future, two so-called "Doubly Special Relativity" theories have been suggested -- the first by Amelino-Camelia (DSR1) and the second by Smolin and Magueijo (DSR2). These theories contain two fundamental scales -- the speed of light and an energy usually taken to be Ep. The symmetry group is still the Lorentz group, but in both cases acting nonlinearly on the energy-momentum sector. Accordi…
We construct a general relativistic conservation law for linear and angular momentum for matter and gravitational fields in a finite volume of space that does not rely on any spacetime symmetries. This work builds on our previous construction of a general relativistic energy conservation law with the same features. Our approach uses the Brown and York quasilocal stress-energy-momentum tensor for matter and gravitational fields, plus the concept of a rigid quasilocal frame (RQF) introduced in previous work. The RQF approach allows us to construct, in a generic spacetime, frames of reference who…
According to Yang \& Mills (1954), a {\it conserved} current and a related rigid (`global') symmetry lie at the foundations of gauge theory. When the rigid symmetry is extended to a {\it local} one, a so-called gauge symmetry, a new interaction emerges as gauge potential $A$; its field strength is $F\sim {\rm curl} A$. In gravity, the conservation of the energy-momentum current of matter and the rigid translation symmetry in the Minkowski space of special relativity lie at the foundations of a gravitational gauge theory. If the translation invariance is made local, a gravitational potential $\…
The central equations in classical general relativity are the Einstein Field Equations, which accurately describe not only the generation of pseudo-Riemannian curvature by matter and radiation manifesting as gravitational effects, but more importantly mass-energy dynamics, evolution and distribution on the space-time manifold. Herein, we introduce a geometric phase in general relativity corresponding to Schwarzschild black hole information content. This quantity appropriately satisfies a local conservation law subject to minimal coupling, with other desirable properties such as the quantizatio…
We review the Lagrangian formulation of Noether symmetries (as well as "generalized Noether symmetries") in the framework of Calculus of Variations in Jet Bundles, with a special attention to so-called "Natural Theories" and "Gauge-Natural Theories", that include all relevant Field Theories and physical applications (from Mechanics to General Relativity, to Gauge Theories, Supersymmetric Theories, Spinors and so on). It is discussed how the use of Poincare'-Cartan forms and decompositions of natural (or gauge-natural) variational operators give rise to notions such as "generators of Noether sy…
A new, general, field theoretic approach to the derivation of asymptotic conservation laws is presented. In this approach asymptotic conservation laws are constructed directly from the field equations according to a universal prescription which does not rely upon the existence of Noether identities or any Lagrangian or Hamiltonian formalisms. The resulting general expressions of the conservation laws enjoy important invariance properties and synthesize all known asymptotic conservation laws, such as the ADM energy in general relativity.
A direct method for obtaining the differential conservation laws in the field theory from the principle of stationary action is proposed. The method is based on a variation of field functions through small local transformation of a special kind. The action of general theory of relativity are considered.
An idealized "test" object in general relativity moves along a geodesic. However, if the object has a finite mass, this will create additional curvature in the spacetime, causing it to deviate from geodesic motion. If the mass is nonetheless sufficiently small, such an effect is usually treated perturbatively and is known as the gravitational self-force due to the object. This issue is still an open problem in gravitational physics today, motivated not only by basic foundational interest, but also by the need for its direct application in gravitational-wave astronomy. In particular, the observ…
Starting from the continuity equation and Ohm's law, it is demonstrated that electrons incident upon the center of a copper sphere will reappear instantly on the surface of the sphere. Consequences of such superluminal phenomena for general relativity will be discussed.
We discuss inelastic collisions of two rotating disks by using the conservation laws for baryonic mass and angular momentum. In particular, we formulate conditions for the formation of a new disk after the collision and calculate the total energy loss to obtain upper limits for the emitted gravitational energy.
The debate on conservation laws in general relativity eighty years ago is reviewed and restudied. The physical meaning of the identities in the conservation laws for matter plus gravitational field is reexamined and new interpretations for gravitational wave are given. The conclusions of these studies are distinct from the prevalent views, it can be demonstrated that gravitational wave does not transmit energy (and momentum) but only transmits informations. An experimental test is offered to decide which conservation laws are correct.
Backgrounds are pervasive in almost every application of general relativity. Here we consider the Lagrangian formulation of general relativity for large perturbations with respect to a curved background spacetime. We show that Noether's theorem combined with Belinfante's "symmetrization" method applied to the group of displacements provide a conserved vector, a "superpotential" and a energy-momentum that are independent of any divergence added to the Hilbert Lagrangian of the perturbations. The energy-momentum is symmetrical and divergenceless only on backgrounds that are Einstein spaces in th…
We derive conservation laws for collisions of self-gravitating $n$-branes (or $n$-dimensional shells) in an $(n+2)$ dimensional spacetime including induced gravity on the brane. Previous work has shown how geometrical identities in general relativity enforce conservation of energy-momentum at collisions. The inclusion of induced gravity terms introduces a gravitational self-energy on the brane which permits energy-momentum conservation of matter fields on the brane to be broken, so long as the total energy-momentum, including induced gravity terms, is conserved. We give simple examples with tw…
We present the Lagrangian whose corresponding action is the trace K action for General Relativity. Although this Lagrangian is second order in the derivatives, it has no second order time derivatives and its behaviour at space infinity in the asymptotically flat case is identical to other alternative Lagrangians for General Relativity, like the gamma-gamma Lagrangian used by Einstein. We develop some elements of the variational principle for field theories with boundaries, and apply them to second order Lagrangians, where we stablish the conditions -- proposition 1 -- for the conservation of t…
This paper has three main aims: first, to give a pedagogical introduction to Noether's two theorems and their implications for energy conservation in general relativity, which was a central point of discussion between Hilbert, Klein, Noether and Einstein. Second, it introduces and compares two proposals for gravitational energy and momentum, one of which is very influential in physics: and, so far as I know, neither of the two has been discussed in the philosophical literature. Third, it assesses these proposals in connection with recent philosophical discussions of energy and momentum in gene…
Quasilocal definitions of stress-energy-momentum---that is, in the form of boundary densities (in lieu of local volume densities)---have proven generally very useful in formulating and applying conservation laws in general relativity. In this essay, we take a first basic look into applying these to cosmology, specifically using the Brown-York quasilocal stress-energy-momentum tensor for matter and gravity combined. We compute this tensor and present some simple results for a flat FLRW spacetime with a perfect fluid matter source.
The existence of conservation laws is one of the most important requirement of physical theories. Some of them, like energy conservation, knows no experimental exception. However, the generalization of these conservation laws to curved space presents many challenges. The implementation of conservation laws in the General Relativity theory is revised, and the possibility of the generalization of the usual expression is discussed. The Rastall's theory of gravity, which considers a modification of the usual conservation of the energy-momentum tensor, is discussed in more detail. Some applications…
Ten conservation laws in useful polynomial form are derived from a Cartan form and Exterior Differential System (EDS) for the tetrad equations of vacuum relativity. The Noether construction of conservation laws for well posed EDS is introduced first, and an illustration given, deriving 15 conservation laws of the free field Maxwell Equations from symmetries of its EDS. The Maxwell EDS and tetrad gravity EDS have parallel structures, with their numbers of dependent variables, numbers of generating 2-forms and generating 3-forms, and Cartan character tables all in the ratio of 1 to 4. They have …
The Rastall's theory is a modification of the General Relativity theory leading to a different expression for the conservation law in the matter sector compared with the usual one. It has been argued recently that such a theory may have applications to the dark energy problem, since a pressureless fluid may lead to an accelerated universe. In the present work we confront the Rastall's theory with the power spectrum data. The results indicate a configuration that essentially reduces the Rastall's theory to General Relativity, unless the non-usual conservation law refers to a scalar field, situa…
We briefly recall the problem of defining conserved quantities such as energy in general relativity, and the solution by introducing a symmetric background. We apply the general formalism to perturbed Robertson-Walker spacetimes with de Sitter geometry as background. We relate the obtained conserved quantities to Traschen's integral constraints and mention a few applications in cosmology.
In recent work we constructed completely general conservation laws for energy and linear and angular momentum of extended systems in general relativity based on the notion of a rigid quasilocal frame (RQF). We argued at a fundamental level that these RQF conservation laws are superior to conservation laws based on the local stress-energy-momentum tensor of matter because (1) they do not rely on spacetime symmetries and (2) they properly account for both matter and gravitational effects. Moreover, they provide simple, exact, operational expressions for fluxes of gravitational energy and linear …
The memory effect is a prediction of general relativity on the same footing as the existence of gravitational waves. The memory effect is understood at future null infinity as a transition induced by null radiation from a Poincaré vacuum to another vacuum. Those are related by a supertranslation, which is a fundamental symmetry of asymptotically flat spacetimes. In this essay, I argue that finite supertranslation diffeomorphisms should be extended into the bulk spacetime consistently with canonical charge conservation. It then leads to fascinating geometrical features of gravitational Poincaré…
The field-theoretical approach is reviewed. Perturbations in general relativity as well as in an arbitrary $D$-dimensional metric theory are studied on a background, which is a solution (arbitrary) of the theory. Lagrangian for perturbations is defined, and field equations for perturbations are derived from the variational principle. These equations are exact and equivalent to the equations in the standard formulation, but can be approximate also. The field-theoretical description is invariant under gauge (inner) transformations, which can be presented both in exact and approximate forms. Foll…
Starting from the Ashtekar Hamiltonian variables for general relativity, the self-dual Einstein equations (SDE) may be rewritten as evolution equations for three divergence free vector fields given on a three dimensional surface with a fixed volume element. From this general form of the SDE, it is shown how they may be interpreted as the field equations for a two dimensional field theory. It is further shown that these equations imply an infinite number of non-local conserved currents. A specific way of writing the vector fields allows an identification of the full SDE with those of the two di…
Emmy Noether proved two deep theorems, and their converses, on the connection between symmetries and conservation laws. Because these theorems are not in the mainstream of her scholarly work, which was the development of modern abstract algebra, it is of some historical interest to examine how she came to make these discoveries. The present paper is an historical account of the circumstances in which she discovered and proved these theorems which physicists refer to collectively as Noether's Theorem. The work was done soon after Hilbert's discovery of the variational principle which gives the …
In this paper, we make a close comparison of a covariant definition of an energy/entropy in general relativity, recently proposed by a collaboration including the present authors, with existing definitions of energies such as the one from the pseudo-tensor and the quasi-local energy. We show that existing definitions of energies in general relativity are conserved charges from the Noether's 2nd theorem for the general coordinate transformation, whose conservations are merely identities implied by the local symmetry and always hold without using equations of motion. Thus none of existing defini…
We study equal and unequal-mass neutron star mergers by means of new numerical relativity simulations in which the general relativistic hydrodynamics solver employs an algorithm that guarantees mass conservation across the refinement levels of the computational mesh. We consider eight binary configurations with total mass $M=2.7\,M_\odot$, mass-ratios $q=1$ and $q=1.16$, and four different equation of states (EOSs), and one configuration with a stiff EOS, $M=2.5M_\odot$ and $q=1.5$. We focus on the post-merger dynamics and study the merger remnant, dynamical ejecta and the postmerger gravitati…
E. Noether's general analysis of conservation laws has to be completed in a Lagrangian theory with local gauge invariance. Bulk charges are replaced by fluxes of superpotentials. Gauge invariant bulk charges may subsist when distinguished one-dimensional subgroups are present. As a first illustration we propose a new {\it Affine action} that reduces to General Relativity upon gauge fixing the dilatation (Weyl 1918 like) part of the connection and elimination of auxiliary fields. It allows a comparison of most gravity superpotentials and we discuss their selection by the choice of boundary cond…
