The M$ø$ller tetrad gravitational energy-momentum expression was recently evaluated for a small vacuum region using orthonormal frames adapted to Riemann normal coordinates. However the result was not proportional to the Bel-Robinson tensor $B_{αβμν}$. Treating a modified quasilocal expressions in a similar way, we found one unique combination that gives a multiple of $B_{αβμν}$ which provides a non-negative gravitational energy-momentum in the small sphere approximation. Moreover, in addition to $B_{αβμν}$, we found a certain tensor $S_{αβμν}+K_{αβμν}$ which gives the same "energy-momentum" density in vacuum. Using this tensor combination, we obtained an infinite set of solutions that provides a positive gravitational energy within the same limit.
Expression for the Witten-Nester 4-spinor 3-form of the Hamiltonian density of gravitational field in the asymptotically flat space-time in terms of the Sommers-Sen spinors, direct with a certain orthonormal three-frame connect, is obtained. A direct connection between the one and the ADM Hamiltonian density in the Sen-Witten frame is established on this basis.
In this thesis, I examine in detail the properties of rigid quasilocal frames (RQF), which have been proposed as a geometrically natural way to define spatially extended reference frames in general relativity. I also explore their usefulness, in particular, as a tool for constructing completely general conservation laws that do not rely on the presence of spacetime symmetries and include both matter and gravitational contributions without the need for any ad hoc structures such as pseudotensors. In doing so, I show how the RQF approach affords a deeper understanding of the nature of gravitatio…
Asymptotically flat gravitating systems have 10 conserved quantities, which lack proper local densities. It has been hoped that the teleparallel equivalent of Einstein's GR (TEGR, aka GR${}_{||}$) could solve this gravitational energy-momentum localization problem. Meanwhile a new idea: quasilocal quantities, has come into favor. The earlier quasilocal investigations focused on energy-momentum. Recently we considered quasilocal angular momentum for the teleparallel theory and found that the popular expression (unlike our ``covariant-symplectic'' one) gives the correct result only in a certain …
Early energy-momentum investigations for gravitating systems gave reference frame dependent pseudotensors; later the quasilocal idea was developed. Quasilocal energy-momentum can be determined by the Hamiltonian boundary term, which also identifies the variables to be held fixed on the boundary. We show that a pseudotensor corresponds to a Hamiltonian boundary term. Hence they are quasilocal and acceptable; each is the energy-momentum density for a definite physical situation with certain boundary conditions. These conditions are identified for well-known pseudotensors.
Chen and Nester proposed four boundary expressions for the quasilocal quantities using the covariant Hamiltonian formalism. Based on these four expressions, there is a simple generalization that one can consider, so that a two parameter set of boundary expressions can be constructed. Using these modified expressions, a nice result for gravitational energy-momentum can be obtained in holonomic frames.
We present a method for computing the flux of energy through a closed surface containing a gravitating system. This method, which is based on the quasilocal formalism of Brown and York, is illustrated by two applications: a calculation of (i) the energy flux, via gravitational waves, through a surface near infinity and (ii) the tidal heating in the local asymptotic frame of a body interacting with an external tidal field. The second application represents the first use of the quasilocal formalism to study a non-stationary spacetime and shows how such methods can be used to study tidal effects …
We introduce the notion of a rigid quasilocal frame (RQF) as a geometrically natural way to define a "system" in general relativity. An RQF is defined as a two-parameter family of timelike worldlines comprising the worldtube boundary of the history of a finite spatial volume, with the rigidity conditions that the congruence of worldlines is expansion-free (constant size) and shear-free (constant shape). This definition of a system is anticipated to yield simple, exact geometrical insights into the problem of motion in general relativity. It begins by answering the questions what is in motion (…
Recently Brown and York have devised a new method for defining quasilocal energy in general relativity. Their method yields expressions for the quasilocal energy and momentum surface densities associated with the two-boundary of a spacelike slice of a spatially bounded spacetime. These expressions are essentially Arnowitt-Deser-Misner variables, but with canonical conjugacy defined with respect to the time history of the two-boundary. This paper introduces Ashtekar-type variables on the time history of the two-boundary and shows that these variables lead to elegant alternative expressions for …
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 …
We investigate the thermodynamics of Einstein-Maxwell(-Dilaton) theory for an asymptotically flat spacetime in a quasilocal frame. We firstly define a quasilocal thermodynamic potential via the Euclidean on-shell action and formulate a quasilocal Smarr relation from Eulerian theorem. Then we calculate quasilocal energy and surface pressure by employing Brown-York quasilocal method along with Mann-Marolf counterterm and find entropy from the quasilocal thermodynamic potential. These quasilocal variables are consistent with Tolman temperature and the entropy in a quasilocal frame turns out to be…
We argue that conservation laws based on the local matter-only stress-energy-momentum tensor (characterized by energy and momentum per unit volume) cannot adequately explain a wide variety of even very simple physical phenomena because they fail to properly account for gravitational effects. We construct a general quasi}local conservation law based on the Brown and York total (matter plus gravity) stress-energy-momentum tensor (characterized by energy and momentum per unit area), and argue that it does properly account for gravitational effects. As a simple example of the explanatory power of …
From a covariant Hamiltonian formulation, by using symplectic ideas, we obtain certain covariant boundary expressions for the quasilocal quantities of general relativity and other geometric gravity theories. The contribution from each of the independent dynamic geometric variables (the frame, metric or connection) has two possible covariant forms associated with the selected type of boundary condition. The quasilocal expressions also depend on a reference value for each dynamic variable and a displacement vector field. Integrating over a closed two surface with suitable choices for the vector …
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…
The Hamiltonian of a gravitational system defined in a region with boundary is quantized. The classical Hamiltonian, and starting point for the regularization, is required by functional differentiablity of the Hamiltonian constraint. The boundary term is the quasilocal energy of the system and becomes the ADM mass in asymptopia. The quantization is carried out within the framework of canonical quantization using spin networks. The result is a gauge invariant, well-defined operator on the Hilbert space induced from the state space on the whole spatial manifold. The spectrum is computed. An alte…
We study the duality of quasilocal energy and charges with non-orthogonal boundaries in the (2+1)-dimensional low-energy string theory. Quasilocal quantities shown in the previous work and some new variables arisen from considering the non-orthogonal boundaries as well are presented, and the boost relations between those quantities are discussed. Moreover, we show that the dual properties of quasilocal variables such as quasilocal energy density, momentum densities, surface stress densities, dilaton pressure densities, and Neuve-Schwarz(NS) charge density, are still valid in the moving observe…
Traditional approaches to energy-momentum localization led to reference frame dependent pseudotensors. The more modern idea is quasilocal energy-momentum. We take a Hamiltonian approach. The Hamiltonian boundary term gives not only the quasilocal values but also boundary conditions via the Hamiltonian variation boundary principle. Selecting a Hamiltonian boundary term involves several choices. We found that superpotentials can serve as Hamiltonian boundary terms, consequently pseudotensors are actually quasilocal and legitimate. Various Hamiltonian boundary term quasilocal expressions are cons…
In a recent paper [D. Philipp, V. Perlick, D. Puetzfeld, E. Hackmann, and C. Lämmerzahl, Phys. Rev. D 95, 104037 (2017)], a definition of a general-relativistic geoid, restricted to stationary spacetimes, was presented in terms of "isochronometric surfaces". In this note, we explicate how this definition is just a special, highly restrictive case of our earlier formulation of the general-relativistic geoid using quasilocal frames, which is valid for generic, non-stationary spacetimes [M. Oltean, R.J. Epp, P.L. McGrath, and R.B. Mann, Class. Quantum Grav. 33, 105001 (2016)]. Moreover, like the …
The notion of a rigid quasilocal frame (RQF) provides a geometrically natural way to define a system in general relativity, and a new way to analyze the problem of motion. An RQF is defined as a two-parameter family of timelike worldlines comprising the boundary (topologically R x S^2) of the history of a finite spatial volume, with the rigidity conditions that the congruence of worldlines be expansion- and shear-free. In other words, the size and shape of the system do not change. In previous work, such systems in Minkowski space were shown to admit precisely the same six degrees of freedom o…
Spin-polarised cylindrically symmetric solution are shown not to be compatible with teleparallel gravity.This can be done in two distinct manners.The first is to show that not all components of the orthonormal tetrad (OT).It is argue however that this result maybe done by a bad choice of the spin distribution along the cylinder.A cylindrically symmetric Riemannian solution is obtained which represents a conical geometry of defects.
Carlip has shown that the entropy of the three-dimensional black hole has its origin in the statistical mechanics of microscopic states living at the horizon. Beginning with a certain orthonormal frame action, and applying similar methods, I show that an analogous result extends to the (Euclidean) black hole in any spacetime dimension. However, this approach still faces many interesting challenges, both technical and conceptual.
By adopting Nester's higher dimensional special orthonormal frames (HSOF) the tetrad equations for vacuum gravity are put into first order symmetric hyperbolic (FOSH) form with constant coefficients, independent of any time slicing or coordinate specialization.
We develop, in the context of general relativity, the notion of a geoid -- a surface of constant "gravitational potential". In particular, we show how this idea naturally emerges as a specific choice of a previously proposed, more general and operationally useful construction called a quasilocal frame -- that is, a choice of a two-parameter family of timelike worldlines comprising the worldtube boundary of the history of a finite spatial volume. We study the geometric properties of these geoid quasilocal frames, and construct solutions for them in some simple spacetimes. We then compare these …
The equations of the 1+3 orthonormal frame approach are explicitly presented and discussed. Natural choices of local coordinates are mentioned. A dimensionless formulation is subsequently given. It is demonstrated how one can obtain a number of interesting problems by specializing the general equations. In particular, equation systems for ``silent'' dust cosmological models also containing magnetic Maxwell fields, locally rotationally symmetric spacetime geometries and spatially homogeneous cosmological models are presented. We show that while the 3-Cotton--York tensor is zero for Szekeres dus…
Within a first-order framework, we comprehensively examine the role played by boundary conditions in the canonical formulation of a completely general two-dimensional gravity model. Our analysis particularly elucidates the perennial themes of mass and energy. The gravity models for which our arguments are valid include theories with dynamical torsion and so-called generalized dilaton theories (GDTs). Our analysis of the canonical action principle (i) provides a rigorous correspondence between the most general first-order two-dimensional Einstein-Cartan model (ECM) and GDT and (ii) allows us to…
Doubly special relativity (DSR) is an effective model for encoding quantum gravity in flat spacetime. To incorporate DSR into general relativity, one could use "Gravity's rainbow", where the spacetime background felt by a test particle would depend on its energy. For a black hole, there are two natural orthonormal frames, the stationary one hovering above it and freely falling one along geodesics. Since the rainbow metric is the metric that the radiated particles "see", a more natural orthonormal frame is the one anchored to the particles. And the cases with the stationary orthonormal frame ha…
In previous paper, we present an SO(3) Wu-Yang-like Kaluza-Klein dyon so- lution satisfies the Einstein equation in the seven-dimensional spacetimes. In this note, we will show an alternative approach using an orthonormal frame, the Cartan's structure equations, and calculating the affine spin connection one-form,curvature tensor and Ricci tensor. The results from these two different methods are coincident.
The purpose of the classical Einstein and Landau-Lifshitz pseudotensors is for determining the gravitational energy. Neither of them can guarantee a positive energy in holonomic frames. In the small sphere approximation, it has been required that the quasilocal expression for the gravitational energy-momentum density should be proportional to the Bel-Robinson tensor $B_{αβμν}$. However, we propose a new tensor $V_{αβμν}$ which is the sum of certain tensors $S_{αβμν}$ and $K_{αβμν}$, it has certain properties so that it gives the same gravitational "energy-momentum" content as $B_{αβμν}$ does. …
Classical black holes and event horizons are highly non-local objects, defined in terms of the causal past of future null infinity. Alternative, (quasi)local definitions are often used in mathematical, quantum, and numerical relativity. These include apparent, trapping, isolated, and dynamical horizons, all of which are closely associated to two-surfaces of zero outward null expansion. In this paper we show that three-surfaces which can be foliated with such two-surfaces are suitable boundaries in both a quasilocal action and a phase space formulation of general relativity. The resulting forma…
The energy-momentum for a gravitating system can be considered by the tetard teleparalle gauge current in orthonormal frames. Whereas the Einstein pseudotensor used holonomic frames. Tetrad expression itself gives a better result for gravitational energy than Einstein's. Inspired by an idea of Deser, we found a gravitational energy expression which enjoys the positive energy property by combining the tetrad expression and the Einstein pseudotensor, i.e., the connection coefficient has a form appropriate to a suitable intermediate between orthonormal and holonomic frames.
A quasilocal framework of stationary and dynamical untrapped hypersurfaces is introduced to generalize the notions of energy and angular momentum of isolated and dynamical trapping horizons to general strong gravitating systems.
We discuss some specializations of the frames of flat orthonormal frame bundles over geometries of indefinite signature, and the resulting symmetries of families of embedded Riemannian or pseudo-Riemannian geometries. The specializations are closed sets of linear constraints on the connection 1-forms of the framing. The embeddings can be isometric, as in minimal surfaces or Regge-Teitelboim gravity, or torsion-free, as in Einstein vacuum gravity. Involutive exterior differential systems are given, and their Cartan character tables calculated to express the well-posedness of the underlying part…
The deformation equation of a spacelike submanifold with an arbitrary codimension is given by a general construction without using local frames. In the case of codimension-1, this equation reduces to the evolution equation of the extrinsic curvature of a spacelike hypersurface. In the more interesting case of codimension-2, after selecting a local null frame, this deformation equation reduces to the well known (cross) focusing equations. We show how the thermodynamics of trapping horizons is related to these deformation equations in two different formalisms: with and without introducing quasil…
The basic and fundamental aspects of General Relativity are in general analysed in mathematical level of coordinate basis or holonomic frame by several authors in the literature. However, for many purposes it is more convenient to use a general basis, often called in four dimensions, a tetrad or vierbein, very useful in a local frame with orthonormal basis or pseudo-orthonormal basis. This text presents an introduction to non-coordinate basis and the two Cartan's structure equations that are mathematical implements in Riemannian geometry that facilitate the calculation of curvature tensors. Th…
Doubly special relativity (DSR) is an effective model for encoding quantum gravity in flat spacetime. To incorporate DSR into general relativity, one could use "Gravity's rainbow", where the spacetime background felt by a test particle would depend on its energy. In this scenario, one could rewrite the rainbow metric $g_{μν}\left( E\right) $ in terms of some orthonormal frame fields and use the modified equivalence principle to determine the energy dependence of $g_{μν}\left( E\right) $. Obviously, the form of $g_{μν}\left( E\right) $ depends on the choice of the orthonormal frame. For a stati…
Three natural classes of orthonormal frames, namely Frenet-Serret, Fermi-Walker and parallel transported frames, exist along any timelike world line in spacetime. Their relationships are investigated for timelike circular orbits in stationary axisymmetric spacetimes, and illustrated for black hole spacetimes.
Considering gravitational collapse of matter, it is important problem to clarify what kind of conditions leads to the formation of naked singularity. For this purpose, we apply the 1+3 orthonormal frame formalism introduced by Uggla \textit{et al.} to spherically symmetric gravitational collapse of perfect fluid. This formalism allows us to construct an autonomous system of evolution and constraint equations for scale-invariant dynamical variables normalized by the volume expansion rate of the timelike orthonormal frame vector. We investigate the asymptotic evolution of such dynamical variable…
Moller's tetrad gravitational energy-momentum "tensor" is evaluated for a small vacuum region using an orthonormal frame adapted to Riemann normal coordinates. We find that it does satisfy the highly desired property of being a positive multiple of the Bel-Robinson tensor.
We use an orthonormal frame approach to provide a general framework for the first order hyperbolic reduction of the Einstein equations coupled to a fairly generic class of matter models. Our analysis covers the special cases of dust and perfect fluid. We also provide a discussion of self-gravitating elastic matter. The frame is Fermi-Walker propagated and coordinates are chosen such as to satisfy the Lagrange condition. We show the propagation of the constraints of the Einstein-matter system.
A class of gauges for the Einstein vacuum equations is introduced, along with three symmetric hyperbolic systems. The first implies the local realizability of the gauge. The second is the dynamical subset of the field equations. The third is used to show that the constraints propagate. The gauges are for an orthonormal frame formalism, with first order, quadratically nonlinear equations. The unknowns are 16 frame components and 28 connection components. After gauge-fixing, a total of 33 remain.
We present two families of exterior differential systems (EDS) for causal embeddings of orthonormal frame bundles over Riemannian spaces of dimension q = 2,3,4,5.. into orthonormal frame bundles over flat spaces of higher dimension. We calculate Cartan characters showing that these EDS are dynamical field theories. The first family includes a new non-isometric embedding EDS for classical Einstein vacuum relativity (q = 4). The second, generated only by 2-forms, is a family of classical "stringy" or Kaluza-type (q = 5) integrable systems. Cartan forms are found for all these dynamical systems.
A one-loop correction of the quasilocal energy in the Schwarzschild background, with flat space as a reference metric, is performed by means of a variational procedure in the Hamiltonian framework. We examine the graviton sector in momentum space, in the lowest possible state. An application to the black hole pair creation via the Casimir energy is presented. Implications on the foam-like scenario are discussed.
We present two families of exterior differential systems (EDS) for non-isometric embeddings of orthonormal frame bundles over Riemannian spaces of dimension q = 2, 3, 4, 5.... into orthonormal frame bundles over flat spaces of sufficiently higher dimension. We have calculated Cartan characters showing that these EDS satisfy Cartan's test and are well-posed dynamical field theories. The first family includes a constant-coefficient (cc) EDS for classical Einstein vacuum relativity (q = 4). The second family is generated only by cc 2-forms, so these are integrable (but nonlinear) systems of parti…
