We apply the 1+1+2 covariant approach to describe a general static and spherically symmetric relativistic stellar object which contains two interacting fluids. We then use the 1+1+2 equations to derive the corresponding Tolman-Oppenheimer-Volkoff (TOV) equations in covariant form in the isotropic, non-interacting case. These equations are used to obtain new exact solutions by means of direct resolution and reconstruction techniques. Finally, we show that the generating theorem known for the single fluid case can also be used to obtain two-fluid solutions from single fluid ones.
Phase transitions with spontaneous symmetry breaking and vector order parameter are considered in multidimensional theory of general relativity. Covariant equations, describing the gravitational properties of topological defects, are derived. The topological defects are classified in accordance with the symmetry of the covariant derivative of the vector order parameter. The abilities of the derived equations are demonstrated in application to the brane world concept. New solutions of the Einstein equations with a transverse vector order parameter are presented. In the vicinity of phase transit…
Traditional derivations of general relativity from the graviton degrees of freedom assume space-time Lorentz covariance as an axiom. In this essay, we survey recent evidence that general relativity is the unique spatially-covariant effective field theory of the transverse, traceless graviton degrees of freedom. The Lorentz covariance of general relativity, having not been assumed in our analysis, is thus plausibly interpreted as an accidental or emergent symmetry of the gravitational sector. From this point of view, Lorentz covariance is a necessary feature of low-energy graviton dynamics, not…
We derive a coordinate-independent formulation of the post-1-Newtonian approximation to general relativity. This formulation is a generalization of the Newton-Cartan geometric formulation of Newtonian gravity. It involves several fields and a connection, but no spacetime metric at the fundamental level. We show that the usual coordinate-dependent equations of post-Newtonian gravity are recovered when one specializes to asymptotically flat spacetimes and to appropriate classes of coordinates.
We propose a covariant entropy bound in gravitational theories beyond general relativity (GR), using Wald-Jacobson-Myers entropy instead of Bekenstein-Hawking entropy. We first extend the proof of the bound known in 4-dimensional GR to D-dimensional GR, f(R) gravity and canonical scalar-tensor theory. We then consider Einstein-Gauss-Bonnet (EGB) gravity as a more non-trivial example and, under a set of reasonable assumptions, prove the bound in the GR branch of spherically symmetric configurations. As a corollary, it is shown that under the null and dominant energy conditions, the generalized …
We present Hamilton's equations for the teleparallel equivalent of general relativity (TEGR), which is a reformulation of general relativity based on a curvatureless, metric compatible, and torsionful connection. For this, we consider the Hamiltonian for TEGR obtained through the vector, antisymmetric, symmetric and trace-free, and trace irreducible decomposition of the phase space variables. We present the Hamiltonian for TEGR in the covariant formalism for the first time in the literature, by considering a spin connection depending on Lorentz matrices. We introduce the mathematical formalism…
Creating a modified Belinfante/Rosenfeld procedure, Mathisson-Papapetrou-like equations are derived by which a comparison of General-Covariant Continuum Physics with General Relativity Theory becomes possible.
An action principle of singular hypersurfaces in general relativity and scalar-tensor type theories of gravity in the Einstein frame is presented without assuming any symmetry. The action principle is manifestly doubly covariant in the sense that coordinate systems on and off a hypersurface are disentangled and can be independently specified. It is shown that, including variation of the metric, the position of the hypersurface and matter fields, the variational principle gives the correct set of equations of motion: the Einstein equation off the hypersurface, Israel's junction condition in a d…
The covariant Hamilton-Jacobi equation for the Teleparallel Equivalent of General Relativity is derived based on the analysis of the second-class constraints within the covariant Hamiltonian theory of De Donder-Weyl according to the constraints algorithm developed by Kanatchikov.
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 non-relativistic covariant framework for fields is extended to investigate fields and fluids on scale covariant curved backgrounds. The scale covariant Newton-Cartan background is constructed using the localization of spacetime symmetries of non-relativistic fields in flat space. Following this, we provide a Weyl covariant formalism which can be used to study scale invariant fluids. By considering ideal fluids as an example, we describe its thermodynamic and hydrodynamic properties and explicitly demonstrate that it satisfies the local second law of thermodynamics. As a further application…
Numerical integration of the field equations in bimetric relativity is necessary to obtain solutions describing realistic systems. Thus, it is crucial to recast the equations as a well-posed problem. In general relativity, under certain assumptions, the covariant BSSN formulation is a strongly hyperbolic formulation of the Einstein equations, hence its Cauchy problem is well-posed. In this paper, we establish the covariant BSSN formulation of the bimetric field equations. It shares many features with the corresponding formulation in general relativity, but there are a few fundamental differenc…
We introduce a naturally-defined totally invariant spacetime energy expression for general relativity incorporating the contribution from gravity. The extension links seamlessly to the action integral for the gravitational field. The demand that the general expression for arbitrary systems reduces to the Tolman integral in the case of stationary bounded distributions, leads to the matter-localized Ricci integral for energy-momentum in support of the energy localization hypothesis. The role of the observer is addressed and as an extension of the special relativistic case, the field of observers…
We present a manifestly Lorentz-covariant description of the phase space of general relativity with the Immirzi parameter. This formulation emerges after solving the second-class constraints arising in the canonical analysis of the Holst action. We show that the new canonical variables give rise to other Lorentz-covariant parametrizations of the phase space via canonical transformations. The resulting form of the first-class constraints in terms of new variables is given. In the time gauge, these variables and the constraints become those found by Barbero.
A general covariant extension of Einstein\'{}s field equations is considered with a view to Numerical Relativity applications. The basic variables are taken to be the metric tensor and an additional four-vector $Z_μ$. Einstein's solutions are recovered when the additional four-vector vanishes, so that the energy and momentum constraints amount to the covariant algebraic condition $Z_μ=0$. The extended field equations can be supplemented by suitable coordinate conditions in order to provide symmetric hyperbolic evolution systems: this is actually the case for either harmonic coordinates or norm…
We investigate the covariant Hamiltonian symplectic structure of General Relativity for spatially bounded regions of spacetime with a fixed time-flow vector. For existence of a well-defined Hamiltonian variational principle taking into account a spatial boundary, it is necessary to modify the standard Arnowitt-Deser-Misner Hamiltonian by adding a boundary term whose form depends on the spatial boundary conditions for the gravitational field. The most general mathematically allowed boundary conditions and corresponding boundary terms are shown to be determined by solving a certain equation obta…
I illustrate a simple hamiltonian formulation of general relativity, derived from the work of Esposito, Gionti and Stornaiolo, which is manifestly 4d generally covariant and is defined over a finite dimensional space. The spacetime coordinates drop out of the formalism, reflecting the fact that they are not related to observability. The formulation can be interpreted in terms of Toller's reference system transformations, and provides a physical interpretation for the spinnetwork to spinnetwork transition amplitudes computable in principle in loop quantum gravity and in the spin foam models.
The problem of formulating a manifest covariant Hamiltonian theory of General Relativity in the presence of source fields is addressed, by extending the so-called "DeDonder-Weyl" formalism to the treatment of classical fields in curved space-time. The theory is based on a synchronous variational principle for the Einstein equation, formulated in terms of superabundant variables. The technique permits one to determine the continuum covariant Hamiltonian structure associated with the Einstein equation. The corresponding continuum Poisson bracket representation is also determined. The theory reli…
A general covariant extension of Einstein's field equations is considered with a view to Numerical Relativity applications. The basic variables are taken to be the metric tensor and an additional four-vector. The extended field equations, when supplemented by suitable coordinate conditions, determine the time evolution of all these variables without any constraint. Einstein's solutions are recovered when the additional four-vector vanishes, so that the energy and momentum constraints hold true. The extended system is well posed when using the natural extension of either harmonic coordinates or…
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 revisit the manifestly covariant large $c$ expansion of General Relativity, $c$ being the speed of light. Assuming the relativistic connection has no pole in $c^{-2}$, this expansion is known to reproduce Newton-Cartan gravity and a covariant version of Post-Newtonian corrections to it. We show that relaxing this assumption leads to the inclusion of twistless torsion in the effective non-relativistic theory. We argue that the resulting TTNC theory is an effective description of a non-relativistic regime of General Relativity that extends Newtonian physics by including strong gravitational t…
The elliptic system of equations, which is general-covariant and locally SU(2)-covariant, is investigated. The new condition of the Dirichlet problem solvability and the condition of zeros absence for solutions are obtained for this system, which contains in particular case the Sen-Witten equation. On this basis it is proved the existence of the wide class of hypersurfaces, in all points of which there exists a correspondence between the Sen-Witten spinor field and three-frame, which generalizes the Nester orthoframe. The Nester special orthoframe also exists on a certain subclass containing n…
The variation of the energy for a gravitational system is directly defined from the Hamiltonian field equations of General Relativity. When the variation of the energy is written in a covariant form it splits into two (covariant) contributions: one of them is the Komar energy, while the other is the so-called covariant ADM correction term. When specific boundary conditions are analyzed one sees that the Komar energy is related to the gravitational heat while the ADM correction term plays the role of the Helmholtz free energy. These properties allow to establish, inside a classical geometric fr…
The oft-neglected issue of the causal structure in the flat spacetime approach to Einstein's theory of gravity is considered. Consistency requires that the flat metric's null cone be respected, but this does not happen automatically. After reviewing the history of this problem, we introduce a generalized eigenvector formalism to give a kinematic description of the relation between the two null cones, based on the Segre' classification of symmetric rank 2 tensors with respect to a Lorentzian metric. Then we propose a method to enforce special relativistic causality by using the naive gauge free…
Diffeomorphism-induced symmetry transformations and time evolution are distinct operations in generally covariant theories formulated in phase space. Time is not frozen. Diffeomorphism invariants are consequently not necessarily constants of the motion. Time-dependent invariants arise through the choice of an intrinsic time, or equivalently through the imposition of time-dependent gauge fixation conditions. One example of such a time-dependent gauge fixing is the Komar-Bergmann use of Weyl curvature scalars in general relativity. An analogous gauge fixing is also imposed for the relativistic f…
In this short review, we discuss the approach of the commutator algebra of covariant derivative to analyse the gravitational theories, starting from the standard Einstein's general theory of relativity and focusing on the Rastall theory. After that, we discuss the important role of the torsion in this mathematical framework. In the Appendix of the paper we analyse the importance of the nascent gravitational wave astronomy as a tool to discriminate among the general theory of relativity and alternative theories of gravity.
In our previous paper, we have presented a covariant BRST quantization of unimodular gravity which may account for the smallness of the cosmological constant, and have shown that the physical degrees of freedom in the theory are the same as general relativity. The formulation has been given by using rank-2 antisymmetric tensor fields for both ghosts and antighosts. Here we give an alternative formulation using a vector field for the antighost but keeping the same structure for the ghosts. This gives a significantly simpler covariant quantization with less ghosts and no tripole modes in the gho…
Hamiltonian gravity, relying on arbitrary choices of "space," can obscure spacetime symmetries. We present an alternative, manifestly spacetime covariant formulation that nonetheless distinguishes between "spatial" and "temporal" variables. The key is viewing dynamical fields from the perspective of a field of observers -- a unit timelike vector field that also transforms under local Lorentz transformations. On one hand, all fields are spacetime fields, covariant under spacetime symmeties. On the other, when the observer field is normal to a spatial foliation, the fields automatically fall int…
We give a fully covariant energy momentum stress tensor for the gravitational field which is easily physically motivated, and which leads to a very general derivation of the Einstein equation for gravity. We do not need to assume any property of the surce matter fields' energy momentum stress tensor other than symmetry. We give a physical motivation for this choice using laser light pressure. As a consequence of our derivation, the energy momentum stress tensor for the total source matter and fields must be divergence free, when spacetime is 4 dimensional. Moreoverr, if the total source matter…
The different forms of the Hamiltonian formulations of linearized General Relativity/spin-two theories are discussed in order to show their similarities and differences. It is demonstrated that in the linear model, non-covariant modifications to the initial covariant Lagrangian (similar to those modifications used in full gravity) are in fact unnecessary. The Hamiltonians and the constraints are different in these two formulations but the structure of the constraint algebra and the gauge invariance derived from it are the same. It is shown that these equivalent Hamiltonian formulations are rel…
The Weyssenhoff fluid is a perfect fluid with spin where the spin of the matter fields is the source of torsion in an Einstein-Cartan framework. Obukhov and Korotky showed that this fluid can be described as an effective fluid with spin in general relativity. A dynamical analysis of such a fluid is performed in a gauge invariant manner using the 1+3 covariant approach. This yields the propagation and constraint equations for the set of dynamical variables. A verification of these equations is performed for the special case of irrotational flow with zero peculiar acceleration by evolving the co…
We give an SU(2) covariant representation of the constraints of Euclidean general relativity in the Ashtekar variables. The guiding principle is the use of triads to transform all free spatial indices into SU(2) indices. A central role is played by a special covariant derivative. The Gauss, diffeomorphism and Hamiltonian constraints become purely algebraic restrictions on the curvature and the torsion associated with this connection. We introduce coordinates on the jet space of the dynamical fields which cleanly separate the constraint and gauge directions from the true physical directions. Th…
This thesis is a review of backreaction, which is a possible alternative to dark energy and modified gravity. We describe in detail the 3+1 covariant formalism and Frobenius' theorem. We present the averaging procedure developed by Buchert, the Buchert equations and the generalization of these equations to the case of general matter. We then generalize to arbitrary number of spatial dimensions. We focus on the case of 2+1 dimensions, where the relation between the topology and the geometry of a surface imposes a global constraint on backreaction.
We construct a generally-covariant formulation of non-equilibrium thermodynamics in General Relativity. We find covariant entropic forces arising from gradients of the entropy density, and a corresponding non-conservation of the energy momentum tensor in terms of these forces. We also provide a Hamiltonian formulation of General Relativity in the context of non-equilibrium phenomena and write the Raychaudhuri equations for a congruence of geodesics. We find that a fluid satisfying the strong energy condition could avoid collapse for a positive and sufficiently large entropic-force contribution…
We carry out the canonical analysis of a covariant version of unimodular gravity in terms of the connection representation. We then proceed to quantize this theory by implementing the Dirac procedure. We confirm whether and how the Kodama state, which is a solution of quantum general relativity, can be extended into covariant unimodular gravity. Finally, we discuss the difference of quantum states between covariant unimodular gravity, the original unimodular gravity, and general relativity.
We argue that there is nothing puzzling in the fact that the Hamiltonian formulation of a covariant theory, General Relativity, after a non-covariant change of field variables is not canonically related to the formulation based on the original variable, the metric tensor. Were such a puzzle to be "solved" it would lead to the conclusion that a covariant theory can be converted into a non-covariant one in many different ways and without consequence. The non-canonicity of transformations from covariant to non-covariant variables shows the need to work in the original variables so as to be able t…
We use the formalism developed by Wald and Zoupas to derive explicit covariant expressions for the charges and fluxes associated with the Bondi-Metzner-Sachs symmetries at null infinity in asymptotically flat spacetimes in vacuum general relativity. Our expressions hold in non-stationary regions of null infinity, are local and covariant, conformally-invariant, and are independent of the choice of foliation of null infinity and of the chosen extension of the symmetries away from null infinity. While similar expressions have appeared previously in the literature in Bondi-Sachs coordinates (to wh…
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.
It is shown that Schwarzschild black hole and de Sitter solutions exist as exact solutions of a recently proposed relativistic covariant formulation of (power-counting) renormalizable gravity with a fluid. The formulation without a fluid is also presented here. The stability of the solutions is studied and their corresponding entropies are computed, by using the covariant Wald method. The area law is shown to hold both for the Schwarzschild and for the de Sitter solutions found, confirming that, for the $β=1$ case, one is dealing with a minimal modification of GR.
For a static, perfect fluid sphere with a general equation of state, we obtain the relativistic equation of hydrostatic equilibrium, namely the Tolman-Oppenheimer-Volkov equation, as the thermodynamical equilibrium in the microcanonical, as well as the canonical, ensemble. We find that the stability condition determined by the second variation of entropy coincides with the dynamical stability condition derived by variations to first order in the dynamical Einstein's equations. Thus, we show the equivalence of microcanonical thermodynamical stability with linear dynamical stability for a static…
We consider various forms of action allowing us to describe a perfect fluid in the framework of General Relativity. First we consider a potential motion without pressure. Starting from an action in terms of the current density vector built similarly to the action of a single particle, we build a series of equivalent actions through various changes of variables. The results are then generalized to the cases of motion with vorticity and the presence of pressure. The obtained forms of action are compared to the previously known variants. We discuss the possibilities to use the obtained results to…
This note is to bring to the reader's attention the fact that general relativity and quantum mechanics differ from each other in one main aspect. General relativity is based on the diffeomorphism covariant formulation of the laws of physics while quantum mechanics is constructed such that its fundamental laws remain invariant to a change of topology. It is the goal of this paper to show that in order to obtain a complete description of quantum gravity one has to extend the principle of diffeomorphism invariance from general relativity in the sense of quantum mechanics i.e. the laws of physics …
We show the first simple, systematic and direct approach to decoupling gravitational sources in general relativity. As a direct application, a robust and simple way to generate anisotropic solutions for self-gravitating systems from perfect fluid solutions is presented.
The scalar-tensor theories have become popular recently in particular in connection with attempts to explain present accelerated expansion of the universe, but they have been considered as a natural extension of general relativity long time ago. The Horndeski scalar-tensor theory involving four invariantly defined Lagrangians is a natural choice since it implies field equations involving at most second derivatives. Following the formalisms of defining covariant global quantities and conservation laws for perturbations of spacetimes in standard general relativity we extend these methods to the …
We introduce a Lorentz-covariant description of tachyons, free of inconsistencies. Our approach is based on an appropriate extension of the special relativity beyond the light barrier, owing to the freedom of synchronization of distant clocks.
Unimodular gravity (UG) is an important theory which may explain the smallness of the cosmological constant. To get insight into the covariant quantization of UG, we discuss the BRST quantization of General Relativity (GR) with a cosmological constant in the unimodular gauge. We develop a novel way to gauge fix the transverse diffeomorphism (TDiff) and then further to fulfill the unimodular gauge. This process requires the introduction of an additional pair of BRST doublets which decouple from the physical sector together with the other three pairs of BRST doublets for the TDiff. We show that …
In continuum physics is presupposed that general-relativistic balance equations are valid which are created from the Lorentz-covariant ones by application of the equivalence principle. Consequently, the question arises, how to make these general-covariant balances compatible with Einstein's field equations. The compatibility conditions are derived by performing a modified Belinfante-Rosenfeld symmetrization for the non-symmetric and not divergence-free general-relativistic energy-momentum tensor. The procedure results in the Mathisson-Papapetrou equations.
Cyclic universes with bouncing solutions are candidates for solving the big bang initial singularity problem. Here we seek bouncing solutions in a modified Gauss-Bonnet gravity theory, of the type $R+f(G)$, where $R$ is the Ricci scalar, $G$ is the Gauss-Bonnet term, and $f$ some function of it. In finding such a bouncing solution we resort to a technique that reduces the order of the differential equations of the $R+f(G)$ theory to second order equations. As general relativity is a theory whose equations are of second order, this order reduction technique enables one to find solutions which a…
