In this work we use generalized deformed gauge groups for investigation of symmetry of general relativity (GR). GR is formulated in generalized reference frames, which are represented by (anholonomic in general case) affine frame fields. The general principle of relativity is extended to the requirement of invariance of the theory with respect to transitions between generalized reference frames, that is, with respect to the group $GL^g$ of local linear transformations of affine frame fields. GR is interpreted as the gauge theory of the gauge group of translations $T^g_M$, and therefore is invariant under the space-time diffeomorphisms. The groups $GL^g$ and $T^g_M$ are united into group $S^g_M$, which is their semidirect product and is the complete symmetry group of the general relativity in an affine frame (GRAF).
The consequence of $GL^g$-invariance of GRAF is the Palatini equation, which in the absence of torsion goes into the metricity condition, and vice versa, that is, is fulfilled identically in the Riemannian space. The consequence of the $T^g_M$-invariance of GRAF is representation of the Einstein equation in the superpotential form, that is, in the form of dynamic Maxwell equations (or Young-Mills equations). Deformation of the group $S^g_M$ leads to renormalisation of energy-momentum of the gravitation field. At the end we show that by limiting admissible reference frames (by $GL^g$-gauge fixing) from GRAF, in addition to Einstein gravity, one can obtain other local equivalent formulations of GR: general relativity in an orthonormal frame or teleparallel equivalent of general relativity, dilaton gravity, unimodular gravity, etc.
The role of the SO(2,1) symmetry in General Relativity is analyzed. Cosmological solutions of Einstein field equations invariant with respect to a space-like Lie algebra G_r, with r between 3 and 6 and containing so(2,1) as a subalgebra, are also classified.
Within the framework of projective geometry, we investigate kinematics and symmetry in $(α,β)$ spacetime-one special types of Finsler spacetime. The projectively flat $(α,β)$ spacetime with constant flag curvature is divided into four types. The symmetry in type A-Riemann spacetime with constant sectional curvature is just the one in de Sitter special relativity. The symmetry in type B-locally Minkowski spacetime is just the one in very special relativity. It is found that type C-Funk spacetime and type D-scaled Berwald's metric spacetime both possess the Lorentz group as its isometric group. …
From the principle of relativity with two universal invariant parameters $c$ and $l$, 24 possible kinematical (including geometrical and static) algebras can be obtained. Each algebra is of 10 dimensional, generating the symmetry of a 4 dimensional homogeneous space-time or a pure space. In addition to the ordinary Poincaré algebra, there is another Poincaré algebra among the 24 algebras. New 4d geometries with the new Poincaré symmetry are presented. The motion of free particles on one of the new space-times is discussed.
We investigate the interplay and connections between symmetry properties of equations, the interpretation of coordinates, the construction of observables, and the existence of physical relativity principles in spacetime theories. Using the refined notion of an event as a ``point-coincidence'' between scalar fields that completely characterise a spacetime model, we also propose a natural generalisation of the relational local observables that does not require the existence of four everywhere invertible scalar fields. The collection of all point-coincidences forms in generic situations a four-di…
A new class of time-symmetric solutions to the initial value constraints of vacuum General Relativity is introduced. These data are globally regular, asymptotically flat (with possibly several asymptotic ends) and in general have no isometries, but a $U(1)\times U(1)$ group of conformal isometries. After decomposing the Lichnerowicz conformal factor in a double Fourier series on the group orbits, the solutions are given in terms of a countable family of uncoupled ODEs on the orbit space.
With a more relaxed perspective on what constitutes a relativity symmetry mathematically, we revisit the notion of possible relativity or kinematic symmetries mutually connected through Lie algebra contractions. We focus on the contractions of an $SO(m,n)$ symmetry as a relativity symmetry on an $m+n$ dimension geometric arena, which generalizes the notion of spacetime, and discuss systematically contractions that reduce the dimension one at a one, aiming at going one step beyond what has been discussed in the literature. Our key results are five different contractions of a Galilean-type symme…
The observed accelerated cosmic expansion is problematic in that it seems to require an otherwise unobserved dark energy for its origin. A possible alternative explanation has been recently given, which attempts to account for this expansion in terms of a hypothesized matter-anti-matter repulsion. This repulsion or anti-gravity is derived by applying the CPT theorem to general relativity. We show that this proposal cannot work for two reasons: 1) it incorrectly predicts the behavior of photons and 2) the CPT transformation itself is not consistently applied.
In this paper, we have searched the existence of the similarity solution for plane symmetric inhomogeneous cosmological models in general relativity. The matter source consists of perfect fluid with proportionality relation between expansion scalar and shear scalar. The isovector fields of Einstein's field equation for the models under consideration are derived. A new class of exact solutions of Einstein's field equation have been obtained for inhomogeneous space-time. The physical behaviors and geometric aspects of the derived models have been discussed in detail.
We propose a generalization of the isometry transformations to the geometric context of the field theories with spin where the local frames are explicitly involved. We define the external symmetry transformations as isometries combined with suitable tetrad gauge transformations and we show that these form a group which is locally isomorphic with the isometry one. We point out that the symmetry transformations that leave invariant the equations of the fields with spin have generators with specific spin terms which represent new physical observables. The examples we present are the generators of…
Gravitational properties of a hedge-hog type topological defect in two extra dimensions are considered in General Relativity employing a vector as the order parameter. All previous considerations were done using the order parameter in the form of a multiplet in a target space of scalar fields. The difference of these two approaches is analyzed and demonstrated in detail. Regular solutions of the Einstein equations are studied analytically and numerically. It is shown that the existence of a negative cosmological constant is sufficient for the spontaneous symmetry breaking of the initially plai…
We give a derivation of general relativity and the gauge principle that is novel in presupposing neither spacetime nor the relativity principle. We consider a class of actions defined on superspace with two key properties. The first is 3-coordinate invariance. This is the only postulated symmetry, and it leads to the standard momentum constraint. The second property is that the Lagrangian is constructed from a `local' square root of an expression quadratic in the velocities, `local' because it is taken before integration over 3-space. It gives rise to quadratic constraints that do not correspo…
We reformulate the general theory of relativity in the language of Riemann-Cartan geometry. We start from the assumption that the space-time can be described as a non-Riemannian manifold, which, in addition to the metric field, is endowed with torsion. In this new framework, the gravitational field is represented not only by the metric, but also by the torsion, which is completely determined by a geometric scalar field. We show that in this formulation general relativity has a new kind of invariance, whose invariance group consists of a set of conformal and gauge transformations, called Cartan…
Quantization of systems with constraints can be carried on with several methods. In the Dirac formulation the classical generators of gauge transformations are required to annihilate physical quantum states to ensure their gauge invariance. Carrying on BRST symmetry it is possible to get a condition on physical states which, differently from the Dirac method, requires them to be invariant under the BRST transformation. Employing this method for the action of general relativity expressed in terms of the spin connection and tetrad fields with path integral methods, we construct the generator of …
Investigations of the possibility that some novel ``quantum" properties of spacetime might induce a modification dispersion relation focused at first on scenarios with Planck-scale violations of Lorentz symmetry. More recently several studies have considered the possibility of a ``doubly special relativity", in which the modification of the dispersion relation emerges from a framework with both the Planck scale and the speed-of-light scale as characteristic scales of a deformation of the Lorentz transformations. For the schemes with broken Lorentz symmetry at the Planck scale there is a large …
In this paper we review two symmetry axioms of special relativity and their connections to each other together with their role in some famous predictions of relativity theory, such as time dilation, length contraction, and the twin paradox. We also discuss briefly counterparts of these symmetry axioms in general relativity and formulate a conjecture, namely that without them the axioms of general relativity would capture general relativistic spacetimes only up to conformal equivalence.
Using a BRST treatment, we show that the equivalence of General Relativity and Shape Dynamics can be extended to a theory that respects the BRST-symmetries of General Relativity as well as the ones of an extended version of Shape Dynamics. This version of Shape Dynamics implements local spatial Weyl transformations as well as a local and abstract analogue of special conformal transformations. Standard effective field theory arguments suggest that the definition of a gravity theory should implement this duality between General Relativity and Shape Dynamics, thus the name "Doubly General Relativ…
Quantum relativity as a generalized, or rather deformed, version of Einstein relativity with a linear realization on a classical six-geometry beyond the familiar setting of space-time offer a new framework to think about the quantum space-time structure. The formulation requires two deformations to be implemented through imposing two fundamental invariants. We take them to be the independent Planck mass and Planck length. Together, they gives the quantum $\hbar$. The scheme leads to {\small \boldmath\protect SO(2,4)} as the relativity symmetry. The quantum world has an AdS$_5$ `classical' geom…
Gravitational properties of a hedge-hog type topological defect in two extra dimensions are considered in General Relativity employing a vector as the order parameter. The developed macroscopic theory of phase transitions with spontaneous symmetry breaking is applied to the analysis of possible "thick" brane structures. The previous considerations were done using the order parameter in the form of a multiplet in a target space of scalar fields. The difference of these two approaches is analyzed and demonstrated in detail. There are two different symmetries of regular solutions of Einstein equa…
In this paper we provide a possible realization of Penrose's idea of nonlinear gravitons using a new description of nonmetric general relativity. In the addressal of issues surrounding the normalizability of the Kodama state and its reliability as a ground state for gravity, we expand the theory in fluctuations about the Kodama state. This produces a theory of complex gravity with a well-defined Hilbert space structure, whose quantization we carry out both at the linearized level and in the full nonlinear theory. The results of this paper demonstrate the preservation of the physical degrees of…
We develop a method for constructing of the basic functions with witch to expand small perturbations of space-time in General Relativity. The method allows to obtain the tensor harmonics for perturbations of the background space-time admitting an arbitrary group of isometry, and to split the linearized Einstein equations into the irreducible combinations. The essential point of the work is the construction of the generalized Casimir operator for the underlying group, which is defined not only on vector but also on tensor fields. It is used to construct the basic functions for spaces of tensor …
The Killing equation for a general Finsler space is set up. It is showed that the Killing equation of $(α,β)$ space can be divided into two parts. One is the same with Killing equation of a Riemannian metric, another equation can be regarded as a constraint. The solutions of Killing equations present explicitly the isometric symmetry of Finsler space. We find that the isometric group of a special case of $(α,β)$ space is the same with the symmetry of Very Special Relativity (VSR). The Killing vectors of Finsler-Funk space are given. Unlike Riemannian constant curvature space, the 4 dimensional…
Deformed Special Relativity (DSR) is obtained by imposing a maximal energy to Special Relativity and deforming the Lorentz symmetry (more exactly the Poincaré symmetry) to accommodate this requirement. One can apply the same procedure deforming the Galilean symmetry in order to impose a maximal speed (the speed of light). This leads to a non-commutative space structure, to the expected deformations of composition of speed and conservation of energy-momentum. In doing so, one runs into most of the ambiguities that one stumbles onto in the DSR context. However, this time, Special Relativity is t…
We assume a GL(4,R) space-time symmetry which is spontaneously broken to SO(3,1). We carry out the coset construction of the effective theory for the non-linearly realized broken symmetry in terms of the Goldstone fields and matter fields transforming linearly under the unbroken Lorentz subgroup. We then identify functions of the Goldstone and matter fields that transform linearly also under the broken symmetry. Expressed in terms of these quantities the effective theory reproduces the vierbein formalism of General Relativity with general coordinate invariance being automatically realized non-…
We continue our investigation of the configuration space of general relativity begun in I (gr-qc/9411009). Here we examine the Hamiltonian constraint when the spatial geometry is momentarily static (MS). We show that MS configurations satisfy both the positive quasi-local mass (QLM) theorem and its converse. We derive an analytical expression for the spatial metric in the neighborhood of a generic singularity. The corresponding curvature singularity shows up in the traceless component of the Ricci tensor. We show that if the energy density of matter is monotonically decreasing, the geometry ca…
Gravitational waves with a space-translation Killing field are considered. In this case, the 4-dimensional Einstein vacuum equations are equivalent to the 3-dimensional Einstein equations with certain matter sources. This interplay between 4- and 3- dimensional general relativity can be exploited effectively to analyze issues pertaining to 4 dimensions in terms of the 3-dimensional structures. An example is provided by the asymptotic structure at null infinity: While these space-times fail to be asymptotically flat in 4 dimensions, they can admit a regular completion at null infinity in 3 dime…
According to general relativity, the interaction of a matter field with gravitation requires the simultaneous introduction of a tetrad field, which is a field related to translations, and a spin connection, which is a field assuming values in the Lie algebra of the Lorentz group. These two fields, however, are not independent. By analyzing the constraint between them, it is concluded that the relevant local symmetry group behind general relativity is provided by the Lorentz group. Furthermore, it is shown that the minimal coupling prescription obtained from the Lorentz covariant derivative coi…
The relativistic conception of space and time is challenged by the quantum nature of physical observables. It has been known for a long time that Poincaré symmetry of field theory can be extended to the larger conformal symmetry. We use these symmetries to define quantum observables associated with positions in space-time, in the spirit of Einstein theory of relativity. This conception of localisation may be applied to massive as well as massless fields. Localisation observables are defined as to obey Lorentz covariant commutations relations and in particular include a time observable conjugat…
Locality of interactions is an essential ingredient of Special Relativity. Recently, a new framework under the name of relative locality \cite{AmelinoCamelia:2011bm} has been proposed as a way to consider Planckian modifications of the relativistic dynamics of particles. We note in this paper that the loss of absolute locality is a general feature of theories beyond Special Relativity with an implementation of a relativity principle. We give an explicit construction of such an implementation and compare it both with the previously mentioned framework of relative locality and the so-called Doub…
Like Euclid, Riemann and Lobachevsky geometries on an almost equal footing, based on the principle of relativity of maximum symmetry proposed by Lu and the postulate on invariant universal constants, dS/AdS SR can be set up on an almost equal footing with Einstein's SR. For dS-case, there is a coin-like relation: A law of inertia in Beltrami atlas with Beltrami time simultaneity for the PoR on one side. The proper-time simultaneity and a RW-like dS-space with entropy and an accelerated expanding S^3 fitting the cosmological principle on another. If our universe is asymptotic to the RW-like dS-…
Within the framework of general relativity, in some cases at least, it is a delicate and interesting question just what it means to say that an extended body is or is not "rotating". It is so for two reasons. First, one can easily think of different criteria of rotation. Though they agree if the background spacetime structure is sufficiently simple, they do not do so in general. Second, none of the criteria fully answers to our classical intuitions. Each one exhibits some feature or other that violates those intuitions in a significant and interesting way. The principal goal of the paper is to…
Using a suitable rescaling of the independent variable, a Lagrangian is found for the nonlinear Vialov equation ruling the longitudinal profiles of glaciers and ice caps in the shallow ice approximation. This leads to a formal analogy between the (rescaled) Vialov equation and the Friedmann equation of relativistic cosmology, which is explored. This context provides a new symmetry of the (rescaled) Vialov equation and gives, at least formally, all its solutions using a generating function, which is the Nye profile for the degenerate case of perfectly plastic ice.
A black hole solution of Einstein's field equations with cylindrical symmetry is found. Using the Hamiltonian formulation one is able to define mass and angular momentum for the cylindrical black hole through the corresponding and equivalent three dimensional theory. The causal structure is analyzed. Comments: revised version.
It is shown, for the self-consistent system of scalar, electro-magnetic and gravitational fields in general relativity, that the equations of motion admit a special kind of solutions with spherical or cylindrical symmetry. For these solutions, the physical fields vanish and the space-time is flat outside of the critical sphere or cylinder. Therefore, the mass and the electric charge of these configurations are zero. The principal difference between droplet-like solutions with spherical symmetry and those with cylindrical one has been established. In the first case there exists a possibility of…
The present work aims to search for an implementation of a new symmetry in the space-time by introducing the idea of an invariant minimum speed scale ($V$). Such a lowest limit $V$, being unattainable by the particles, represents a fundamental and preferred reference frame connected to a universal background field (a vacuum energy) that breaks Lorentz symmetry. So there emerges a new principle of symmetry in the space-time at the subatomic level for very low energies close to the background frame ($v\approx V$), providing a fundamental understanding for the uncertainty principle, i.e., the unc…
The problem of finding an appropriate geometrical/physical index for measuring a degree of inhomogeneity for a given space-time manifold is posed. Interrelations with the problem of understanding the gravitational/informational entropy are pointed out. An approach based on the notion of approximate symmetry is proposed. A number of related results on definitions of approximate symmetries known from literature are briefly reviewed with emphasis on their geometrical/physical content. A definition of a Killing-like symmetry is given and a classification theorem for all possible averaged space-tim…
We develop a gauge invariant canonical perturbation scheme for perturbations around symmetry reduced sectors in generally covariant theories, such as general relativity. The central objects of investigation are gauge invariant observables which encode the dynamics of the system. We apply this scheme to perturbations around a homogeneous and isotropic sector (cosmology) of general relativity. The background variables of this homogeneous and isotropic sector are treated fully dynamically which allows us to approximate the observables to arbitrary high order in a self--consistent and fully gauge …
It might seem that a choice of a time coordinate in Hamiltonian formulations of general relativity breaks the full four-dimensional diffeomorphism covariance of the theory. This is not the case. We construct explicitly the complete set of gauge generators for Ashtekar's formulation of canonical gravity. The requirement of projectability of the Legendre map from configuration-velocity space to phase space renders the symmetry group a gauge transformation group on configuration-velocity variables. Yet there is a sense in which the full four-dimensional diffeomorphism group survives. Symmetry gen…
The quantum gauge general relativity is proposed in the framework of quantum gauge theory of gravity. It is formulated based on gauge principle which states that the correct symmetry for gravitational interactions should be gravitational gauge symmetry. The gravitational gauge group is studied in the paper. Then gravitational gauge interactions of pure gravitational gauge field is studied. It is found that the field equation of gravitational gauge field is just the Einstein's field equation. After that, the gravitational interactions of scalar field, Dirac field and vector fields are studied, …
The gravitational behavior of antimatter is still unknown. While we may be confident that antimatter is self-attractive, the interaction between matter and antimatter might be either attractive or repulsive. We investigate this issue on theoretical grounds. Starting from the CPT invariance of physical laws, we transform matter into antimatter in the equations of both electrodynamics and gravitation. In the former case, the result is the well-known change of sign of the electric charge. In the latter, we find that the gravitational interaction between matter and antimatter is a mutual repulsion…
We investigate the effects of general relativity upon the non-axisymmetric ``bar'' mode secular instability of rapidly rotating stars, i.e. the relativistic and compressible analog of the transition from Maclaurin spheroids to Jacobi ellipsoids. Our method consists in perturbing a stationary axisymmetric configuration, constructed by a 2-D general relativistic numerical code, and taking into account only the dominant terms in the non-axisymmetric part of the 3-D relativistic equations. For a polytropic equation of state, we have determined, as a function of the degree of relativity, the critic…
We present a set of three PDEs for three real scalars that are equivalent to the full Einstein equations without any symmetry assumptions. The main variables in this formulation are null surfaces and a conformal factor. Furthermore, for asymptotically flat spacetimes the free data (representing gravitational radiation) enters as the source term in the resulting equations. This could be important for an asymptotic quantization procedure.
Barbour's interpretation of Mach's principle led him to postulate that gravity should be formulated as a dynamical theory of spatial conformal geometry, or in his terminology, "shapes." Recently, it was shown that the dynamics of General Relativity can indeed be formulated as the dynamics of shapes. This new Shape Dynamics theory, unlike earlier proposals by Barbour and his collaborators, implements local spatial conformal invariance as a gauge symmetry that replaces refoliation invariance in General Relativity. It is the purpose of this paper to answer frequent questions about (new) Shape Dyn…
This research aims to consider a new principle of symmetry in the space-time by means of the elimination of the classical idea of rest including a universal minimum limit of speed in the subatomic world. Such a limit, unattainable by the particles, represents a preferred inertial reference frame associated with a cosmological background field that breaks Lorentz symmetry. So there emerges a modified relativistic dynamics with a minimum speed related to the Planck length of the early universe, which leads us to search for a quantum gravity at low energies.
We investigate inertial frames in the absence of Lorentz invariance, reconsidering the usual group structure implied by the relativity principle. We abandon the relativity principle, discarding the group structure for the transformations between inertial frames, while requiring these transformations to be at least linear (to preserve homogeneity). In theories with a preferred frame (aether), the set of transformations between inertial frames forms a groupoid/pseudogroup instead of a group, a characteristic essential to evading the von Ignatowsky theorems. In order to understand the dynamics, w…
Firstly, we present a reformulation of the standard canonical approach to spherically symmetric systems in which the radial gauge is imposed. This is done via the gauge unfixing technique, which serves as the exposition in the context of the radial gauge. Secondly, we apply the same techniques to the full theory, without assuming spherical symmetry, resulting in a reduced phase space description of general relativity. The canonical structure of the theory is analyzed.
We find new classes of exact solutions of the initial momentum constraint for vacuum Einstein's equations. Considered data are either invariant under a continuous symmetry or they are assumed to have the exterior curvature tensor of a simple form. In general the mean curvature $H$ is non-constant and $g$ is not conformally flat. In the generic case with the symmetry we obtain general solution in an explicit form. In other cases solutions are given up to quadrature. We also find a class of explicit solutions without symmetries which generalizes data induced by the Kerr metric or other metrics r…
I discuss the (2,2)-formalism of general relativity based on the (2,2)-fibration of a generic 4-dimensional spacetime of the Lorentzian signature. In this formalism general relativity is describable as a Yang-Mills gauge theory defined on the (1+1)-dimensional base manifold, whose local gauge symmetry is the group of the diffeomorphisms of the 2-dimensional fibre manifold. After presenting the Einstein's field equations in this formalism, I solve them for spherically symmetric case to obtain the Schwarzschild solution. Then I discuss possible applications of this formalism.
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…
The paper aims to introduce a new symmetry principle in the space-time geometry through the elimination of the classical idea of rest and by including a universal minimum limit of speed in the subatomic world. Such a limit, unattainable by particles, represents a preferred reference frame associated with a universal background field that breaks Lorentz symmetry. Thus the structure of space-time is extended due to the presence of a vacuum energy density, which leads to a negative pressure at cosmological scales. The tiny values of the cosmological constant and the vacuum energy density shall be…
In general relativity, the masslessness of gravitons can be traced to symmetry under diffeomorphisms. Here we consider another possibility, whereby the masslessness arises from spontaneous violation of Lorentz symmetry.
Tiny violations of the Lorentz symmetry of relativity and the associated discrete CPT symmetry could emerge in a consistent theory of quantum gravity such as string theory. Recent evidence for linear polarization in gamma-ray bursts improves existing sensitivities to Lorentz and CPT violation involving photons by factors ranging from ten to a million.
This paper gives a summary of the author's works concerning the emergent general relativity in a particular class of tensor models, which possess Gaussian classical solutions. In general, a classical solution in a tensor model may be physically regarded as a background space, and small fluctuations about the solution as emergent fields on the space. The numerical analyses of the tensor models possessing Gaussian classical background solutions have shown that the low-lying long-wavelength fluctuations around the backgrounds are in one-to-one correspondence with the geometric fluctuations on fla…
