No New Symmetries of the Vacuum Einstein Equations (original) (raw)
Symmetry and Integrability of Classical Field Equations
arXiv (Cornell University), 2008
A number of characteristics of integrable nonlinear partial differential equations (PDE's) for classical fields are reviewed, such as Bäcklund transformations, Lax pairs, and infinite sequences of conservation laws. An algebraic approach to the symmetry problem of PDE's is described, suitable for PDE's the solutions of which are non-scalar fields (e.g., are in the form of matrices). A method is proposed which, in certain cases, may "unify" the symmetry and integrability properties of a nonlinear PDE. Application is made to the self-dual Yang-Mills equation and the Ernst equation of General Relativity.
ON GENERAL SOLUTIONS OF EINSTEIN EQUATIONS
International Journal of Geometric Methods in Modern Physics, 2011
equivalent to arXiv: 0909.3949v1 [gr-qc]; an extended/modified variant published in IJTP 49 (2010) 884-913, equivalent to arXiv: 0909.3949v4 [gr-qc]; on June 20, 2011, moderators arXiv.org accepted to provide a different number to this "short" variant in physics.gen-ph)
On General Solutions for Field Equations in Einstein and Higher Dimension Gravity
2009
We prove that the Einstein equations can be solved in a very general form for arbitrary spacetime dimensions and various types of vacuum and non-vacuum cases following a geometric method of anholonomic frame deformations for constructing exact solutions in gravity. The main idea of this method is to introduce on (pseudo) Riemannian manifolds an alternative (to the Levi-Civita connection) metric compatible linear connection which is also completely defined by the same metric structure. Such a canonically distinguished connection is with nontrivial torsion which is induced by some nonholonomy frame coefficients and generic off-diagonal terms of metrics. It is possible to define certain classes of adapted frames of reference when the Einstein equations for such an alternative connection transform into a system of partial differential equations which can be integrated in very general forms. Imposing nonholonomic constraints on generalized metrics and connections and adapted frames (sele...
New Supplementary Conditions for a Non-Linear Field Theory: General Relativity
1999
The Einstein theory of general relativity provides a peculiar example of classical field theory ruled by non-linear partial differential equations. A number of supplementary conditions (more frequently called gauge conditions) have also been considered in the literature. In the present paper, starting from the de Donder gauge, which is not conformally invariant but is the gravitational counterpart of the Lorenz gauge, we consider, led by geometric structures on vector bundles, a new family of gauges in general relativity, which involve fifth-order covariant derivatives of metric perturbations. A review of recent results by the authors is presented: restrictions on the general form of the metric on the vector bundle of symmetric rank-two tensor fields over space-time; admissibility of such gauges in the case of linearized theory about flat Euclidean space; generalization to a suitable class of curved Riemannian backgrounds, by solving an integral equation; non-local construction of conformally invariant gauges.
International Journal of Engineering Science, 2003
Using the symmetry reduction approach we have herein examined, under continuous groups of transformations, the invariance of Einstein exterior equations for stationary axisymmetric and rotating case, in conventional and nonconventional forms, that is a coupled system of nonlinear partial differential equations of second order. More specifically, the said technique yields the invariant transformation that reduces the given system of partial differential equations to a system of nonlinear ordinary differential equations (nlodes) which, in the case of conventional form, is reduced to a single nlode of second order. The first integral of the resulting nlode has been obtained via invariant-variational principle and NoetherÕs theorem and involves an integration constant. Depending upon the choice of the arbitrary constant two different forms of the exact solutions are indicated. The generalized forms of Weyl and Schwarzschild solutions for the case of no spin have also been deduced as particular cases. Investigation of nonconventional form of Einstein exterior equations has not only led to the recovery of solutions obtained through conventional form but it also yields physically important asymptotically flat solutions. In a particular case, a single third order nlode has been derived which evidently opens up the possibility of finding many further interesting solutions of the exterior field equations.
Reformulation of the symmetries of first-order general relativity
We report a new internal gauge symmetry of the n-dimensional Palatini action with cosmological term (n > 3) that is the generalization of three-dimensional local translations. This symmetry is obtained through the direct application of the converse of Noether's second theorem on the theory under consideration. We show that diffeomorphisms can be expressed as linear combinations of it and local Lorentz transformations with field-dependent parameters up to terms involving the variational derivatives of the action. As a result, the new internal symmetry together with local Lorentz transformations can be adopted as the fundamental gauge symmetries of general relativity. Although their gauge algebra is open in general, it allows us to recover, without resorting to the equations of motion, the very well-known Lie algebra satisfied by translations and Lorentz transformations in three dimensions. Finally, we report the analog of the new gauge symmetry for the Holst action, finding th...
How Extra Symmetries Affect Solutions in General Relativity
Universe
To get exact solutions to Einstein’s field equations in general relativity, one has to impose some symmetry requirements. Otherwise, the equations are too difficult to solve. However, sometimes, the imposition of too much extra symmetry can cause the problem to become somewhat trivial. As a typical example to illustrate this, the effects of conharmonic flatness are studied and applied to Friedmann–Lemaitre–Robertson–Walker spacetime. Hence, we need to impose some symmetry to make the problem tractable, but not too much so as to make it too simple.
Exact Solutions with Noncommutative Symmetries in Einstein and Gauge Gravity
2003
We present new classes of exact solutions with noncommutative symmetries constructed in vacuum Einstein gravity (in general, with nonzero cosmological constant), five dimensional (5D) gravity and (anti) de Sitter gauge gravity. Such solutions are generated by anholonomic frame transforms and parametrized by generic off-diagonal metrics. For certain particular cases, the new classes of metrics have explicit limits with Killing symmetries but, in general, they may be characterized by certain anholonomic noncommutative matrix geometries. We argue that different classes of noncommutative symmetries can be induced by exact solutions of the field equations in 'commutative' gravity modeled by a corresponding moving real and complex frame geometry. We analyze two classes of black ellipsoid solutions (in the vacuum case and with cosmological constant) in 4D gravity and construct the analytic extensions of metrics for certain classes of associated frames with complex valued coefficients. The third class of solutions describes 5D wormholes which can be extended to complex metrics in complex gravity models defined by noncommutative geometric structures. The anholonomic noncommutative symmetries of such objects are analyzed. We also present a descriptive account how the Einstein gravity can be related to gauge models of gravity and their noncommutative extensions and discuss such constructions in relation to the Seiberg-Witten map for the gauge gravity. Finally, we consider a formalism of vielbeins deformations subjected to noncommutative symmetries in order to generate solutions for noncommutative gravity models with Moyal (star) product.
Nonlinear realizations of space-time symmetries. Scalar and tensor gravity
Annals of Physics, 1971
The study of conformal group symmetries within the framework of nonlinear realizations is extended and reexpressed in terms of metric tensors and connections on spacetime. The standard Vierbein formalism of general relativity is then reinterpreted in terms of nonlinear realizations of the group GL(4, R). Throughout we emphasise the connection between massless Goldstone bosons and the preferred fields of nonlinear realizations.
Generalised symmetries in linear gravity
Linearised gravity has a global symmetry under which the graviton is shifted by a symmetric tensor satisfying a certain flatness condition. There is also a dual symmetry that can be associated with a global shift symmetry of the dual graviton theory. The corresponding conserved charges are shown to satisfy a centrally-extended algebra. We discuss the gauging of these global symmetries, finding an obstruction to the simultaneous gauging of both symmetries which we interpret as a mixed 't Hooft anomaly for the ungauged theory. We discuss the implications of this, analogous to those resulting from a similar structure in Maxwell theory, and interpret the graviton and dual graviton as Nambu-Goldstone modes for these shift symmetries.
A new symmetry of the relativistic wave equation A new symmetry of the relativistic wave equation
In this paper we show that there exists a new symmetry in the relativistic wave equation for a scalar field in arbitrary dimensions. This symmetry is related to redefinitions of the metric tensor which implement a map between non-equivalent manifolds. It is possible to interpret these transformations as a generalization of the conformal transformations. In addition, one can show that this set of manifolds together with the transformation connecting its metrics forms a group. As long as the scalar field dynamics is invariant under these transformations, there immediately appears an ambiguity concerning the definition of the underlying background geometry.
Structure of the space of solutions of Einstein field equations - A new variables approach
On a wave-particle in closed and open isotropic universes J. Math. Phys. 52, 013508 On a class of global characteristic problems for the Einstein vacuum equations with small initial data Following Ashtekar's new Hamiltonian formulation of general relativity ["Anew Hamiltonian formulation of general relativity," Syracuse University preprint, 1986; Phys. Rev. ] on the structure of the space of solutions of vacuum Einstein equations in the case of space-times admitting a compact Cauchy hypersurface are rederived and extended.
New Exact Solutions for Static Axially Symmetric Einstein Vacuum Equations
2016
The Einstein static vacuum solutions with axially symmetry have been considered. The symmetry group analysis (isovector fields) method reduces the Einstein equations to ordinary differential equations (ODEs). These equations can be solved analytically with the aid of Mathematica programm. Some symmetry transformations and many new exact (similarity) solutions of Einstein vacuum equations are obtained.
Classical and Quantum Gravity, 2002
We derive a generic identity which holds for the metric (i.e. variational) energymomentum tensor under any field transformation in any generally covariant classical Lagrangian field theory. The identity determines the conditions under which a symmetry of the Lagrangian is also a symmetry of the energy-momentum tensor. It turns out that the stress tensor acquires the symmetry if the Lagrangian has the symmetry in a generic curved spacetime. In this sense a field theory in flat spacetime is not self-contained. When the identity is applied to the gauge invariant spin-two field in Minkowski space, we obtain an alternative and direct derivation of a known no-go theorem: a linear gauge invariant spin-2 field, which is dynamically equivalent to linearized General Relativity, cannot have a gauge invariant metric energy-momentum tensor. This implies that attempts to define the notion of gravitational energy density in terms of the metric energy-momentum tensor in a field-theoretical formulation of gravity must fail.
About the Symmetry of General Relativity
Journal of Geometry and Symmetry in Physics, 2020
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 g GL of local linear transformations of affine frame fields. GR is interpreted as the gauge theory of the gauge group of translations g M T , and therefore is invariant under the space-time diffeomorphisms. The groups g GL and g M T are united into group g M S , which is their semidirect product and is the complete symmetry group of the general relativity in an affine frame (GRAF). The consequence of g GL -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 g M T -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 g M S leads to renormalisation of energy-momentum of the gravitation field. At the end we show that by limiting admissible reference frames (by g GL -gauge fixing) from GRAF, in addition to Einstein gravity, one can obtain other local equivalent formulations of GR: general relativity in an orthonormal frame, dilaton gravity, unimodular gravity, etc.
On the universality of Einstein equations
General Relativity and Gravitation, 1987
It is proved that a Lagrangian field theory based on a linear connection in space-time is equivalertt to Einstein's general relativity interacting with additional matter fields.
A new characterization of half-flat solutions to Einstein's equation
Communications in Mathematical Physics, 1988
A 3-f 1 formulation of complex Einstein's equation is first obtained on a real 4-manifold M, topologically Σ x R, where Σ is an arbitrary 3-manifold. The resulting constraint and evolution equations are then simplified by using variables that capture the (anti-) self dual part of the 4-dimensional Weyl curvature. As a result, to obtain a vacuum self-dual solution, one has just to solve one constraint and one "evolution" equation on a field of triads on Σ: DivVί α = 0 and ^-£ ijk [F j? F k ] α , with i = l,2,3, where Div denotes divergence with respect to a fixed, non-dynamical volume element. If the triad is real, the resulting self-dual metric is real and positive definite. This characterization of self-dual solutions in terms of triads appears to be particularly well suited for analysing the issues of exact integrability of the (anti-) self-dual Einstein system. Finally, although the use of a 3 +1 decomposition seems artificial from a strict mathematical viewpoint, as David C. Robinson has recently shown, the resulting triad description is closely related to the hyperkahler geometry that (anti-) self-dual vacuum solutions naturally admit.