Lie derivatives of various geometrical and physical quantities define symmetries and conformal symmetries in general relativity. Thus we obtain motions, collineations, conformal motions and conformal collineations. These symmetries are used not only to find new solutions of Einstein's field equations but to classify the spaces also. Different classification schemes are presented here. Relationships between these symmetries are discussed and illustrating examples are presented.
Matter collineations of spherically Symmetric Lorentzian Manifolds are considered. These are investigated when the energy-momentum tensor is non-degenerate and also when it is degenerate. We have classified spacetimes admitting higher symmetries and spacetimes admitting SO(3) as the maximal isometry group. For the non-degenerate case, we obtain either {\it four}, {\it six}, {\it seven} or {\it ten} independent matter collineations in which {\it four} are isometries and the rest are proper. The results of the previous paper [1] are recovered as a special case. It is worth noting that we have al…
Given the class of Finsler spaces with Lorentzian signature $(M,L)$ on a manifold $M$ endowed with a timelike vector field $\mathcal{X}$ satisfying $g_{(x,y)}(\mathcal{X},\mathcal{X})<0$ at any point $(x,y)$ of the slit tangent bundle, a pseudo-Riemannian metric defined on $M$ of signature $n-1$ is associated to the fundamental tensor $g$. Furthermore, an affine, torsion free connection is associated to the Chern connection determined by $L$. The definition of the average connection does not make use of $\mathcal{X}$. Therefore, there is no direct relation between these two averaged objects.
We present a definition of null G-structures on Lorentzian manifolds and investigate their geometric properties. This definition includes the Robinson structure on 4-dimensional black holes as well as the null structures that appear in all supersymmetric solutions of supergravity theories. We also identify the induced geometry on some null hypersurfaces and that on the orbit spaces of null geodesic congruences in such Lorentzian manifolds. We give the algebra of diffeomorphisms that preserves a null G-structure and demonstrate that in some cases it interpolates between the BMS algebra of an as…
The essential features of a quantum group deformation of classical symmetries of General Relativity in the case with non-vanishing cosmological constant $Λ$ are presented. We fully describe (anti-)de Sitter non-commutative spacetimes and curved momentum spaces in (1+1) and (2+1) dimensions arising from the $κ$-deformed quantum group symmetries. These non-commutative spacetimes are introduced semiclassically by means of a canonical Poisson structure, the Sklyanin bracket, depending on the classical $r$-matrix defining the $κ$-deformation, while curved momentum spaces are defined as orbits gener…
The notion of relativistic observer is confronted with Naveira's classification of (pseudo-)Riemannian almost-product structures on space-time manifolds. Some physical properties and their geometrical counterparts are shortly discussed.
We review the intimate connection between (super-)gravity close to a spacelike singularity (the "BKL-limit") and the theory of Lorentzian Kac-Moody algebras. We show that in this limit the gravitational theory can be reformulated in terms of billiard motion in a region of hyperbolic space, revealing that the dynamics is completely determined by a (possibly infinite) sequence of reflections, which are elements of a Lorentzian Coxeter group. Such Coxeter groups are the Weyl groups of infinite-dimensional Kac-Moody algebras, suggesting that these algebras yield symmetries of gravitational theorie…
We review the {\it Noether Symmetry Approach} as a geometric criterion to select theories of gravity. Specifically, we deal with Noether Symmetries to solve the field equations of given gravity theories. The method allows to find out exact solutions, but also to constrain arbitrary functions in the action. Specific cosmological models are taken into account.
We show that the conservation laws for the geodesic equation which are associated to affine symmetries can be obtained from symmetries of the Lagrangian for affinely parametrized geodesics according to Noether's theorem, in contrast to claims found in the literature. In particular, using Aminova's classification of affine motions of Lorentzian manifolds, we show in detail how affine motions define generalized symmetries of the geodesic Lagrangian. We compute all infinitesimal proper affine symmetries and the corresponding geodesic conservation laws for all homogeneous solutions to the Einstein…
We write the Dirac equation in curved 4-dimensional Lorentzian spacetime using concepts from the analysis of partial differential equations as opposed to geometric concepts.
Recent developments in quantum gravity have shown the Lorentzian treatment to be a fruitful approach towards the emergence of macroscopic spacetimes. In this paper, we discuss another related aspect of the Lorentzian treatment: we argue that collective quantum coherence may provide a simple mechanism for highlighting symmetric configurations over generic non-symmetric ones. After presenting the general framework of the mechanism, we show the phenomenon in some concrete simple examples in the randomly connected tensor network, which is tightly related to a certain model of quantum gravity, i.e.…
The \SN (SN) equation is recast on purely geometrical grounds, namely in terms of Bargmann structures over $(\d+1)$-dimensional Newton-Cartan (NC) spacetimes. Its maximal group of invariance, which we call the SN group, is determined as the group of conformal Bargmann automorphisms that preserve the coupled Schrödinger and NC gravitational field equations. Canonical unitary representations of the SN group are worked out, helping us recover, in particular, a very specific occurrence of dilations with dynamical exponent $z=(\d+2)/3$.
We study the global structure of Lorentzian manifolds with partial sectional curvature bounds. In particular, we prove completeness theorems for homogeneous and isotropic cosmologies as well as static spherically symmetric spacetimes. The latter result is used to rigorously prove the absence of static spherically symmetric black holes in more than three dimensions. The proofs of these new results are preceded by a detailed exposition of the local aspects of sectional curvature bounds for Lorentzian manifolds, which extends and strengthens previous constructions.
In the tensorial group field theory (TGFT) approach to quantum gravity, the basic quanta of the theory correspond to discrete building blocks of geometry. It is expected that their collective dynamics gives rise to continuum spacetime at a coarse grained level, via a process involving a phase transition. In this work we show for the first time how phase transitions for realistic TGFT models can be realized using Landau-Ginzburg mean-field theory. More precisely, we consider models generating 4-dimensional Lorentzian triangulations formed by spacelike tetrahedra whose quantum geometry is encode…
Manifolds with exceptional holonomy play an important role in string theory, supergravity and M-theory. It is explained how one can find the holonomy algebra of an arbitrary Riemannian or Lorentzian manifold. Using the de~Rham and Wu decompositions, this problem is reduced to the case of locally indecomposable manifolds. In the case of locally indecomposable Riemannian manifolds, it is known that the holonomy algebra can be found from the analysis of special geometric structures on the manifold. If the holonomy algebra $\mathfrak{g}\subset\mathfrak{so}(1,n-1)$ of a locally indecomposable Loren…
We consider Lorentzian manifolds as examples of partially ordered measure spaces, sets endowed with compatible partial order relations and measures, in this case given by the causal structure and the volume element defined by each Lorentzian metric. This places the structure normally used to describe spacetime in geometrical theories of gravity in a more general context, which includes the locally finite partially ordered sets of the causal set approach to quantum gravity. We then introduce a function characterizing the closeness between any two partially ordered measure spaces and show that, …
This thesis presents a framework in which to explore kinematical symmetries beyond the standard Lorentzian case. This framework consists of an algebraic classification, a geometric classification, and a derivation of the geometric properties required to define physical theories on the classified spacetime geometries. The work completed in substantiating this framework for kinematical, super-kinematical, and super-Bargmann symmetries constitutes the body of this thesis. To this end, the classification of kinematical Lie algebras in spatial dimension $D = 3$, as presented in [3, 4], is reviewed;…
It is known that General Relativity ({\bf GR}) uses Lorentzian Manifold $(M_4;g)$ as a geometrical model of the physical space-time. $M_4$ means here a four-dimensional differentiable manifold endowed with Lorentzian metric $g$. The metric $g$ satisfies Einstein equations. Since the 1970s many authors have tried to generalize this geometrical model of the physical space-time by introducing torsion and even more general metric-affine geometry. In this paper we discuss status of torsion in the theory of gravity. At first, we emphasize that up to now we have no experimental evidence for the exist…
In this thesis, we study the one parameter point transformations which leave invariant the differential equations. In particular we study the Lie and the Noether point symmetries of second order differential equations. We establish a new geometric method which relates the point symmetries of the differential equations with the collineations of the underlying manifold where the motion occurs. This geometric method is applied in order the two and three dimensional Newtonian dynamical systems to be classified in relation to the point symmetries; to generalize the Newtonian Kepler-Ermakov system i…
We give a covariant definition of closeness between (time oriented) Lorentzian metrics on a manifold M, using a family of functions which measure the difference in volume form on one hand and the difference in causal structure relative to a volume scale on the other hand. These functions will distinguish two geometric properties of the Alexandrov sets $ A(p,q), \tilde{A} (p,q) $ relative to two space time points $q$ and $p$ and metrics $g$ and $ \tilde{g} $. It will be shown that this family generates uniformities and consequently a topology on the space of Lorentzian metrics which is Hausdorf…
Following Geroch, Traschen, Mars and Senovilla, we consider Lorentzian manifolds with distributional curvature tensor. Such manifolds represent spacetimes of general relativity that possibly contain gravitational waves, shock waves, and other singular patterns. We aim here at providing a comprehensive and geometric (i.e., coordinate-free) framework. First, we determine the minimal assumptions required on the metric tensor in order to give a rigorous meaning to the spacetime curvature within the framework of distribution theory. This leads us to a direct derivation of the jump relations associa…
