In this paper the application of the $M$-projective curvature tensor in the general theory of relativity has been studied. Firstly, we have proved that an $M$-projectively flat quasi-Einstein spacetime is of a special class with respect to an associated symmetric tensor field, followed by the theorem that a spacetime with vanishing $M$-projective curvature tensor is a spacetime of quasi-constant curvature. Then we have proved that an $M$-projectively flat quasi-Einstein spacetime is infinitesimally spatially isotropic relative to the unit timelike vector field $ξ$. In the next section we have proved that an $M$-projectively flat Ricci semi-symmetric quasi-Einstein spacetime satisfying a definite condition is an $N(\frac{2l-m}{6})$-quasi Einstein spacetime. In the last section, we have firstly proved that an $M$-projectively flat perfect fluid spacetime with torse-forming vector field $ξ$ satisfying Einstein field equation with cosmological constant represents an inflation, then we have found out the curvature of such spacetime, followed by proving the theorem that the spacetime also becomes semi-symmetric under these conditions. Lastly, we have found out the square of the length of the Ricci tensor in this type of spacetime and also proved that if an $M$-projectively flat perfect fluid spacetime satisfying Einstein field equation with cosmological constant, with torse-forming vector field $ξ$ admits a symmetric $(0,2)$ tensor $α$ parallel to $\nabla$ then either $λ= \frac{k}{2}(p-σ)$ or $α$ is a constant multiple of $g$.
The paper aims to investigate curvature inheritance symmetry in M-projectively flat spacetimes. It is shown that the curvature inheritance symmetry in M-projectively flat spacetime is a conformal motion. We have proved that M- projective curvature tensor follows the symmetry inheritance property along a vector field $ξ$, when spacetime admits the conditions of both curvature inheritance symmetry and conformal motion or motion along the vector field $ξ$. Also, we have derived some results for M-projectively flat spacetime with perfect fluid following the Einstein field equations with a cosmolog…
A class of projective actions of the orthogonal group on the projective space is being studied. It is shown that the Fock--Lorentz, and Magueijo--Smolin transformations known as Doubly Special Relativity are such transformations. The formalism easily lead to new type transformations.
We give a critical analysis of projective relativity theory. Examining Kaluza's own intention and the following development by Klein, Jordan, Pauli, Thiry, Ludwig and others, we conclude that projective relativity was abused in its own terms. Much more in the case of newer higher dimensional Kaluza--Klein theories with non-Abelian gauge groups. Reviewing the projective formulation of the Jordan isomorphy theorem yields some hints how one can proceed in a different direction. We can interpret the condition R5_{μν}=0 not as a field equation in a 5-dimensional Riemannian space, e.g. as vacuum Ein…
Projective invariance is a symmetry of the Palatini version of General Relativity which is not present in the metric formulation. The fact that the Riemann tensor changes nontrivially under projective transformations implies that, unlike in the usual metric approach, in the Palatini formulation this tensor is subject to a gauge freedom, which allows some ambiguities even in its scalar contractions. In this sense, we show that for the Schwarzschild solution there exists a projective gauge in which the (affine) Kretschmann scalar, $K\equiv {R^α}_{βμν}{R_α}^{βμν}$, can be set to vanish everywhere…
We outline unimodular conformal and projective relativity (UCPR), an extension of unimodular relativity in which the conformal and projective structures play central roles. Under $SL(4,\mathbb{R})$ symmetry group, the pseudo-Riemannian metric naturally decomposes into a four-volume element and a conformal metric; and the affine connection decomposes into a one-form and a trace-free projective connection. In UCPR, these four space-time structures are treated as independent fields that have clear physical interpretations. A Palatini-type variational principle for the usual general relativity Lag…
Consider a manifold with boundary, and such that the interior is equipped with a pseudo-Riemannian metric. We prove that, under mild asymptotic non-vanishing conditions on the scalar curvature, if the Levi-Civita connection of the interior does not extend to the boundary (because for example the interior is complete) whereas its projective structure does, then the metric is projectively compact of order 2; this order is a measure of volume growth toward infinity. The result implies a host of results including that the metric satisfies asymptotic Einstein conditions, and induces a canonical con…
A study of proper Weyl projective curvature collineations in FRW k=0 space-time is given using the rank of Weyl projective curvature matrix and direct integration techniques. It is shown that a very special class of the above space-time admits proper Weyl projective curvature collineation.
The quantum phase leads to projective representations of symmetry groups in quantum mechanics. The projective representations are equivalent to the unitary representations of the central extension of the group. A celebrated example is Wigner's formulation of special relativistic quantum mechanics as the projective representations of the inhomogeneous Lorentz group. However, Wigner's formulation makes no mention of the Weyl-Heisenberg group and the hermitian representation of its algebra that are the Heisenberg commutation relations fundamental to quantum physics. We put aside the relativistic …
We give a justification for generalizing the notion of the energy momentum stress tensor in general relativity which results in the Jacobi curvature tensor being the embodiment of total energy momentum stress including gravity in general relativity.
If universal quantum interaction is really connected with the coset structure of deformations of quantum states then the curvature of projective Hilbert state space should be observable. I discuss some approach to the measurement of curvature-dependent values.
Some concepts of real and complex projective geometry are applied to the fundamental physical notions that relate to Minkowski space and the Lorentz group. In particular, it is shown that the transition from an infinite speed of propagation for light waves to a finite one entails the replacement of a hyperplane at infinity with a light cone and the replacement of an affine hyperplane - or rest space - with a proper time hyperboloid. The transition from the metric theory of electromagnetism to the pre-metric theory is discussed in the context of complex projective geometry, and ultimately it is…
We discuss a set of relations, set in the form of results, conjectures and problems, between the L^{2}-norm of the Ricci curvature of a 3-manifold, the scalar curvature and the volume radius. We illustrate the scope of these relations with potential applications to the Einstein Constant Mean Curvature flow (or GR seen as a geometric flow of constant mean curvature), but we believe the framework has it own geometric interest.
We present a new evolution system for Einstein's field equations which is based on tetrad fields and conformally compactified hyperboloidal spatial hypersurfaces which reach future null infinity. The boost freedom in the choice of the tetrad is fixed by requiring that its timelike leg be orthogonal to the foliation, which consists of constant mean curvature slices. The rotational freedom in the tetrad is fixed by the 3D Nester gauge. With these conditions, the field equations reduce naturally to a first-order constrained symmetric hyperbolic evolution system which is coupled to elliptic equati…
We review curvature-based hyperbolic forms of the evolution part of the Cauchy problem of General Relativity that we have obtained recently. We emphasize first order symmetrizable hyperbolic systems possessing only physical characteristics.
We present a practical application of parallel symbolic computation in General Relativity: the calculation of curvature invariants for large dimension. We discuss the structure of the calculations, an implementation of the technique and scaling of the computation with spacetime dimension for various invariants.
The equations of General Relativity are recast in the form of a wave equation for the Weyl tensor. This allows to reformulate gravitational wave theory in terms of curvature waves, rather than metric waves. The existence of two transverse polarization states for curvature waves is proven and in the linearized approximation the quadrupole formula is rederived. A perturbative scheme to extend the linearized result to the non-linear regime is outlined.
By using a projective connection over the space of two-dimensional affine connections, we are able to show that the metric interaction of Polyakov 2D gravity with a coadjoint element arises naturally through the projective Ricci tensor. Through the curvature invariants of Thomas-Whitehead, we are able to define an action that could describe dynamics to the projective connection. We discuss implications of the projective connection in higher dimensions as related to gravitation.
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. …
A general affine connection has enough degrees of freedom to describe the classical gravitational and electromagnetic fields in the metric-affine formulation of gravity. The gravitational field is represented in the Lagrangian by the symmetric part of the Ricci tensor and the classical electromagnetic field can be represented by the tensor of homothetic curvature. The simplest metric-affine Lagrangian that depends on the tensor of homothetic curvature generates the Einstein-Maxwell equations for a massless vector. Metric-affine Lagrangians with matter fields depending on the connection are sub…
We derive a generalized deviation equation -- analogous to the well-known geodesic deviation equation -- for test bodies in General Relativity. Our result encompasses and generalizes previous extensions of the standard geodesic deviation equation. We show how the standard as well as a generalized deviation equation can be used to measure the curvature of spacetime by means of a set of test bodies. In particular, we provide exact solutions for the curvature by using the standard deviation equation as well as its next order generalization.
Characteristic classes in space-time manifolds are discussed for both even- and odd-dimensional spacetimes. In particular, it is shown that the Einstein--Hilbert action is equivalent to a second Chern-class on a modified Poincare bundle in four dimensions. Consequently, the cosmological constant and the trace of an energy-momentum tensor become divisible modulo R/Z.
Chamseddine and Mukhanov recently proposed a modified version of general relativity that implements the idea of a limiting curvature. In the spatially flat, homogeneous, and isotropic sector, their theory turns out to agree with the effective dynamics of the simplest version of loop quantum gravity if one identifies their limiting curvature with a multiple of the Planck curvature. At the same time, it extends to full general relativity without any symmetry assumptions and thus provides an ideal toy model for full loop quantum gravity in the form of a generally covariant effective action known …
We consider the thermodynamic properties of the constant curvature black hole solution recently found by Banados. We show that it is possible to compute the entropy and the quasilocal thermodynamics of the spacetime using the Einstein-Hilbert action of General Relativity. The constant curvature black hole has some unusual properties which have not been seen in other black hole spacetimes. The entropy of the black hole is not associated with the event horizon; rather it is associated with the region between the event horizon and the observer. Further, surfaces of constant internal energy are no…
An experiment in Low Earth Orbit (LEO) is proposed to measure components of the Riemann curvature tensor using atom interferometry. We show that the difference in the quantum phase $Δϕ$ of an atom that can travel along two intersecting geodesics is given by $mR_{0i0j}/\hbar$ times the spacetime volume contained within the geodesics. Our expression for $Δϕ$ also holds for gravitational waves in the long wavelength limit.
We clarify some points about the systems considered by Sota, Suzuki and Maeda in Class. Quantum Grav. 13, 1241 (1996). Contrary to the authors' claim for a non-homoclinic kind of chaos, we show the chaotic cases they considered are homoclinic in origin. The power of local criteria to predict chaos is once more questioned. We find that their local, curvature--based criterion is neither necessary nor sufficient for the occurrence of chaos. In fact, we argue that a merit of their search for local criteria applied to General Relativity is just to stress the weakness of locality itself, free of any…
Recent data suggest that the Universe could be positively curved. Combined with an inflationary stage, this might lead to a curvature bounce instead of the Big Bang. The background evolution is presented, as a function of the parameters controlling the cosmic evolution. The primordial tensor spectrum is also calculated and possible observational footprints of the model are underlined. Several potentials are considered and general remarks are made about "naturalness" in this context.
Using the generally covariant magnetofluid formalism for a hot plasma, a new spacetime curvature driven mechanism for generating seed vorticity/magnetic field is presented. The ``battery'' owes its origin to the interaction between gravity and the inhomogeneous plasma thermodynamics. The general relativistic drive is evaluated for two simple cases: seed formation in a simplified model of a hot plasma accreting in stable orbits around a Schwarzschild black hole, and for particles in free fall near the horizon. Some astrophysical applications are suggested.
We propose a new foliation of asymptotically Euclidean initial data sets by 2-spheres of constant spacetime mean curvature (STCMC). The leaves of the foliation have the STCMC-property regardless of the initial data set in which the foliation is constructed which asserts that there is a plethora of STCMC 2-spheres in a neighborhood of spatial infinity of any asymptotically flat spacetime. The STCMC-foliation can be understood as a covariant relativistic generalization of the CMC-foliation suggested by Huisken and Yau. We show that a unique STCMC-foliation exists near infinity of any asymptotica…
Spin 1/2 particles in geodesic trajectories experience no gravitational potential but they still have non-zero couplings to the curvature tensor. The effect of space time curvature on fermions can be parameterized by a vector and a pseudo-vector potential. These apparent CPT violating terms can be measured with satellite based spin-polarized torsion balance and clock comparison experiments. The Earth's curvature effect is of the order of 10^{-37} Gev which is not far from the present bounds of ~10^{-29} Gev on such CPT violating couplings.
In this work we examine what are the cosmological implications of allowing the geometrical curvature density to behave independently from the energy density contents. Using the full data extracted by Planck mission from CMB, combined with BAO and SNIa measurements, we derive, in the light of this approach, new constraints on the cosmological parameters. In particular we determine the behavior of the curvature dark energy degeneracy when allowing a varying equation of state for the latter. We also examine whether this approach could bridge the gap recently found between the Hubble parameter val…
We present a new expression for the Weyl scalar Psi_4 that can be used in numerical relativity to extract the gravitational wave content of a spacetime. The formula relies upon the identification of transverse tetrads, namely the ones in which Psi_1=Psi_3=0. It is well known that tetrads with this property always exist in a general Petrov type I spacetime. A sub-class of these tetrads naturally converges to the Kinnersley tetrad in the limit of Petrov type D spacetime. However, the transverse condition fixes only four of the six parameters coming from the Lorentz group of transformations appli…
We examine an extension of General Relativity with an explicit non-minimal coupling between matter and curvature. The purpose of this work is to present an overview of the implications of the latter to various contexts, ranging from astrophysical matter distributions to a cosmological setting. Various results are discussed, including the impact of this non-minimal coupling on the choice of Lagrangian density, on a mechanism to mimic galactic and cluster dark matter, on the possibility of accounting for the accelerated expansion of the Universe, energy density fluctuations and modifications to …
We study gravitational curvature effects in circular and radial geodesics in static, spherically symmetric space-times, using Fermi normal coordinates. We first set up these coordinates in the general case, and then use this to study effective magnetic fields due to gravitational curvature in the exterior and interior Schwarzschild, Janis-Newman-Winicour, and Bertrand space-times. We show that these fields can be large for specific parameter values in the theories, and thus might have observational significance. We discuss the qualitative differences of the magnetic field for vacuum space-time…
A underlying dynamical structure for both relativity and quantum theory-``superrelativity'' has been proposed in order to overcome the well known incompatibility between these theories. The relationship between curvature of spacetime (gravity) and curvature of the projective Hilbert space of pure quantum states is established as well.
This paper surveys aspects of the convergence and degeneration of Riemannian metrics on a given manifold M - the Cheeger-Gromov theory - and extensions thereof to Ricci curvature in place of full curvature. This theory is then applied to study a collection of different issues in mathematical aapects of General Relativity.
We examine the problem of overcharging extremal and near-extremal black hole solutions of Einstein-Gauss-Bonnet gravity in any dimension, generalizing the result in general relativity. We show that as in the case of general relativity, it is not possible to create a naked singularity by overcharging an extremal black hole in Einstein-Gauss-Bonnet gravity using a charged test particle. Our result suggests that the validity of the cosmic censorship hypothesis transcends beyond general relativity to well motivated higher curvature gravity.
This article proposes some cosmological reflections at the qualitative and conjectural level, suggested by the Fantappie & Arcidiacono projective relativity theory. The difference will firstly be discussed between two types of singularity in this theory: geometric (de Sitter horizon) and physical (big bang, big crunch). The reasons for the existence of geometric singularities are deeply rooted in the principle of inertia and in the principle of relativity, while physical singularities are associated with the creation or destruction of matter. In this framework, quantum mechanics is introduced …
The replacement of the Poincaré-invariant Einstein special relativity by a de Sitter-invariant special relativity produces concomitant changes in all relativistic theories, including general relativity. A crucial change in the latter is that both the background de Sitter curvature and the gravitational dynamical curvature turns out to be included in a single curvature tensor. This means that the cosmological term no longer explicitly appears in Einstein equation, and is consequently not restricted to be constant. In this paper, the Newtonian limit of such theory is obtained, and the ensuing Ne…
Starting from the Lovelock action and its supplementation by the relevant Gibbons-Hawking-York boundary term, the curvature action corresponding to second-order General Relativity is stated in accordance to the topological properties of the space-time manifold $\mathcal{M}$ with metric solutions being interpreted as topological solitons. Furthermore, this is shown to arise naturally from a topological interpretation of Mach's principle, with the appropriate manifestation of general covariance. Mach's principle is again invoked to suggest formulations of the curvature action in alternative elli…
Given a collection of N asymptotically Euclidean ends with zero scalar curvature, we construct a Riemannian manifold with zero scalar curvature and one asymptotically Euclidean end, whose boundary has a neighborhood isometric to the disjoint union of a specified collection of sub-regions of the given ends. An application is the construction of time-symmetric solutions of the constraint equations which model N-body initial data.
Deterministic nature of general relativity is ensured by the strong cosmic censorship conjecture, which asserts that spacetime cannot be extended beyond Cauchy horizon with square integrable connection. Although this conjecture holds true for asymptotically flat black hole spacetimes in general relativity, a potential violation of this conjecture occurs in charged asymptotically de Sitter spacetimes. Since it is expected that Einstein-Hilbert action will involve higher curvature corrections, in this article we have studied whether one can restore faith in the strong cosmic censorship when high…
The conformal method for constructing initial data for Einstein's equations is presented in both the Hamiltonian and Lagrangian picture (extrinsic curvature decomposition and conformal thin sandwich formalism, respectively), and advantages due to the recent introduction of a weight-function in the extrinsic curvature decomposition are discussed. I then describe recent progress in numerical techniques to solve the resulting elliptic equations, and explore innovative approaches toward the construction of astrophysically realistic initial data for binary black hole simulations.
General relativity can be presented in terms of other geometries besides Riemannian. In particular, teleparallel geometry (i.e., curvature vanishes) has some advantages, especially concerning energy-momentum localization and its ``translational gauge theory'' nature. The standard version is metric compatible, with torsion representing the gravitational ``force''. However there are many other possibilities. Here we focus on an interesting alternate extreme: curvature and torsion vanish but the nonmetricity $\nabla g$ does not---it carries the ``gravitational force''. This {\it symmetric telepar…
A family of quasilocal mass definitions that includes as special cases the Hawking mass and the Brown-York ``rest mass'' energy is derived for spacelike 2-surfaces in spacetime. The definitions involve an integral of powers of the norm of the spacetime mean curvature vector of the 2-surface, whose properties are connected with apparent horizons. In particular, for any spacelike 2-surface, the direction of mean curvature is orthogonal (dual in the normal space) to a unique normal direction in which the 2-surface has vanishing expansion in spacetime. The quasilocal mass definitions are obtained …
We will present a complete set of equations, in the form of an Einstein-Bianchi system, that describe the evolution of generic smooth lattices in spacetime. All 20 independent Riemann curvatures will be evolved in parallel with the leg-lengths of the lattice. We will show that the evolution equations for the curvatures forms a hyperbolic system and that the associated constraints are preserved. This work is a generalisation of our previous paper arXiv:1101.3171 on the Einstein-Bianchi system for the Schwarzschild spacetime to general 3+1 vacuum spacetimes.
A new type of gauge quantum theory (superrelativity) has been proposed. This differs from ordinary gauge theories in sense that the affine connection of our theory is constructed from first derivatives of the Fubini-Study metric tensor. That is we have not merely analogy with general relativity but this construction should presumably provide a unification of general relativity and quantum theory. Here we shall discuss the physical meaning of metric properties of the projective Hilbert space and manifestation of its nontrivial physical character.
A regularization procedure, that allows one to relate singularities of curvature to those of the Einstein tensor without some of the shortcomings of previous approaches, is proposed. This regularization is obtained by requiring that (i) the density $|det {\bf g}|^{1/2}G^a_{b}$, associated to the Einstein tensor $G^a_{b}$ of the regularized metric, rather than the Einstein tensor itself, be a distribution and (ii) the regularized metric be a continuous metric with a discontinuous extrinsic curvature across a non-null hypersurface of codimension one. In this paper, the curvature and Einstein ten…
Scalar Gauss-Bonnet gravity is the only theory with quadratic curvature corrections to general relativity whose field equations are of second differential order. This theory allows for nonperturbative dynamical corrections and is therefore one of the most compelling case studies for beyond-general relativity effects in the strong-curvature regime. However, having second-order field equations is not a guarantee for a healthy time evolution in generic configurations. As a first step towards evolving black-hole binaries in this theory, we here derive the 3+1 decomposition of the field equations f…
Gravitational wave observations of compact binaries allow us to test general relativity (and modifications thereof) in the strong and highly-dynamical field regime of gravity. Here we confront two extensions to general relativity, dynamical Chern-Simons and Einstein-dilaton-Gauss-Bonnet theories, against the gravitational wave sources from the GWTC-1 and GWTC-2 catalogs by the LIGO-Virgo Collaboration. By stacking the posterior of individual events, we strengthen the constraint on the square root of the coupling parameter in Einstein-dilaton-Gauss-Bonnet gravity to $\sqrt{α_{\rm \tiny EdGB}} <…
In the paper we show that the general relativity action (and Lagrangian) in recent Einstein-Palatini formulation is equivalent to the action (and Langrangian) of a gauge field. We begin with a bit of information of the Einstein-Palatini (EP) action, then we present how Einstein fields equations can be derived from it. In the next section, we consider Einstein-Palatini action integral for general relativity with a positive cosmological constant $Λ$ in terms of $\hat{F}$, the curvature of the affine connection's pulled back from de Sitter bundle to Lorentz bundle. We will see that in terms of $\…
It is shown that, with some reasonable assumptions, the theory of general relativity can be made compatible with quantum mechanics by using the field equations of general relativity to construct a Robertson-Walker metric for a quantum particle so that the line element of the particle can be transformed entirely to that of the Minkowski spacetime, which is assumed by a quantum observer, and the spacetime dynamics of the particle described by a Minkowski observer takes the form of quantum mechanics. Spacetime structure of a quantum particle may have either positive or negative curvature. However…
