Raychaudhuri Equation Research Articles

Raychaudhuri equation in spacetimes with torsion and nonmetricity

We introduce and develop the 1+3 covariant approach to relativity and cosmology to spacetimes of arbitrary dimensions that have nonzero torsion and do not satisfy the metricity condition. Focusing on timelike observers, we identify and discuss the main differences between their kinematics and those of their counterparts living in standard Riemannian spacetimes. At the centre of our analysis lies the Raychaudhuri equation, which is the fundamental formula monitoring the convergence/divergence, namely the collapse/expansion, of timelike worldline congruences. To the best of our knowledge, we provide the most general expression so far of the Raychaudhuri equation, with applications to an extensive range of non-standard astrophysical and cosmological studies. Assuming that metricity holds, but allowing for nonzero torsion, we recover the results of analogous previous treatments. Focusing on non-metricity alone, we identify a host of effects that depend on the nature of the timelike congruence and on the type of the adopted non-metricity. We also demonstrate that in spaces of high symmetry one can recover the pure-torsion results from their pure non-metricity analogues, and vice-versa, via a simple ansatz between torsion and non-metricity.

Effective Null Raychaudhuri Equation

The effects on Raychaudhuri’s equation of an intrinsically-discrete or particle nature of spacetime are investigated. This is done through the consideration of null congruences emerging from, or converging to, a generic point of spacetime, i.e., in geometric circumstances somehow prototypical of singularity issues. We do this from an effective point of view, that is through a (continuous) description of spacetime modified to embody the existence of an intrinsic discreteness on the small scale, this adding to previous results for non-null congruences. Various expressions for the effective rate of change of expansion are derived. They in particular provide finite values for the limiting effective expansion and its rate of variation when approaching the focal point. Further, this results in a non-vanishing of the limiting cross-sectional area itself of the congruence.

Raychaudhuri and optical equations for null geodesic congruences with torsion

We study null geodesic congruences (NGCs) in the presence of spacetime torsion, recovering and extending results in the literature. Only the highest spin irreducible component of torsion gives a proper acceleration with respect to metric NGCs, but at the same time obstructs abreastness of the geodesics. This means that it is necessary to follow the evolution of the drift term in the optical equations, and not just shear, twist and expansion. We show how the optical equations depend on the non-Riemannian components of the curvature, and how they reduce to the metric ones when the highest spin component of torsion vanishes.

Classical defocussing of world lines in higher dimensions

A five-dimensional gravity theory, motivated by the brane-world picture, with Kaluza scalar in the 5-dimensional metric as g55(r);r=x2+y2+z2, is considered near the possible singularity (small distance scales where gravity is strong) and is shown to give rise to a positive contribution to the Raychaudhuri equation. This inhibits the focusing of world lines and contributes to non-focusing of the worldlines in the 5-dimensional space. It is also shown that the results extend to time dependent cases such as those relevant for black hole interiors and cosmology.

Conformal symmetric Bianchi type-I cosmologies in f(R) gravity

In this study, we investigate the Bianchi type-I cosmologies with string cloud attached to perfect fluid in f(R) gravity. The field equations and their exact solutions for Bianchi type-I cosmologies with string cloud attached to a perfect fluid are found by using the conformal symmetry properties. The obtained solutions under the varied selection of arbitrary constants indicate three cosmological models. Isotropy conditions for obtained cosmological models are investigated for large value of time. Whether or not the string cloud in conformal symmetric Bianchi type-I universe supports the isotropy condition for the large value of time has been investigated. Also, we examine the contracting and decelerating features of the obtained solutions by using Raychaudhuri equation. Finally, geometrical and physical results of the solutions are discussed.

Black hole entropy production and transport coefficients in Lovelock gravity

We study the entropy evolution of black holes in Lovelock gravity by formulating a thermodynamic generalization of null Raychaudhuri equation. We show that the similarity between the expressions of entropy change of the black hole horizon due to perturbation and that of a fluid, which is out of equilibrium, transcends beyond general relativity to the Lovelock class of theories. Exploiting this analogy we find that the shear and bulk viscosities for the black holes in Lovelock theories exactly match with those obtained in the membrane paradigm and also from holographic considerations.

Towards the Raychaudhuri equation beyond general relativity

In general relativity, gravity is universally attractive, a feature embodied by the Raychaudhuri equation which requires that the expansion of a congruence of geodesics is always non-increasing, as long as matter obeys the strong or weak energy conditions. This behavior of geodesics is an important ingredient in general proofs of singularity theorems, which show that many spacetimes are singular in the sense of being geodesically incomplete and suggest that general relativity is itself incomplete. It is possible that alternative theories of gravity, which reduce to general relativity in some limit, can resolve these singularities, so it is of interest to consider how the behavior of geodesics is modified in these frameworks. We compute the leading corrections to the Raychaudhuri equation for the expansion due to models in string theory, braneworld gravity, $f(R)$ theories, and loop quantum cosmology, for cosmological and black hole backgrounds, and show that while in most cases geodesic convergence is reinforced, in a few cases terms representing repulsion arise, weakening geodesic convergence and thereby the conclusions of the singularity theorems.

Congruence Convergence in Pp-Wave Space–Time

We argue that the well-known geodesic completeness property of pp-waves, can be disregarded once the geodesics are extracted as being extended along sets of Brinkmann coordinates. This issue is investigated in the more general context of congruence convergence and we show that the problem leads to diverse issues for non-geodesic congruences. The discussion is mostly based on the null congruence expansion and a generalized Raychaudhuri equation is also provided.

Conditions for defocusing around more general metrics in infinite derivative gravity

Infinite Derivative Gravity is able to resolve the Big Bang curvature singularity present in general relativity by using a simplifying ansatz. We show that it can also avoid the Hawking- Penrose singularity, by allowing defocusing of null rays through the Raychaudhuri equation. This occurs not only in the minimal case where we ignore the matter contribution, but also in the case where matter plays a key role. We investigate the conditions for defocusing for the general case where this ansatz applies and also for more specific metrics, including a general Friedmann-Robertson-Walker (FRW) metric and three specific choices of the scale factor which produce a bouncing FRW universe.

Quantum no-singularity theorem from geometric flows

In this paper, we analyze the classical geometric flow as a dynamical system. We obtain an action for this system, such that its equation of motion is the Raychaudhuri equation. This action will be used to quantize this system. As the Raychaudhuri equation is the basis for deriving the singularity theorems, we will be able to understand the effects and such a quantization will have on the classical singularity theorems. Thus, quantizing the geometric flow, we can demonstrate that a quantum space–time is complete (nonsingular). This is because the existence of a conjugate point is a necessary condition for the occurrence of singularities, and we will be able to demonstrate that such conjugate points cannot occur due to such quantum effects.

The GUP and quantum Raychaudhuri equation

In this paper, we compare the quantum corrections to the Schwarzschild black hole temperature due to quadratic and linear-quadratic generalised uncertainty principle, with the corrections from the quantum Raychaudhuri equation. The reason for this comparison is to connect the deformation parameters β0 and α0 with η which is the parameter that characterises the quantum Raychaudhuri equation. The derived relation between the parameters appears to depend on the relative scale of the system (black hole), which could be read as a beta function equation for the quadratic deformation parameter β0. This study shows a correspondence between the two phenomenological approaches and indicates that quantum Raychaudhuri equation implies the existence of a crystal-like structure of spacetime.

Unified Geometrical Basis for the Generalized Ehlers Identities and Raychaudhuri Equations

It is shown that the timelike, spacelike and null versions of the Ehlers identity, as well as ensuing Raychaudhuri equations, might be all derived within a single geometrical approach based on the definition of the Riemann curvature tensor specified with respect to the corresponding congruence. Still, spacelike and null cases have a number of nontrivial peculiarities deserving special attention.

Weak field equations and generalized FRW cosmology on the tangent Lorentz bundle

We study field equations for a weak anisotropic model on the tangent Lorentz bundle TM of a spacetime manifold. A geometrical extension of general relativity (GR) is considered by introducing the concept of local anisotropy, i.e. a direct dependence of geometrical quantities on observer 4−velocity. In this approach, we consider a metric on TM as the sum of an h-Riemannian metric structure and a weak anisotropic perturbation, field equations with extra terms are obtained for this model. As well, extended Raychaudhuri equations are studied in the framework of Finsler-like extensions. Canonical momentum and mass-shell equation are also generalized in relation to their GR counterparts. Quantization of the mass-shell equation leads to a generalization of the Klein–Gordon equation and dispersion relation for a scalar field. In this model the accelerated expansion of the universe can be attributed to the geometry itself. A cosmological bounce is modeled with the introduction of an anisotropic scalar field. Also, the electromagnetic field equations are directly incorporated in this framework.

Raychaudhuri equation in the Finsler–Randers space-time and generalized scalar-tensor theories

In this work, we obtain the Raychaudhuri equations for various types of Finsler spaces as the Finsler–Randers (FR) space-time and in a generalized geometrical structure of the space-time manifold which contains two fibers that represent two scalar fields [Formula: see text]. We also derive the Klein–Gordon equation for this model. In addition, the energy conditions are studied in a FR cosmology and are correlated with FRW model. Finally, we apply the Raychaudhuri equation for the model [Formula: see text], where M is a FRW-space-time.

Evaporation/Antievaporation and energy conditions in alternative gravity

We discuss the evaporation and antievaporation instabilities of Nariai solution in extended theories of gravity. These phenomena were explicitly shown in several different extensions of General Relativity, suggesting that a universal cause is behind them. We show that evaporation and antievaporation are originated from deformations of energy conditions on the Nariai horizon. Energy conditions get new contributions from the extra propagating degrees of freedom, which can provide extra focalizing or antifocalizing terms in the Raychaudhuri equation. We show the two explicit cases of [Formula: see text]-gravity and Gauss–Bonnet gravity.

Replacing dark energy by silent virialisation

Context.Standard cosmologicalN-body simulations have background scale factor evolution that is decoupled from non-linear structure formation. Prior to gravitational collapse, kinematical backreaction (QD) justifies this approach in a Newtonian context.Aims.However, the final stages of a gravitational collapse event are sudden; a globally imposed smooth expansion rate forces at least one expanding region to suddenly and instantaneously decelerate in compensation for the virialisation event. This is relativistically unrealistic. A more conservative hypothesis is to allow non-collapsed domains to continue their volume evolution according to theQDZel’dovich approximation (QZA). We aim to study the inferred average expansion under this “silent” virialisation hypothesis.Methods.We set standard (MPGRAFIC) EdS 3-torus (T3) cosmologicalN-body initial conditions. UsingRAMSES, we partitioned the volume into domains and called theDTFElibrary to estimate the per-domain initial values of the three invariants of the extrinsic curvature tensor that determine the QZA. We integrated the Raychaudhuri equation in each domain using theINHOMOGlibrary, and adopted the stable clustering hypothesis to represent virialisation (VQZA). We spatially averaged to obtain the effective global scale factor. We adopted an early-epoch–normalised EdS reference-model Hubble constantH1EDS= 37.7km s-1∕Mpc and an effective Hubble constantHeff,0= 67.7km s-1∕Mpc.Results.From 2000 simulations at resolution 2563, we find that reaching a unity effective scale factor at 13.8 Gyr (16% above EdS), occurs for an averaging scale ofL13.8= 2.5−0.4+0.1Mpc∕heff. Relativistically interpreted, this corresponds to strong average negative curvature evolution, with the mean (median) curvature functionalΩRDgrowing from zero to about 1.5–2 by the present. Over 100 realisations, the virialisation fraction and super-EdS expansion correlate strongly at fixed cosmological time.Conclusions.Thus, starting from EdS initial conditions and averaging on a typical non-linear structure formation scale, the VQZA dark-energy–free average expansion matchesΛCDM expansion to first order. The software packages used here are free-licensed.

Initial singularity and pure geometric field theories

In the present article we use a modified version of the geodesic equation, together with a modified version of the Raychaudhuri equation, to study initial singularities. These modified equations are used to account for the effect of the spin-torsion interaction on the existence of initial singularities in cosmological models. Such models are the results of solutions of the field equations of a class of field theories termed pure geometric. The geometric structure used in this study is an absolute parallelism structure satisfying the cosmological principle. It is shown that the existence of initial singularities is subject to some mathematical (geometric) conditions. The scheme suggested for this study can be easily generalized.

Entropy Measures as Geometrical Tools in the Study of Cosmology.

Classical chaos is often characterized as exponential divergence of nearby trajectories. In many interesting cases these trajectories can be identified with geodesic curves. We define here the entropy by with being the distance between two nearby geodesics. We derive an equation for the entropy, which by transformation to a Riccati-type equation becomes similar to the Jacobi equation. We further show that the geodesic equation for a null geodesic in a double-warped spacetime leads to the same entropy equation. By applying a Robertson–Walker metric for a flat three-dimensional Euclidean space expanding as a function of time, we again reach the entropy equation stressing the connection between the chosen entropy measure and time. We finally turn to the Raychaudhuri equation for expansion, which also is a Riccati equation similar to the transformed entropy equation. Those Riccati-type equations have solutions of the same form as the Jacobi equation. The Raychaudhuri equation can be transformed to a harmonic oscillator equation, and it has been shown that the geodesic deviation equation of Jacobi is essentially equivalent to that of a harmonic oscillator. The Raychaudhuri equations are strong geometrical tools in the study of general relativity and cosmology. We suggest a refined entropy measure applicable in cosmology and defined by the average deviation of the geodesics in a congruence.

Spacetime thermodynamics in the presence of torsion

It was shown by Jacobson in 1995 that the Einstein equation can be derived as a local constitutive equation for an equilibrium spacetime thermodynamics. With the aim to understand if such thermodynamical description is an intrinsic property of gravitation, many attempts have been done so far to generalise this treatment to a broader class of gravitational theories. Here we consider the case of the Einstein-Cartan theory as a prototype of theories with non-propagating torsion. In doing so, we study the properties of Killing horizons in the presence of torsion, establish the notion of local causal horizon in Riemann-Cartan spacetimes, and derive the generalised Raychaudhuri equation for this kind of geometries. Then, starting with the entropy that can be associated to these local causal horizons, we derive the Einstein-Cartan equation by implementing the Clausius equation. We outline two ways of proceeding with the derivation depending on whether we take torsion as a geometric field or as a matter field. In both cases we need to add internal entropy production terms to the Clausius equation as the shear and twist cannot be taken to be zero a priori for our setup. This fact implies the necessity of a non-equilibrium thermodynamics treatment for the local causal horizon. Furthermore, it implies that a non-zero twist at the horizon will in general contribute to the Hartle-Hawking tidal heating for black holes with possible implications for future observations.

Covariant kinematics of light in media and a generalized Raychaudhuri equation

There is ongoing interest in adopting various tools and ideas from general relativity for optical applications and the study of light propagation through natural or engineered media. Here, the covariant kinematics of light propagating through arbitrary dielectric media in curved space-times are derived, allowing for analysis and tracing of congruences of light through media that may smoothly vary in character between vacuum, positively refracting, and negatively refracting; or null, timelike, and spacelike with respect to the background metric. The kinematics are then used to generalize the Raychaudhuri equation -- an important tool in general relativity that describes the focus of a congruence. These results will be useful for the analysis of optical devices, particularly those designed using transformation optics, and serve as theoretical tools to study generalized concepts in general relativity.