Geodesic Congruences Research Articles

Supersymmetry of the Robinson-Trautman solution

The Robinson-Trautman solution in the Einstein-Maxwell-Λ system admits a shear-free and twist-free null geodesic congruence with a nonvanishing expansion. Restricting to the case where the Maxwell field is aligned, i.e., the spacetime is algebraically special, we provide an exhaustive classification of supersymmetric Robinson-Trautman spacetimes in the four-dimensional N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 2 gauged supergravity. The differential constraints that arise from the integrability conditions of the Killing spinor equation enable us to systematically reconstruct the metric. We derive the explicit form of the Killing spinor either by directly integrating the Killing spinor equation or by casting the solution into the canonical form of supersymmetric solutions given in [13]. In any case, the supersymmetric Robinson-Trautman solution generically exhibits a naked singularity.

On the existence and avoidance of singularity in [formula omitted] gravity using Raychaudhuri equation

The paper deals with the modified Raychaudhuri equation in the framework of homogeneous and isotropic f(Q) gravity model subject to the parametrization f(Q)=Q+F(Q). Focusing of a congruence of time-like geodesics has been studied via the convergence condition for two choices of F(Q) namely, a polynomial in Q and an exponential form of Q. Finally, possibilities for the avoidance of initial big-bang singularity in both the models have been examined via the signature of Raychaudhuri scalar.

Quantum-gravitational null Raychaudhuri equation

We consider a congruence of null geodesics in the presence of a quantized spacetime metric. The coupling to a quantum metric induces fluctuations in the congruence; we calculate the change in the area of a pencil of geodesics induced by such fluctuations. For the gravitational field in its vacuum state, we find that quantum gravity contributes a correction to the null Raychaudhuri equation which is of the same sign as the classical terms. We thus derive a quantum-gravitational focusing theorem valid for linearized quantum gravity.

Solutions with pure radiation and gyratons in 3D massive gravity theories

We find exact solutions of topologically massive gravity (TMG) and new massive gravity (NMG) in 2 + 1 dimensions (3D) with an arbitrary cosmological constant, pure radiation, and gyratons, i.e. with possibly non-zero T uu and T ux in canonical coordinates. Since any ‘reasonable’ geometry in 3D (i.e. admitting a null geodesic congruence) is either expanding Robinson–Trautman ( Θ≠0 ) or Kundt ( Θ=0 ), we focus on these two classes. Assuming expansions Θ=1/r (‘GR-like’ Robinson–Trautman) or Θ=0 (general Kundt), we systematically integrate the field equations of TMG and NMG and identify new classes of exact solutions. The case of NMG contains an additional assumption of g ux being quadratic in r, which is automatically enforced in TMG as well as in 3D GR. In each case, we reduce the field equations as much as possible and identify new classes of solutions. We also discuss various special subclasses and study some explicit solutions.

Singularity attenuation with quantum‐mechanically revisited metric tensor

AbstractThe space and initial singularities are reexamined in the most reliable solutions to the Einstein's field equations (EFE), that is, the Einstein–Gilbert–Straus (EGS) metric. In discretized Finsler geometry, additional curvatures and thereby geometric structures likely emerge, which are distinct from the conventional spacetime curvatures and geometric structures that the Einstein's theory of general relativity introduced. The generalized fundamental tensor, which is obtained in the Fisleriean geometry, imposes quantum‐mechanically revisions on the Landau–Raychaudhuri evolution equations. The time‐like geodesic congruence in EGS metric is then analyzed, analytically and numerically. The evolution of a family of trajectories whose congruence is defined by the flow lines generated by velocity fields is determined. We conclude that both two types of singularities seem to be attenuated or even regulate. With the singularity attenuation, we refer to the fundamental nature of the additional curvatures at quantum relativistic scales.

Principal Equatorial Null Geodesic Congruences in the Kerr Metric, and Their Quantum Propagators

Principal Equatorial Null Geodesic Congruences in the Kerr Metric, and Their Quantum Propagators

Quantum focusing conjecture and the Page curve

The focusing theorem fails for evaporating black holes because the null energy condition is violated by quantum effects. The quantum focusing conjecture is proposed so that it is satisfied even if the null energy condition is violated. The conjecture states that the derivative of the sum of the area of a cross-section of the null geodesic congruence and the entanglement entropy of matter outside it is non-increasing. Naively, it is expected that the quantum focusing conjecture is violated after the Page time as both the area of the horizon and the entanglement entropy of the Hawking radiation are decreasing. We calculate the entanglement entropy after the Page time by using the island rule, and find the following results: (i) the page time is given by an approximately null surface, (ii) the entanglement entropy is increasing along the outgoing null geodesic even after the Page time, and (iii) the quantum focusing conjecture is not violated.

Timelike geodesic congruence in the simplest solutions of general relativity with quantum-improved metric tensor

A recent quantum-geometrical approach for the possible reconciliation of the principles of general relativity and quantum mechanics extends the fundamental tensor to the relativistic (quantum) scales, in which additional geometric structures and curvatures, i.e., additional sources of gravitation, are emerged. This paper introduces some characteristics of the emerged curvatures and their consequences on regulating the space and initial singularities. We analyze the timelike geodesic congruence in the simplest solutions of the Einstein field equations; the homogeneous, isotropic, and spherically symmetric Schwarzschild, de Sitter–Schwarzschild, and Friedmann–Lemaitre–Robertson–Walker (FLRW) metrics. From the expansion of the trajectory congruence that follows the change in the shape of the volume which keeps the same set of geodesics in the bundle along the flow lines which are generated by the velocity fields, we conclude that the analytical and numerical analyses give an unambiguous signature that the space and initial singularities are attenuated or even regulated, especially in the relativistic (quantum) regime. This means that the regulation of the singularity in Einstein’s general relativity (GR) would not be possible until quantum-mechanical ingredients are properly imposed on it.

Raychaudhuri equation and bouncing cosmology

This work deals with an exhaustive study of bouncing cosmology in the background of homogeneous and isotropic Friedmann–Lemaître–Robertson–Walker spacetime. The geometry of the bouncing point has been studied extensively and used as a tool to classify the models from the point of view of cosmology. Raychaudhuri equation (RE) has been furnished in these models to classify the bouncing point as regular point or singular point. Behavior of time-like geodesic congruence in the neighborhood of the bouncing point has been discussed using the Focusing Theorem which follows as a consequence of the RE. An analogy of the RE with the evolution equation for a linear harmonic oscillator has been made and an oscillatory bouncing model has been discussed in this context.

Rotating black holes embedded in a cosmological background for scalar-tensor theories

We present solutions of DHOST theories describing a rotating black hole embedded in an expanding universe. The solution is constructed by conformal transformation of a stealth Kerr(-de Sitter) black hole. The conformal factor depends explicitly on the scalar field — but not on its derivative — and defines the new theory. The scalar field of the stealth Kerr(-de Sitter) solution depends on time, leading to the time-dependence of the obtained conformal metric, with cosmological asymptotics at large distances. We study the properties of the obtained metric by considering regular null geodesic congruences, and identify trapping black hole and cosmological horizons.

On the visibility of singularities in general relativity and modified gravity theories

We investigate the global causal structure of the end state of a spherically symmetric marginally bound Lemaitre–Tolman–Bondi (LTB) (Lemaitre 1933 Ann. Soc. Sci. Brux. A 53 51, Tolman 1934 Proc. Natl Acad. Sci. USA 20 169–76, Bondi 1947 Mon. Not. Astron. Soc. 107 410) collapsing cloud (which is well studied in general relativity) in the framework of modified gravity having the generalized Lagrangian in the action. Here R is the Ricci scalar, and is a constant. By fixing the functional form of the metric components of the LTB spacetime, using up the available degree of freedom, we realize that the matching surface of the interior and the exterior metric are different for different values of α. This change in the matching surface can alter the causal property of the first central singularity. We depict this by showing a numerical example. Additionally, for a globally naked singularity to have physical relevance, a congruence of null geodesics should escape from such singularity to be visible to an asymptotic observer for infinite time. For this to happen, the first central singularity should be a nodal point. We here give a heuristic method to show that this singularity is a nodal point by considering the above class of theory of gravity, of which general relativity is a particular case.

On the non-linear stability of the Cosmological region of the Schwarzschild-de Sitter spacetime

The non-linear stability of the sub-extremal Schwarzschild-de Sitter spacetime in the stationary region near the conformal boundary is analysed using a technique based on the extended conformal Einstein field equations and a conformal Gaussian gauge. This strategy relies on the observation that the Cosmological stationary region of this exact solution can be covered by a non-intersecting congruence of conformal geodesics. Thus, the future domain of dependence of suitable spacelike hypersurfaces in the Cosmological region of the spacetime can be expressed in terms of a conformal Gaussian gauge. A perturbative argument then allows to prove existence and stability results close to the conformal boundary and away from the asymptotic points where the Cosmological horizon intersects the conformal boundary. In particular, we show that small enough perturbations of initial data for the sub-extremal Schwarzschild-de Sitter spacetime give rise to a solution to the Einstein field equations which is regular at the conformal boundary. The analysis in this article can be regarded as a first step towards a stability argument for perturbation data on the Cosmological horizons.

The Maslov index and some applications to dispersion relations in curved space times

The aim of the present work is to generalize the results given in Osorio Morales and Santillán [Eur. Phys. J. C 82, 353 (2022)] to a generic situation for causal geodesics. It is argued that these results may be of interest for causality issues. Recall that the presence of superluminal signals in a generic space time (M, gμν) does not necessarily imply violations of the principle of causality {[G. M. Shore, Nucl. Phys. B 778, 219 (2007)] and [T. J. Hollowood and G. M. Shore, Phys. Lett. B 655, 67 (2007)]}. In flat spaces, global Lorenz invariance leads to the conclusion that closed time-like curves appear if these signals are present. In a curved space instead, there is only local Poincare invariance, and the presence of closed causal curves may be avoided even in the presence of a superluminal mode, especially when terms violating the strong equivalence principle appear in the action. This implies that the standard analytic properties of the spectral components of these functions are therefore modified, and in particular, the refraction index n(ω) is not analytic in the upper complex ω plane. The emergence of these singularities may also take place for non-superluminal signals due to the breaking of global Lorenz invariance in a generic space time. In the present work, it is argued that the homotopy properties of the Maslov index are useful for studying how the singularities of n(ω) vary when moving along a geodesic congruence. In addition, several conclusions obtained in Shore [Nucl. Phys. B 778, 219 (2007)] and Hollowood and Shore [Phys. Lett. B 655, 67 (2007)] are based on the Penrose limit along a null geodesic, and they are restricted to GR with matter satisfying strong energy conditions. The use of the Maslov index may allow a more intrinsic description of singularities, not relying on that limit, and a generalization of these results about non-analyticity to generic gravity models with general matter content.

Quantum gravity fluctuations in the timelike Raychaudhuri equation

We consider a timelike geodesic congruence in the presence of perturbative quantum fluctuations of the spacetime metric. We calculate the change in the volume of a bundle of geodesics due to such fluctuations and thereby obtain a quantum-gravitationally modified timelike Raychaudhuri equation. Quantum gravity generically increases the convergence of congruences and the production of caustics.

Possible relationship between initial conditions for inflation and past geodesic incompleteness of the inflationary spacetime

According to the Borde-Guth-Vilenkin (BGV) theorem an expanding region of spacetime cannot be extended to the past beyond some boundary ℬ. Therefore, the inflationary universe must have had some kind of beginning. However, the BGW theorem says nothing about the boundary conditions on ℬ, or even about its location. Here we present a single-scalar field model of the Two-Measure Theory, where the non-Riemannian volume element Υ d 4 x is present in the action. As a result of the model dynamics, an upper bound φ 0 of admissible values of the scalar field φ appears, which sets the position of ℬ in the form of a spacelike hypersurface Υ(x) = 0 with a boundary condition: Υ → 0+ as φ → φ 0 -. A detailed study has established that if the initial kinetic energy density ρ kin (in) prevails over initial gradient energy density ρ grad (in) then there is an interval of initial values φ in (min) ≤ φ in < φ 0, where ρ kin (in) and ρ grad (in) cannot exceed the potential energy density and hence the initial conditions necessary for the onset of inflation are satisfied. It is shown that under almost all possible left-handed boundary conditions on ℬ, that is where Υ → 0-, the metric tensor in the Einstein frame has a jump discontinuity on ℬ, so the Christoffel connection coefficients are not defined on the spacelike hypersurface Υ = 0. Thus, if φ in (min) ≤ φ in < φ 0 and ρ kin (in) > ρ grad (in), then there was an inflationary stage in the history of our Universe and the congruence of timelike geodesics cannot be extended to the past beyond the hypersurface Υ = 0.

Graviton noise on tidal forces and geodesic congruences

In this work we continue with our recent study, using the Feynman-Vernon worldline influence action and the Schwinger-Keldysh closed-time-path formalism, to consider the effects of quantum noise of gravitons on the motion of point masses. This effect can be regarded as due to a stochastic tensorial force whose correlator is given by the graviton noise kernel associated with the Hadamard function of the quantized gravitational field. Solving the Langevin equation governing the motion of the separation of two masses, the fluctuations of the separation due to the graviton noise can be obtained for various states of the quantum field. Since this force has the stretching and compressing effects like the tidal force, we can view it as one. We therefore derive the expressions for, and estimate the magnitude of, this tidal force for the cases of the Minkowski and the squeezed vacua. The influence of this force on the evolution of the geodesic congruence through the Raychaudhuri equation is then studied and the effects of quantum graviton noise on the shear and rotation tensors presented.

Geodesic completeness of a ring wormhole

In this work the possible geodesic completeness of an electromagnetic dipole wormhole is studied in detail. The space-time contains a curvature singularity and belongs to a class of solutions to the Einstein-Maxwell equations with a coupled scalar field. Specifically, a numerical analysis is performed to examine congruences of null geodesics that are directed toward the singularity. The results found here show that, depending on the strength of the coupling between the scalar and electromagnetic fields, the wormhole can be either geodesically incomplete or complete. We then focus on those wormholes that are geodesically complete and study the geometry of the neighborhood of their singularity using Kaluza-Klein theory in a five-dimensional space-time. This process allows us to provide a possible explanation of the completeness of geodesics despite unbounded curvature.

The Raychaudhuri equation in inhomogeneous FLRW space-time: A f(R)-gravity model

In general description of the Raychaudhuri equation it is found that this first order non-linear differential equation can be written as a second order linear differential equation in the form of Harmonic Oscillator with varying frequency. Further, the integrability of the Raychaudhuri equation has been studied and also the expansion scalar is obtained in an explicit form. Subsequently, f(R) gravity theory has been studied in the background of inhomogeneous FLRW spacetime with an aim to formulate the Raychaudhuri equation. A congruence of time-like geodesics has been investigated using the Raychaudhuri equation to examine whether the geodesics converge or not and some possible conditions are determined to avoid singularity. Finally, a brief quantum description has been presented.

Twistor space origins of the Newman-Penrose map

Recently, we introduced the “Newman-Penrose map”, a novel correspondence between a certain class of solutions of Einstein’s equations and self-dual solutions of the vacuum Maxwell equations, which we showed was closely related to the classical double copy. Here, we give an alternative definition of this correspondence in terms of quantities that are defined naturally on twistor space, and a shear-free null geodesic congruence on Minkowski space whose twistorial character is articulated by the Kerr theorem. The advantage of this reformulation is that it is purely geometrical in nature, being manifestly invariant under both spacetime diffeomorphisms and projective transformations on twistor space. While the original formulation of the map may be more convenient for most explicit calculations, the twistorial formulation we present here may be of greater theoretical utility.

Geodesic congruences in modified Schwarzschild black holes

We investigate two different kinds of modified Schwarzschild black holes: A regularized Schwarzschild black hole and a quantum deformed Schwarzschild black hole. We study the geodesics and geodesic congruences in these two modified Schwarzschild black holes. In particular, we calculate the expansion of radial timelike and null geodesic congruences. Based on these results, we discuss some similarities and differences between these two kinds of modified Schwarzschild black holes.