Geodesic Congruences Research Articles

A type N radiation field solution with Lambda

An anti-de Sitter background four-dimensional type N solution of the Einstein’s field equations, is presented. The matter-energy content pure radiation field satisfies the null energy condition (NEC), and the metric is free-from curvature divergence. In addition, the metric admits a non-expanding, non-twisting and shear-free geodesic null congruence which is not covariantly constant. The space-time admits closed time-like curves which appear after a certain instant of time in a causally well-behaved manner. Finally, the physical interpretation of the solution, based on the study of the equation of the geodesics deviation, is analyzed.

Light rays and the tidal gravitational pendulum

Null geodesic deviation in classical general relativity is expressed in terms of a scalar function, defined as the invariant magnitude of the connecting vector between neighbouring light rays in a null geodesic congruence projected onto a two-dimensional screen space orthogonal to the rays, where λ is an affine parameter along the rays. We demonstrate that η satisfies a harmonic oscillator-like equation with a λ-dependent frequency, which comprises terms accounting for local matter affecting the congruence and tidal gravitational effects from distant matter or gravitational waves passing through the congruence, represented by the amplitude, of a complex Weyl driving term. Oscillating solutions for η imply the presence of conjugate or focal points along the rays. A polarisation angle, is introduced comprising the orientation of the connecting vector on the screen space and the phase, of the Weyl driving term. Interpreting β as the polarisation of a gravitational wave encountering the light rays, we consider linearly polarised waves in the first instance. A highly non-linear, second-order ordinary differential equation, (the tidal pendulum equation), is then derived, so-called due to its analogy with the equation describing a non-linear, variable-length pendulum oscillating under gravity. The variable pendulum length is represented by the connecting vector magnitude, whilst the acceleration due to gravity in the familiar pendulum formulation is effectively replaced by . A tidal torque interpretation is also developed, where the torque is expressed as a coupling between the moment of inertia of the pendulum and the tidal gravitational field. Precessional effects are briefly discussed. A solution to the tidal pendulum equation in terms of familiar gravitational lensing variables is presented. The potential emergence of chaos in general relativity is discussed in the context of circularly, elliptically or randomly polarised gravitational waves encountering the null congruence.

On twisting type [N] ⊗ [N] Ricci flat complex spacetimes with two homothetic symmetries

In this article, HH spaces of type [N] ⊗ [N] with twisting congruence of null geodesics defined by the 4-fold undotted and dotted Penrose spinors are investigated. It is assumed that these spaces admit two homothetic symmetries. The general form of the homothetic vector fields is found. New coordinates are introduced, which enable us to reduce the HH system of partial differential equations to one ordinary differential equation (ODE) on one holomorphic function. In a special case, this is a second-order ODE and its general solution is explicitly given. In the generic case, one gets rather involved fifth-order ODE.

Conformal geodesics in spherically symmetric vacuum spacetimes with cosmological constant

An analysis of conformal geodesics in the Schwarzschild–de Sitter and Schwarzschild–anti-de Sitter families of spacetimes is given. For both families of spacetimes we show that initial data on a spacelike hypersurface can be given such that the congruence of conformal geodesics arising from this data cover the whole maximal extension of canonical conformal representations of the spacetimes without forming caustic points. For the Schwarzschild–de Sitter family, the resulting congruence can be used to obtain global conformal Gaussian systems of coordinates of the conformal representation. In the case of the Schwarzschild–anti-de Sitter family, the natural parameter of the curves only covers a restricted time span so that these global conformal Gaussian systems do not exist.

A Type D Non-Vacuum Spacetime with Causality Violating Curves, and Its Physical Interpretation

We present a topologically trivial, non-vacuum solution of the Einstein’s field equations in four dimensions, which is regular everywhere. The metric admits circular closed timelike curves, which appear beyond the null curve, and these timelike curves are linearly stable under linear perturbations. Additionally, the spacetime admits null geodesics curve, which are not closed, and the metric is of type D in the Petrov classification scheme. The stress-energy tensor anisotropic fluid satisfy the different energy conditions and a generalization of Equation-of-State parameter of perfect fluid . The metric admits a twisting, shearfree, nonexapnding timelike geodesic congruence. Finally, the physical interpretation of this solution, based on the study of the equation of the geodesics deviation, will be presented.

Sachs\u2019 free data in real connection variables

We discuss the Hamiltonian dynamics of general relativity with real connection variables on a null foliation, and use the Newman-Penrose formalism to shed light on the geometric meaning of the various constraints. We identify the equivalent of Sachs’ constraint-free initial data as projections of connection components related to null rotations, i.e. the translational part of the ISO(2) group stabilising the internal null direction soldered to the hypersurface. A pair of second-class constraints reduces these connection components to the shear of a null geodesic congruence, thus establishing equivalence with the second-order formalism, which we show in details at the level of symplectic potentials. A special feature of the first-order formulation is that Sachs’ propagating equations for the shear, away from the initial hypersurface, are turned into tertiary constraints; their role is to preserve the relation between connection and shear under retarded time evolution. The conversion of wave-like propagating equations into constraints is possible thanks to an algebraic Bianchi identity; the same one that allows one to describe the radiative data at future null infinity in terms of a shear of a (non-geodesic) asymptotic null vector field in the physical spacetime. Finally, we compute the modification to the spin coefficients and the null congruence in the presence of torsion.

Kerr-Newman black hole in the formalism of isolated horizons

The near horizon geometry of general black holes in equilibrium can be conveniently characterized in the formalism of weakly isolated horizons in the form of the Bondi-like expansions (Krishnan B, Class.\ Quantum Grav.\ 29, 205006, 2012). While the intrinsic geometry of the Kerr-Newman black hole has been extensively investigated in the weakly isolated horizon framework, the off-horizon description in the Bondi-like system employed by Krishnan has not been studied. We extend Krishnan's work by explicit, non-perturbative construction of the Bondi-like tetrad in the full Kerr-Newman spacetime. Namely, we construct the Bondi-like tetrad which is parallelly propagated along a nontwisting null geodesic congruence transversal to the horizon and provide all Newman-Penrose scalars associated with this tetrad. This work completes the description of the Kerr-Newman spacetime in the formalism of weakly isolated horizons and is a starting point for the investigation of deformed black holes.

A stationary cylindrically symmetric spacetime which admits CTCs and its physical interpretation

Here we present a stationary cylindrically symmetric spacetime which is free from curvature divergence, and satisfy pure radiation field as matter–energy content with positive constant energy density. The metric admits circular closed timelike curves (CTCs) everywhere outside a finite region and the stability of these curves under small linear perturbations is studied, and found to be linearly stable. The metric is of type N in the Petrov classification scheme, and it has a shearfree geodesic null congruence. The physical interpretation of this solution, based on the study of the equation of the geodesic deviation, will be presented. It is demonstrated that, this solution can be understood as exact transverse gravitational waves with amplitudes Ψ4.

A new twist on the geometry of gravitational plane waves

The geometry of twisted null geodesic congruences in gravitational plane wave spacetimes is explored, with special focus on homogeneous plane waves. The rôle of twist in the relation of the Rosen coordinates adapted to a null congruence with the fundamental Brinkmann coordinates is explained and a generalised form of the Rosen metric describing a gravitational plane wave is derived. The Killing vectors and isometry algebra of homogeneous plane waves (HPWs) are described in both Brinkmann and twisted Rosen form and used to demonstrate the coset space structure of HPWs. The van Vleck-Morette determinant for twisted congruences is evaluated in both Brinkmann and Rosen descriptions. The twisted null congruences of the Ozsváth-Schücking, ‘anti-Mach’ plane wave are investigated in detail. These developments provide the necessary geometric toolkit for future investigations of the rôle of twist in loop effects in quantum field theory in curved spacetime, where gravitational plane waves arise generically as Penrose limits; in string theory, where they are important as string backgrounds; and potentially in the detection of gravitational waves in astronomy.

Optical scalars and congruences of light rays: A link between beams and analytic aberrations

Optical scalars are functions designed to analyze the behavior of geodesic congruences in general relativity. Refracted rays are three-dimensional congruences of light rays and they can be studied with this formalism. In this work we obtain the optical scalars for such congruences: the expansion $\mathrm{\ensuremath{\Theta}}$, the twist $\ensuremath{\omega}$, and the shear $\ensuremath{\kappa}$. Furthermore, we apply this machinery to study the aberrations of wavefronts to establish a link between them and the aberration function $W$.

Newtonian potential and geodesic completeness in infinite derivative gravity

Recent study has shown that a non-singular oscillating potential, a feature of Infinite Derivative Gravity (IDG) theories, matches current experimental data better than the standard GR potential. In this work we show that this non-singular oscillating potential can be given by a wider class of theories which allows the defocusing of null rays, and therefore geodesic completeness. We consolidate the conditions whereby null geodesic congruences may be made past-complete, via the Raychaudhuri Equation, with the requirement of a non-singular Newtonian potential in an IDG theory. In so doing, we examine a class of Newtonian potentials characterised by an additional degree of freedom in the scalar propagator, which returns the familiar potential of General Relativity at large distances.

Geodesic flows in a charged black hole spacetime with quintessence

We investigate the evolution of timelike geodesic congruences, in the background of a charged black hole spacetime surrounded by quintessence. The Raychaudhuri equations for three kinematical quantities namely the expansion scalar, shear and rotation along the geodesic flows in such spacetime are obtained and solved numerically. We have also analysed both the weak and the strong energy conditions for the focussing of timelike geodesic congruences. The effect of the normalisation constant (alpha ) and the equation of state parameter (varepsilon ) on the evolution of the expansion scalar is discussed, for the congruences with and without an initial shear and rotation. It is observed that there always exists a critical value of the initial expansion below which we have focussing with smaller values of the normalisation constant and the equation of state parameter. As the corresponding values of both of these parameters are increased, no geodesic focussing is observed. The results obtained are then compared with those of the Reissner Nordström and Schwarzschild black hole spacetimes as well as their de Sitter black hole analogues accordingly.

Light-cones, almost light-cones and almost-complex light-cones

We point out an unusual relationship among a variety of null geodesic congruences; (a) the generators of ordinary light-cones and (b) certain (related) shear-free but twisting congruences in Minkowski Space-time as well as (c) asymptotically shear-free null geodesic congruences that exist in the neighborhood of Penrose's Scri in Einstein or Einstein-Maxwell asymptotically flat-space-times. We refer to these geodesic congruences respectively as: Lignt-Cones (LCs), as Almost-Complex- Light-Cones, (ACLCs), [though they are real they resemble complex light-cones in complex Minkowski space] and finally to a family of congruences in asymptotically flat-spaces as `Almost Light-Cones', (ALC). The two essential points of resemblance among the three families are: (1) they are all either shear-free or asymptotically shear-free and (2) in each family the individual members of the family can be labeled by the points in a real or complex four-dimensional manifold. As an example, the Minkowski space LCs are labeled by the (real) coordinate value of their apex. In the case of (ACLCs) (complex coordinate values), the congruences will have non-vanishing twist whose magnitude is determined by the imaginary part of the complex coordinate values. In studies of gravitational radiation, Bondi-type of null surfaces and their associated Bondi coordinates have been almost exclusively used for calculations. Some surprising relations arise if, instead of the Bondi coordinates, one uses ALCs and their associated coordinate systems in the analysis of the Einstein-Maxwell equations near Scri. More explicitly and surprisingly, the asymptotic Bianchi Identities expressed in the coordinates of the ALCs, turn directly into many of the standard definitions and relations of classical mechanics.

Causal nature and dynamics of trapping horizons in black hole collapse

In calculations of gravitational collapse to form black holes, trapping horizons (foliated by marginally trapped surfaces) make their first appearance either within the collapsing matter or where it joins on to a vacuum exterior. Those which then move outwards with respect to the matter have been proposed for use in defining black holes, replacing the global concept of an ‘event horizon’ which has some serious drawbacks for practical applications. We here present results from a study of the properties of both outgoing and ingoing trapping horizons, assuming strict spherical symmetry throughout. We have investigated their causal nature (i.e. whether they are spacelike, timelike or null), making contact with the Misner–Sharp–Hernandez formalism, which has often been used for numerical calculations of spherical collapse. We follow two different approaches, one using a geometrical quantity related to expansions of null geodesic congruences, and the other using the horizon velocity measured with respect to the collapsing matter. After an introduction to these concepts, we then implement them within numerical simulations of stellar collapse, revisiting pioneering calculations from the 1960s where some features of the emergence and subsequent behaviour of trapping horizons could already be seen. Our presentation here is aimed firmly at ‘real world’ applications of interest to astrophysicists and includes the effects of pressure, which may be important for the asymptotic behaviour of the ingoing horizon.

Surprising structures hiding in Penrose’s future null infinity

Since the late1950s, almost all discussions of asymptotically flat (Einstein–Maxwell) space-times have taken place in the context of Penrose’s null infinity, In addition, almost all calculations have used the Bondi coordinate and tetrad systems. Beginning with a known asymptotically flat solution to the Einstein–Maxwell equations, we show first, that there are other natural coordinate systems, near (analogous to light-cones in flat-space) that are based on (asymptotically) shear-free null geodesic congruences (analogous to the flat-space case). Using these new coordinates and their associated tetrad, we define the complex dipole moment, (the mass dipole plus i times angular momentum), from the l = 1 harmonic coefficient of a component of the asymptotic Weyl tensor. Second, from this definition, from the Bianchi identities and from the Bondi–Sachs mass and linear momentum, we show that there exists a large number of results—identifications and dynamics—identical to those of classical mechanics and electrodynamics. They include, among many others, , spin, Newton’s second law with the rocket force term (v) and radiation reaction, angular momentum conservation and others. All these relations take place in the rather mysterious H-space rather than in space-time.This leads to the enigma: ‘why do these well known relations of classical mechanics take place in H-space?’ and ‘What is the physical meaning of H-space?’

Type N Einstein space Time Machine spacetime

We present an Einstein space axially symmetry spacetime admitting closed timelike curves (CTCs) which appear after a certain instant of time, i.e., a time machine spacetime. The spacetime is a four-dimensional generalization of flat Misner space in curved spacetime, free-from curvature divergences and belongs to type N in the Petrov classification. The spacetime admits a non-expanding, non-twisting, and shear-free null geodesic congruence.

Effective loop quantum geometry of Schwarzschild interior

The success of loop quantum cosmology to resolve classical singularities of homogeneous models has led to its application to the classical Schwarszchild black hole interior, which takes the form of a homogeneous Kantowski-Sachs model. The first steps of this were done in pure quantum mechanical terms, hinting at the traversable character of the would-be classical singularity, and then others were performed using effective heuristic models capturing quantum effects that allowed a geometrical description closer to the classical one but avoided its singularity. However, the problem of establishing the link between the quantum and effective descriptions was left open. In this work, we propose to fill in this gap by considering the path-integral approach to the loop quantization of the Kantowski-Sachs model corresponding to the Schwarzschild black hole interior. We show that the transition amplitude can be expressed as a path integration over the imaginary exponential of an effective action which just coincides, under some simplifying assumptions, with the heuristic one. Additionally, we further explore the consequences of the effective dynamics. We prove first that such dynamics imply some rather simple bounds for phase-space variables, and in turn, remarkably, in an analytical way, they imply that various phase-space functions that were singular in the classical model are now well behaved. In particular, the expansion rate, its time derivative, and the shear become bounded, and hence the Raychaudhuri equation is finite term by term, thus resolving the singularities of classical geodesic congruences. Moreover, all effective scalar polynomial invariants turn out to be bounded.

Maxwell, Yang–Mills, Weyl and eikonal fields defined by any null shear-free congruence

We show that (specifically scaled) equations of shear-free null geodesic congruences on the Minkowski space-time possess intrinsic self-dual, restricted gauge and algebraic structures. The complex eikonal, Weyl 2-spinor, [Formula: see text] Yang–Mills and complex Maxwell fields, the latter produced by integer-valued electric charges (“elementary” for the Kerr-like congruences), can all be explicitly associated with any shear-free null geodesic congruence. Using twistor variables, we derive the general solution of the equations of the shear-free null geodesic congruence (as a modification of the Kerr theorem) and analyze the corresponding “particle-like” field distributions, with bounded singularities of the associated physical fields. These can be obtained in a straightforward algebraic way and exhibit nontrivial collective dynamics simulating physical interactions.

Defocusing of null rays in infinite derivative gravity

Einstein's General theory of relativity permits spacetime singularities, where null geodesic congruences focus in the presence of matter, which satisfies an appropriate energy condition. In this paper, we provide a minimal defocusing condition for null congruences without assuming any ansatz-dependent background solution. The two important criteria are: (1) an additional scalar degree of freedom, besides the massless graviton must be introduced into the spacetime; and (2) an infinite derivative theory of gravity is required in order to avoid tachyons or ghosts in the graviton propagator. In this regard, our analysis strengthens earlier arguments for constructing non-singular bouncing cosmologies within an infinite derivative theory of gravity, without assuming any ansatz to solve the full equations of motion.

No practical lensing by Lambda: Deflection of light in the Schwarzschild–de Sitter spacetime

Debate persists as to whether the cosmological constant $\Lambda$ can directly modify the power of a gravitational lens. With the aim of reestablishing a consensus on this issue, I conduct a comprehensive analysis of gravitational lensing in the Schwarzschild--de Sitter spacetime, wherein the effects of $\Lambda$ should be most apparent. The effective lensing law is found to be in precise agreement with the $\Lambda=0$ result: $\alpha_\mathrm{eff} = 4m/b_\mathrm{eff}+15\pi m^2/4b_\mathrm{eff}^2 +O(m^3/b_\mathrm{eff}^3)$, where the effective bending angle $\alpha_\mathrm{eff}$ and impact parameter $b_\mathrm{eff}$ are defined by the angles and angular diameter distances measured by a comoving cosmological observer. [These observers follow the timelike geodesic congruence which (i) respects the continuous symmetries of the spacetime and (ii) approaches local isotropy most rapidly at large distance from the lens.] The effective lensing law can be derived using lensed or unlensed angular diameter distances, although the inherent ambiguity of unlensed distances generates an additional uncertainty $O(m^5/\Lambda b_\mathrm{eff}^7)$. I conclude that the cosmological constant does not interfere with the standard gravitational lensing formalism.