Lightlike Geodesics Research Articles

Purely spatial frame-dragging and oblate spherodial refractive index as an optical analog to Kerr spacetime

Abstract This paper presents an optical analogy to the lightlike geodesics in spacetime with frame-dragging effects, particularly in Kerr spacetime. This was done by correspondingly equating parts of the deflection angle expression of the lightlike geodesics to that of the optical analog with gradient refractive index (GRIN). The resulting conditions of analogy separates the GRIN from the frame-dragging effects, yielding an oblate spheroidally symmetric GRIN expression and a new optical Lagrangian which situates such medium in a non-Euclidean background space. Such space is demostrated to possess a purely spatial frame-dragging effect. This analogy might suggest a generalized notion of analog magnetism or provide additional insights to studies on gravitational lensing and construction of optical analogs in the laboratory setting.

Gravitational lensing in a spacetime with cosmic string within the Eddington-inspired Born–Infeld gravity

This study explores the deflection angle of photon rays or light-like geodesics within the framework of Eddington-inspired Born–Infeld (EiBI) gravity background space-time, taking into account the influence of cosmic strings. The primary focus lies in deriving the effective potential of the system applicable to both null and time-like geodesics, as well as determining the angle of deflection for light-like geodesics. Our analysis shows that the presence of cosmic strings induces modifications in these physical quantities, leading to shifts in their respective values.

Wave equations in the Kerr–de Sitter spacetime: The full subextremal range

We prove that solutions to linear wave equations in a subextremal Kerr–de Sitter spacetime have asymptotic expansions in quasinormal modes up to a decay order given by the normally hyperbolic trapping, extending the result of the second named author (2013). The main novelties are a different way of obtaining a Fredholm setup that defines the quasinormal modes and a new analysis of the trapping of lightlike geodesics in the Kerr–de Sitter spacetime, both of which apply in the full subextremal range. In particular, this reduces the question of decay for solutions to wave equations to the question of mode stability.

Particle motion and tidal force in a non-vacuum-charged naked singularity

We investigate the gravitational field of a charged, non-vacuum, non-rotating, spherically symmetric body of mass M assuming a static solution to the Einstein–Maxwell field equations. We show the characteristics of perihelion precession of orbits in the case of charged naked singularity (CNS) spacetime. Here we discuss some novel features of light-like geodesics in this spacetime. We also discuss the comparative study of tidal force in the null singularity spacetime and charged naked singularity spacetime.

Timelike surfaces in [formula omitted] with a canonical null direction

Let N be a Lorentz surface and let ∂t be a unit vector field on the Lorentz manifold N×R tangent to R. A timelike surface Σ in N×R is said to have a canonical null direction with respect to ∂t if the projection ∂t⊤ on the tangent space of Σ gives a lightlike vector field. We prove that these surfaces are minimal and ruled, whose rulings are lightlike geodesics and lines of curvature. Other results include the fact that these surfaces are flat if they are totally geodesic. Furthermore, let us assume that the surface Σ is the graph of a function f:N→R. Then it has a canonical null direction if and only if the gradient of f is a lightlike vector field on N. Another property is that the Gaussian curvature of Σ coincides with the Gaussian curvature of N and the sectional curvature of N×R, along tangent planes to Σ. In particular such surfaces in S12×R have constant curvature. We prove, using isothermal coordinates, that functions on N whose gradient is lightlike locally always exist. We give several examples of such surfaces.

Gravitational lensing by Morris-Thorne–type wormhole space-time with cosmic strings

In this paper, we study the deflection angle of photon light in Morris-Thorne–type wormhole space-time with a cosmic string. It is shown that the angle of deflection of photon light as well as the effective potential (time-like or light-like geodesics) of the system is influenced by the cosmic string and is higher compared to the known result.

Precession of bounded orbits and shadow in quantum black hole spacetime

The geodesics around compact objects are one of the best tools to understand the geometrical properties of the central gravity source. In this paper, we study timelike and lightlike geodesics of a quantum black hole proposed in [E. Binetti et al., Phys. Rev. D 106, 046006 (2022).], which does not require committing to a specific model of quantum gravity. We analyze the quantum correction on the related obsrevables: periapsis precession of bounded orbits and angular shadow size. We find that, compared to the classical case, the quantum correction will enhance the periapsis precession as well as the black hole shadow. This effect is more significant for a smaller quantum black hole.

A Gutzwiller Trace Formula for Dirac Operators on a Stationary Spacetime

A Duistermaat–Guillemin–Gutzwiller trace formula for Dirac-type operators on a globally hyperbolic spatially compact stationary spacetime is achieved by generalising the recent construction by Strohmaier and Zelditch (Adv Math 376:107434, 2021) to a vector bundle setting. We have analysed the spectrum of the Lie derivative with respect to a global timelike Killing vector field on the solution space of the Dirac equation and found that it consists of discrete real eigenvalues. The distributional trace of the time evolution operator has singularities at the periods of induced Killing flow on the space of lightlike geodesics. This gives rise to the Weyl law asymptotic at the vanishing period.

All noncommutative spaces of κ-Poincaré geodesics

Noncommutative spaces of geodesics provide an alternative way of introducing noncommutative relativistic kinematics endowed with quantum group symmetry. In this paper we present explicitly the seven noncommutative spaces of time-, space- and light-like geodesics that can be constructed from the time-, space- and light-versions of the κ-Poincaré quantum symmetry in (3 + 1) dimensions. Remarkably enough, only for the light-like (or null-plane) κ-Poincaré deformation the three types of noncommutative spaces of geodesics can be constructed, while for the time-like and space-like deformations both the quantum time-like and space-like geodesics can be defined, but not the light-like one. This obstruction comes from the constraint imposed by the coisotropy condition for the corresponding deformation with respect to the isotropy subalgebra associated to the given space of geodesics, since all these quantum spaces are constructed as quantizations of the corresponding classical coisotropic Poisson homogeneous spaces. The known quantum space of geodesics on the light cone is given by a five-dimensional homogeneous quadratic algebra, and the six nocommutative spaces of time-like and space-like geodesics are explicitly obtained as six-dimensional nonlinear algebras. Five out of these six spaces are here presented for the first time, and Darboux generators for all of them are found, thus showing that the quantum deformation parameter κ −1 plays exactly the same algebraic role on quantum geodesics as the Planck constant ℏ plays in the usual phase space description of quantum mechanics.

Propagation time delay and frame dragging effects of lightlike geodesics in the timing of a pulsar orbiting SgrA*

ABSTRACTTiming a pulsar in a close orbit around the supermassive black hole SgrA* at the centre of the Milky Way would open the window for an accurate determination of the black hole parameters and for new tests of general relativity and alternative modified gravity theories. An important relativistic effect which has to be taken into account in the timing model is the propagation delay of the pulses in the gravitational field of the black hole. Due to the extreme mass ratio of the pulsar and the supermassive back hole we use the test particle limit to derive an exact analytical formula for the propagation delay of lightlike geodesics in a Kerr space–time, and deduce a relativistic formula for the corresponding frame dragging effect on the arrival time. As an illustration, we treat an edge-on orbit in which the frame dragging effect on the emitted lightlike geodesics is expected to be maximal. We compare our formula for the propagation time delay with Post-Newtonian approaches, and in particular with the frame dragging terms derived in previous works by Wex & Kopeikin and Rafikov & Lai. Our approach correctly identifies the asymmetry of the frame dragging delay with respect to superior conjunction, avoids singularities in the time delay, and indicates that in the Post-Newtonian approach frame dragging effects on the lightlike pulses are generally slightly overestimated.

Parametrized black holes: scattering investigation

We study the scattering of light-like geodesics and massless scalar waves by a static Konoplya–Zhidenko black hole, considering the case that the parametrized black hole solution contains a single deformation parameter. By performing a geodesic analysis, we compute the classical differential scattering cross section and probe the influence of the deformation parameter on null trajectories. Moreover, we investigate the propagation of a massless scalar field in the vicinity of the static Konoplya–Zhidenko black hole and use the plane waves formalism to compute the differential scattering cross section. We confront our numerical results in the backward direction with the glory approximation, finding excellent agreement. We compare the results for the deformed black hole with the Schwarzschild case, finding that the additional parameter has an important role in the behavior of the scattering process for moderate-to-high scattering angles.

Some integral geometry problems for wave equations

We consider the Cauchy problem and the source problem for normally hyperbolic operators on the Minkowski spacetime, and study the determination of solutions from their integrals along light-like geodesics. For the Cauchy problem, we give a new proof of the stable determination result obtained by Vasy and Wang (2021 Commun. Math. Phys. 384 503–32). For the source problem, we obtain stable determination for sources with space-like singularities. Our proof is based on the microlocal analysis of the normal operator of the light ray transform composed with the parametrix for strictly hyperbolic operators.

The peeling theorem with arbitrary cosmological constant

A method for deriving the asymptotic behaviour of any physical field is presented. This leads to a geometrically meaningful derivation of the peeling properties for arbitrary values of the cosmological constant. Application to the outstanding case of the physical Weyl tensor provides the explicit form of all terms that determine its asymptotic behaviour along arbitrary lightlike geodesics. The results follow under the assumption of a conformal completion à la Penrose. The only freedom available is the choice of a null vector at the conformal boundary of the space–time (which determines the lightlike geodesic arriving there).

Instantons and no wormholes in AdS3×S3×CY2

We study supergravity instantons sourced by axion (and saxion) fields in the Euclidean ${\mathrm{AdS}}_{3}\ifmmode\times\else\texttimes\fi{}{S}^{3}\ifmmode\times\else\texttimes\fi{}{\mathrm{CY}}_{2}$ vacua of IIB supergravity. Such instantons are described by geodesic curves on the moduli space; the timelike geodesics can describe Euclidean wormholes, the lightlike geodesics describe (generalizations of) D instantons, and spacelike geodesics are subextremal versions thereof. We perform a concrete classification of such geodesics and find that, despite earlier claims, the wormholes fail to be regular. A subclass of the lightlike geodesics is supersymmetric and, up to dualities, lifts to Euclidean strings wrapping 2-cycles in the ${\mathrm{CY}}_{2}$. The dual of these instantons is expected to be worldsheet instantons of the D1--D5 conformal field theory.

Gravitational lensing in the charged NUT–de Sitter spacetime

It is a long standing open question if a gravitomagnetic charge, the gravitational analogon to a hypothetical magnetic charge in electrodynamics, exists in nature. It naturally occurs in certain exact solutions to Einstein's electrovacuum field equations with cosmological constant. The charged NUT-de Sitter metric is such a solution. It describes a black hole with electric and gravitomagnetic charges and a cosmological constant. In this paper we will address the question how we can observe the gravitomagnetic charge using gravitational lensing. For this purpose we first solve the equations of motion for lightlike geodesics using Legendre's canonical forms of the elliptic integrals and Jacobi's elliptic functions. We fix a stationary observer in the domain of outer communication and introduce an orthonormal tetrad. The orthonormal tetrad relates the direction under which the observer detects a light ray to its latitude-longitude coordinates on the observer's celestial sphere. In this parameterisation we rederive the angular radius of the shadow, formulate a lens map, discuss the redshift and the travel time. We also discuss relevant differences with respect to spherically symmetric and static spacetimes and how we can use them to determine if an astrophysical black hole has a gravitomagnetic charge.

Photon sphere and shadow of a time-dependent black hole described by a Vaidya metric

In this paper we derive exact analytical formulas for the evolution of the photon sphere and for the angular radius of the shadow in a special Vaidya space-time. The Vaidya metric describes a spherically symmetric object that gains or loses mass, depending on a mass function $m(v)$ that can be freely chosen. Here we consider the case that $m(v)$ is a linearly increasing or decreasing function. The first case can serve as a simple model for an accreting black hole, the second case for a (Hawking) radiating black hole. With a linear mass function the Vaidya metric admits a conformal Killing vector field which, together with the spherical symmetry, gives us enough constants of motion for analytically calculating the lightlike geodesics. Both in the accreting and in the radiating case, we first calculate the lightlike geodesics, the photon sphere, the angular radius of the shadow, and the redshift of light in coordinates in which the metric is manifestly conformally static, then we analyze the photon sphere and the shadow in the original Eddington-Finkelstein-like Vaidya coordinates.

Morse-theoretical analysis of lensing in wormhole spacetimes

We consider a class of stationary and axisymmetric wormhole spacetimes that is closely related to, but not identical with, the class of Teo wormholes. We fix a point $p$ (observation event) and a timelike curve $\ensuremath{\gamma}$ (worldline of a light source), and we characterize the set of all past-oriented lightlike geodesics from $p$ to $\ensuremath{\gamma}$. As any such geodesic corresponds to an image of the light source on the observer's sky, this allows us to investigate the lensing properties of the wormhole. As a main result, we prove with the help of Morse theory that, under very mild conditions on $\ensuremath{\gamma}$, the observer always sees infinitely many images of $\ensuremath{\gamma}$. Moreover, we study some qualitative features of the lightlike geodesics with the help of two potentials that determine the sum of centrifugal and Coriolis forces of observers in circular motion for the case that the observers' velocity approaches the velocity of light. We exemplify the general results with two specific wormhole spacetimes.

Cosmological simulations of number counts

In this paper we present for the first time the angular power spectra C ℓ(z,z') for number counts from relativisticN-body simulations. We use the relativistic N-body code gevolution with its exact integration of lightlike geodesics which include all relativistic scalar contributions to the number counts. We compare our non-perturbative numerical results with the results from class using the hmcode approximation for the non-linear matter power spectrum. We find that this simple description is excellent for both, the density and the convergence.On the other hand, the current implementation of redshift-space distortions in Boltzmann codes is not accurate.We also find that the largest contribution to the unequal-redshift power spectra is the cross-correlation of the density and the lensing contribution to the number counts, especially for redshift bins that are far apart. Correlatingthe number counts with the convergence map we find that the signal is dominated by the lensing-lensing term when the convergence field redshift is not higher than the number counts one, while it is dominated by the density-lensing term in the opposite case.In the present study, the issue of galaxy bias is deliberately left aside by considering only unbiased samples of matter particles from the simulations.

A Note on Spacelike Submanifolds Through Light Cones in Lorentzian Space Forms

We analyse the intrinsic and extrinsic geometry of spacelike submanifolds in light cones $$\Lambda _{c}(p)$$ of de Sitter and anti-de Sitter spacetimes. Every light cone $$\Lambda _{c}(p)$$ contains the lightlike geodesics starting from p and, essentially, it coincides with the horisms $$E^{+}(p)\cup E^{-}(p)$$ . The analysis works by means of an explicit correspondence with the spacelike submanifolds through the light cone in the Lorentz–Minkowski spacetime. In particular, a characterization of totally umbilical compact surfaces through light cones in de Sitter and anti-de Sitter is shown and we obtain an estimation of the first eigenvalue of the Laplace operator on a compact spacelike surface in a light cone.

Anisotropic conformal invariance of lightlike geodesics in pseudo-Finsler manifolds * *This work is a result of the activity developed within the framework of the Programme in Support of Excellence Groups of the Región de Murcia, Spain, by Fundación Séneca, Science and Technology Agency of the Región de Murcia. The first author was partially supported by MICINN/FEDER project with reference PGC2018-097046-B-I00

In this paper, we prove that lightlike geodesics of a pseudo-Finsler manifold and its focal points are preserved up to reparametrization by anisotropic conformal changes, using the Chern connection and the anisotropic calculus developed in [, ] and the fact that geodesics are critical points of the energy functional and Jacobi fields, the kernel of its index form. This result has applications to the study of Finsler spacetimes.