Original Spacetime Research Articles

A Riemannian geometric approach for timelike and null spacetime geodesics

The geodesic motion in a Lorentzian spacetime can be described by trajectories in a 3-dimensional Riemannian metric. In this article we present a generalized Jacobi metric obtained from projecting a Lorentzian metric over the directions of its Killing vectors. The resulting Riemannian metric inherits the geodesics for asymptotically flat spacetimes including the stationary and axisymmetric ones. The method allows us to find Riemannian metrics in three and two dimensions plus the radial geodesic equation when we project over three different directions. The 3-dimensional Riemannian metric reduces to the Jacobi metric when static, spherically symmetric and asymptotically flat spacetimes are considered. However, it can be calculated for a larger variety of metrics in any number of dimensions. We show that the geodesics of the original spacetime metrics are inherited by the projected Riemannian metric. We calculate the Gaussian and geodesic curvatures of the resulting 2-dimensional metric, we study its near horizon and asymptotic limit. We also show that this technique can be used for studying the violation of the strong cosmic censorship conjecture in the context of general relativity.

Enhanced Approaches to Gravitational Lensing: A Regular Charged Black Hole in Weak Field Limit

Gravitational lensing, an integral aspect of general relativity, profoundly impacts our understanding of black holes (BH’s). The presence of a centered singularity resulting from gravitational collapse is a fundamental characteristic of a BH Nonetheless, the exploitation of non-linear electrodynamics (NLED) divulges the emphatic concept of non-singularity. Our analysis primarily focuses on calculating the deflection angle in the weak field approximation, deliberately excluding the original spacetime configuration. The photon’s effective geometry is typically linked to NLED. Instead, a photon traverses an effective null geodesic that aligns with the geometry of a regular metric. The findings reveal an additional term in the deflection angle compared to the Reissner-Nordström (RN) metric, evidenced by a shifted positional displacement of the tertiary image associated with a regular BH This study succinctly encapsulates these findings, positioning them within the extensive terminology ambient to BH’s singularity and GL, thus edifying the scientific memoir with precise astrophysical insights.

On the symmetries of singular limits of spacetimes

We consider spacetime metrics with a given (but quite generic) dependence on a dimensionful parameter such that in the 0 and ∞ limits of that parameter the metric becomes singular. We study the isometry groups of the original spacetime metrics and of the singular metrics that arise in the limits and the corresponding symmetries of the motion of p-branes evolving in them, showing how the Killing vectors and their Lie algebras can be found in general. We illustrate our general results with several examples which include limits of anti-de Sitter spacetime in which the holographic screen is one of the singular metrics and of pp-waves.

Optical appearance of a thin-shell wormhole with a Hayward profile

The optical properties of a thin-shell wormhole (TSW) with a Hayward profile is investigated. Adopting the ray-tracing method, we demonstrate that the TSW’s contralateral spacetime is capable of reflecting a significant portion of light back to the observer spacetime. We analyze the effective potential, light deflection, and azimuthal angle of the TSW and find that these quantities are affected by the mass ratio of the black holes (BHs). Specifically, if the mass of the contralateral spacetime BH is greater than that of the original spacetime BH, and the impact parameter satisfies the condition Hb_{textrm{c2}}<b_{1}<b_{textrm{c1}}, the trajectory of the photon exhibits round-trip characteristics. Assuming the presence of a thin accretion disk surrounding the observing spacetime BH, our results indicate that the image formed by the TSW exhibits additional photon rings and a lensing band compared to an image produced by a BH alone.

Glue-on AdS holography for Toverline{T} -deformed CFTs

The Toverline{T} deformation is a solvable irrelevant deformation whose properties depend on the sign of the deformation parameter μ. In particular, Toverline{T} -deformed CFTs with μ < 0 have been proposed to be holographically dual to Einstein gravity where the metric satisfies Dirichlet boundary conditions at a finite cutoff surface. In this paper, we put forward a holographic proposal for Toverline{T} -deformed CFTs with μ > 0, in which case the bulk geometry is constructed by gluing a patch of AdS3 to the original spacetime. As evidence, we show that the Toverline{T} trace flow equation, the spectrum on the cylinder, and the partition function on the torus and the sphere, among other results, can all be reproduced from bulk calculations in glue-on AdS3.

A novel probe of Einstein-Hilbert action: Dynamic upgradation of metric parameters

The Einstein-Hilbert (EH) action is peculiar in many ways. Some of the Peculiar features have already been highlighted in literature. In the present article, we have discussed some peculiar features of EH action which has not been discussed earlier. It is well-known that there are several ways of decomposing the EH action into the bulk and the surface part with different underlying motivations. We provide a review on all of these decompositions. Then, we attempt to study the static coordinate as a limiting case of a time-dependent coordinate via dynamic upgradation of the constant metric parameters. Firstly, we study the consequences when the constant parameters, present in a static and spherically symmetric (SSS) metric, are promoted to the time dependent variables, which allows us to incorporate the time-dependence in the static coordinate. We find that, in every sets of decomposition, the expression of the bulk term remains invariant, whereas the surface term changes by a total derivative term. Finally, when we obliterate the time dependence of the metric parameters, we find that the expression of the Ricci-scalar (or the EH action) does not go back to its original value. Instead, we find that the curvature becomes singular on the horizon, which implies a topological change from the original spacetime.

Holographic Fisher information metric in Schrödinger spacetime

In this paper, we study the Fisher information metric on the space of the coupling constants on both sides of the duality between non-relativistic dipole field theories and string theory in Schrödinger spacetime. We consider the following setup. In the gauge theory side one can deform a given conformal field theory by a proper scalar operator and compute the quantum information metric via the two-point correlation function between two such operators. On the string side the deformation corresponds to a scalar field probing the background. In the large N limit of the theory the probing can be done without backreaction on the original spacetime, thus one can construct a perturbative scheme for the calculation of the dual holographic Fisher information metric as shown by [1]. Considering the asymptotic behaviour of the holographic Fisher information metric close to the boundary of the Schrödinger spacetime we show that its divergence structure exactly matches its dual quantum counterpart up to the leading order, thus extending the holographic setup up to the non-relativistic case. One should note that the existence of other terms is not seen from the boundary theory to this level of approximation. Their behaviour near the boundary, however, is pointing what kind of information from the boundary theory is missing to be able to reconstruct the bulk. Obviously, more work is needed to refine and elucidate their meaning and interrelations in holographic setup.

Relativistic spacetime crystals.

Periodic space crystals are well established and widely used in physical sciences. Time crystals have been increasingly explored more recently, where time is disconnected from space. Periodic relativistic spacetime crystals on the other hand need to account for the mixing of space and time in special relativity through Lorentz transformation, and have been listed only in 2D. This work shows that there exists a transformation between the conventional Minkowski spacetime (MS) and what is referred to here as renormalized blended spacetime (RBS); they are shown to be equivalent descriptions of relativistic physics in flat spacetime. There are two elements to this reformulation of MS, namely, blending and renormalization. When observers in two inertial frames adopt each other's clocks as their own, while retaining their original space coordinates, the observers become blended. This process reformulates the Lorentz boosts into Euclidean rotations while retaining the original spacetime hyperbola describing worldlines of constant spacetime length from the origin. By renormalizing the blended coordinates with an appropriate factor that is a function of the relative velocities between the various frames, the hyperbola is transformed into a Euclidean circle. With these two steps, one obtains the RBS coordinates complete with new light lines, but now with a Euclidean construction. One can now enumerate the RBS point and space groups in various dimensions with their mapping to the well known space crystal groups. The RBS point group for flat isotropic RBS spacetime is identified to be that of cylinders in various dimensions: mm2 which is that of a rectangle in 2D, (∞/m)m which is that of a cylinder in 3D, and that of a hypercylinder in 4D. An antisymmetry operation is introduced that can swap between space-like and time-like directions, leading to color spacetime groups. The formalism reveals RBS symmetries that are not readily apparent in the conventional MS formulation. Mathematica script is provided for plotting the MS and RBS geometries discussed in the work.

Scattering Theory for the Charged Klein–Gordon Equation in the Exterior De Sitter–Reissner–Nordström Spacetime

We show asymptotic completeness for the charged Klein–Gordon equation in the exterior De Sitter–Reissner–Nordström spacetime when the product of the charge of the black with the charge of the Klein-Gordon field is small enough. We then interpret scattering as asymptotic transports along principal null geodesics in a Kaluza–Klein extension of the original spacetime.

The Quantum Coherence Induced by Geometric Curvature of Gauge Field in Non-equilibrium Quantum Dynamics

The study of non-equilibrium quantum dynamics receives attentions recently. However, the nature and effects of nonequilibirumness such as detailed-balance breaking and the relationship to the underlying intrinsic geometry is still unclear. In this study, we uncover that a gauge field will be induced by the nonequilibriumness in the coherence representation and furthermore its internal geometrical curvature is directly related to the degree of detailed balance breaking. The nonequilibriumness of the quantum system induces an intrinsic geometric curvature which can enhance the quantum coherence, leading to the possibility of the space time origin for the non-local quantum correlations or the possibility of curved space time emergence from nonequilibrium quantum dynamics. We also uncovered that the internal curvature of the gauge field provides a bridge to connect the generalized quantum fluctuation dissipation theorem to the fluctuation theorem and time irreversibility of the quantum dynamics. The quantum time irreversibility is due to the path dependent factor along any particular path in an internal curved space, which is analogous to the Wilson lines (or Wilson loops) in Abelian gauge theory. We also found that the steady state quantum coherence disappears when the non-trivial internal curvature vanishes for the quantum system coupled with environments. When the curvature is relatively small indicating weak detailed balance breaking, the coherence increases as curvature increases. The internal curvature can provide a general and direct quantitative measure of the detail-balance breaking for any quantum/classical non-equilibrium systems, even without knowing the underlying steady state distribution or the steady state flux. Using an example of two harmonic oscillators coupled to two environments with different temperatures, we explicitly show the dependence of the internal curvature and quantum coherence.

Global spacetime structure of compactified inflationary universe

We investigate the global spacetime structure of torus de Sitter universe, which is exact de Sitter space with torus identification based on the flat chart. We show that past incomplete null geodesics in torus de Sitter universe are locally extendible. Then we give an extension of torus de Sitter universe so that at least one of the past incomplete null geodesics in the original spacetime becomes complete. However, we find that extended torus de Sitter universe has two ill behaviors. The first one is a closed causal curve. The second one is so called quasi regular singularity, which means that there is no global, consistent extension of spacetime where all curves become complete, nevertheless each curve is locally extensible.

Aspects of (4 + 4)-Kaluza–Klein type theory

We develop a type of Kaluza–Klein formalism in (4 + 4)-dimensions. In the framework of this formalism, we obtain a new kind of Schwarzschild metric solution that, via Kruskal–Szequeres, can be interpreted as mirror black and white holes. We found that this new type of mirror black and white hole solution in (3 + 1)-dimensions supports the idea that original spacetime can be extended to a (4 + 4)-signature. Using octonions, we also discuss linearized gravity in (4 + 4)-dimensions.

Memory, Penrose limits and the geometry of gravitational shockwaves and gyratons

The geometric description of gravitational memory for strong gravitational waves is developed, with particular focus on shockwaves and their spinning analogues, gyratons. Memory, which may be of position or velocity-encoded type, characterises the residual separation of neighbouring ‘detector’ geodesics following the passage of a gravitational wave burst, and retains information on the nature of the wave source. Here, it is shown how memory is encoded in the Penrose limit of the original gravitational wave spacetime and a new ‘timelike Penrose limit’ is introduced to complement the original plane wave limit appropriate to null congruences. A detailed analysis of memory is presented for timelike and null geodesic congruences in impulsive and extended gravitational shockwaves of Aichelburg-Sexl type, and for gyratons. Potential applications to gravitational wave astronomy and to quantum gravity, especially infra-red structure and ultra-high energy scattering, are briefly mentioned.

A generalised Garfinkle–Vachaspati transform

The Garfinkle-Vachaspati transform is a deformation of a metric in terms of a null, hypersurface orthogonal, Killing vector $k^\mu$. We explore a generalisation of this deformation in type IIB supergravity taking motivation from certain studies of the D1-D5 system. We consider solutions of minimal six-dimensional supergravity admitting null Killing vector $k^\mu$ trivially lifted to type IIB supergravity by the addition of four-torus directions. The torus directions provide covariantly constant spacelike vectors $l^\mu$. We show that the original solution can be deformed as $g_{\mu \nu} \to g_{\mu \nu} + 2 \Phi k_{(\mu}l_{\nu)}, C_{\mu \nu} \to C_{\mu \nu} - 2 \Phi k_{[\mu}l_{\nu]}$, provided the two-form supporting the original spacetime satisfies $i_k (dC) = - d k$, and where $\Phi$ satisfies the equation of a minimal massless scalar field on the original spacetime. We show that the condition $i_k (dC) = - d k$ is satisfied by all supersymmetric solutions admitting null Killing vector. Hence all supersymmetric solutions of minimal six-dimensional supergravity can be deformed via this method. As an example of our approach, we work out the deformation on a class of D1-D5-P geometries with orbifolds. We show that the deformed spacetimes are smooth and identify their CFT description. Using Bena-Warner formalism, we also express the deformed solutions in other duality frames.

Szekeres universes with homogeneous scalar fields

We consider the existence of an “inflaton” described by an homogeneous scalar field in the Szekeres cosmological metric. The gravitational field equations are reduced to two families of solutions which describe the homogeneous Kantowski–Sachs spacetime and an inhomogeneous FLRW(-like) spacetime with spatial curvature a constant. The main differences with the original Szekeres spacetimes containing only pressure-free matter are discussed. We investigate the stability of the two families of solution by studying the critical points of the field equations. We find that there exist stable solutions which describe accelerating spatially-flat FLRW geometries.

Ultrarelativistic boost of a black hole in the magnetic universe of Levi-Civita–Bertotti–Robinson

We consider an exact Einstein-Maxwell solution constructed by Alekseev and Garcia which describes a Schwarzschild black hole immersed in the magnetic universe of Levi-Civita, Bertotti and Robinson (LCBR). After reviewing the basic properties of this spacetime, we study the ultrarelativistic limit in which the black hole is boosted to the speed of light, while sending its mass to zero. This results in a non-expanding impulsive wave traveling in the LCBR universe. The wave front is a 2-sphere carrying two null point particles at its poles -- a remnant of the structure of the original static spacetime. It is also shown that the obtained line-element belongs to the Kundt class of spacetimes, and the relation with a known family of exact gravitational waves of finite duration propagating in the LCBR background is clarified. In the limit of a vanishing electromagnetic field, one point particle is pushed away to infinity and the single-particle Aichelburg-Sexl pp-wave propagating in Minkowski space is recovered.

Pre-big Bang Geometric Extensions of Inflationary Cosmologies

Robertson-Walker spacetimes within a large class are geometrically extended to larger cosmologies that include spacetime points with zero and negative cosmological times. In the extended cosmologies, the big bang is lightlike, and though singular, it inherits some geometric structure from the original spacetime. Spacelike geodesics are continuous across the cosmological time zero submanifold which is parameterized by the radius of Fermi space slices, i.e, by the proper distances along spacelike geodesics from a comoving observer to the big bang. The continuous extension of the metric, and the continuously differentiable extension of the leading Fermi metric coefficient $g_{\tau\tau}$ of the observer, restrict the geometry of spacetime points with pre-big bang cosmological time coordinates. In our extensions the big bang is two dimensional in a certain sense, consistent with some findings in quantum gravity.

Generalized Vaidya solutions and Misner-Sharp mass for n -dimensional massive gravity

Dynamical solutions are always of interest to people in gravity theories. We derive a series of generalized Vaidya solutions in the $n$-dimensional de Rham-Gabadadze-Tolley (dRGT) massive gravity with a singular reference metric. Similar to the case of the Einstein gravity, the generalized Vaidya solution can describe shining/absorbing stars. Moreover, we also find a more general Vaidya-like solution by introducing a more generic matter field than the pure radiation in the original Vaidya spacetime. As a result, the above generalized Vaidya solution is naturally included in this Vaidya-like solution as a special case. We investigate the thermodynamics for this Vaidya-like spacetime by using the unified first law, and present the generalized Misner-Sharp mass. Our results show that the generalized Minser-Sharp mass does exist in this spacetime. In addition, the usual Clausius relation $\delta Q= TdS$ holds on the apparent horizon, which implicates that the massive gravity is in a thermodynamic equilibrium state. We find that the work density vanishes for the generalized Vaidya solution, while it appears in the more general Vaidya-like solution. Furthermore, the covariant generalized Minser-Sharp mass in the $n$-dimensional de Rham-Gabadadze-Tolley massive gravity is also derived by taking a general metric ansatz into account.

Dualities and geometrical invariants for static and spherically symmetric spacetimes

In this work, we consider spinless particles in curved spacetime and symmetries related to extended isometries. We search for solutions of a generalized Killing equation whose structure entails a general class of Killing tensors. The conserved quantities along particle’s geodesic are associated with a dual description of the spacetime metric. In the Hamiltonian formalism, some conserved quantities generate a dual description of the metric. The Killing tensors belonging to the conserved objects imply in a nontrivial class of dual metrics even for a Schwarzschild metric in the original spacetime. From these metrics, we construct geometrical invariants for classes of dual spacetimes to explore their singularity structure. A nontrivial singularity behavior is obtained in the dual sector.

The Ehlers–Geroch theorem on geodesic motion in general relativity

We provide a detailed and rigorous proof of (a generalized version of) the Ehlers–Geroch theorem on geodesic motion in metric theories of gravity: we assume that (M, g) is a spacetime satisfying an averaged form of the dominant energy condition and some further technical assumptions indicated in the bulk of this paper. Then, a small body which is allowed to deform the original spacetime metric g moves, nonetheless, along a geodesic of (M, g).