Spacetime Metric Research Articles

Relativity without miracles

It has been claimed, recently, that the fact that all the non-gravitational fields are locally Poincare invariant and that these invariances coincide, in a certain regime, with the symmetries of the spacetime metric is miraculous in general relativity (GR). In this paper I show that, in the context of GR, it is possible to account for these so-called miracles of relativity. The way to do so involves integrating the realisation that the gravitational field equations (the Einstein field equation in GR) impose constraints on the behaviour of matter in a novel interpretation of the equivalence principle, which dictates the determination of local inertial frames through gravitational interaction. This proposed explanation of the miracles can also deal with the problematic cases for attempts at explaining them in the context of the standard geometrical perspective on relativity theory.

An optical perspective on the theory of relativity -Ⅰ: Basic concepts and the equivalence principle

This article reviews the process by which Einstein founded the theory of relativity, considers Einstein's perspective on physics and examines the relationship between our new perspective and Einstein's perspective. The basic concepts involved in our new perspective are given, including trajectory coordinates, reference space-time and the refractive index. These concepts form the basis of subsequent articles. As a part of putting forward these basic concepts, a simple and intuitive explanation of the relativistic effect is given and a new form of the space-time metric is given by using a refractive index. This new form of space-time metric enables us to explain the difference and connection between special relativity and general relativity and provides a straightforward way of describing the equivalence principle.

Neutrino oscillations in gravitational and cosmological backgrounds

We use the eikonal approximation in order to calculate the additional phase shift between two neutrino mass eigenstates during their propagation in a background of gravitational wave or scalar perturbations in the flat and the FRW spacetime metric. We comment on the dependence of the results on the characteristics of the perturbations, give some order-of-magnitude estimates, and find that, although small, the resulting phase difference persists for large redshifts, up to the validity of our approximations.

Nonlinear stability of self-gravitating irrotational Chaplygin fluids in a FLRW geometry

We analyze the global nonlinear stability of FLRW (Friedmann-Lemaître-Robertson-Walker) spacetimes in the presence of an irrotational perfect fluid. We assume that the fluid is governed by the so-called (generalized) Chaplygin equation of state p=−A2ρα relating the pressure to the mass-energy density, in which A>0 and α∈(0,1] are constants. We express the Einstein equations in wave gauge as a system of coupled nonlinear wave equations and, after performing a conformal transformation, we analyze the global behavior of solutions toward the future. Under small perturbations, the (3+1)-spacetime metric, the mass-energy density, and the velocity vector describing the geometry and fluid unknowns remain globally close to a reference FLRW solution. Our analysis provides also the precise asymptotic behavior of the perturbed solutions toward the future.

Derivation of Lighthill’s Eighth Power Law of an Aeroacoustic Quadrupole in Acoustic Spacetime

Acoustic spacetime is a four-dimensional manifold analogue to the relativistic spacetime with the reference speed of light replaced by the speed of sound. It has been established primarily for the indirect studies of relativistic phenomena by means of their better understood acoustic analogues. More recently, it has also been used for the analytical treatment of sound propagation in various uniform and non-uniform flows of the background fluid. In this paper the analogy is extended and utilized to derive Lighthill’s eight power law for sound generation of an aeroacoustic quadrupole. Adding to the existing analogue theory, propagating sound waves are described in terms of a weak perturbation of the background acoustic spacetime metric. The obtained result proves that the acoustic analogy can be extended to cover both weak perturbation of the fluid due to the sound waves and certain sound generation mechanisms, at least in incompressible low Mach number flows.

On the symmetry and conservation law classification of the de Sitter–Schwarzschild metric and the corresponding wave and Klein–Gordon equations

The main purpose of this work is to focus on a discussion of Lie symmetries admitted by de Sitter–Schwarzschild spacetime metric, and the corresponding wave or Klein–Gordon equations constructed in the de Sitter–Schwarzschild geometry. The obtained symmetries are classified and the variational (Noether) conservation laws associated with these symmetries via the natural Lagrangians are obtained. In the case of the metric, we obtain additional variational ones when compared with the Killing vectors leading to additional conservation laws and for the wave and Klein–Gordon equations, the variational symmetries involve less tedious calculations as far as invariance studies are concerned.

(2+1) -dimensional sonic black hole from a spin-orbit-coupled Bose-Einstein condensate and its analog Hawking radiation

We study the properties of a $2+1$ dimensional Sonic black hole (SBH) that can be realised, in a quasi-two-dimensional two-component spin-orbit coupled Bose-Einstein condensate (BEC). The corresponding equation for phase fluctuations in the total density mode that describes phonon field in the hydrodynamic approximation is described by a scalar field equation in $2+1$ dimension whose space-time metric is significantly different from that of the SBH realised from a single component BEC that was studied experimentally, and, theoretically meticulously in literature. Given the breakdown of the irrotationality constraint of the velocity field in such spin-orbit coupled BEC, we study in detail how the time evolution of such condensate impacts the various properties of the resulting SBH. By time evolving the condensate in a suitably created laser-induced potential, we show that such a sonic black hole is formed, in an annular region bounded by inner and outer event horizon as well as elliptical ergo-surfaces. We observe amplifying density modulation due to the formation of such sonic horizons and show how they change the nature of analogue Hawking radiation emitted from such sonic black hole by evaluating the density-density correlation at different times, using the truncated Wigner approximation (TWA) for different values of spin-orbit coupling parameters. We finally investigate the thermal nature of such analogue Hawking radiation.

Hamilton–Jacobi Equation for a Charged Test Particle in the Stäckel Space of Type (2.0)

All electromagnetic potentials and space–time metrics of Stäckel spaces of type (2.0) in which the Hamilton–Jacobi equation for a charged test particle can be integrated by the method of complete separation of variables are found. Complete sets of motion integrals, as well as complete sets of killing vector and tensor fields, are constructed. The results can be used when studying solutions of field equations in the theory of gravity.

Bulk reconstruction of metrics with a compact space asymptotically

Holographic duality implies that the geometric properties of the gravitational bulk theory should be encoded in the dual field theory. These naturally include the metric on dimensions that become compact near the conformal boundary, as is the case for any asymptotically locally AdSn × mathbbm{S} k spacetime. Almost all previous work on metric reconstruction ignores these dimensions and would thus at most apply to dimensionally-reduced metrics. In this work, we generalize the approach to bulk reconstruction using light-cone cuts and propose a prescription to obtain the full higher-dimensional metric of generic spacetimes up to an overall conformal factor. We first extend the definition of light-cone cuts to include information about the asymptotic compact dimensions, and show that the full conformal metric can be recovered from these extended cuts. We then give a prescription for obtaining these extended cuts from the dual field theory. The location of the usual cuts can still be obtained from bulk-point singularities of correlators, and the new information in the extended cut can be extracted by using appropriate combinations of operators dual to Kaluza-Klein modes of the higher-dimensional bulk fields.

Light bending by a slowly rotating source in quadratic theories of gravity

In this paper we study the light bending caused by a slowly rotating source in the context of quadratic theories of gravity, in which the Einstein--Hilbert action is extended by additional terms quadratic in the curvature tensors. The deflection angle is computed employing the method based on the Gauss--Bonnet theorem and working in the approximation of a weak lens; also, we assume that the source and observer are at an infinite distance. The formalism presented is very general and applies to any spacetime metric in the limit of weak gravitational field and slow rotation. We find the explicit formula for the deflection angle for several local and nonlocal theories, and also discuss some phenomenological implications.

The Spinor-Tensor Gravity of the Classical Dirac Field

In this work, with the help of the quantum hydrodynamic formalism, the gravitational equation associated with the classical Dirac field is derived. The hydrodynamic representation of the Dirac equation described by the evolution of four mass densities, subject to the theory-defined quantum potential, has been generalized to the curved space-time in the covariant form. Thence, the metric of space-time has been defined by imposing the minimum action principle. The derived gravity shows the spontaneous emergence of the “cosmological” gravity tensor (CGT), a generalization of the classical cosmological constant (CC), as a part of the energy-impulse tensor density (EITD). Even if the classical cosmological constant is set to zero, the CGT is non-zero, allowing a stable quantum vacuum (out of the collapsed branched polymer phase). The theory shows that in the classical macroscopic limit, the general relativity equation is recovered. In the perturbative approach, the CGT leads to a second-order correction to Newtonian gravity that takes contribution from the space where the mass is localized (and the space-time is curvilinear), while it tends to zero as the space-time approaches the flat vacuum, leading, as a means, to an overall cosmological constant that may possibly be compatible with the astronomical observations. The Dirac field gravity shows analogies with the modified Brans–Dicke gravity, where each spinor term brings an effective gravity constant G divided by its field squared. The work shows that in order to obtain the classical minimum action principle and the general relativity limit of the macroscopic classical scale, quantum decoherence is necessary.

Blackhole in nonlocal gravity: comparing metric from Newmann\u2013Janis algorithm with slowly rotating solution

The strong gravitational field near massive blackhole is an interesting regime to test General Relativity (GR) and modified gravity theories. The knowledge of spacetime metric around a blackhole is a primary step for such tests. Solving field equations for rotating blackhole is extremely challenging task for the most modified gravity theories. Though the derivation of Kerr metric of GR is also demanding job, the magical Newmann–Janis algorithm does it without actually solving Einstein equation for rotating blackhole. Due to this notable success of Newmann–Janis algorithm in the case of Kerr metric, it has been being used to obtain rotating blackhole solution in modified gravity theories. In this work, we derive the spacetime metric for the external region of a rotating blackhole in a nonlocal gravity theory using Newmann–Janis algorithm. We also derive metric for a slowly rotating blackhole by perturbatively solving field equations of the theory. We discuss the applicability of Newmann–Janis algorithm to nonlocal gravity by comparing slow rotation limit of the metric obtained through Newmann–Janis algorithm with slowly rotating solution of the field equation.

Properties of a recent quantum extension of the Kruskal geometry

Recently, it was shown that, in an effective description motivated by loop quantum gravity, singularities of the Kruskal spacetime are naturally resolved [A. Ashtekar, J. Olmedo and P. Singh, Phys. Rev. Lett. 121 (2018) 241301; A. Ashtekar, J. Olmedo and P. Singh, Phys. Rev. D 98 (2018) 126003]. In this paper, we explore a few properties of this quantum corrected effective metric. In particular, we (i) calculate the Hawking temperature associated with the horizon of the effective geometry and show that the quantum correction to the temperature is completely negligible for macroscopic black holes, just as one would hope; (ii) discuss the subtleties associated with the asymptotic properties of the spacetime metric, and show that the metric is asymptotically flat in a precise sense; (iii) analyze the asymptotic fall-off of curvature; and, (iv) show that the ADM energy is well defined (and agrees with that determined by the horizon area), even though the curvature falls off less rapidly than in the standard asymptotically flat context.

Extended galactic rotational velocity profiles in f(R) gravity background

An attempt has been made to explore the geometric effects of f(R) action on the galactic dynamics under the weak field approximation. The rotational velocity is calculated beyond the Einstein’s geometric theory of gravity. It is inspired by the cosmological geometric relation obtained in the power-law f(R) gravity model in vacuum. We analyse the action with a small positive deviation from the Einstein–Hilbert gravity action (taking R as f(R)propto R^{1+delta }) at the galactic scales for the explanation of the flatness paradox associated with the clustered galactic dark matter. We obtain the contribution of a dynamical f(R) cosmological background geometry on accelerating the test mass. Furthermore, the integrated effective acceleration of the test mass due to a massive spherically symmetric source in f(R) background is calculated via the study of geodesics for the suitable spacetime metric and an equation for the effective rotational velocity has been developed. We test the viability of the proposed model by tracing the motion of a test mass far from the disk of galactic matter for smaller delta . The possible galactic rotational velocity curves in f(R) background are discussed for the formula obtained with delta<< 1. We also obtain constraints on delta O(10^{-6}) confirmed by observations.

The scattering amplitude of stringy hadrons: strings with opposite charges on their endpoints

In this note we describe hadrons: mesons and baryons as strings with electric charges on their endpoints. We consider here only the neutral system with opposite charges, coupled to an external constant electromagnetic field. We derive certain classical solutions including rotating folded open strings. We write down the mode expansion and canonically quantize the system. The OPEs associated with such strings are determined. We rederive the non-commutativity of the zero modes and the fact that the charges modify the spacetime metric. We show that the quantum worldsheet energy momentum tensor on the boundary is affected by the endpoint charges and differs from the corresponding Noether current. We determine the generalization of the Veneziano scattering amplitude for such strings in the critical dimension. Phenomenological implications are addressed and in particular we show that the external magnetic field can be tuned so that the amplitude vanishes for particular kinematic setups. We discuss the generalization of such strings to non-critical four dimensional spacetime. In particular we renormalize the divergence of the Polchinski-Strominger effective action associated with the rotating folded string.

Structure of the singular ring in Kerr-like metrics

The Kerr geometry is believed to represent the exterior spacetime of astrophysical black holes. We here re-analyze the geometry of Kerr-like metrics (Kerr, Kerr-Newman, Kerr-de Sitter, and Kerr-anti de Sitter), paying particular attention to the region near the singular set. We find that, although the Kretschmann scalar vanishes at the singular set along a given direction, a certain combination of curvature invariants diverges regardless of the direction of approach. We also find that the two-dimensional geometry induced by the spacetime metric on the orbits of the isometry group also possesses a singularity regardless of the direction of approach. Likewise, the two-dimensional geometry in the directions orthogonal to the isometry orbits is $C^{2}$-divergent, but extends continuously at the singular set as a cone with opening angle $4\pi$. We conclude by showing that tidal forces lead to infinite stresses on neighboring geodesics that approach the singular set, destroying any such observers in finite proper time. Those geodesics that come in from infinity and do not hit the singular set but approach it are found to need tremendous energy to get close to the singular set, experiencing an acceleration transversal to the equatorial plane which grows without bound when the minimal distance of approach goes to zero. While establishing these results, we also present an alternative description of some other known properties, as well as introducing toroidal coordinates that provide a hands-on description of the double-covering for the geometry near the singular set.

Space-Time: from Indefinite at the Locality to Generality at the Infinity

The article presents physical and theoretical generalizations of the space-time metrics, based on invariant coordinate time transformations over the intervals of the observed periodic emission of pulsars. Synchronization of spatial reference systems by the observed passage of the wave front of the coherent radiation of a pulsar corresponds to the axiomatic matching of coordinate time scales defined by the observed rotation parameters of the pulsar with the initial coordinates of spatial reference frames within the homogeneous Galileo space of the Galaxy. As a result, all physical processes in such coordinates systems, including the observed coherent radiation of a pulsar, are indistinguishable from each of them. The estimated span of minimum recognizable on the pulsar time scale is 10−12–10−14 s in the extent from several years to several decades and then gradually decreases, reaching limit values of the order of 10−16–10−17 s in the extent compared to the lifetime of a pulsar. The relative size of the resolved interval is within approximately 10−19–10−29.

Probing beyond-Kerr spacetimes with inspiral-ringdown corrections to gravitational waves

Gravitational waves from the explosive merger of distant black holes are encoded with details regarding the complex extreme-gravity spacetime present at their source. Famously described by the Kerr spacetime metric for rotating black holes in general relativity, what if effects beyond this theory are present? One way to efficiently test this hypothesis is to first obtain a metric which parametrically deviates from the Kerr metric in a model-independent way. Given such a metric, one can then predict the ensuing corrections to both the inspiral and ringdown portions of the gravitational waveform for black holes present in the new spacetime. With these tools in hand, one can then test gravitational wave signals for such effects by two different methods, (i) inspiral-merger-ringdown consistency test, and (ii) parameterized test. In this paper, we demonstrate the exact recipe one needs to do just this. We first derive parameterized corrections to the waveform inspiral, ringdown, and remnant properties for a generic non-Kerr spacetime and apply this to two example beyond-Kerr spacetimes each parameterized by a single non-Kerr parameter. We then predict the beyond-Kerr parameter magnitudes required in an observed gravitational wave signal to be statistically inconsistent with the Kerr case in general relativity. We find that the two methods give very similar bounds. The constraints found with existing gravitational-wave events are comparable to those from x-ray observations, while future gravitational-wave observations using Cosmic Explorer (Laser Interferometer Space Antenna) can improve such bounds by two (three) orders of magnitude.

The relativistic heat equation on flat Friedman spacetime

The exact solutions of the relativistic hyperbolic heat equation on a flat Friedman spacetime metric are obtained. These solutions are obtained in three cases of flat Friedman model which correspond to radiation, matter and dark energy-dominated universe models. In particular, in the dark energy-dominated universe case, a closed form solution is obtained for which, using appropriate boundary conditions, an interesting exact solution is derived. This solution suggests that the temperature of the universe drops as it expands by a scale of an exponential function.

General expansion of time transfer functions in optical spacetime

When dealing with highly accurate modeling of time and frequency transfers into arbitrarily moving dielectrics medium, it may be convenient to work with Gordon's optical spacetime metric rather than the usual physical spacetime metric. Additionally, an accurate modeling of the geodesic evolution of observable quantities (e.g., the range and the Doppler) requires us to know the reception or the emission time transfer functions. In the physical spacetime, these functions can be derived to any post-Minkowskian orders through a recursive procedure. In this work, we show that the time transfer functions can be determined to any order in Gordon's optical spacetime as well. The exact integral forms of the gravitational, the refractive, and the coupling contributions are recursively derived. The expression of the time transfer function is given within the post-linear approximation assuming a stationary optical spacetime covered with geocentric celestial reference system coordinates. The light-dragging effect due to the steady rotation of the neutral atmosphere of the Earth is found to be at the threshold of visibility in many experiments involving accurate modeling of the time and frequency transfers.