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This paper explains why spacetime singularities do not constitute a breakdown of physical laws, and points out that the difference between the metrics at singularities and those outside of singularities is factual, rather than nomological.

We study the spectrum $\{\lambda_j(m)\}_{j=1}^{\infty}$ of a timelike Killing vector field $Z$ acting as a differential operator $D_Z$ on the Hilbert space of solutions of the massive Klein-Gordon equation $(\Box_g + m^2) u = 0$ on a globally hyperbolic stationary spacetime $(M, g)$ with compact Cauchy hypersurface. The inverse mass $m^{-1}$ is formally like the Planck constant in a Schr\"odinger equation, and we give Weyl asymptotics as $m \to \infty$ for the number $$N_{\nu, C}(m)= \# \{j \mid \frac{\lambda_j(m)}{m} \in [\nu - \frac{C}{m}, \nu + \frac{C}{m} ]\}$$ for a given $C > 0$. The semi-classical mass asymptotics are governed by the dynamics of the Killing flow $e^{tZ} $ on the hypersurface in the space of mass $1$ geodesics $\gamma$ where $\langle \dot{\gamma}, Z \rangle= \nu$., Comment: to appear in Indagationes Mathematicae, special volume in honour of Hans Duistermaat

We present a novel theorem regarding light rings in a stationary spacetime with an ergoregion. We prove that any stationary, axisymmetric, and asymptotically flat spacetime in 1 + 3 dimensions with an ergoregion must have at least one light ring outside the ergoregion. A possible extension of the proof for asymptotically de-Sitter and anti-de-Sitter spherically symmetric black holes is also discussed., Comment: 5 pages, no figure

In this paper, we investigate the deflection of a charged particle moving in the equatorial plane of Kerr-Newman spacetime, focusing on weak field limit. To this end, we use the Jacobi geometry, which can be described in three equivalent forms, namely Randers-Finsler metric, Zermelo navigation problem, and $(n+1)$-dimensional stationtary spacetime picture. Based on Randers data and Gauss-Bonnet theorem, we utilize osculating Riemannian manifold method and the generalized Jacobi metric method to study the deflection angle, respectively. In the $(n+1)$-dimensional spacetime picture, the motion of charged particle follows the null geodesic, and thus we use the standard geodesic method to calculate the deflection angle. Three methods lead to the same second-order deflection angle, which is obtained for the first time. The result shows that the black hole spin $a$ affects the deflection of charged particles both gravitationally and magnetically at the leading order (order $\mathcal{O}([M]^2/b^2)$). When $qQ/E 2M$, the opposite happens, and the ray is divergently deflected by the lens. We also showed that the effect of the magnetic charge of the dyonic Kerr-Newman black hole on the deflection angle is independent of the particle's charge., 14 pages, 1 figure, to appear in PRD

We consider the dynamics of a charged particle (e.g., an electron) oscillating in a laser field in flat spacetime and describe it in terms of the variable mass metric. By applying Einstein's equivalence principle, we show that, after representing the electron motion in a time-dependent manner, the variable mass metric takes the form of the Friedmann–Lemaitre–Robertson–Walker metric. We quantize a pseudo-scalar field in this spacetime and derive the production rate of electrically neutral, spinless particles. We show that this approach can provide an alternative experimental method to axion searches.

It is shown that the redshift between two Cauchy surfaces in a globally hyperbolic spacetime equals the ratio of the associated contact forms on the space of light rays of that spacetime., 9 pages, 1 figure

It has been argued that space-time must be inextendible—that it must be “as large as it can be” in some sense. Here, we register some skepticism with respect to this position.

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 Ψ 4 .

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.

A new technique is introduced to study the completeness of inextensible electromagnetic trajectories in an n ( ≥ 2 ) -dimensional stationary spacetime. Sufficient conditions on an electromagnetic field on a stationary spacetime are imposed to ensure that the associated ( n + 1 ) -dimensional Kaluza–Klein bundle spacetime is itself stationary. The problem is then reduced to the geodesic completeness of the corresponding Kaluza–Klein bundle spacetime. Applications are given specially to the case of a standard static spacetime.

We establish the notions of f-stability and strong f-stability concerning closed spacelike hypersurfaces immersed with constant f-mean curvature in a conformally stationary spacetime endowed with a conformal timelike vector field V and a weight function f. When V is closed, with the aid of the f-Laplacian of a suitable support function, we characterize f-stable closed spacelike hypersurfaces through the analysis of the first eigenvalue of the Jacobi operator associated to the corresponding variational problem. Furthermore, we obtain sufficient conditions which assure that a strongly f-stable closed spacelike hypersurface must be either f-maximal or isometric to a leaf orthogonal to V.

In this paper, we have investigated the spectroscopy of the rotating Kaluza-Klein spacetime by applying Bohr–Sommerfeld quantization rule and the first law of thermodynamics. we derived the expression of the adiabatic invariant quantity in the dragged–Painleve coordinate system. Then, via revisited adiabatic invariant quantity, we derive the area and entropy spectra of the spacetime. We obtained the area spectrum of the Kaluza-Klein spacetime is ${\Delta } A=8\pi {l_{P}^{2}}$ , and the entropy spectrum is ΔS = 2π. This result is consistent with the Bekenstein’s original result, which imply the entropy and horizon area are discrete and equidistant for the spacetime.

The sonic point/photon surface correspondence is thoroughly investigated in a general setting. First, we investigate a sonic point of a transonic steady perfect fluid flow in a general stationary spacetime, particularly focusing on the radiation fluid. The necessary conditions that the flow must satisfy at a sonic point are derived as conditions for the kinematical quantities of the congruence of streamlines in analogy with the de Laval nozzle equation in fluid mechanics. We compare the conditions for a sonic point with the notion of a photon surface, which can be defined as a timelike totally umbilical hypersurface. As a result, we find that, for the realization of the sonic point/photon surface correspondence, the speed of sound $v_{\rm s}$ must be given by $1/\sqrt{d}$ with $d$ being the spatial dimension of the spacetime. For the radiation fluid ($v_{\rm s}=1/\sqrt{d}$), we confirm that a part of the conditions is shared by the sonic point and the photon surface. However, in general, a Bondi surface, a set of sonic points, does not necessarily coincide with a photon surface. Additional assumptions, such as a spatial symmetry, are essential to the realization of the sonic point/photon surface correspondence in all known examples., Comment: 34 pages

Embargado hasta 20/02/2021 A Lorentz manifold (M, g) is said to be a conformally stationary spacetime if it is endowed with a globally defined conformal timelike vector field K, whereas it is a pp-wave when there is a globally defined parallel lightlike vector field K on M. The study of rigidity and non-existence results for spacelike hypersurfaces with constant mean curvature in these spaces has been considered in the last years by several authors. In this note we unify some known techniques and results related to both problems by considering spacelike hypersurfaces in a spacetime (M, g) endowed with a globally defined conformal causal vector field. This wide family of Lorentz manifolds not only includes previous cases, but it also includes new families of spacetimes.

Causal concept for the general black hole shadow is investigated, instead of the photon sphere. We define several `wandering null geodesics' as complete null geodesics accompanied by repetitive conjugate points, which would correspond to null geodesics on the photon sphere in Schwarzschild spacetime. We also define a `wandering set', that is, a set of totally wandering null geodesics as a counterpart of the photon sphere, and moreover, a truncated wandering null geodesic to symbolically discuss its formation. Then we examine the existence of a wandering null geodesic in general black hole spacetimes mainly in terms of Weyl focusing. We will see the essence of the black hole shadow is not the stationary cycling of the photon orbits which is the concept only available in a stationary spacetime, but their accumulation. A wandering null geodesic implies that this accumulation will be occur somewhere in an asymptotically flat spacetime., The title and some statements are corrected. This is the version which will be published in CQG

We investigated the distinction between two kinds of "Complexity equals Action"(CA) conjecture counting methods which are separately provided by Brown $ et\, al. $ and Lehner $et\, al.$ separately. For the late-time CA complexity growth rate, we show that the difference between two counting methods only comes from the boundary term of the segments on the horizon. However, both counting methods give the identical late-time result. Our proof is general, independent of the underlying theories of higher curvature gravity as well as the explicit stationary spacetime background. To be specific, we calculate the late-time action growth rate in SAdS black hole for F(Ricci) gravity, and show that these two methods actually give the same result. Moreover, by using the Iyer-Wald formalism, we find that the full action rate within the WDW patch can be expressed as some boundary integrations, and the final contribution only comes from the boundary on singularity. Although the definitions of the mass of black hole has been modified in F(Ricci) gravity, its late-time result has the same form with that of SAdS black hole in Einstein gravity., Comment: 7 pages, 1 figure, this paper have been accepted by Physical Review D

In this paper, we analyze trajectories of spacelike curves that are critical points of a Lagrangian depending on its total torsion. We focus on two important families of spacetimes, generalized Robertson–Walker and standard static spacetimes. For the former, we show that such trajectories are those with a constant curvature. For the latter, we also obtain a characterization in terms of the curvature of the trajectory but in this case measured with an appropriate conformal metric.

In this paper we demonstrate that the “pure” spacetime and electromagnetic spacetime are bound together. The “pure” spacetime and electromagnetic spacetime all behave as the wave in character and furthermore, both of them change at the same speed. Based on the understanding of the “pure” spacetime and electromagnetic spacetime, we give out the reasonable explanation why the speed of light keeps constant and how the gravity is created among the matters. In addition, some practical applications of the concepts developed in this work are proposed.

Within the framework of the general theory of relativity, we investigate all possible astrophysical and geometrical effects induced by self-gravitating extreme matter in the form of ideal fluids with limiting and superlimiting equations of state or in the form of strongly nonlinear physical fields. Attention is directed to the possibility of the formation of wormholes and other unusual astrophysical objects.

In this rather short communication we hope to draw attention to a new and rather exciting finding which we think represents an important and non-trivial insight into the current dark energy and dark matter cutting edge research [1-11]. In short we found that a combination of the basic invariant of Einstein D = 4 and Kaluza D = 5 spacetime manifold is the simplest and optimal setting to perform computation [7, 8] and gain insight into the major cosmological problem of dark matter, accelerated cosmic expansion, missing dark energy of the universe [1] as well as the corresponding measurable ordinary energy density which was found, to the surprise of the entire scientific community, to be a mere 4.5% of what was expected [1, 2] based on some of our otherwise most cherished and trusted theories [1-4]. In particular we must stress the importance of the physical insight gained about the source of the difference between the three distinct categories of energy as afforded by the new spacetime picture [7, 8]. Roughly speaking our new conception of spacetime considers a 5 dimensional Kaluza spacetime manifold effectively encasing the 4 dimensional Einstein spacetime [7-9].

Quantum particles are assumed to have a path constituting a random fluctuation super imposed on a classical one resulting in a golden mean spiral propagating in spacetime. Consequently, the dimension of the path of the quantum particle is given by one plus the random Cantor set Zitterbewegung, i.e. 1+O where O is the golden mean Hausdorff dimension of a random Cantor set. Proceeding in this way, we can derive the basic topological invariants of the corresponding spacetime which turned out to be that of E-infinity spacetime 4+O3 as well as a fractal Witten’s M-theory 11+O5. Setting O3 and O5 equal zero, we retrieve Einstein’s spacetime and Witten’s M-theory spacetime respectively where O3 is the latent Casimir topological pressure of spacetime and O5 is Hardy’s quantum entanglement of the same.

The spectral scheme for spacetime geometry is a totally new framework for quantitatively describing the spacetime geometry in terms of the spectra of a certain elliptic operator (typically the Laplacian operator) on the space in question. The central idea of the framework can be symbolically stated as ``Let us hear the shape of the Universe!" There are several advantages of this framework compared to the traditional geometrical description in terms of the Riemannian metric. After sketching the basics of the spectral scheme, we give a new formula for the Einstein-Hilbert action, which is a central quantity for the general relativity theory, in terms of the spectral scheme. We then pay attention to its application to the quantum universes and see how the quantum fluctuations of spacetimes can be effectively described in terms of the spectral scheme. Key words: General relativity, spacetime structures, spectral scheme, cosmology.

The object of the present paper is to study a perfect fluid K¨ahlerspacetime. A perfect fluid K¨ahler spacetime satisfying the Einstein field equation with a cosmological term has been studied and the existence of killingand conformal killing vectors have been discussed. Certain results related to sectional curvature for pseudo projectively flat perfect fluid K¨ahler spacetime have been obtained. Dust model for perfect fluid K¨ahler spacetime has also been studied.

We prove the higher dimensional charged Penrose inequality conjecture on a spacetime where the initial geometry is spherically symmetric using the definition of a quasi-local mass in higher dimensional spacetime. Some physical assumptions on the initial spacetime and on its hypersurface have been taken in order to avoid the real singularity. We also impose two physical conditions on matter such as the dominant energy condition and the localization of the matter distribution.

We find an explicit solution of the source free Maxwell equations in a generalized Kerr-NUT-(A)dS spacetime in all dimensions. This solution is obtained as a linear combination of the closed conformal Killing-Yano tensor $h_{ab}$, which is present in such a spacetime, and a derivative of the primary Killing vector, associated with $h_{ab}$. For the vanishing cosmological constant the obtained solution reduces to the Wald's electromagnetic field generated from the primary Killing vector., Comment: 4 pages, no figures v2: added references

We introduce and analyze the concept of in nitesimal relative position vector eld between \in nitesimally nearby" observers, showing the equivalence between di erent de nitions. Through the Fermi-Walker derivative of in nitesimal relative position vector elds along an observer in a reference frame, we characterize spacetimes admitting an umbilic foliation. Su cient and necessary conditions for those spacetimes to be a conformally stationary spacetime are given. Finally, the important class of cosmological models known as generalized Robertson-Walker spacetimes is characterized

In this paper, we study the weak gravitational deflection of relativistic massive particles for a receiver and source at finite distance from the lens in stationary, axisymmetric and asymptotically flat spacetimes. For this purpose, we extend the generalized optical metric method to the generalized Jacobi metric method by using the Jacobi-Maupertuis Randers-Finsler metric. More specifically,we apply the Gauss-Bonnet theorem to the generalized Jacobi metric space and then obtain an expression for calculating the deflection angle, which is related to Gaussian curvature of generalized optical metric and geodesic curvature of particles orbit. In particular, the finite-distance correction to the deflection angle of signal with general velocity in the the Kerr black hole and Teo wormhole spacetimes are considered. Our results cover the previous work of the deflection angle of light, as well as the deflection angle of massive particles in the limit for the receive and source at infinite distance from the lens object. In Kerr black hole spacetime, we compared the effects due to the black hole spin, the finite-distance of source or receiver, and the relativistic velocity in microlensings and lensing by galaxies. It is found in these cases, the effect of BH spin is usually a few orders larger than that of the finite-distance and relativistic velocity, while the relative size of the latter two could vary according to the particle velocity, source or observer distance and other lensing parameters., Comment: 16 pages, 4 figures

It is (or should be) well known that specification of a heat bath requires both a temperature and a 4-velocity, the rest frame of the heat bath. In static spacetimes there is a very natural and unique candidate for the 4-velocity of the heat bath, the normalized timelike Killing vector. However in stationary non-static spacetimes the situation is considerably more subtle, and several different "natural" 4-velocity fields suitable for characterizing the rest frame of a heat bath can be defined - thus Buchdahl's 1949 analysis for the Tolman temperature gradient in a stationary spacetime is only part of the story. In particular, the heat bath most suitable for describing the Hawking radiation from a rotating black hole is best described in terms of a gradient flow normal to the spacelike hypersurfaces, not in terms of Killing vectors., 7 pages

In this work we study the scalar curvature S of a spacelike hypersurface M immersed in a Generalized Robertson Walker (GRW) spacetime M ‾ = I × f F . Specifically, we express S in terms of the Ricci curvature of the fiber F , the warping function f and the shape operator of the immersion. As a consequence, under natural hypothesis we provide several characterizations of the spacelike slices, as well as the totally umbilical spacelike hypersurfaces, of a GRW spacetime. Moreover, we particularize our study to important GRW spacetimes as Einstein GRW spacetimes, static GRW spacetimes, the Einstein de Sitter spacetime, the steady state spacetime, and spatially parabolic GRW spacetimes. Moreover, we prove that there is no complete totally geodesic spacelike hypersurfaces in a family of Generalized Robertson Walker spacetimes which includes the steady state spacetime.

The four-dimensional character of Einstein’s spacetime is generally accepted in mainstream physics as beyond reasonable doubt correct. However the real problem is when we require scale invariance and that this spacetime be four-dimensional on all scales. It is true that on our classical scale, the 4D decouples into 3D plus one time dimension and that on very large scale only the curvature of spacetime becomes noticeable. However the critical problem is that such spacetime must remain 4D no matter how small the scale we are probing is. This is something of crucial importance for quantum physics. The present work addresses this basic, natural and logical requirement and shows how many contradictory results and shortcomings of relativity and quantum gravity could be eliminated when we “complete” Einstein’s spacetime in such a geometrical gauge invariant way. Concurrently the work serves also as a review of the vast Literature on E-Infinity theory used here.

Dark energy is explained using familiar notions and concepts used in quantum field theory, string theory and the exact mathematical theory of spacetime. The main result of the present work is first a new mathematical definition of pre-quantum spacetime (QST) as a multiset made of infinitely many empty Cantor sets connected to pre-quantum wave empty set (QW) and the pre-quantum particle (QP) zero set via the cobordism equation ∂(QW) = (QP)U(QST). Second, and in turn, this new path of reasoning is used to validate the quantum splitting of Einstein’s E = mc2 into the sum of the ordinary energy E = mc2/22 of the quantum particle and the dark energy E = mc2(21/22) of the quantum wave, used predominantly to explain the observed accelerated expansion of the universe.

Recently, non-standard Lagrangians have gained a growing importance in theoretical physics and in the theory of non-linear differential equations. However, their formulations and implications in general relativity are still in their infancies despite some advances in contemporary cosmology. The main aim of this paper is to fill the gap. Though non-standard Lagrangians may be defined by a multitude form, in this paper, we considered the exponential type. One basic feature of exponential non-standard Lagrangians concerns the modified Euler-Lagrange equation obtained from the standard variational analysis. Accordingly, when applied to spacetime geometries, one unsurprisingly expects modified geodesic equations. However, when taking into account the time-like paths parameterization constraint, remarkably, it was observed that mutually discrete gravity and discrete spacetime emerge in the theory. Two different independent cases were obtained: A geometrical manifold with new spacetime coordinates augmented by a metric signature change and a geometrical manifold characterized by a discretized spacetime metric. Both cases give raise to Einstein’s field equations yet the gravity is discretized and originated from “spacetime discreteness”. A number of mathematical and physical implications of these results were discussed though this paper and perspectives are given accordingly.

The aim of this paper is to investigate some geometric and physical properties of the generalized quasi Einstein spacetime G(Q E)4 under certain conditions. Firstly, we prove the existence of G(Q E)4 by constructing a non trivial example. Then it is proved that the G(Q E)4 spacetime with the conditions \(\mathcal {B}\cdot S=L_{S}Q(g,S)\), where \(\mathcal {B}\) denotes the Ricci tensor or the concircular curvature tensor is an \(N(\frac {a-b}{3})\)-quasi Einstein spacetime and in a G(Q E)4 spacetime with C ⋅ S = 0, where C is the conformal curvature tensor, a − b is an eigenvalue of the Ricci operator. Then, we deal with the Ricci recurrent G(Q E)4 spacetime and prove that in this spacetime, the acceleration vector and the vorticity tensor vanish; but this spacetime has the non-vanishing expansion scalar and the shear tensor. Moreover, it is shown that every Ricci recurrent G(Q E)4 is Weyl compatible, purely electric spacetime and its possible Petrov types are I or D.

A new proposal for the notion of spacetime in a relativistic generalization of special relativity based on a modification of the composition law of momenta is presented. Locality of interactions is the principle which defines the spacetime structure for a system of particles. The formulation based on $\kappa$-Poincar\'e Hopf algebra is shown to be contained in this framework as a particular example., Comment: 14 pages