In this paper, we describe in detail a scheme for the construction of highly accurate numerical solutions to Einstein's field equations in five and six spacetime dimensions corresponding to non-uniform black strings. The scheme consists of a sophistically adapted multi-domain pseudo-spectral method which incorporates a detailed understanding of the solution's behavior at the domain boundaries and at critical points. In particular, the five-dimensional case is exceedingly demanding as logarithmic terms appear which need to be treated with special care. The scheme resolves these issues and permits the investigation of unprecedentedly strong deformations of the black string horizon. As a consequence, we are able to study in detail the critical regime in phase diagrams displaying characteristic thermodynamic quantities such as mass and entropy. Our results show typical spiral curves in such diagrams which provides a strong support of previous numerical works.
We study the class of vacuum (Ricci flat) six-dimensional spacetimes admitting a non-degenerate multiple Weyl aligned null direction l, thus being of Weyl type II or more special. Subject to an additional assumption on the asymptotic fall-off of the Weyl tensor, we prove that these spacetimes can be completely classified in terms of the two eigenvalues of the (asymptotic) twist matrix of l and of a discrete parameter $U^0=\pm 1/2, 0$. All solutions turn out to be Kerr-Schild spacetimes of type D and reduce to a family of "generalized" Myers-Perry metrics (which include limits and analytic cont…
Benefiting from the index spinorial formalism, the Killing spinor equation is integrated in six-dimensional spacetimes. The integrability conditions for the existence of a Killing spinor are worked out and the Killing spinors are classified in two algebraic types, in the first type the scalar curvature of the spacetime must be negative, while in the second type the spacetime must be an Einstein manifold. In addition, the equations that define Killing-Yano (KY) and closed conformal Killing-Yano (CCKY) tensors are expressed in the index notation and, as consequence, all non-vanishing KY and CCKY…
In the spacetime of a $d$-dimensional static tense brane black hole we elaborate the mechanism by which massive scalar fields decay. The metric of a six-dimensional black hole pierced by a topological defect is especially interesting. It corresponds to a black hole residing on a tensional 3-brane embedded in a six-dimensional spacetime, and this solution has gained importance due to the planned accelerator experiments. It happened that the intermediate asymptotic behaviour of the fields in question was determined by an oscillatory inverse power-law. We confirm our investigations by numerical c…
Global existence to the coupled Einstein-Maxwell system which rules the dynamics of a kind of charged matter with a pseudo-tensor of pressure is proved, in Bianchi I-VIII spacetimes. We study the geodesics completeness, the asymptotic behavior, the positivity conditions, and we prove that the problem is well-posed in the sense of Hadamard.
On type II(D) Einstein spacetimes in six dimensions
Institute of Mathematics, Czech Academy of Sciences（捷克科学院数学研究所）；Institute of Theoretical Physics, Faculty of Mathematics and Physics, Charles University（查理大学数学物理学院理论物理研究所）
The objective of the present paper is to study 4-dimensional almost pseudo Ricci symmetric perfect fluid spacetimes $(APRS)_4$. We show that a Robertson-Walker spacetime is $(APRS)_4$ and vice versa under certain condition imposed on its scale factor. Some popular toy models of $F(R)$-gravity are also studied under the current setting and various energy conditions are investigated.
Based on the analogue spacetime programme, and many other ideas currently mooted in "quantum gravity", there is considerable ongoing speculation that the usual pseudo-Riemannian (Lorentzian) manifolds of general relativity might eventually be modified at short distances. Two specific modifications that are often advocated are the adoption of Finsler geometries (or more specifically, pseudo-Finsler spacetimes) and the possibility of birefringence (or more generally, multi-refringence). We have investigated the possibility of whether it is possible to usefully and cleanly deal with these two pos…
The relativistic Dirac equation in four-dimensional spacetime reveals a coherent relation between the dimensions of spacetime and the degrees of freedom of fermionic spinors. A massless Dirac fermion generates new symmetries corresponding to chirality spin and charge spin as well as conformal scaling transformations. With the introduction of intrinsic W-parity, a massless Dirac fermion can be treated as a Majorana-type or Weyl-type spinor in a six-dimensional spacetime that reflects the intrinsic quantum numbers of chirality spin. A generalized Dirac equation is obtained in the six-dimensional…
Current physics is faced with the fundamental problem of unifying quantum theory and general relativity, which would have resulted in quantum gravity. The main effort to construct the latter has been bent on quantizing spacetime structure, in particular metric. Meanwhile, taking account of the indeterministic aspect of the quantum description of matter, which manifests itself in quantum jumps, essentially affects classical spacetime structure and the Einstein equation. Quantum jumps give rise to a family of sets of simultaneous events, which implies the existence of universal cosmological time…
In this article we construct and discuss several aspects of the two-component spinorial formalism for six-dimensional spacetimes, in which chiral spinors are represented by objects with two quaternionic components and the spin group is identified with $SL(2;\mathbb{H})$, which is a double covering for the Lorentz group in six dimensions. We present the fundamental representations of this group and show how vectors, bivectors, and 3-vectors are represented in such spinorial formalism. We also complexify the spacetime, so that other signatures can be tackled. We argue that, in general, objects b…
The Kerr theorem is revisited as part of the twistor program in six dimensions. The relationship between pure spinors and integrable 3-planes is investigated. The real condition for Lorentzian spacetimes is seen to induce a projective property in the space of solutions, reminiscent of the quaternionic structure of the 6-dimensional Lorentz group. The twistor equation (or Killing spinor equations generically) also has an interpretation as integrable null planes and a family of Einstein spacetimes with this property are presented in the Kerr-Schild fashion.
Finsler spacetimes have become increasingly popular within the theoretical physics community over the last two decades. Because physicists need to use pseudo-Finsler structures to describe propagation}of signals, there will be nonzero null vectors in both the tangent and cotangent spaces --- this causes significant problems in that many of the mathematical results normally obtained for "usual" (Euclidean signature) Finsler structures either do not apply, or require significant modifications to their formulation and/or proof. We shall first provide a few basic definitions, explicitly demonstrat…
We formulate the concept of time machine structure for spacetimes exhibiting a compactely constructed region with closed timelike curves. After reviewing essential properties of the pseudo Schwarzschild spacetime introduced by A. Ori, we present an analysis of its geodesics analogous to the one conducted in the case of the Schwarzschild spacetime. We conclude that the pseudo Schwarzschild spacetime is geodesically incomplete and not extendible to a complete spacetime. We then introduce a rotating generalization of the pseudo Schwarzschild metric, which we call the the pseudo Kerr spacetime. We…
We investigate the geometrical and physical structures of a pseudo-symmetric spacetime $(PS)_4$ with timelike vector under the condition of conformal flatness. We classify it into two possible types: constant Ricci scalar and closed velocity vector. We further study this spacetime as a solution of $F(R)$-gravity theory and show that the pressure and energy density of the effective cosmological perfect fluid separately possess certain typical relations with the geometry and the gravity sector of the theory. Based on this result, some observational and cosmological analyses are done for the $F(R…
While it is generally agreed that the nature of spacetime must be drastically different at the Planck scale, it has been a common practice to assume that spacetime is endowed with a full pseudo-Riemannian geometry regardless of the physical fields present or the length scale at which the geometry is probed, and to adopt this assumption in theories of cosmology, particle physics, quantum gravity, etc. Following Einstein's view that the mathematical description of spacetime ought to be physically motivated, we initiate a discussion on the validity of this assumption, and propose that the full ps…
It is reasonably well-known that birefringent crystal optics can to some extent be described by the use of pseudo-Finslerian spacetimes (an extension of pseudo-Riemannian spacetime). What is less commonly appreciated is that there are two separate and quite disjoint pseudo-Finsler structures for the two photon polarizations, and further, that there are separate tangent-space pseudo-Finsler structures defined by the group velocity and co-tangent-space pseudo-co-Finsler structures defined by the phase velocity. The inter-connections between these four separate pseudo-Finsler structures are rathe…
We calculate the Casimir force for a fermionic quantum field in a piston geometry with three parallel plates. The fermion satisfies bag boundary conditions on the plates and the spacetime is assumed to have compact extra dimensions. The calculation is performed in the cases where the extra space has toroidal and spherical topology. We are mainly interested in the case in which the fermion is coupled non-trivially to an extra dimensional defect, with a torus extra dimensional topological background. We found that in certain limits, the Casimir force corresponding to the defect-fermion system an…
A new class of solutions for Einstein's field equations representing a static spherically symmetric anisotropic distribution of matter is obtained on the background of pseudo-spheroidal spacetime. We have prescribed the bounds of the model parameters $ k $ and $ p_0 $ on the basis of the elementary criteria for physical acceptability, viz., regularity, stability and energy conditions. By taking the values of model parameters from the prescribed bounds, we have shown that our model is compatible with the observational data of a wide variety of compact stars like 4U 1820-30, PSR J1903+327, 4U 16…
A new class of exact solutions of Einstein's field equations representing anisotropic distribution of matter on pseudo-spheroidal spacetime is obtained. The parameters appearing in the model are restricted through physical requirements of the model. It is found that the models given in the present work is compatible with observational data of a wide variety of compact objects like 4U 1820-30, PSR J1903+327, 4U 1608-52, Vela X-1, PSR J1614-2230, SMC X-4, Cen X-3. A particular model of pulsar PSR J1614-2230 is studied in detail and found that it satisfies all physical requirements needed for phy…
Pseudo-Newtonian gravitational potential describing the gravitational field of static and spherically symmetric black holes in the universe with a repulsive cosmological constant is introduced. In order to demonstrate the accuracy of the pseudo-Newtonian approach, the related effective potential for test-particle motion is constructed and compared with its general relativistic counterpart given by the Schwarzschild-de Sitter geometry. The results indicate that such an approach could be useful in applications of developed Newtonian theories of accretion discs in astrophysically interesting situ…
Conformally compactified (3+1)-dimensional Minkowski spacetime may be identified with the projective light cone in (4+2)-dimensional spacetime. In the latter spacetime the special conformal group acts via rotations and boosts, and conformal inversion acts via reflection in a single coordinate. Hexaspherical coordinates facilitate dimensional reduction of Maxwell electromagnetic field strength tensors to (3+1) from (4 + 2) dimensions. Here we focus on the operation of conformal inversion in different coordinatizations, and write some useful equations. We then write a conformal invariant and a p…
The Bardeen solution corresponding to Einstein field equations with a cosmological constant is a regular black hole. The main goal of this manuscript is to investigate the geometric structures in terms of curvature conditions admitted by this spacetime. It is found that this spacetime is pseudosymmetric and possess several kinds of pseudosymmetries. Also, it is a manifold of pseudosymmetry Weyl curvature and the difference tensor C.R-R.C linearly depends on the tensors Q(g;C) and Q(S;C). It is interesting to note that such a spacetime is weakly generalized recurrent manifold and satisfies spec…
We give six arguments that the Planck scale should be viewed as a fundamental minimum or boundary for the classical concept of spacetime, beyond which quantum effects cannot be neglected and the basic nature of spacetime must be reconsidered. The arguments are elementary, heuristic, and plausible, and as much as possible rely on only general principles of quantum theory and gravity theory. The paper is primarily pedagogical, and its main goal is to give physics students, non-specialists, engineers etc. an awareness and appreciation of the Planck scale and the role it should play in present and…
Pseudo-Newtonian gravitational potential introduced in spherically symmetric black-hole spacetimes with a repulsive cosmological constant is tested for equilibrium toroidal configurations of barotropic perfect fluid orbiting the black holes. Shapes and potential depths are determined for the marginally stable barotropic tori with uniform distribution of specific angular momentum, using both the pseudo-Newtonian and fully relativistic approach. For the adiabatic (isoentropic) perfect fluid, temperature profiles, mass-density and pressure profiles and total masses of pseudo-Newtonian and relativ…
One of the deepest insights from the general theory of relativity is the relational nature of spacetime. While it is a generally agreed on that the nature of spacetime must be drastically different at the Planck scale, it has been a common practice to assume a full pseudo-Riemannian geometry of spacetime no matter what the physical fields present are, and to apply such geometry to cosmology. In the spirit of preserving the relational description of spacetime, we initiate a discussion on the applicability of general relativity all the way to Planck scale, and propose the possibility that the fu…
Using the SL(2,R)-duality in a dimensionally reduced spacetime in (the bosonic sector of) five-dimensional minimal supergravity, we construct general Kaluza-Klein black hole solutions which carry six independent charges, its mass, angular momentum along four dimensions, electric and magnetic charges of the Maxwell fields in addition to Kaluza-Klein electric and magnetic monopole charges.
By employing a pseudo-orthonormal coordinate-free approach, the Dirac equation for particles in the Kerr--Newman spacetime is separated into its radial and angular parts. In the massless case to which a special attention is given, the general Heun-type equations turn into their confluent form. We show how one recovers some results previously obtained in literature, by other means.
This paper is devoted to the study of curvature properties of Hayward black hole (briefly, HBH) spacetime, which is a solution of Einstein field equations (briefly, EFE) having non-vanishing cosmological constant. We have proved that the HBH spacetime is an Einstein manifold of level $2$, $2$-quasi Einstein, generalized quasi-Einstein and Roter type manifold. Also, it is shown that the nature of the HBH spacetime is pseudosymmetric and it obeys several types of pseudosymmetries, such as, pseudosymmetry due to concircular, conformal and conharmonic curvature (i.e., $F\cdot F=\mathcal{L}Q(g,F)$ …
In this paper we study classical general relativistic static wormhole configurations with pseudo-spherical symmetry. We show that in addition to the hyperbolic wormhole solutions discussed by Lobo and Mimoso in the Ref. Phys.\ Rev.\ D {\bf 82}, 044034 (2010), there exists another wormhole class, which is truly pseudo-spherical counterpart of spherical Morris-Thorne wormhole (contrary to the Lobo-Mimoso wormhole class), since all constraints originally defined by Morris and Thorne for spherically symmetric wormholes are satisfied. We show that, for both classes of hyperbolic wormholes the energ…
We show the dynamical stability of a six-dimensional braneworld solution with warped flux compactification recently found by the authors. We consider linear perturbations around this background spacetime, assuming the axisymmetry in the extra dimensions. The perturbations are expanded by scalar-, vector- and tensor-type harmonics of the four-dimensional Minkoswki spacetime and we analyze each type separately. It is found that there is no unstable mode in each sector and that there are zero modes only in the tensor sector, corresponding to the four-dimensional gravitons. We also obtain the firs…
Analogue spacetimes are powerful models for probing the fundamental physical aspects of geometry - while one is most typically interested in ultimately reproducing the pseudo-Riemannian geometries of interest in general relativity and cosmology, analogue models can also provide useful physical probes of more general geometries such as pseudo-Finsler spacetimes. In this chapter we shall see how a 2-component Bose-Einstein condensate can be used to model a specific class of pseudo-Finsler geometries, and after suitable tuning of parameters, both bi-metric pseudo-Riemannian geometries and standar…
Confined Klein-Gordon oscillators in Minkowski spacetime and a pseudo-Minkowski spacetime with a space-like dislocation: PDM KG-oscillators, isospectrality and invariance
We first calculate equations of motion for particles in the Kerr-Sen-de Sitter black hole spacetime. Then in the eikonal regime, we analytically obtain the quasi-normal resonant modes of massless neutral scalar field perturbation and find its imaginary part to be characterized by the surface gravity of the near-extremal Kerr-Sen-de Sitter black hole with Cauchy horizon approaching the event horizon. We further show that the Penrose strong cosmic censorship conjecture is thus respected in this spacetime with dilaton scalar field and axion pseudoscalar field.
We consider spacetime initiated by a finite-sized initial boundary as a generalization of the Hartle-Hawking no-boundary state. We study entanglement entropy of matter state prepared by such spacetime. We find that the entanglement entropy for large subregion is given either by the initial state entanglement or the entanglement island, preventing the entropy to grow arbitrarily large. Consequently, the entanglement entropy is always bounded from above by the boundary area of the island, leading to an entropy bound in terms of the island. The island $I$ is located in the analytically continued …
We give an introductory account of the AdS/CFT correspondence in the 1/2-BPS sector of ${\cal N}=4$ super Yang-Mills theory.Six of the dimensions of the string theory are emergent in the Yang-Mills theory. In this article we suggest how these dimensions and local physics in these dimensions emerge. The discussion is aimed at non-experts.
In this paper we describe an algorithm to determine the vectors normal to a space-time V4 embedded in a pseudo-Euclidean manifold M4+n. An application of this algorithm is given considering the Schwarzchild space-time geometry embedded in a 6 dimensional pseudo-Euclidean manifold, using the algebraic computing system REDUCE.
The idea that spacetime has to be replaced by Clifford space (C-space) is explored. Quantum field theory (QFT) and string theory are generalized to C-space. It is shown how one can solve the cosmological constant problem and formulate string theory without central terms in the Virasoro algebra by exploiting the peculiar pseudo-Euclidean signature of C-space and the Jackiw definition of the vacuum state. As an introduction into the subject, a toy model of the harmonic oscillator in pseudo-Euclidean space is studied.
We show that the generalization of Doubly Special Relativity to a curved de Sitter background can be obtained starting from a six-dimensional spacetime on which quadratic constraints on position and momentum coordinates are imposed.
The Johannsen-Psaltis spacetime describes a rotating black hole with parametric deviations from the Kerr metric. By construction this spacetime explicitly violates the no-hair theorems. Rotating black hole solutions in any modified theory of gravity could be written in terms of the Johannsen-Psaltis metric. We examined the accretion of gas onto a black hole described by the static limit of this spacetime. We employed a potential that generalises the Paczynski-Wiita potential to the static Johannsen-Psaltis metric. Our analysis utilised a recent pseudo-Newtonian formulation of the dynamics arou…
We prove that higher dimensional Einstein spacetimes which possess a geodesic, non-degenerate double Weyl aligned null direction (WAND) $\ell$ must additionally possess a second double WAND (thus being of type D) if either: (a) the Weyl tensor obeys $C_{abc[d}\ell_{e]}\ell^c=0$ ($\LeftrightarrowΦ_{ij}=0$, i.e., the Weyl type is II(abd)); (b) $\ell$ is twistfree. Some comments about an extension of the Goldberg-Sachs theorem to six dimensions are also made.
A generalization from the usual $5$-dimensional two-brane Randall-Sundrum (RS) model to a $6$-dimensional multi-brane RS model is presented. The extra dimensions are extended from one to two; correspondingly the single-variable warp function is generalized to be a double-variable function, to represent the two extra dimensions. In the analysis of the Einstein equation we have two remarkable discoveries. One is that, when branes are absent, the cosmological parameter distributed in the two extra dimensions acts as a function describing a family of circles. These circles are not artificially add…
We show that a partially massless graviton can propagate on a large set of spacetimes which are not Einstein spacetimes. Starting from a recently constructed theory for a massive graviton that propagates the correct number of degrees of freedom on an arbitrary spacetime, we first give the full explicit form of the scalar constraint responsible for the absence of a sixth degree of freedom. We then spell out generic conditions for the constraint to be identically satisfied, so that there is a scalar gauge symmetry which makes the graviton partially massless. These simplify if one assumes that sp…
Inspired by the Lifshitz gravity as a theory with anisotropic scaling behavior, we suggest a new $(n+1)-$dimensional metric in which the time and spatial coordinates scale anisotropically as $(t,r,θ_{i})\,\to (λ^{z}t,λ^{-1}r,λ^{x_i}\,θ_{i})$. Due to the anisotropic scaling dimension of the spatial coordinates, this spacetime does not support the full Schrödinger symmetry group. We look for the analytical solution of Gauss-Bonnet gravity in the context of the mentioned geometry. We show that Gauss-Bonnet gravity admits an analytical solution provided that the constants of the theory are properl…
General higher-dimensional rotating black hole spacetimes of any dimensions admit the Killing and Killing-Yano tensors, which generate the hidden symmetries just as in four-dimensional Kerr spacetime. We study these properties of the black holes using the formalism of supersymmetric mechanics of pseudo-classical spinning point particles. We present two nontrivial supercharges, corresponding to the Killing-Yano and conformal Killing-Yano tensors of the second rank. We demonstrate that an unusual extended Poisson-Dirac algebra of these supercharges results in two independent Killing tensors in s…
In the present article a new class of exact solutions of Einstein's field equations for charged anisotropic distribution is obtained on the background of pseudo-spheroidal spacetime characterized by the metric potential $g_{rr}=\frac{1+K\frac{r^2}{R^2}}{1+\frac{r^2}{R^2}}$, where $K$ and $R$ are geometric parameters of the spacetime. The radial pressure $p_r$ and electric field intensity $E$ are taken in the form $8πp_r = \frac{K-1}{R^2}\frac{\left(1-\frac{r^2}{R^2} \right)}{\left(1+K\frac{r^2}{R^2} \right)^2}$ and $E^2 = \frac{α(K-1)\frac{r^2}{R^2}}{R^2\left(1+K\frac{r^2}{R^2} \right)^2}$. Th…
Employing a pseudo-orthonormal coordinate-free approach, the solutions to the Klein--Gordon and Dirac equations for particles in Melvin spacetime are derived in terms of Heun's biconfluent functions.
Black hole spectroscopy is a powerful tool to probe the Kerr nature of astrophysical compact objects and their environment. The observation of multiple ringdown modes in gravitational waveforms could soon lead to high-precision gravitational spectroscopy, so it is critical to understand if the quasinormal mode spectrum is stable against perturbations. It was recently shown that the pseudospectrum can shed light on the spectral stability of black hole quasinormal modes. We study the pseudospectrum of Reissner-Nordström spacetimes and we find a spectral instability of scalar and gravitoelectric …
We analyse the response of an arbitrarily-accelerated Unruh-DeWitt detector coupled to a massless scalar field in Minkowski spacetimes of dimensions up to six, working within first-order perturbation theory and assuming a smooth switch-on and switch-off. We express the total transition probability as a manifestly finite and regulator-free integral formula. In the sharp switching limit, the transition probability diverges in dimensions greater than three but the transition rate remains finite up to dimension five. In dimension six, the transition rate remains finite in the sharp switching limit…
Pseudo-spectral methods in gravitational lensing: Spin precession in loop quantum black hole spacetimes
School of Physics and Astronomy, China West Normal University（西华师范大学物理与电子工程学院）；School of Mathematics and Physics, Yancheng Institute of Technology（盐城工学院数理学院）
The dynamics of pseudo-classical spinning particles in spacetime of gravitational plane waves of general polarization and harmonic profile is studied. The resulting equations of motion are solved exactly and the results are compared with those of the other approaches. The relative accelerations of nearby particles is also calculated.
