Bound states of spin-half particles in a static gravitational field close to the black hole field

We consider the bound-state energy levels of a spin-1/2 fermion in the gravitational field of a near-black hole object. In the limit that the metric of the body becomes singular, all binding energies tend to the rest-mass energy (i.e. total energy approaches zero). We present calculations of the ground state energy for three specific interior metrics (Florides, Soffel and Schwarzschild) for which the spectrum collapses and becomes quasi-continuous in the singular metric limit. The lack of zero or negative energy states prior to this limit being reached prevents particle pair production occurring. Therefore, in contrast to the Coulomb case, no pairs are produced in the non-singular static metric. For the Florides and Soffel metrics the singularity occurs in the black hole limit, while for the Schwarzschild interior metric it corresponds to infinite pressure at the centre. The behaviour of the energy level spectrum is discussed in the context of the semi-classical approximation and using general properties of the metric.     

Einstein equations and the joining of discontinuous metrics

We consider the problem of joining of metrics when these are not continuous across the joining (hyper-) surface. We confine ourselves to static, spherically symmetric metrics which join without requiring gradients of a δ-function in the energy-momentum tensor. It is found that a surface tension is always associated in cases where the metrics are discontinuous. In some cases, the joined metrics satisfy Einstein equations (in the sense of distributions) while, in others, the surface tension associated with the limiting discontinuous metric depends on the interpolating functions used to produce it suggesting sensitivity to short distance effects beyond general relativity.     

Flat Information Geometries in Black Hole Thermodynamics

The Hessian of either the entropy or the energy function can be regarded as a metric on a Gibbs surface. For two parameter families of asymptotically flat black holes in arbitrary dimension one or the other of these metrics are flat, and the state space is a flat wedge. The mathematical reason for this is traced back to the scale invariance of the Einstein–Maxwell equations. The picture of state space that we obtain makes some properties such as the occurence of divergent specific heats transparent.Supported by VR.     

Energy levels of a scalar particle in a static gravitational field close to the black hole limit

The bound-state energy levels of a scalar particle in the gravitational field of finite-sized objects with interiors described by the Florides and Schwarzschild metrics are found. For these metrics, bound states with zero energy (where the binding energy is equal to the rest mass of the scalar particle) only exist when a singularity occurs in the metric. Therefore, in contrast to the Coulomb case, no pairs are produced in the non-singular static metric. For the Florides metric the singularity occurs in the black hole limit, while for the Schwarzschild interior metric it corresponds to infinite pressure at the center. Moreover, the energy spectrum is shown to become quasi-continuous as the metric becomes singular.     

On Spin(7) holonomy metric based on SU(3)/U(1): II

We continue the investigation of Spin(7) holonomy metric of cohomogeneity one with the principal orbit SU(3)/U(1). A special choice of U(1) embedding in SU(3) allows more general metric ansatz with five metric functions. There are two possible singular orbits in the first-order system of Spin(7) instanton equation. One is the flag manifold SU(3)/T² also known as the twistor space of CP(2) and the other is CP(2) itself. Imposing a set of algebraic constraints, we find a two-parameter family of exact solutions which have SU(4) holonomy and are asymptotically conical. There are two types of asymptotically locally conical (ALC) metrics in our model, which are distinguished by the choice of S¹ circle whose radius stabilizes at infinity. We show that this choice of M theory circle selects one of the possible singular orbits mentioned above. Numerical analyses of solutions near the singular orbit and in the asymptotic region support the existence of two families of ALC Spin(7) metrics: one family consists of deformations of the Calabi hyper-Kähler metric, the other is a new family of metrics on a line bundle over the twistor space of CP(2).     

Boundary conditions and non-equilibrium thermodynamics

The theory of non-equilibrium thermodynamics is applied to a system of two immiscible fluids and their interface. A singular energy density at the interface, which is related to the phenomenon of surface tension, is taken into account. Furthermore the momentum and the heat currents are allowed to be singular at the interface. Using the conservation laws and the Gibbs' relation for the surface, an expression for the singular entropy production density at the interface is obtained. The linear phenomenological laws between fluxes and thermodynamic forces occurring in this singular entropy production density are given. Some of these linear laws are boundary conditions for the solution of the differential equations governing the evolution of the state variables in the bulk.     

Energy Multipliers for Perturbations of the Schwarzschild Metric

We consider the wave equations associated to metrics close to the Schwarzschild metric. We investigate spacelike energy multipliers likely to yield local decay of solutions to these wave equations, in the spirit of Morawetz. For rotationally invariant metrics, we obtain multipliers giving a control of the solutions having finitely many vanishing spherical harmonics. The structure of these multipliers is closely related to the photosphere of the metric. For Kerr metrics, in contrast, we display a region, which we call the intersphere region, where no energy inequality with the required properties can exist.     

An exact solution for uniformly accelerated particles in general relativity

Recently an exact solution of Einstein's empty-space equations referring to four uniformly accelerated particles was given. The relation of this to static axially symmetric metrics of the Weyl and Einstein-Rosen classes is investigated in the present paper. A physical interpretation of the singularity along half of the axis of symmetry of the uniformly accelerated metric in Weyl's form is given. An exact solution corresponding to an expanding (contracting) singular null surface is obtained by a limiting process from that for uniformly accelerated particles.     

Energy conditions and conservation laws in LTB metric via Noether symmetries

In this paper, we have investigated Noether symmetries in Lemaitre–Tolman–Bondi (LTB) metric. Using the Lagrangian associated with the LTB metric, the set of determining equations for Noether symmetries is obtained and then integrated in several cases. It is shown that the LTB metric can be classified in to eight distinct classes corresponding to Noether algebra of dimension 4, 5, 6, 7, 8, 9, 11 and 17. The obtained Noether symmetries are compared with Killing and homothetic vectors. The well known Noether’s theorem is used to find the expressions for conservation laws in each case. Moreover, it is shown that most of the obtained metrics are anisotropic or perfect fluid models which satisfy certain energy conditions and the equation of state.     

Magnetic fields and the Weyl tensor in the early universe

We have solved the Einstein–Maxwell equations for a class of metrics with constant spatial curvature by considering only a primordial magnetic field as source. We assume a slight modification of the Tolman averaging relations so that the energy–momentum tensor of this field possesses an anisotropic pressure component. This inhomogeneous magnetic universe is isotropic and its time evolution is guided by the usual Friedmann equations. In the case of a flat universe, the space-time metric is free of singularities (except the well-known initial singularity at \(\text {t} = 0\) ). It is shown that the anisotropic pressure of our model has a straightforward relation to the Weyl tensor. We then analyze the effect of this new ingredient on the motion of test particles and on the geodesic deviation of the cosmic fluid.     

初态对线型分子体系量子速度极限度量的影响

量子速度极限(QSL)的实用性研究关系到更高效量子技术的实现,研究不同分子体系中QSL问题可为基于分子体系的量子信息技术提供理论支持.采用代数方法讨论了不同的初始态对QSL度量方式的影响,研究发现初始态和分子参数均会影响QSL的度量方式,对分子体系无论Fock态还是相干态,量子Fisher信息度量方式优于Wigner-Yanase信息度量方式.广义几何QSL度量更适合描述强相干态下的分子动力学演化.     

Energy and angular momentum of charged rotating black holes

We show that the pseudotensors of Einstein, Tolman, Landau and Lifshitz, Papapetrou, and Weinberg essentially coincide for any Kerr-Schild metric if calculations are carried out in Kerr-Schild Cartesian coordinates. This generalizes a previous result by Gürses and Gürsey that dealt only with the pseudotensors of Einstein and Landau-Lifshitz. We compute exactly the energy and angular momentum distributions for the Kerr-Newman metric in Kerr-Schild Cartesian coordinates and compare the results with those obtained by using different definitions of quasilocal mass, which unlike pseudotensors do not agree for all Kerr-Schild metrics.     

<Emphasis Type="Italic">D</Emphasis>-Dimensional Metrics with <Emphasis Type="Italic">D</Emphasis>−3 Symmetries

Symmetry transformations in a space of D-dimensional vacuum metrics with D?3 commuting Killing vectors are studied. We solve directly the Einstein equations in the Maison formulation under additional assumptions. We show that the Reissner-Nordström solution is related by the symmetry transformation to a particular case of the 5-dimensional Gross-Perry metric and the 5-dimensional plane wave solution is related to the Gross-Perry-Sorkin metric.     

Circularly Symmetric Static Metric in Three Dimensions and Its Killing Symmetry

In this note we consider a general three-dimensional circularly symmetric static metric and investigate possible Killing symmetries it possesses.We then summarize the work by addressing Einstein equations and briefly discuss their implications on the energy momentum content of some of the metrics that arise in the process.     

Finding Collineations of Kimura Metrics

Kimura investigated static spherically symmetric metrics and found several to have quadratic first integrals. We use REDUCE and the package Dimsym to seek collineations for these metrics. For one metric we find that three proper projective collineations exist, two of which are associated with the two irreducible quadratic first integrals found by Kimura. The third projective collineation is found to have a reducible quadratic first integral. We also find that this metric admits two conformal motions and that the resulting reducible conformal Killing tensors also lead to Kimura's quadratic integrals. We demonstrate that when a Killing tensor is known for a metric we can seek an associated collineation by solving first order equations that give the Killing tensor in terms of the collineation rather than the second order determining equations for collineations. We report less interesting results for other Kimura metrics.     

Analytical Solutions of the Gravitational Field Equations in de Sitter and Anti-de Sitter Spacetimes

The generalized Laplace partial differential equation, describing gravitational fields, is investigated in de Sitter spacetime from several metric approaches—such as the Riemann, Beltrami, Börner-Dürr, and Prasad metrics—and analytical solutions of the derived Riccati radial differential equations are explicitly obtained. All angular differential equations trivially have solutions given by the spherical harmonics and all radial differential equations can be written as Riccati ordinary differential equations, which analytical solutions involve hypergeometric and Bessel functions. In particular, the radial differential equations predict the behavior of the gravitational field in de Sitter and anti-de Sitter spacetimes, and can shed new light on the investigations of quasinormal modes of perturbations of electromagnetic and gravitational fields in black hole neighborhood. The discussion concerning the geometry of de Sitter and anti-de Sitter spacetimes is not complete without mentioning how the wave equation behaves on such a background. It will prove convenient to begin with a discussion of the Laplace equation on hyperbolic space, partly since this is of interest in itself and also because the wave equation can be investigated by means of an analytic continuation from the hyperbolic space. We also solve the Laplace equation associated to the Prasad metric. After introducing the so called internal and external spaces—corresponding to the symmetry groups SO(3,2) and SO(4,1) respectively—we show that both radial differential equations can be led to Riccati ordinary differential equations, which solutions are given in terms of associated Legendre functions. For the Prasad metric with the radius of the universe independent of the parametrization, the internal and external metrics are shown to be of AdS-Schwarzschild-like type, and also the radial field equations arising are shown to be equivalent to Riccati equations whose solutions can be written in terms of generalized Laguerre polynomials and hypergeometric confluent functions.     

Kerr–Schild metrics and gravitational radiation

We describe conditions assuring that the Kerr–Schild type solutions of Einstein's equations with pure radiation fields are asymptotically flat at future null infinity. Such metrics cannot describe “true” gravitational radiation from bounded sources—it is shown that the Bondi news function vanishes identically. We obtain formulae for the total energy and angular momentum at ℐ. As an example we consider a non-stationary generalization of the Kerr metric given by Vaidya and Patel. Angular momentum and total energy are expressed in closed form as functions of retarded time.     

On the Fefferman class of metrics associated with a three-dimensional CR space

We derive the explicit forms of Fefferman's metric for a Cauchy-Riemann space admitting a solution of the tangential Cauchy-Riemann equation and of the corresponding Weyl tensor. We show that its Petrov type is 0 in the case of the hyperquadric or N in all other cases, and that the Fefferman class of metrics does not contain any nonflat solution of Einstein's vacuum equations with cosmological constant.Work supported in part by the Polish Ministry of Science and Higher Education, Research Problem CPBP 01.03.     

Variational equations on mixed Riemannian–Lorentzian metrics

A class of elliptic–hyperbolic equations is placed in the context of a geometric variational theory, in which the change of type is viewed as a change in the character of an underlying metric. A fundamental example of a metric which changes in this way is the extended projective disc, which is Riemannian at ordinary points, Lorentzian at ideal points, and singular on the absolute. Harmonic fields on such a metric can be interpreted as the hodograph image of extremal surfaces in Minkowski 3-space. This suggests an approach to generalized Plateau problems in three-dimensional space-time via Hodge theory on the extended projective disc. Analogous variational problems arise on Riemannian–Lorentzian flow metrics in fiber bundles (twisted nonlinear Hodge equations), and on certain Riemannian–Lorentzian manifolds which occur in relativity and quantum cosmology. The examples surveyed come with natural gauge theories and Hodge dualities. This paper is mainly a review, but some technical extensions are proven.     

Dynamic field theory and equations of motion in cosmology

We discuss a field-theoretical approach based on general-relativistic variational principle to derive the covariant field equations and hydrodynamic equations of motion of baryonic matter governed by cosmological perturbations of dark matter and dark energy. The action depends on the gravitational and matter Lagrangian. The gravitational Lagrangian depends on the metric tensor and its first and second derivatives. The matter Lagrangian includes dark matter, dark energy and the ordinary baryonic matter which plays the role of a bare perturbation. The total Lagrangian is expanded in an asymptotic Taylor series around the background cosmological manifold defined as a solution of Einstein’s equations in the form of the Friedmann–Lemaître–Robertson–Walker (FLRW) metric tensor. The small parameter of the decomposition is the magnitude of the metric tensor perturbation. Each term of the series expansion is gauge-invariant and all of them together form a basis for the successive post-Friedmannian approximations around the background metric. The approximation scheme is covariant and the asymptotic nature of the Lagrangian decomposition does not require the post-Friedmannian perturbations to be small though computationally it works the most effectively when the perturbed metric is close enough to the background FLRW metric. The temporal evolution of the background metric is governed by dark matter and dark energy and we associate the large scale inhomogeneities in these two components as those generated by the primordial cosmological perturbations with an effective matter density contrast δρ/ρ≤1$\delta \rho /\rho \le 1$. The small scale inhomogeneities are generated by the condensations of baryonic matter considered as the bare perturbations of the background manifold that admits δρ/ρ?1$\delta \rho /\rho ?1$. Mathematically, the large scale perturbations are given by the homogeneous solution of the linearized field equations while the small scale perturbations are described by a particular solution of these equations with the bare stress–energy tensor of the baryonic matter. We explicitly work out the covariant field equations of the successive post-Friedmannian approximations of Einstein’s equations in cosmology and derive equations of motion of large and small scale inhomogeneities of dark matter and dark energy. We apply these equations to derive the post-Friedmannian equations of motion of baryonic matter comprising stars, galaxies and their clusters.