Conformally stationary Lorentzian tori with no conjugate points are flat

In this paper, the quantum corrections to the kinematics of geometry, specifically geodesics, are presented. This is done by employing the path integral over the geodesics. Interestingly, the geodesics do not see any modifications in this framework. However for the distances, it is demonstrated that these quantum corrections exhibit distinct behaviors for time-like, light-like, and space-like geodesics. For time-like geodesics, the maximum correction is the Planck length, which disappears when the classical separation vanishes. The light-like geodesics do not exhibit quantum corrections, meaning that the causal light cone remains the same in both classical and quantum frameworks under certain conditions. The quantum corrections for space-like geodesics impose a minimum on space-like separation, potentially playing a role in removing singularities by preventing null congruences from being closer than the Planck length. This framework also explores the correspondence between space-like/time-like geodesics and quantum/statistical physics.

The topology and geometry of the space of null geodesics N of a space-time M are used to study the causal structure of the space-time itself. In particular, the question of whether the topology of N is Hausdorff or admits a compatible manifold structure carries information on the global structure of M, and the transversality properties of the intersections of skies of points tell whether the points are conjugate points on a null geodesic.

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.

In a recent paper, a new conformally flat metric was introduced, describing an expanding scalar field in a spherically symmetric geometry. The spacetime can be interpreted as a Schwarzschild-like model with an apparent horizon surrounding the curvature singularity. For the above metric, we present the complete conformal Lie algebra consisting of a six-dimensional subalgebra of isometries (Killing Vector Fields or KVFs) and nine proper conformal vector fields (CVFs). An interesting aspect of our findings is that there exists a gradient (proper) conformal symmetry (i.e., its bivector Fab vanishes) which verifies the importance of gradient symmetries in constructing viable cosmological models. In addition, the 9-dimensional conformal algebra implies the existence of constants of motion along null geodesics that allow us to determine the complete solution of null geodesic equations.

It is shown explicitly that when the characteristic vector field that defines a Gödel-type metric is also a Killing vector, there always exist closed timelike or null curves in spacetimes described by such a metric. For these geometries, the geodesic curves are also shown to be characterized by a lower-dimensional Lorentz force equation for a charged point particle in the relevant Riemannian background. Moreover, two explicit examples are given for which timelike and null geodesics can never be closed.

We investigate the perturbation theory of a fixed-norm, timelike\nLorentz-violating vector field. After consistently quantizing the vector field\nto put constraints on its parameters, we compute the primordial spectra of\nperturbations generated by inflation in the presence of this vector field. We\nfind that its perturbations are sourced by the perturbations of the inflaton;\nwithout the inflaton perturbation the vector field perturbations decay away\nleaving no primordial spectra of perturbations. Since the inflaton perturbation\ndoes not have a spin-1 component, the vector field generically does not\ngenerate any spin-1 ``vector-type'' perturbations. Nevertheless, it will modify\nthe amplitude of both the spin-0 ``scalar-type'' and spin-2 ``tensor-type''\nperturbation spectra, leading to violations of the inflationary consistency\nrelationship.\n

Although it originated in the study of 4-dimensional spacetimes, the Newman-Penrose formalism is also an effective tool in dimension three, provided that a distinguished vector field is present. Here we show how a 3-dimensional version of the Newman-Penrose formalism can be used to study both the local and global geometry of Lorentzian 3-manifolds. Globally, we find obstructions to Lorentzian metrics generalizing those of constant curvature; locally, we classify Lorentzian 3-manifolds that admit a timelike Killing vector field. These results have appeared in [1] and [3], the latter joint with R. Ream.KeywordsLorentzian geometryRicci curvatureKilling vector fieldsLocal classificationGlobal obstructions

We consider spherically symmetric spacetimes with an outer trapping horizon. Such spacetimes are generalizations of spherically symmetric black hole spacetimes where the central mass can vary with time, like in black hole collapse or black hole evaporation. While these spacetimes possess in general no timelike Killing vector field, they admit a Kodama vector field which in some ways provides a replacement. The Kodama vector field allows the definition of a surface gravity of the outer trapping horizon. Spherically symmetric spacelike cross sections of the outer trapping horizon define in- and outgoing lightlike congruences. We investigate a scaling limit of Hadamard 2-point functions of a quantum field on the spacetime onto the ingoing lightlike congruence. The scaling limit 2-point function has a universal form and a thermal spectrum with respect to the time parameter of the Kodama flow, where the inverse temperature beta = 2pi /kappa is related to the surface gravity kappa of the horizon cross section in the same way as in the Hawking effect for an asymptotically static black hole. Similarly, the tunnelling probability that can be obtained in the scaling limit between in- and outgoing Fourier modes with respect to the time parameter of the Kodama flow shows a thermal distribution with the same inverse temperature, determined by the surface gravity. This can be seen as a local counterpart of the Hawking effect for a dynamical horizon in the scaling limit. Moreover, the scaling limit 2-point function allows it to define a scaling limit theory, a quantum field theory on the ingoing lightlike congruence emanating from a horizon cross section. The scaling limit 2-point function as well as the 2-point functions of coherent states of the scaling limit theory is correlation-free with respect to separation along the horizon cross section; therefore, their relative entropies behave proportional to the cross-sectional area. We thus obtain a proportionality of the relative entropy of coherent states of the scaling limit theory and the area of the horizon cross section with respect to which the scaling limit is defined. Thereby, we establish a local counterpart, and microscopic interpretation in the setting of quantum field theory on curved spacetimes, of the dynamical laws of outer trapping horizons, derived by Hayward and others in generalizing the laws of black hole dynamics originally shown for stationary black holes by Bardeen, Carter and Hawking.

Let $(M^n,g)$ be an $n$-dimensional compact connected Riemannian manifold with smooth boundary. We show that the presence of a nontrivial conformal gradient vector field on $M$, with an appropriate control on the Ricci curvature makes $M$ to be isometric to a hemisphere of $\mathbb{S}^{n}$. We also prove that if an Einstein manifold admits nonzero conformal gradient vector field, then its scalar curvature is positive and it is isometric to a hemisphere of $\mathbb{S}^{n}$. Furthermore, we prove that if $ M $ admits a nontrivial conformal vector field and has constant scalar curvature, then the scalar curvature is positive. Finally, a suitable control on the energy of a conformal vector field implies that $M$ is isometric to a hemisphere $\mathbb{S}^n_+$.

A non-extreme dilatonic charged (by two “color electric” charges) black hole solution is examined within a four-dimensional gravity model that incorporates two scalar (dilaton) fields and two Abelian vector fields. The scalar and vector fields interact through exponential terms containing two dilatonic coupling vectors. The solution is characterized by a dimensionless parameter a(0<a<2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(0< a < 2)$$\\end{document}, which is a specific function of dilatonic coupling vectors. The paper presents solutions for timelike and null circular geodesics that may play a crucial role in different astrophysical scenarios, including quasinormal modes of various test fields in the eikonal approximation. For a=1/2,1,3/2,2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$a = 1/2,1, 3/2,2 $$\\end{document}, the radii of the innermost stable circular orbit are presented and analyzed.

We consider a class of Lorentz-violating theories of gravity involving a timelike unit vector field (the aether) coupled to a metric, two examples being Einstein-aether theory and Ho\v{r}ava gravity. The action always includes the Ricci scalar of the metric and the invariants quadratic in covariant derivatives of the aether, but the theories differ in how the aether is constructed from other fields, and whether those fields are varied in the action. Fields that are not varied define background structures breaking diffeomorphsim invariance, including threadings, folations, and clocks, which generally produce novel degrees of freedom arising from the violation of what would otherwise be initial value constraints. The principal aims of this paper are to survey the nature of the theories that arise, and to understand the consequences of breaking diffeomorphism invariance in this setting. In a companion paper [arXiv:1504.03305], we address some of the phenomenology of the "ponderable aether" case in which the presence of a background clock endows the aether with a variable internal energy density that behaves in some respects like dark matter.

Semi-classical mass asymptotics on stationary spacetimes

The existence of Killing vectors in conformally flat perfect fluid spacetimes in general relativity is considered. In particular Killing vectors which are neither orthogonal nor parallel to the fluid velocity vector are considered and stationary fields in which the fluid velocity vector is not parallel to the timelike Killing vector field are shown to exist. This class of solutions is shown to include several stationary (but non-static) axisymmetric fields, thus providing counter-examples to a theorem of Collinson (1976). In the case when the fluid is non-expanding, the number of spacelike Killing vectors is shown to depend on the rank of four functions of time which appear in the metric. Some examples of stationary but non-static fields are presented in closed form.

We study focal points and Maslov index of a horizontal geodesic γ : I → M in the total space of a semi-Riemannian submersion π : M → B by determining an explicit relation with the corresponding objects along the projected geodesic \({\pi\circ\gamma:I\to B}\) in the base space. We use this result to calculate the focal Maslov index of a (spacelike) geodesic in a stationary spacetime which is orthogonal to a timelike Killing vector field.

We prove in this paper a divergence theorem for symmetric (0,2)-tensors on a semi-Riemannian manifold with boundary. We obtain a generalization of results obtained by Ünal in [9, Acta Appl. Math. 40(1995)] and E. García-Río and D. N. Kupeli in [4, Proceeding of the Third World Congress of Nonlinear Analysts, Part 5 (Catania, 2000). Nonlinear Anal. 47 (5) 2995-3004, 2001]. As a tool, we use an induced volume form on the degenerate boundary by introducing a star like operator. A vanishing theorem for gradient timelike Killing vector fields on Einstein semi-Riemannian manifolds is obtained.