Curvature and Killing Vector Fields on Lorentzian 3-Manifolds

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.

We obtain some generalized Minkowski type integral formulas for compact Riemannian (resp., spacelike) hypersurfaces in Riemannian (resp., Lorentzian) manifolds in the presence of an arbitrary vector field that we assume to be timelike in the case where the ambient space is Lorentzian. Some of these formulas generalize existing formulas in the case of conformal and Killing vector fields. We apply these integral formulas to obtain interesting results concerning the characterization of such hypersurfaces in some particular cases such as when the ambient space is Einstein admitting an arbitrary (in particular, conformal or Killing) vector field, and when the hypersurface has a constant mean curvature.

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

Geometric properties of almost pure metric plastic pseudo-Riemannian manifolds

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.

In this chapter, we introduce the second fundamental form of a map between semi-Riemannian manifolds. The second fundamental form of a map is used largely in the analysis of certain properties of maps between manifolds in the literature. However, in this chapter, we study the second fundamental form of a map only as much as we need to carry out our work on semi-Riemannian maps. We devote the first section to the definition and basic properties of the second fundamental form of a map between semi-Riemannian manifolds. In section 2, we define affine and harmonic maps between semi-Riemannian manifolds and derive certain properties of affine maps. Then in section 3, we define the divergence of a vector field along a map between semi-Riemannian manifolds and generalize the semi-Riemannian divergence theorem to such vector fields. Finally in section 4, we derive the Bochner identity of a map between semi-Riemannian manifolds which will be used in showing a Riemannian map to be affine under certain curvature conditions. Throughout this chapter, and in all following chapters, let M 1 M 2 and M be manifolds with dimensions n 1, n 2 and n, respectively. Also let (M l, g 1), (M 2, g 2 and (M, g) be semi-Riemannian manifolds of indices v l, v 2 and v with Levi-Civita connections \(\mathop \nabla \limits^1 ,\mathop \nabla \limits^2 \) and ∇, respectively, unless otherwise stated.

The Noether-charge realization and the Hamiltonian realization for the $\diff({\cal M})$ algebra in diffeomorphism invariant gravitational theories are studied in a covariant formalism. For the Killing vector fields, the Nother-charge realization leads to the mass formula as an entire vanishing Noether charge for the vacuum black hole spacetimes in general relativity and the corresponding first law of the black hole mechanics. It is analyzed in which sense the Hamiltonian functionals form the $\diff({\cal M})$ algebra under the Poisson bracket and shown how the Noether charges with respect to the diffeomorphism generated by vector fields and their variations in general relativity form this algebra. The asymptotic behaviors of vector fields generating diffeomorphism of the manifold with boundaries are discussed. In order to get more precise estimation for the "central extension" of the algebra, it is analyzed in the Newman-Penrose formalism and shown that the "central extension" for a large class of vector fields is always zero on the Killing horizon. It is also checked whether the Virasoro algebra may be picked up by choosing the vector fields near the horizon. The conclusion is unfortunately negative.

Some relations between the causal character of projective vector fields and curvature on a Lorentzian manifold M are studied. As a consequence, obstructions to the existence of such vector fields are found. Affine, homothetic and Killing vector fields are considered specifically.

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.

Publisher Summary This chapter presents the results on harmonic and killing vector fields to the case of Riemannian spaces with boundary. The harmonic tensors on such Riemannian spaces are studied, and the relations between curvature and relative Betti numbers in Riemannian spaces with boundary are discussed. If, in an M with boundary B, the Ricci curvature is positive (negative) definite and the second fundamental form is negative (positive) semi-definite, then there does not exist a harmonic (Killing, conformal Eilling) vectoy field tangential to B other than the zero vector field. If, in an M with boundary B, the Ricci curvature is positive (negative) definite and the mean curvature of B is negative (positive) or zero, then there does not exist a harmonic (Killing, conformal Killing) vector field normal to B other than the zero vector field. If an M admits a Killing vector field normal to the boundary B and vanishing only on a nowhere dense set of points on B, then the boundary is totally geodesic.

Elliptic Metrics on Lorentz Manifolds

Semi-classical mass asymptotics on stationary spacetimes

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.

Using the relationship between the entropy and the Euler characteristic, and the usual decomposition of spin connection, an entropy density is introduced to describe the inner structure of the entropy of RN black holes. It is pointed out that the entropy of RN black holes is determined by the singularities of the timelike Killing vector field of RN spacetime, and that these singularities carry the topological numbers, Hopf indices and Brouwer degrees, naturally, which are topological invariants. Taking account of the physical meaning of entropy in statistics, the entropy and its density of RN black holes are modified and they are given by the Hopf indices merely.

In General Relativity in Hamiltonian form, change has seemed to be missing, defined only asymptotically, or otherwise obscured at best, because the Hamiltonian is a sum of first-class constraints and a boundary term and thus supposedly generates gauge transformations. By construing change as essential time dependence (lack of a time-like Killing vector field), one can find change locally in vacuum GR in the Hamiltonian formulation just where it should be. But what if spinors are present? This paper is motivated by the tendency in space-time philosophy tends to slight fermionic/spinorial matter, the tendency in Hamiltonian GR to misplace changes of time coordinate, and the tendency in treatments of the Einstein-Dirac equation to include a gratuitous local Lorentz gauge symmetry along with the physically significant coordinate freedom. Spatial dependence is dropped in most of the paper, both restricting the physical situation and largely fixing the spatial coordinates. In the interest of including all and only the coordinate freedom, the Einstein-Dirac equation is investigated using the Schwinger time gauge and Kibble-Deser symmetric triad condition are employed as a 3+1 version of the DeWitt-Ogievetsky-Polubarinov nonlinear group realization formalism that dispenses with a tetrad and local Lorentz gauge freedom. Change is the lack of a time-like stronger-than-Killing field for which the Lie derivative of the metric-spinor complex vanishes. An appropriate 3+1-friendly form of the Rosenfeld-Anderson-Bergmann-Castellani gauge generator G, a tuned sum of first class-constraints, is shown to change the canonical Lagrangian by a total derivative, implying the preservation of Hamilton’s equations. Given the essential presence of second-class constraints with spinors and their lack of resemblance to a gauge theory (unlike, say, massive photons), it is useful to have an explicit physically interesting example. This gauge generator implements changes of time coordinate for solutions of the equations of motion, showing that the gauge generator makes sense even with spinors.