Stationary Lorentzian Manifolds Research Articles

Boundary conditions and infrared divergences

We review the procedure to construct quasi-free ground states, for real scalar fields whose dynamics is dictated by the Klein-Gordon equation, on standard static Lorentzian manifolds with a time-like boundary. We observe that, depending on the assigned boundary condition of Robin type, this procedure does not always lead to the existence of a suitable bi-distribution ω2∈D′(M×M) due to the presence of infrared divergences. As a concrete example we consider a Bertotti-Robinson spacetime in two different coordinate patches. In one case we show that infrared divergences do not occur only for Dirichlet boundary conditions as one might expect a priori, while, in the other case, we prove that they occur only when Neumann boundary conditions are imposed at the time-like boundary.

An index theorem on asymptotically static spacetimes with compact Cauchy surface

We consider the Dirac operator on asymptotically static Lorentzian manifolds with an odd-dimensional compact Cauchy surface. We prove that if Atiyah-Patodi-Singer boundary conditions are imposed at infinite times then the Dirac operator is Fredholm. This generalizes a theorem due to B\"ar-Strohmaier in the case of finite times, and we also show that the corresponding index formula extends to the infinite setting. Furthermore, we demonstrate the existence of a Fredholm inverse which is at the same time a Feynman parametrix in the sense of Duistermaat-H\"ormander. The proof combines methods from time-dependent scattering theory with a variant of Egorov's theorem for pseudo-differential hyperbolic systems.

The qualitative behavior for approximate Dirac-harmonic maps into stationary Lorentzian manifolds

The qualitative behavior for approximate Dirac-harmonic maps into stationary Lorentzian manifolds

Regularity for Dirac-harmonic maps into certain pseudo-Riemannian manifolds

We show the smoothness of weakly Dirac-harmonic maps from a closed spin Riemann surface into stationary Lorentzian manifolds, and obtain a regularity theorem for a class of critical elliptic systems without anti-symmetry structures.

The Light Ray Transform in Stationary and Static Lorentzian Geometries

Given a Lorentzian manifold, the light ray transform of a function is its integrals along null geodesics. This paper is concerned with the injectivity of the light ray transform on functions and tensors, up to the natural gauge for the problem. First, we study the injectivity of the light ray transform of a scalar function on a globally hyperbolic stationary Lorentzian manifold and prove injectivity holds if either a convex foliation condition is satisfied on a Cauchy surface on the manifold or the manifold is real analytic and null geodesics do not have cut points. Next, we consider the light ray transform on tensor fields of arbitrary rank in the more restrictive class of static Lorentzian manifolds and show that if the geodesic ray transform on tensors defined on the spatial part of the manifold is injective up to the natural gauge, then the light ray transform on tensors is also injective up to its natural gauge. Finally, we provide applications of our results to some inverse problems about recovery of coefficients for hyperbolic partial differential equations from boundary data.

Fundamental solutions for the wave operator on static Lorentzian manifolds with timelike boundary

We consider the wave operator on static, Lorentzian manifolds with timelike boundary and we discuss the existence of advanced and retarded fundamental solutions in terms of boundary conditions. By means of spectral calculus we prove that answering this question is equivalent to studying the self-adjoint extensions of an associated elliptic operator on a Riemannian manifold with boundary $(M,g)$. The latter is diffeomorphic to any, constant time hypersurface of the underlying background. In turn, assuming that $(M,g)$ is of bounded geometry, this problem can be tackled within the framework of boundary triples. These consist of the assignment of two surjective, trace operators from the domain of the adjoint of the elliptic operator into an auxiliary Hilbert space $\mathsf{h}$, which is the third datum of the triple. Self-adjoint extensions of the underlying elliptic operator are in one-to-one correspondence with self-adjoint operators $\Theta$ on $\mathsf{h}$. On the one hand, we show that, for a natural choice of boundary triple, each $\Theta$ can be interpreted as the assignment of a boundary condition for the original wave operator. On the other hand, we prove that, for each such $\Theta$, there exists a unique advanced and retarded fundamental solution. In addition, we prove that these share the same structural property of the counterparts associated to the wave operator on a globally hyperbolic spacetime.

Null Geodesies in Static Warped Braneworld

We consider spherically symmetric spacetimes embedded in a 5-dimensional static Lorentzian manifold. Studying the null geodesic equations in warped spacetimes bounded by static 3-branes, we discuss possible relations of particles following the geodesics with the gravitational waves detected recently.

Bubbling analysis for approximate Lorentzian harmonic maps from Riemann surfaces

For a sequence of approximate harmonic maps (u_n,v_n) (meaning that they satisfy the harmonic system up to controlled error terms) from a compact Riemann surface with smooth boundary to a standard static Lorentzian manifold with bounded energy, we prove that identities for the Lorentzian energy hold during the blow-up process. In particular, in the special case where the Lorentzian target metric is of the form g_N -beta dt^2 for some Riemannian metric g_N and some positive function beta on N, we prove that such identities also hold for the positive energy (obtained by changing the sign of the negative part of the Lorentzian energy) and there is no neck between the limit map and the bubbles. As an application, we complete the blow-up picture of singularities for a harmonic map flow into a standard static Lorentzian manifold. We prove that the energy identities of the flow hold at both finite and infinite singular times. Moreover, the no neck property of the flow at infinite singular time is true.

Energy identity for harmonic maps into standard stationary Lorentzian manifolds

For a harmonic map from a closed Riemann surface into a standard stationary Lorentzian manifold, we prove that its Hopf differential is holomorphic. Moreover, we prove that for a sequence of such maps with their energy uniformly bounded, the Lorentzian energy identity holds during the blow-up process.

Compact conformally stationary Lorentzian manifolds with no causal conjugate points

A sequence of integral inequalities for any compact conformally stationary Lorentzian manifold with no conjugate points along its causal geodesics is obtained. If the equality holds for one of them, the Lorentzian manifold must be flat. As an application, several classification results for such manifolds are proved.

Pseudo Focal Points Along Lorentzian Geodesics and Morse Index

Abstract Given a Lorentzian manifold (M, g), a geodesic γ in M and a timelike Jacobi field γ along γ, we introduce a special class of instants along γ that we call γ- pseudo conjugate (or focal relatively to some initial orthogonal submanifold). We prove that the γ-pseudo conjugate instants form a finite set, and their number equals the Morse index of (a suitable restriction of) the index form. This gives a Riemannian-like Morse index theorem. As special cases of the theory, we will consider geodesics in stationary and static Lorentzian manifolds, where the Jacobi field γ is obtained as the restriction of a globally defined timelike Killing vector field.

Genericity of nondegenerate critical points and Morse geodesic functionals

We consider a family of variational problems on a Hilbert manifold parameterized by an open subset of a Banach manifold, and we discuss the genericity of the nondegeneracy condition for the critical points. Using classical techniques, we prove an abstract genericity result that employs the infinite dimensional Sard―Smale theorem, along the lines of an analogous result of B. White [29]. Applications are given by proving the genericity of metrics without degenerate geodesics between fixed endpoints in general (non compact) semi-Riemannian manifolds, in orthogonally split semi-Riemannian manifolds and in globally hyperbolic Lorentzian manifolds. We discuss the genericity property also in stationary Lorentzian manifolds.

On a Gromoll-Meyer type theorem in globally hyperbolic stationary spacetimes

Following the lines of the celebrated Riemannian result of Gromoll and Meyer, we use infinite dimensional equivariant Morse theory to establish the existence of infinitely many geometrically distinct closed geodesics in a class of globally hyperbolic stationary Lorentzian manifolds.

Iteration of closed geodesics in stationary Lorentzian manifolds

Following the lines of Bott in (Commun Pure Appl Math 9:171–206, 1956), we study the Morse index of the iterates of a closed geodesic in stationary Lorentzian manifolds, or, more generally, of a closed Lorentzian geodesic that admits a timelike periodic Jacobi field. Given one such closed geodesic γ, we prove the existence of a locally constant integer valued map Λγ on the unit circle with the property that the Morse index of the iterated γN is equal, up to a correction term εγ∈{0,1}, to the sum of the values of Λγ at the N-th roots of unity. The discontinuities of Λγ occur at a finite number of points of the unit circle, that are special eigenvalues of the linearized Poincaré map of γ. We discuss some applications of the theory.

Lightlike rays in stationary spacetimes with critical growth

The aim of this paper is to extend some previous results on the existence of lightlike geodesics joining a point to a line to the case of stationary Lorentzian manifolds whose metric coefficients have an optimal growth.

Orthogonal trajectories on stationary spacetimes under intrinsic assumptions

Using global variational methods and coordinate free assumptions, we obtain existence and multiplicity results on stationary Lorentzian manifolds for solutions to the Lorentz force equation joining two spacelike submanifolds. Some examples and applications are provided.

Maslov index and Morse theory for the relativistic Lorentz force equation

We study the Jacobi equation for fixed endpoints solutions of the Lorentz force equation on a Lorentzian manifold. The flow of the Jacobi equation along each solution preserves the so-called twisted symplectic form, and the corresponding curve in the symplectic group determines an integer valued homology class called the Maslov index of the solution. We introduce the notion of F-conjugate plane for each conjugate instant; the restriction of the spacetime metric to the F-conjugate plane is used to compute the Maslov index, which is given by a sort of algebraic count of the conjugate instants. For a stationary Lorentzian manifold and an exact electromagnetic field admitting a potential vector field preserving the flow of the Killing vector field, we introduce a constrained action functional having finite Morse index and whose critical points are fixed endpoints solution of the Lorentz force equation. We prove that the value of this Morse index equals the Maslov index and we prove the Morse relations for the solutions of the Lorentz force equation in a static spacetime.

Exact solutions of Dirac equation on a 2D gravitational background

We obtain classes of two-dimensional static Lorentzian manifolds, which through the supersymmetric formalism of quantum mechanics admit the exact solvability of Dirac equation on these curved backgrounds. Specially in the case of a modified supersymmetric harmonic oscillator the wave function and energy spectrum of Dirac equation is given explicitly.

Geodesics in Static Lorentzian Manifolds with Critical Quadratic Behavior

Abstract The aim of this paper is t o study the geodesic connectedness of a complete static Lorentzian manifold (M.〈·, ·〉L) such that. if K is its irrotational timelike Killing vector field, the growth of β = -〈K, K〉L is (at most) quadratic. This is shown to be equivalent to a critical variational problem on a Riemannian manifold, where some geometrical interpretations yield the optimal results. Multiplicity of connecting geodesics is studied, especially in the timelike case, where the restrictions for the number of causal connecting geodesics are stressed. Extensions to the non-complete case, including discussions on pseudoconvexity, are also given.

Classification of Cylindrically Symmetric Static Spacetimes According to Their Ricci Collineations

A complete classification of cylindrically symmetric static Lorentzian manifolds according to their Ricci collineations (RCs) is provided. The Lie algebras of RCs for the non-degenerate Ricci tensor have dimensions 3 to 10, excluding 8 and 9. For the degenerate tensor the algebra is mostly but not always infinite dimensional; there are cases of 10-, 5-, 4- and 3-dimensional algebras. The RCs are compared with the Killing vectors (KVs) and homothetic motions (HMs). The (non-linear) constraints corresponding to the Lie algebras are solved to construct examples which include some exact solutions admitting proper RCs. Their physical interpretation is given. The classification of plane symmetric static spacetimes emerges as a special case of this classification when the cylinder is unfolded.