Geodesics in Static Lorentzian Manifolds with Critical Quadratic Behavior

2003 ◽  
Vol 3 (4) ◽  
Author(s):  
R. Bartolo ◽  
A.M. Candela ◽  
J.L. Flores ◽  
M. Sánchez
Keyword(s):  
Lorentzian Manifold ◽  
Lorentzian Manifolds ◽  
Geodesic Connectedness

AbstractThe aim of this paper is t o study the geodesic connectedness of a complete static Lorentzian manifold (M.〈·, ·〉

scholarly journals RECOVERY OF ZEROTH ORDER COEFFICIENTS IN NON-LINEAR WAVE EQUATIONS

2020 ◽  
pp. 1-27
Author(s):  
Ali Feizmohammadi ◽  
Lauri Oksanen
Keyword(s):  
Wave Equations ◽  
Model Problem ◽  
Wave Interaction ◽  
Lorentzian Manifold ◽  
Gaussian Beams ◽  
Linear Wave ◽  
Lorentzian Manifolds ◽  
Solution Map ◽  
Globally Hyperbolic ◽  
Linear Wave Equations

This paper is concerned with the resolution of an inverse problem related to the recovery of a function $V$ from the source to solution map of the semi-linear equation $(\Box _{g}+V)u+u^{3}=0$ on a globally hyperbolic Lorentzian manifold $({\mathcal{M}},g)$ . We first study the simpler model problem, where $({\mathcal{M}},g)$ is the Minkowski space, and prove the unique recovery of $V$ through the use of geometric optics and a three-fold wave interaction arising from the cubic non-linearity. Subsequently, the result is generalized to globally hyperbolic Lorentzian manifolds by using Gaussian beams.

scholarly journals Curvature Estimates in Asymptotically Flat Lorentzian Manifolds

2005 ◽  
Vol 57 (4) ◽  
pp. 708-723 ◽  
Author(s):  
Felix Finster ◽  
Margarita Kraus
Keyword(s):  
Fundamental Form ◽  
Curvature Tensor ◽  
Lorentzian Manifold ◽  
Second Fundamental Form ◽  
Riemannian Curvature ◽  
Lorentzian Manifolds ◽  
Asymptotically Flat ◽  
Riemannian Curvature Tensor ◽  
Curvature Estimates

AbstractWe consider an asymptotically flat Lorentzian manifold of dimension (1, 3). An inequality is derived which bounds the Riemannian curvature tensor in terms of the ADM energy in the general case with second fundamental form. The inequality quantifies in which sense the Lorentzianmanifold becomes flat in the limit when the ADM energy tends to zero.

Some characterizations of Lorentzian manifolds

2019 ◽  
Vol 16 (01) ◽  
pp. 1950016
Author(s):  
Uday Chand De ◽  
Young Jin Suh
Keyword(s):  
Curvature Tensor ◽  
Lorentzian Manifold ◽  
Lorentzian Manifolds ◽  
Conformal Curvature ◽  
Conformal Curvature Tensor

Generalized Robertson–Walker (GRW) spacetime is the generalization of the Robertson–Walker (RW) spacetime and a further generalization of GRW spacetime is the twisted spacetime. In this paper, we generalize the results of the paper [C. A. Mantica, Y. J. Suh and U. C. De, A note on generalized Robertson–Walker spacetimes, Int. J. Geom. Methods Mod. Phys. 13 (2016), Article Id: 1650079, 9 pp., doi: 101142/s0219887816500791 ]. We prove that a Ricci simple Lorentzian manifold with vanishing quasi-conformal curvature tensor is a RW spacetime. Further, we prove that a Ricci simple Lorentzian manifold with harmonic quasi-conformal curvature tensor is a GRW spacetime. As a consequence, we obtain several corollaries. Finally, we have cited some examples of the obtained results.

Compact null hypersurfaces in Lorentzian manifolds

2021 ◽  
Vol 21 (2) ◽  
pp. 251-263
Author(s):  
C. Atindogbé ◽  
M. Gutiérrez ◽  
R. Hounnonkpe
Keyword(s):  
Global Symmetries ◽  
Lorentzian Manifold ◽  
Lorentzian Manifolds ◽  
Geometric Properties ◽  
Totally Geodesic ◽  
Null Hypersurfaces ◽  
Ambient Manifold ◽  
The Family

Abstract We show how the topological and geometric properties of the family of null hypersurfaces in a Lorentzian manifold are related with the properties of the ambient manifold itself. In particular, we focus in how the presence of global symmetries and curvature conditions restrict the existence of compact null hypersurfaces. We use these results to show the influence on the existence of compact totally umbilic null hypersurfaceswhich are not totally geodesic. Finally we describe the restrictions that they impose in causality theory.

scholarly journals Pseudo Focal Points Along Lorentzian Geodesics and Morse Index

2010 ◽  
Vol 10 (1) ◽  
Author(s):  
Miguel Ángel Javaloyes ◽  
Antonio Masiello ◽  
Paolo Piccione
Keyword(s):  
Morse Index ◽  
Killing Vector ◽  
Index Theorem ◽  
Lorentzian Manifold ◽  
Jacobi Field ◽  
Focal Points ◽  
Lorentzian Manifolds ◽  
Special Cases ◽  
Finite Set ◽  
Suitable Restriction

AbstractGiven 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.

scholarly journals On pseudo H-symmetric Lorentzian manifolds with applications to relativity

Filomat ◽  
2020 ◽  
Vol 34 (10) ◽  
pp. 3287-3297
Author(s):  
Uday De ◽  
Young Suh ◽  
Sudhakar Chaubey ◽  
Sameh Shenawy
Keyword(s):  
Curvature Tensor ◽  
Lorentzian Manifold ◽  
Lorentzian Manifolds ◽  
Lorentzian Metric ◽  
Projective Curvature ◽  
Almost Product Manifold ◽  
New Type ◽  
Curvature Tensors ◽  
Fluid Spacetimes

In this paper, we introduce a new type of curvature tensor named H-curvature tensor of type (1, 3) which is a linear combination of conformal and projective curvature tensors. First we deduce some basic geometric properties of H-curvature tensor. It is shown that a H-flat Lorentzian manifold is an almost product manifold. Then we study pseudo H-symmetric manifolds (PHS)n (n > 3) which recovers some known structures on Lorentzian manifolds. Also, we provide several interesting results. Among others, we prove that if an Einstein (PHS)n is a pseudosymmetric (PS)n, then the scalar curvature of the manifold vanishes and conversely. Moreover, we deal with pseudo H-symmetric perfect fluid spacetimes and obtain several interesting results. Also, we present some results of the spacetime satisfying divergence free H-curvature tensor. Finally, we construct a non-trivial Lorentzian metric of (PHS)4.

Existence of geodesics in static Lorentzian manifolds with convex boundary

2000 ◽  
Vol 130 (1) ◽  
pp. 189-215 ◽  
Author(s):  
P. Piccione
Keyword(s):  
Variational Principle ◽  
Fixed Points ◽  
Smooth Boundary ◽  
Differentiable Manifold ◽  
Lorentzian Manifold ◽  
Multiplicity Result ◽  
Lorentzian Manifolds ◽  
Global Hyperbolicity ◽  
Convex Boundary ◽  
Convexity Assumption

We study some global geometric properties of a static Lorentzian manifold Λ embedded in a differentiable manifold M, with possibly non-smooth boundary ∂Λ. We prove a variational principle for geodesics in static manifolds, and using this principle we establish the existence of geodesics that do not touch ∂Λ and that join two fixed points of Λ. The results are obtained under a suitable completeness assumption for Λ that generalizes the property of global hyperbolicity, and a weak convexity assumption on ∂Λ. Moreover, under a non-triviality assumption on the topology of Λ, we also get a multiplicity result for geodesics in Λ joining two fixed points.

scholarly journals Reconstruction of Lorentzian Manifolds from Boundary Light Observation Sets

2017 ◽  
Vol 2019 (22) ◽  
pp. 6949-6987
Author(s):  
Peter Hintz ◽  
Gunther Uhlmann
Keyword(s):  
Open Subset ◽  
Lorentzian Manifold ◽  
Conformal Structure ◽  
Conjugate Points ◽  
Lorentzian Manifolds ◽  
Convexity Assumption ◽  
Nonempty Boundary ◽  
Light Rays ◽  
Snell’S Law ◽  
Snell's Law

Abstract On a time-oriented Lorentzian manifold (M, g) with nonempty boundary satisfying a convexity assumption, we show that the topological, differentiable, and conformal structure of suitable subsets S ⊂ M of sources is uniquely determined by measurements of the intersection of future light cones from points in S with a fixed open subset of the boundary of M; here, light rays are reflected at ∂M according to Snell’s law. Our proof is constructive, and allows for interior conjugate points as well as multiply reflected and self-intersecting light cones.

Null hypersurfaces in Lorentzian manifolds II

1981 ◽  
Vol 89 (3) ◽  
pp. 525-532 ◽  
Author(s):  
K. Katsuno
Keyword(s):  
Lorentzian Manifold ◽  
Parameter Family ◽  
Lorentzian Manifolds ◽  
Geometrical Properties ◽  
Null Hypersurfaces

This paper is a continuation of (8), and is concerned with geometrical properties of special null hypersurfaces. In particular, on a one-parameter family of null hypersurfaces in four-dimensional Lorentzian manifoldV4, we consider the relation between their normal and the Debever vectors, especially repeated ones. Throughout this paper, the same notations as those in (8) are used.

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