# Category: Lorentzian manifolds

Causal structure
In mathematical physics, the causal structure of a Lorentzian manifold describes the causal relationships between points in the manifold.
Clifton–Pohl torus
In geometry, the Clifton–Pohl torus is an example of a compact Lorentzian manifold that is not geodesically complete. While every compact Riemannian manifold is also geodesically complete (by the Hopf
Kretschmann scalar
In the theory of Lorentzian manifolds, particularly in the context of applications to general relativity, the Kretschmann scalar is a quadratic scalar invariant. It was introduced by Erich Kretschmann
Gullstrand–Painlevé coordinates
Gullstrand–Painlevé coordinates are a particular set of coordinates for the Schwarzschild metric – a solution to the Einstein field equations which describes a black hole. The ingoing coordinates are
Timelike homotopy
On a Lorentzian manifold, certain curves are distinguished as timelike. A timelike homotopy between two timelike curves is a homotopy such that each intermediate curve is timelike. No closed timelike
Null hypersurface
In relativity and in pseudo-Riemannian geometry, a null hypersurface is a hypersurface whose normal vector at every point is a null vector (has zero length with respect to the local metric tensor). A
Gravitational singularity
A gravitational singularity, spacetime singularity or simply singularity is a condition in which gravity is so intense that spacetime itself breaks down catastrophically. As such, a singularity is by
Bel decomposition
In semi-Riemannian geometry, the Bel decomposition, taken with respect to a specific timelike congruence, is a way of breaking up the Riemann tensor of a pseudo-Riemannian manifold into lower order te
McVittie metric
In the general theory of relativity, the McVittie metric is the exact solution of Einstein's field equations describing a black hole or massive object immersed in an expanding cosmological spacetime.
Asymptotically flat spacetime
An asymptotically flat spacetime is a Lorentzian manifold in which, roughly speaking, the curvature vanishes at large distances from some region, so that at large distances, the geometry becomes indis
Cauchy surface
In the mathematical field of Lorentzian geometry, a Cauchy surface is a certain kind of submanifold of a Lorentzian manifold. In the application of Lorentzian geometry to the physics of general relati
Gaussian polar coordinates
In the theory of Lorentzian manifolds, spherically symmetric spacetimes admit a family of nested round spheres. In each of these spheres, every point can be carried to any other by an appropriate rota
Light cone
In special and general relativity, a light cone (or "null cone") is the path that a flash of light, emanating from a single event (localized to a single point in space and a single moment in time) and
Static spacetime
In general relativity, a spacetime is said to be static if it does not change over time and is also irrotational. It is a special case of a stationary spacetime, which is the geometry of a stationary
Spacetime topology
Spacetime topology is the topological structure of spacetime, a topic studied primarily in general relativity. This physical theory models gravitation as the curvature of a four dimensional Lorentzian
Causality conditions
In the study of Lorentzian manifold spacetimes there exists a hierarchy of causality conditions which are important in proving mathematical theorems about the global structure of such manifolds. These
Vanishing scalar invariant spacetime
In mathematical physics, vanishing scalar invariant (VSI) spacetimes are Lorentzian manifolds with all polynomial curvature invariants of all orders vanishing. Although the only Riemannian manifold wi
Wormhole
A wormhole (Einstein-Rosen bridge) is a speculative structure connecting disparate points in spacetime, and is based on a special solution of the Einstein field equations. A wormhole can be visualized
Topological censorship
The topological censorship theorem (if valid) states that general relativity does not allow an observer to probe the topology of spacetime: any topological structure collapses too quickly to allow lig
Minkowski space
In mathematical physics, Minkowski space (or Minkowski spacetime) (/mɪŋˈkɔːfski, -ˈkɒf-/) is a combination of three-dimensional Euclidean space and time into a four-dimensional manifold where the spac
Congruence (general relativity)
In general relativity, a congruence (more properly, a congruence of curves) is the set of integral curves of a (nowhere vanishing) vector field in a four-dimensional Lorentzian manifold which is inter
Christoffel symbols
In mathematics and physics, the Christoffel symbols are an array of numbers describing a metric connection. The metric connection is a specialization of the affine connection to surfaces or other mani
Classification of electromagnetic fields
In differential geometry and theoretical physics, the classification of electromagnetic fields is a pointwise classification of bivectors at each point of a Lorentzian manifold. It is used in the stud
Timelike simply connected
Suppose a Lorentzian manifold contains a closed timelike curve (CTC). No CTC can be continuously deformed as a CTC (is timelike homotopic) to a point, as that point would not be causally well behaved.
Pseudo-Euclidean space
In mathematics and theoretical physics, a pseudo-Euclidean space is a finite-dimensional real n-space together with a non-degenerate quadratic form q. Such a quadratic form can, given a suitable choic
Bondi–Metzner–Sachs group
In gravitational theory, the Bondi–Metzner–Sachs (BMS) group, or the Bondi–van der Burg–Metzner–Sachs group, is an asymptotic symmetry group of asymptotically flat, Lorentzian spacetimes at null (i.e.
Penrose diagram
In theoretical physics, a Penrose diagram (named after mathematical physicist Roger Penrose) is a two-dimensional diagram capturing the causal relations between different points in spacetime through a
Schwarzschild coordinates
In the theory of Lorentzian manifolds, spherically symmetric spacetimes admit a family of nested round spheres. In such a spacetime, a particularly important kind of coordinate chart is the Schwarzsch
Spherically symmetric spacetime
In physics, spherically symmetric spacetimes are commonly used to obtain analytic and numerical solutions to Einstein's field equations in the presence of radially moving matter or energy. Because sph
Stationary spacetime
In general relativity, specifically in the Einstein field equations, a spacetime is said to be stationary if it admits a Killing vector that is asymptotically timelike.
Alcubierre drive
The Alcubierre drive ([alˈkubie:re]) is a speculative warp drive idea according to which a spacecraft could achieve apparent faster-than-light travel by contracting space in front of it and expanding
Closed timelike curve
In mathematical physics, a closed timelike curve (CTC) is a world line in a Lorentzian manifold, of a material particle in spacetime, that is "closed", returning to its starting point. This possibilit
Spacetime symmetries
Spacetime symmetries are features of spacetime that can be described as exhibiting some form of symmetry. The role of symmetry in physics is important in simplifying solutions to many problems. Spacet
Geroch's splitting theorem
In the theory of causal structure on Lorentzian manifolds, Geroch's theorem or Geroch's splitting theorem (first proved by Robert Geroch) gives a topological characterization of globally hyperbolic sp
Isotropic coordinates
In the theory of Lorentzian manifolds, spherically symmetric spacetimes admit a family of nested round spheres. There are several different types of coordinate chart which are adapted to this family o
Kruskal–Szekeres coordinates
In general relativity, Kruskal–Szekeres coordinates, named after Martin Kruskal and George Szekeres, are a coordinate system for the Schwarzschild geometry for a black hole. These coordinates have the
Pseudo-Riemannian manifold
In differential geometry, a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a differentiable manifold with a metric tensor that is everywhere nondegenerate. This is a generaliza
Kundt spacetime
In mathematical physics, Kundt spacetimes are Lorentzian manifolds admitting a geodesic null congruence with vanishing optical scalars (expansion, twist and shear). A well known member of Kundt class