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 light cone is an example.
An alternative characterization is that the tangent space at every point of a hypersurface contains a nonzero vector such that the metric applied to such a vector and any vector in the tangent space is zero. Another way of saying this is that the pullback of the metric onto the tangent space is degenerate.
For a Lorentzian metric, all the vectors in such a tangent space are space-like except in one direction, in which they are null. Physically, there is exactly one lightlike worldline contained in a null hypersurface through each point that corresponds to the worldline of a particle moving at the speed of light, and no contained worldlines that are time-like. Examples of null hypersurfaces include a light cone, a Killing horizon, and the event horizon of a black hole.
and in pseudo-Riemannian geometry, a nullhypersurface is a hypersurface whose normal vector at every point is a null vector (has zero length with respect...
In geometry, a hypersurface is a generalization of the concepts of hyperplane, plane curve, and surface. A hypersurface is a manifold or an algebraic variety...
dynamic Einstein field equations. Mathematically a Killing horizon is a nullhypersurface defined by the vanishing of the norm of a Killing vector field (both...
{\theta }}{\hat {\omega }}_{ab}\;.} For a geodesic null congruence restricted on a nullhypersurface, we have ( 20 ) k c ∇ c θ = θ ^ , λ = − 1 2 θ ^ 2...
(quadric hypersurface in higher dimensions), is a generalization of conic sections (ellipses, parabolas, and hyperbolas). It is a hypersurface (of dimension...
saddle point for a smooth function (whose graph is a curve, surface or hypersurface) is a stationary point such that the curve/surface/etc. in the neighborhood...
details. A moving wave surface in special relativity may be regarded as a hypersurface (a 3D subspace) in spacetime, formed by all the events passed by the...
boundary has the topology of the form ∂Σ ~ S2, and if, furthermore, the hypersurfaces Σ are all spacelike, then the region Ω contains a quasi-permanent intrauniverse...
immersed, nowhere spacelike hypersurface S of (M, g) such that, for every point p∈S and every null vector k∈TpS, there exists a null geodesic γ {\displaystyle...
spherically symmetric and nonrotating star which is either emitting or absorbing null dusts. It is named after the Indian physicist Prahalad Chunnilal Vaidya and...
unit vector field with vanishing vorticity, or equivalently, which is hypersurface orthogonal. For example, this situation can arise in studying the world...
turns out that this is possible, in which case we say the congruence is hypersurface orthogonal, if and only if the vorticity vector vanishes identically...
surface approach has one plot a hypersurface rather than a curve and then measure the hypervolume under that hypersurface. Every possible decision rule...
of time (more precisely, everywhere on a spacelike three-dimensional hypersurface, called the Cauchy surface). Failure of the cosmic censorship hypothesis...
Periodic solutions of Hamilton's equations on tori Sergiu Klainerman, Nullhypersurfaces and curvature estimates in general relativity Bruce Kleiner, Singular...
Alternatively, it can be defined as the dimensions of a maximal positive and null subspace. By Sylvester's law of inertia these numbers do not depend on the...
an initial data set (M, g, k) if there is a (necessarily spacelike) hypersurface embedding of M into M, together with a continuous unit normal vector...
time between nearby hypersurfaces, βi is the shift vector that relates the spatial coordinate systems on different hypersurfaces, γij is a positive-definite...
resultant space of a map from points in an n-dimensional base space Rp,q to null vectors in Rp+1,q+1. This allows operations on the base space, including...
geometry. Because the Rindler observers are vorticity-free, they are also hypersurface orthogonal. The orthogonal spatial hyperslices are t = t 0 {\displaystyle...
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