Search results
Results from the WOW.Com Content Network
In vector calculus, a complex lamellar vector field is a vector field which is orthogonal to a family of surfaces. In the broader context of differential geometry, complex lamellar vector fields are more often called hypersurface-orthogonal vector fields. They can be characterized in a number of different ways, many of which involve the curl.
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 of dimension n − 1, which is embedded in an ambient space of dimension n, generally a Euclidean space, an affine space or a projective space. [1]
In a four-dimensional spacetime manifold, a hypersurface is a three-dimensional submanifold that can be either timelike, spacelike, or null. A particular hyper-surface can be selected either by imposing a constraint on the coordinates =, or by giving parametric equations,
In general relativity, the Raychaudhuri equation, or Landau–Raychaudhuri equation, [1] is a fundamental result describing the motion of nearby bits of matter.. The equation is important as a fundamental lemma for the Penrose–Hawking singularity theorems and for the study of exact solutions in general relativity, but has independent interest, since it offers a simple and general validation ...
Rindler chart, for = in equation , plotted on a Minkowski diagram. The dashed lines are the Rindler horizons The dashed lines are the Rindler horizons The worldline of a body in hyperbolic motion having constant proper acceleration α {\displaystyle \alpha } in the X {\displaystyle X} -direction as a function of proper time τ {\displaystyle ...
A hypersurface of X defined by the equation F(x) = c is called a characteristic hypersurface at x if σ P ( x , d F ( x ) ) = 0. {\displaystyle \sigma _{P}(x,dF(x))=0.} Invariantly, a characteristic hypersurface is a hypersurface whose conormal bundle is in the characteristic set of P .
Thus, while we found a spatial hypersurface, it is orthogonal to the world lines of only some our Langevin observers. Because the obstruction from the Frobenius theorem can be understood in terms of the failure of the vector fields p → 2 , p → 3 {\displaystyle {\vec {p}}_{2},\,{\vec {p}}_{3}} to form a Lie algebra , this obstruction is ...
In relativistic cosmology, Weyl's postulate stipulates that in the Friedmann model of the universe (a fluid cosmological model), the wordlines of fluid particles (modeling galaxies) should be hypersurface orthogonal. Meaning, they should form a 3-bundle of non-intersecting geodesics orthogonal to a series of spacelike hypersurfaces (hyperslices ...