Relation With Vector Fields
The integral curves of the vector field are a family of non-intersecting parameterized curves which fill up the spacetime. The congruence consists of the curves themselves, without reference to a particular parameterization. Many distinct vector fields can give rise to the same congruence of curves, since if is a nowhere vanishing scalar function, then and give rise to the same congruence.
However, in a Lorentzian manifold, we have a metric tensor, which picks out a preferred vector field among the vector fields which are everywhere parallel to a given timelike or spacelike vector field, namely the field of tangent vectors to the curves. These are respectively timelike or spacelike unit vector fields.
Read more about this topic: Congruence (general Relativity)
Famous quotes containing the words relation with, relation and/or fields:
“[Man’s] life consists in a relation with all things: stone, earth, trees, flowers, water, insects, fishes, birds, creatures, sun, rainbow, children, women, other men. But his greatest and final relation is with the sun.”
—D.H. (David Herbert)
“It would be disingenuous, however, not to point out that some things are considered as morally certain, that is, as having sufficient certainty for application to ordinary life, even though they may be uncertain in relation to the absolute power of God.”
—René Descartes (1596–1650)
“I don’t know why I ever come in here. The flies get the best of everything.”
—Otis Criblecoblis, U.S. screenwriter. W.C. Fields (W.C. Fields)