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:
“There is a constant in the average American imagination and taste, for which the past must be preserved and celebrated in full-scale authentic copy; a philosophy of immortality as duplication. It dominates the relation with the self, with the past, not infrequently with the present, always with History and, even, with the European tradition.”
—Umberto Eco (b. 1932)
“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)
“It is not those who till the fields who eat fine rice, nor those who rear the silkworms who wear fine silks.”
—Chinese proverb.