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:
“To criticize is to appreciate, to appropriate, to take intellectual possession, to establish in fine a relation with the criticized thing and to make it one’s own.”
—Henry James (1843–1916)
“There is undoubtedly something religious about it: everyone believes that they are special, that they are chosen, that they have a special relation with fate. Here is the test: you turn over card after card to see in which way that is true. If you can defy the odds, you may be saved. And when you are cleaned out, the last penny gone, you are enlightened at last, free perhaps, exhilarated like an ascetic by the falling away of the material world.”
—Andrei Codrescu (b. 1947)
“It matters little comparatively whether the fields fill the farmer’s barn. The true husbandman will cease from anxiety, as the squirrels manifest no concern whether the woods will bear chestnuts this year or not, and finish his labor with every day, relinquishing all claim to the produce of his fields, and sacrificing in his mind not only his first but his last fruits also.”
—Henry David Thoreau (1817–1862)