Congruence

Congruence (symbol: ≅) is the state achieved by coming together, the state of agreement. The Latin congruō meaning “I meet together, I agree”. As an abstract term, congruence means similarity between objects. Congruence, as opposed to approximation, is a relation which implies a species of equivalence.

Mathematics

Congruence (general relativity)

In general relativity, a congruence (more properly, a congruence of curves) is the set of integral curves of a (nowhere vanishing) vector field in a four-dimensional Lorentzian manifold which is interpreted physically as a model of spacetime. Often this manifold will be taken to be an exact or approximate solution to the Einstein field equation.

Types of congruences

Congruences generated by nowhere vanishing timelike, null, or spacelike vector fields are called timelike, null, or spacelike respectively.

A congruence is called a geodesic congruence if the tangent vector field has vanishing covariant derivative, .

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.

Congruence (manifolds)

In the theory of smooth manifolds, a congruence is the set of integral curves defined by a nonvanishing vector field defined on the manifold.

Congruences are an important concept in general relativity, and are also important in parts of Riemannian geometry.

A motivational example

The idea of a congruence is probably better explained by giving an example than by a definition. Consider the smooth manifold R². Vector fields can be specified as first order linear partial differential operators, such as

These correspond to a system of first order linear ordinary differential equations, in this case

where dot denotes a derivative with respect to some (dummy) parameter. The solutions of such systems are families of parameterized curves, in this case

This family is what is often called a congruence of curves, or just congruence for short.

This particular example happens to have two singularities, where the vector field vanishes. These are fixed points of the flow. (A flow is a one-dimensional group of diffeomorphisms; a flow defines an action by the one-dimensional Lie group R, having locally nice geometric properties.) These two singularities correspond to two points, rather than two curves. In this example, the other integral curves are all simple closed curves. Many flows are considerably more complicated than this. To avoid complications arising from the presence of singularities, usually one requires the vector field to be nonvanishing.