Конгруэнции в общей теории относительности: геометрия и физическая интерпретация.

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.

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 it admits a tangent vector field with vanishing covariant derivative, .

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.

In general relativity, a timelike congruence in a four dimensional Lorentzian manifold can be interpreted as a family of world lines of certain ideal observers in our spacetime. In particular, a timelike geodesic congruence can be interpreted as a family of free falling test particles. Null congruences are also important, particularly null geodesic congruences, which can be interpreted as a family of freely propagating light rays. Warning: the world line of a pulse of light moving in a fiber optic cable would not in general be a null geodesic, and light in the very early universe (the radiation dominated epoch) was not freely propagating. The world line of a radar pulse sent from Earth past the Sun to Venus would however be modeled as a null geodesic arc. In dimensions other than four, the relationship between null geodesics and "light" no longer holds: If "light" is defined as the solution to the Laplacian wave equation, then the propagator has both null and time like components in odd space time dimensions and is no longer a pure Dirac delta function in even space time dimensions greater than four.

Describing the mutual motion of the test particles in a null geodesic congruence in a spacetime such as the Schwarzschild vacuum or FRW dust is a very important problem in general relativity. It is solved by defining certain kinematical quantities which completely describe how the integral curves in a congruence may converge (diverge) or twist about one another. It should be stressed that the kinematical decomposition we are about to describe is pure mathematics valid for any Lorentzian manifold. However, the physical interpretation in terms of test particles and tidal accelerations (for timelike geodesic congruences) or pencils of light rays (for null geodesic congruences) is valid only for general relativity (similar interpretations may be valid in closely related theories).