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.
Many distinct vector fields can give rise to the same congruence of curves, since if
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.
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).
Then the components of our vector field are now scalar functions given in tensor notation by writing
, observe that the equation: means that the term in parentheses at left is the transverse part of
This orthogonality relation holds only when X is a timelike unit vector of a Lorentzian Manifold.
, we have: Because the vorticity tensor is antisymmetric, its diagonal components vanish, so it is automatically traceless (and we can replace it with a three-dimensional vector, although we shall not do this).
In the case of a timelike geodesic congruence, the last term vanishes identically.
), and vorticity tensor of a timelike geodesic congruence have the following intuitive meaning: See the citations and links below for justification of these claims.
By the Ricci identity (which is often used as the definition of the Riemann tensor), we can write: By plugging the kinematical decomposition into the left-hand side, we can establish relations between the curvature tensor and the kinematical behavior of timelike congruences (geodesic or not).
In particular, according to the Bel decomposition of the Riemann tensor, taken with respect to our timelike unit vector field, the electrogravitic tensor (or tidal tensor) is defined by: The Ricci identity now gives: Plugging in the kinematical decomposition we can eventually obtain: Here, overdots denote differentiation with respect to proper time, counted off along our timelike congruence (i.e. we take the covariant derivative with respect to the vector field X).
This can be regarded as a description of how one can determine the tidal tensor from observations of a single timelike congruence.
and also to set: Now from the Ricci identity for the tidal tensor we have: But: so we have: By plugging in the definition of
Then (observing that the projection tensor can be used to lower indices of purely spatial quantities), we have: or By elementary linear algebra, it is easily verified that if
The trace of the tidal tensor can also be written: It is sometimes called the Raychaudhuri scalar; needless to say, it vanishes identically in the case of a vacuum solution.