Congruence (general relativity)


In general relativity, a congruence is the set of integral curves of a 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 it admits a tangent vector field with 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.
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.

Physical interpretation

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 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.

Kinematical description

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 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 or pencils of light rays is valid only for general relativity.

The kinematical decomposition of a timelike congruence

Consider the timelike congruence generated by some timelike unit vector field X, which we should think of as a first order linear partial differential operator. Then the components of our vector field are now scalar functions given in tensor notation by writing, where f is an arbitrary smooth function.
The acceleration vector is the covariant derivative ; we can write its components in tensor notation as
Next, 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. It does not hold in more general setting. Write
for the projection tensor which projects tensors into their transverse parts; for example, the transverse part of a vector is the part orthogonal to. This tensor can be seen as the metric tensor of the hypersurface whose tangent vectors are orthogonal to X. Thus we have shown that
Next, we decompose this into its symmetric and antisymmetric parts,
Here,
are known as the expansion tensor and vorticity tensor respectively.
Because these tensors live in the spatial hyperplane elements orthogonal to, we may think of them as three-dimensional second rank tensors. This can be expressed more rigorously using the notion of Fermi Derivative. Therefore, we can decompose the expansion tensor into its traceless part plus a trace part. Writing the trace as, we have
Because the vorticity tensor is antisymmetric, its diagonal components vanish, so it is automatically traceless. Therefore, we now have
This is the desired kinematical decomposition. In the case of a timelike geodesic congruence, the last term vanishes identically.
The expansion scalar, shear tensor, and vorticity tensor of a timelike geodesic congruence have the following intuitive meaning:
  1. the expansion scalar represents the fractional rate at which the volume of a small initially spherical cloud of test particles changes with respect to proper time of the particle at the center of the cloud,
  2. the shear tensor represents any tendency of the initial sphere to become distorted into an ellipsoidal shape,
  3. the vorticity tensor represents any tendency of the initial sphere to rotate; the vorticity vanishes if and only if the world lines in the congruence are everywhere orthogonal to the spatial hypersurfaces in some foliation of the spacetime, in which case, for a suitable coordinate chart, each hyperslice can be considered as a surface of 'constant time'.
See the citations and links below for justification of these claims.

Curvature and timelike congruences

By the Ricci identity, 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. These relations can be used in two ways, both very important:
  1. we can experimentally determine the curvature tensor of a spacetime from detailed observations of the kinematical behavior of any timelike congruence,
  2. we can obtain evolution equations for the pieces of the kinematical decomposition which exhibit direct curvature coupling.
In the famous slogan of John Archibald Wheeler,

Spacetime tells matter how to move; matter tells spacetime how to curve.

We now see how to precisely quantify the first part of this assertion; the Einstein field equation quantifies the second part.
In particular, according to the Bel decomposition of the Riemann tensor, taken with respect to our timelike unit vector field, the electrogravitic 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. This can be regarded as a description of how one can determine the tidal tensor from observations of a single timelike congruence.

Evolution equations

In this section, we turn to the problem of obtaining evolution equations.
It will be convenient to write the acceleration vector as 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 and taking respectively the diagonal part, the traceless symmetric part, and the antisymmetric part of this equation, we obtain the desired evolution equations for the expansion scalar, the shear tensor, and the vorticity tensor.
Consider first the easier case when the acceleration vector vanishes. Then, we have
or
By elementary linear algebra, it is easily verified that if are respectively three dimensional symmetric and antisymmetric linear operators, then is symmetric while is antisymmetric, so by lowering an index, the corresponding combinations in parentheses above are symmetric and antisymmetric respectively. Therefore, taking the trace gives Raychaudhuri's equation :
Taking the traceless symmetric part gives
and taking the antisymmetric part gives
Here,
are quadratic invariants which are never negative, so that are well-defined real invariants. Te 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.