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).
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 \vec{X} f = f_{,a} \, X^a, where f is an arbitrary smooth function. The
acceleration vector is the
covariant derivative \nabla_{\vec{X}} \vec{X}; we can write its components in tensor notation as: :\dot{X}^a = {X^a}_{;b} X^b Next, using {X_b} {X^b} = -1 , observe that the equation: :\left( \dot{X}^a \, X_b + {X^a}_{;b} \right) \, X^b = {X^a}_{;b} \, X^b - \dot{X}^a = 0 means that the term in parentheses at left is the
transverse part of {X^a}_{;b}. 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: :h_{ab} = g_{ab} + X_a \, X_b for the
projection tensor which projects tensors into their transverse parts; for example, the transverse part of a vector is the part
orthogonal to \vec{X}. This tensor can be seen as the metric tensor of the hypersurface whose tangent vectors are orthogonal to X. Thus, we have shown that: :\dot{X}_a \, X_b + X_{a;b} = {h^m}_a \, {h^n}_b X_{m; n} Next, we decompose this into its symmetric and antisymmetric parts: :\dot{X}_a \, X_b + X_{a;b} = \theta_{ab} + \omega_{ab} Here: :\theta_{ab} = {h^m}_a \, {h^n}_b X_{(m;n)} :\omega_{ab} = {h^m}_a \, {h^n}_b X_{[m;n]} are known as the
expansion tensor and
vorticity tensor respectively. Because these tensors live in the spatial
hyperplane elements orthogonal to \vec{X}, 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 \theta, we have: :\theta_{ab} = \sigma_{ab} + \frac{1}{3} \, \theta \, h_{ab} 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). Therefore, we now have: :X_{a;b} = \sigma_{ab} + \omega_{ab} + \frac{1}{3} \, \theta \, h_{ab} - \dot{X}_a \, X_b This is the desired
kinematical decomposition. In the case of a timelike
geodesic congruence, the last term vanishes identically. The expansion scalar, shear tensor ( \sigma_{ab} ), and vorticity tensor of a timelike geodesic congruence have the following intuitive meaning: • 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, • The shear tensor represents any tendency of the initial sphere to become distorted into an ellipsoidal shape, • 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 (which is often used as the definition of the
Riemann tensor), we can write: :X_{a;bn} - X_{a;nb} = R_{ambn} \, X^m 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). These relations can be used in two ways, both very important: • We can (in principle)
experimentally determine the curvature tensor of a spacetime from detailed observations of the kinematical behavior of any timelike congruence (geodesic or not), • We can obtain
evolution equations for the pieces of the kinematical decomposition (
expansion scalar,
shear tensor, and
vorticity tensor) 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 (or
tidal tensor) is defined by: :E[\vec{X}]_{ab} = R_{ambn} \, X^m \, X^n The Ricci identity now gives: :\left( X_{a:bn}-X_{a:nb} \right) \, X^n = E[\vec{X}]_{ab} Plugging in the kinematical decomposition we can eventually obtain: : \begin{align} E[\vec{X}]_{ab} &= \frac{2}{3} \, \theta \, \sigma_{ab} - \sigma_{am} \, {\sigma^m}_b -\omega_{am} \, {\omega^m}_b \\ & - \frac{1}{3} \left( \dot{\theta} + \frac{\theta^2}{3} \right) \, h_{ab} - {h^m}_a \, {h^n}_b \, \left( \dot{\sigma}_{mn} - \dot{X}_{(m;n)} \right) - \dot{X}_a \, \dot{X}_b \\ \end{align} 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.
Evolution equations In this section, we turn to the problem of obtaining
evolution equations (also called
propagation equations or
propagation formulae). It will be convenient to write the acceleration vector as \dot{X}^a = W^a and also to set: :J_{ab} = X_{a:b} = \frac{\theta}{3} \, h_{ab} + \sigma_{ab} + \omega_{ab} - \dot{X}_a \, X_b Now from the Ricci identity for the tidal tensor we have: :\dot{J}_{ab} = J_{an;b} \, X^n - E[\vec{X}]_{ab} But: :\left( J_{an} \, X^n \right)_{;b} = J_{an;b} \, X^n + J_{an} \, {X^n}_{;b} = J_{an;b} \, X^n + J_{am} \, {J^m}_b so we have: :\dot{J}_{ab} = -J_{am} \, {J^m}_b - {E[\vec{X}]}_{ab} + W_{a;b} By plugging in the definition of J_{ab} 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 (observing that the
projection tensor can be used to lower indices of purely spatial quantities), we have: :J_{am} \, {J^m}_b = \frac{\theta^2}{9} \, h_{ab} + \frac{2 \theta}{3} \, \left( \sigma_{ab} + \omega_{ab} \right) + \left (\sigma_{am} \, {\sigma^m}_b + \omega_{am} \, {\omega^m}_b \right) + \left (\sigma_{am} \, {\omega^m}_b + \omega_{am} \, {\sigma^m}_b \right) or :\dot{J}_{ab} = -\frac{\theta^2}{9} \, h_{ab} - \frac{2 \theta}{3} \, \left( \sigma_{ab} + \omega_{ab} \right) -\left (\sigma_{am} \, {\sigma^m}_b + \omega_{am} \, {\omega^m}_b \right) -\left(\sigma_{am} \, {\omega^m}_b + \omega_{am} \, {\sigma^m}_b \right) - {E[\vec{X}]}_{ab} By elementary linear algebra, it is easily verified that if \Sigma, \Omega are respectively three dimensional symmetric and antisymmetric linear operators, then \Sigma^2 + \Omega^2 is symmetric while \Sigma \, \Omega + \Omega \, \Sigma 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 (for timelike geodesics): :\dot{\theta} = \omega^2 - \sigma^2 - \frac{\theta^2}{3} - {E[\vec{X}]^m}_m Taking the traceless symmetric part gives: :\dot{\sigma}_{ab} = -\frac{2\theta}{3} \, \sigma_{ab} -\left( \sigma_{am} \, {\sigma^m}_b + \omega_{am} \, {\omega^m}_b \right) - {E[\vec{X}]}_{ab} + \frac{\sigma^2 - \omega^2 + {E[\vec{X}]^m}_m}{3} \, h_{ab} and taking the antisymmetric part gives: :\dot{\omega}_{ab} = -\frac{2\theta}{3} \, \omega_{ab} -\left (\sigma_{am} \, {\omega^m}_b + \omega_{am} \, {\sigma^m}_b \right) Here: :\sigma^2 = \sigma_{mn} \, \sigma^{mn}, \; \omega^2 = \omega_{mn} \, \omega^{mn} are quadratic invariants which are never negative, so that \sigma, \omega are well-defined real invariants. The trace of the tidal tensor can also be written: :{E[\vec{X}]^a}_a = R_{mn} \, X^m \, X^n It is sometimes called the
Raychaudhuri scalar; needless to say, it vanishes identically in the case of a
vacuum solution.
See also •
congruence (manifolds) •
expansion scalar •
expansion tensor •
shear tensor •
vorticity tensor •
Raychaudhuri equation ==References==