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BRS: Timelike Congruences

  1. Sep 19, 2010 #1

    Chris Hillman

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    Science Advisor

    I. Overview

    Sorry, I hope to come back and write this later!

    [EDIT: I won't have time to write it before the 24 hour limit is up, so look somewhere below for a part I appearing out of order. Sorry!]
     
    Last edited: Sep 20, 2010
  2. jcsd
  3. Sep 19, 2010 #2

    Chris Hillman

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    Science Advisor

    BRS: Timelike Congruences. II. Decomposition

    Begin by recalling from linear algebra that if we have some vector in some euclidean inner product space [itex]\vec{w}[/itex], the orthoprojection onto the one-dimensional subspace spanned by this vector is given in the formalism of matrix algebra by
    [tex]
    P = \frac{1}{w^T \, w} \; w \, w^T
    [/tex]
    For example, if
    [tex]
    w = \left[ \begin{array}{c} 1 \\ -2 \\ 3 \end{array} \right]
    [/tex]
    we find
    [tex]
    P = \frac{1}{14} \;
    \left[ \begin{array}{ccc}
    1 & -2 & 3 \\
    -2 & 4 & -6 \\
    3 & -6 & 9
    \end{array} \right]
    [/tex]
    where you can see at a glance that the columns are scalar multiples of [itex]\vec{w}[/itex]. Therefore, P has rank one, meaning that its image is a one-dimensional vector space, as expected.

    What about projections onto the subspace orthogonal to the one-dimensional subspace spanned by [itex]\vec{w}[/itex]? Very simple: the projection matrix we need is, in the formalism of matrix algebra
    [tex]
    Q = I - P
    [/tex]
    where I is the identity matrix. So in our example
    [tex]
    Q = \frac{1}{14} \;
    \left[ \begin{array}{ccc}
    13 & 2 & -3 \\
    2 & 10 & 6 \\
    -3 & 6 & 5
    \end{array} \right]
    [/tex]
    where each column is orthogonal to [itex]\vec{w}[/itex], and where Q has rank two, meaning that it projects to a two-dimensional subspace of our three-dimensional euclidean inner product space, to wit: the orthogonal complement of the one-dimensional subspace spanned by [itex]\vec{w}[/itex].

    (Warning: in US mathematics, a notational shift mysteriously occurred in the middle of the 20th century, from the early 20th century German tradition of writing vectors as row matrices to writing them as column matrices, i.e. from matrices acting on vectors from the right to matrices acting on vectors from the left. This gives some indication of the formidable difficulties faced by historians struggling to relate nineteenth and early twentieth century math/physics papers to modern textbooks! To change to the older convention, take the transpose of all equations and use [itex](AB)^T = B^T \, A^T[/itex].)

    Note that if [itex]\vec{w}[/itex] is a unit vector, [itex]w^T \, w = 1[/itex], so we can ignore the normalizing scalar factor in the formula for the projection operator.

    In the formalism of tensor algebra, linear operators appear as second rank mixed tensors, and our projection operator (for the orthogonal subspace) takes the form
    [tex]
    {Q^a}_b = {\delta^a}_b - w^a \, w_b
    [/tex]
    where [itex]\vec{w}[/itex] is a unit vector.

    (Out of deference to the tastes of the intended readers, I plan to use coordinate bases in this thread, except when discussing specific examples, when I will compute using coordinate bases and only change to the frame field components--- an operation which can be carried out at each tangent space using only the techniques from elementary linear algebra for a change of linear basis!--- at the very end of each computation.)

    By the way, it is useful to know that orthoprojection operators enjoy some nice algebraic properties
    [tex]
    P^2 = P = P^T, \; \; Q^2 = Q = Q^T
    [/tex]

    We know how to project vectors and covectors [itex]\alpha[/itex], what about tensors such as [itex]\alpha \otimes \beta[/itex]? Well, we do the obvious thing:
    [tex]
    {Q_a}^m \, {Q_b}^n T_{mn}
    [/tex]
    projects a second rank tensor into the hyperplane element.

    In Lorentzian inner product spaces, analogous formulas hold, except that we need to use the Lorentzian transpose rather than the euclidean version. The Lorentzian version arises from the notion of "adjoint" which is appropriate for [itex]\operatorname{diag}(-1,1,1,1)[/itex] rather than the identity matrix.

    Putting together the pieces, given a timelike unit vector [itex]\vec{X}[/itex] in a Lorentzian manifold (signature -+++ according to the LLSC), we see that the tangent space to each event, the projection tensor for the spatial hyperslice orthogonal to [itex]\vec{X}[/itex] must take the form
    [tex]
    h_{ab} = g_{ab} + X_a \, X_b
    [/tex]
    where the change of sign is due to the fact that our normal vector is timelike. Then, at each event, [itex]h_{ab}[/itex] functions like the metric tensor restricted to the spatial hyperplane element. In terms of tensor algebra ("index gymnastics" style), that means we can use it to raise and lower indices in tensors which live in the hyperplane element, for each tangent space.

    (As we'll see (sort of), for a general [itex]\vec{X}[/itex] no family of orthogonal hyperslices will exist, but if one does, on each hyperslice [itex]h_{ab}[/itex] goes give its metric tensor as a Riemannian three-manifold.)

    Now consider the special case of a geodesic unit timelike vector field, i.e. one whose integral curves are proper time parameterized timelike geodesics, and for which the acceleration vector
    [tex]
    \nabla_{\vec{u}} \vec{u} = 0
    [/tex]
    Consider two "nearby" curves in the congruence. Suppose we have a vector [itex]\vec{\xi}[/itex] connecting two such curves at a pair of events; in fact, suppose it points from an event on one curve, the fiducial curve, to an event on the other curve, and is orthogonal to the tangent to the fidicual curve at the initial event. Let us evolve this vector forward "in time" along the fiducial curve by Lie dragging
    [tex]
    {\cal L}_{\vec{u}} \vec{\xi} = 0
    [/tex]
    Then [itex]\vec{\xi}[/itex], now defined along the fiducial curve, is called the deviation vector, and in terms of "index gymnastics" we have such expressions as
    [tex]
    \begin{array}{rcl}
    u^a \, u_a & = & -1 \\
    \dot{u}^a & = & {u^a}_{;b} \, u^b = 0 \\
    \xi_a \, u^b & = & 0 \\
    {u^a}_{;b} \, \xi^b & = & {\xi^a}_{;b} \, u^b
    \end{array}
    [/tex]
    (the second illustrates the "dot notation" and also expresses the componentwise form of the fact that the acceleration vanishes, the third says that [itex]\vec{\xi}[/itex] is orthogonal to our fiducial curve, and the fourth expresses formal properties of the Lie and covariant derivatives.

    Let us write [itex]{J\left[\vec{u}\right]}_{ab} = u_{a ;b}[/itex] and let us call [itex]{J^a}_b[/itex] the Jacobi operator. Then
    [tex]
    {\xi^a}_{;b} u^b = {J^a}_b \xi^b
    [/tex]
    or in operator-like notation
    [tex]
    \dot{\vec{\xi}} = {\cal J}\left[\vec{u}\right] \, \vec{\xi}
    [/tex]
    That is, the proper time derivative along our fiducial curve of our deviation vector is given by letting the Jacobi operator act on the deviation vector.

    You might recall from Jacobi's investigations of curvature that the second proper time derivative along our fiducial curve of the deviation vector is given by
    [tex]
    \ddot{\xi}^a = {{E\left[\vec{u}\right]}^a}_b \, \xi^a
    [/tex]
    where [itex]{E\left[\vec{u}\right]}_{ab}[/itex] is the tidal tensor for [itex]\vec{u}[/itex]. Or, in operator-like notation,
    [tex]
    \ddot{\vec{\xi}} = {\cal E}\left[\vec{u}\right] \, \vec{\xi}
    [/tex]
    The tidal tensor is also called the electroriemann tensor, or electric part of the Riemann tensor, because it is one piece in the three part Bel decomposition of the Riemann tensor; it is easily obtained from the Riemann tensor using its definition
    [tex]
    {E\left[\vec{X}\right]}_{ab} = R_{ambn} \, X^m \, X^n
    [/tex]
    where [itex]\vec{X}[/itex] can be any timelike vector field. (Since no derivation is involved, this formula actually makes sense eventwise.)

    This suggests that it will be rewarding to examine more closely the Jacobi tensor
    [tex]
    {J\left[\vec{X}\right]_{ab} = X_{a;b}
    [/tex]
    where now we allow [itex]\vec{X}[/itex] to be any timelike vector field.

    (Warning: "Jacobi tensor" is sometimes used for a different fourth rank tensor; I am just making up a name here for convenience. And note that so far this is all just motivation, so no need to worry too much about details.)

    Accordingly, let us now suppose [itex]\vec{X}[/itex] is an arbitrary timelike unit vector field, giving an arbitrary timelike congruence of (proper time parameterized) curves. Then
    [tex]
    {h_a}^m \, {h_b}^n J_{mn} = J_{ab} + \dot{X}_a \, X_b
    [/tex]
    gives a useful formula for the projection of [itex]J_{ab}[/itex] into the orthogonal hyperplane elements, in each tangent space. To see why it's true, note that
    [tex]
    \left( {X^a}_{;b} + \dot{X}^a \, X_b \right) \; X^b
    = {X^a}_{;b} \, X^b - \dot{X}^a = 0
    [/tex]
    Thus, [itex]{X^a}_{;b} + \dot{X}^a \, X_b[/itex] is orthogonal to [itex]\vec{X}[/itex], i.e. lives in the hyperplane element. This shows that one effect of nonzero acceleration is to "break" the Jacobi tensor out of the hyperplane element, but [itex]J_{ab} + \dot{X}_a \, X_b[/itex] always lives in the hyperplane element, so it is effectively a three-dimensional tensor. That means we can apply a very useful operation which is standard for any three-dimensional tensor: decompose it additively into three pieces each of which has coordinate-free geometric meaning:
    • scalar part (a scalar multiple of [itex]h_{ab}[/itex])
    • traceless symmetric part
    • anti-symmetric part
    Writing [itex]K_{ab} = J_{ab} + \dot{X}_a \, X_b[/itex], these parts are
    • [itex] \frac{K}{3} \, h_{ab}[/itex], where [itex]K = {K^m}_m[/itex] is the trace (an invariant of [itex]K_{ab}[/itex]),
    • [itex] K_{(ab)} - \frac{K}{3} \, h_{ab} [/itex]
    • [itex] K_{[ab]}[/itex]
    The sum of the first two parts is the symmetric part [itex]K_{(ab)}[/itex], and
    [tex]
    X_{a;b} + \dot{X} \, X_b
    = J_{ab} + \dot{X} \, X_b
    = K_{ab}
    = K_{(ab)}+K_{[ab]}
    [/tex]
    Because these quantities are all defined purely from our timelike unit vector field [itex]\vec{X}[/itex], we can give the quantities in the decomposition special names and emphasize that they are defined in terms of [itex]\vec{X}[/itex]:
    [tex]
    X_{a;b} + \dot{X} \, X_b
    = {\theta\left[\vec{X}\right]}_{ab}+
    {\omega\left[\vec{X}\right]}_{ab}
    [/tex]
    where the pieces are called the expansion tensor and the vorticity tensor of [itex]\vec{X}[/itex] respectively. As we saw, we can break down the expansion tensor into the scalar part, determined by the trace (the expansion scalar) and the traceless symmetric part, which we call the shear tensor.

    These names arise from the intuitive role they play in allied topics in hydrodynamics and the theory of linear elasticity. I won't try to explain those connections here, but later I'll try to explain the intuitive meaning of these quantities.

    What if we do things in the other order? What if, instead of projecting and then decomposing, we decompose [itex]J_{ab}[/itex] (a four dimensional creature) into its symmetric and antisymmetric parts and then project those? We get the same thing, so that we can write
    [tex]
    \begin{array}{rcl}
    {\theta\left[\vec{X}\right]}_{ab} & = & {h^m}_a \, {h^n}_b \, X_{(m;n)} \\
    {\omega\left[\vec{X}\right]}_{ab} & = & {h^m}_a \, {h^n}_b \, X_{[m;n]}
    \end{array}
    [/tex]
    The trace of the expansion tensor, i.e. the expansion scalar, is simply the covariant divergence of [itex]\vec{X}[/itex]:
    [tex]
    \theta \left[ \vec{X} \right] = {X^m}_{;m}
    [/tex]
    Using this, we can form the traceless symmetric part from the expansion tensor to define the shear tensor
    [tex]
    {\sigma\left[\vec{X}\right]}_{ab} =
    {\theta\left[\vec{X}\right]}_{ab}
    -\frac{\theta\left[\vec{X}\right]}{3} \, h_{ab}
    [/tex]

    The end result is that we can write, for a timelike unit vector field [itex]\vec{X}[/itex], i.e. the field of tangent vectors for a (proper time parameterized) congruence of timelike curves, not neccessarily geodesics:
    [tex]
    {h_a}^m \, {h_b}^n \, X_{m;n}
    = X_{a;b} + \dot{X}_a \, X_b
    = \frac{ \theta\left[\vec{X}\right]}{3} \, h_{ab}
    + {\sigma\left[\vec{X}\right]}_{ab}
    + {\omega\left[\vec{X}\right]}_{ab}
    [/tex]
    Here, on the RHS we have
    • the scalar part, which is defined using the expansion scalar of the congruence,
    • the traceless symmetric part or shear tensor of the congruence,
    • the symmetric part or expansion tensor of the congruence is the sum of the scalar and traceless symmetric parts,
      [tex]
      {\theta\left[\vec{X}\right]}_{ab} =
      \frac{\theta\left[\vec{X}\right]}{3} \, h_{ab}
      + {\sigma\left[\vec{X}\right]}_{ab}
      [/tex]
    • the antisymmetric part or vorticity tensor),
    Also,
    • the projection tensor to the spatial hyperplanes orthogonal (in each tangent space) to the vectors [itex]\vec{X}[/itex] is
      [tex]
      h_{ab} = g_{ab} + X_a \, X_b
      [/tex]
    • the components of the acceleration vector [itex]\nabla_{\vec{X}} \vec{X}[/itex] of the congruence are
      [tex]
      \dot{X}^a = X_{a;b} \, X^b
      [/tex]
    All the ingredients of this decomposition are defined in a coordinate-free manner, although we have expressed them using the coordinate basis obtained from an arbitrary local coordinate chart. I have yet to attempt to explain their intuitive meaning, but let me note here without proof some important facts which may help to adumbrate the intuition I hope to later develop:
    • Frobenius showed that the vorticity tensor of the congruence vanishes if and only if the congruence is hypersurface orthogonal, i.e. the Lorentzian manifold where our congruence lives admits a (unique) family of spatial hyperslices which are everywhere orthogonal to the (tangent vectors to the) curves in the congruence,
    • the expansion scalar of the congruence (the covariant divergence of its tangent vector field) arises in many other ways; for example, it gives the proper time logarithmic derivative of the (four-dimensional) volume form evolved along [itex]\vec{X}[/itex].

    In the special case where our unit timelike vector field, i.e. the field of tangent vectors of a congruence of proper time parameterized timelike curves, is geodesic, i.e. is the field of tangent vectors of a congruence of proper time parameterized timelike geodesic curves, all terms involving the acceleration vector drop out. In addition, if we are only interested in this case, we need not worry about projecting before or after decomposing the Jacobi tensor into its symmetric and antisymmetric parts, because in this case, at each event the Jacobi tensor already "lives" in the desired spatial hyperplane element in the tangent space at that event. In this case, we can interpret the curves in the congruence as the world lines of a family of inertial observers, and since no forces act on these observers, we may speak of the kinematic decomposition of the congruence.
     
    Last edited: Sep 19, 2010
  4. Sep 20, 2010 #3

    Chris Hillman

    User Avatar
    Science Advisor

    BRS: Timelike Congruences. III. Evolution of the Decomposition

    It might seem that timelike congruences can have arbitrarily varied expansion scalars, shear tensors, and vorticity tensors, but this is not true. It turns out that the question of which properties timelike congruences in a given Lorentzian manifold can possess is constrained by the properties of the Riemann tensor of our manifold. Even better, the mere fact that a timelike congruence with certain properties exists can provide information about the nature of the Riemann tensor. This is very useful in gtr.

    In this section, I propose to study a very natural question: how does the decomposition of a timelike congruence evolve wrt proper time as we move along the proper time parameterized curves in the congruence?

    Because the quantities in the decomposition of a timelike congruence are computed using first covariant derivatives, and because in general
    [tex]
    {\dot{Y}^{ab}}_{cde} = {Y^{ab}}_{cde;m} \, X^m
    [/tex]
    (where the dot notation hides the fact that we have in mind a particular timelike unit vector field [itex]\vec{X}[/itex], the field of tangent vectors to the curves in a particular timelike congruence), to answer this question we will need to investigate the second covariant derivative of [itex]\vec{X}[/itex] along itself. This is where curvature comes in: the Ricci identity says
    [tex]
    X_{a;nb} - X_{a;bn} = X^m \, R_{ambn}
    [/tex]
    or, rearranging terms,
    [tex]
    X_{a;nb} = X_{a;bn} + X^m \, R_{ambn}
    [/tex]
    Thus
    [tex]
    \begin{array}{rcl}
    \dot{J}_{ab} &= & X_{a;bn} \, X^n \\
    & = & X_{a;nb} \, X^n - R_{ambn} \, X^m \, X^n \\
    & = & X_{a;nb} \, X^n - {E\left[\vec{X}\right]}_{ab} \\
    & = & J_{an;b} \, X^n - {E\left[\vec{X}\right]}_{ab}
    \end{array}
    [/tex]
    But
    [tex]
    \begin{array}{rcl}
    \dot{X}_{a;b} & = & \left( J_{an} \, X^n \right)_{;b} \\
    & = & J_{an;b} \, X^n + J_{an} \, {X^n}_{;b} \\
    & = & J_{an;b} \, X^n + J_{an} \, {J^n}_{b}
    \end{array}
    [/tex]
    so
    [tex]
    \dot{J}_{ab} = \dot{X}_{a;b} - J_{an} \, {J^n}_b
    - {E\left[\vec{X}\right]}_{ab}
    [/tex]

    For simplicity, let's momentarily restrict to the case of a timelike geodesic congruence, so that we can write
    [tex]
    J_{ab} = \frac{\theta}{3} \, h_{ab} + \sigma_{ab} + \omega_{ab}
    [/tex]
    where I am dropping the brackets which reminded us that all quantities appearing in this decomposition are defined in terms of a particular congruence. Expanding [itex] J_{an} \, {J^n}_b[/itex] and collecting terms gives
    [tex]
    J_{an} \, {J^n}_b
    = \frac{\theta^2}{9} \, h_{ab}
    \, + \, \frac{2 \, \theta}{3}
    \; \left( \sigma_{ab} + \omega_{ab} \right)
    \, + \, \sigma_{am} \, {\sigma^m}_b
    \, + \, \omega_{am} \, {\omega^m}_b
    \, + \, \sigma_{am} \, {\omega^m}_b
    \, + \, \omega_{am} \, {\sigma^m}_b
    [/tex]
    The trace is
    [tex]
    \begin{array}{rcl}
    J_{am} \, J^{ma} & = & \frac{\theta^2}{3}
    \, + \, \sigma_{am} \, \sigma^{ma}
    \, + \, \omega_{am} \, \omega^{ma}
    \, + \, \sigma_{am} \, \omega^{ma}
    \, + \, \omega_{am} \, \sigma^{ma} \\
    & = & \frac{\theta^2}{3}
    \, + \, \sigma_{am} \, \sigma^{am}
    \, - \, \omega_{am} \, \omega^{am}
    \, - \, \sigma_{am} \, \omega^{am}
    \, + \, \omega_{am} \, \sigma^{am} \\
    & = & \frac{\theta^2}{3} + \sigma^2 - \omega^2
    \end{array}
    [/tex]
    where we have used the symmetry and tracelessness of the shear tensor and the antisymmetry of the vorticity tensor, and where
    [tex]
    \begin{array}{rcl}
    \sigma^2 & = & \sigma_{mn} \, \sigma^{mn} \\
    \omega^2 & = & \omega_{mn} \, \omega^{mn}
    \end{array}
    [/tex]
    are quadratic invariants of the shear and vorticity tensors of our congruence. (Warning: some authors introduce a factor of 1/2 into these defintions!) Notice that because these are invariants of tensors which live in a three dimensional euclidean inner product space, they are non-negative.

    Thus, we have shown that for timelike geodesic congruences,
    [tex]
    \dot{\theta} =
    - \, \frac{\theta^2}{3} \, - \, \sigma^2 \, + \, \omega^2
    \, - \, {E_m}^{m}
    [/tex]
    More generally, restoring the acceleration terms, you can verify that we obtain
    [tex]
    \dot{\theta} =
    - \, \frac{\theta^2}{3} \, - \, \sigma^2 \, + \, \omega^2
    \, - \, {E_m}^{m} + {\dot{X}^m}_{;m}
    [/tex]
    where the last term is the covariant divergence of the acceleration vector of our timelike unit vector field. This is known as the Raychaudhuri formula. Each term in this formula is a scalar invariant partially characterizing our congruence, which has a geometric meaning independent of which coordinate chart we use. This formula is true and makes sense independently of gtr, and even independently of physics, but it turns out that in the context gtr, each term in the Raychaudhuri formula also has a physical meaning.

    When studying the Raychaudhuri formula, it is useful to notice that
    [tex]
    \begin{array}{rcl}
    {E^a}_a & = & {R^a}_{man} X^m X^n \\
    & = & R_{mn} \, X^m \, X^n \\
    & = & \left( G_{mn} - \frac{G}{2} \, g_{mn} \right) \, X^m X^n \\
    & = & 8 \pi \, \left( T_{mn} - \frac{T}{2} \, g_{mn} \right) \, X^m X^n
    \end{array}
    [/tex]
    Thus, the trace of the tidal tensor is independent of the Weyl curvature tensor! (Note well: the tidal tensor is the electroriemann tensor, not the electroweyl tensor!)

    Now we can easily obtain expressions for the trace of tidal tensor (electroriemann tensor) corresponding to various types of matter tensors. In particular:
    • in a vacuum region, the trace of the electroriemann tensor vanishes,
    • in an electrovacuum region, taking a congruence appropriately aligned with the field (usually there will be many which work),
      [tex]
      {E^m}_m = 8 \pi \, \epsilon
      [/tex]
      where [itex]\epsilon[/itex] is the energy density of the EM field.
    • in a perfect fluid region,
      [tex]
      {E^m}_m = 4 \pi \, \left( \rho + 3 p \right)
      [/tex]
      where [itex]\rho[/itex] is the mass-energy density and p is the pressure.
    Indeed, if we know the form of the matter tensor, we can usually say more. For example, consider the congruence of world lines of the fluid elements in a perfect fluid. The matter tensor is
    [tex]
    T^{ab} = ( \rho + p ) \, X^a \, X^b + p \, g^{ab}
    [/tex]
    and the identity [itex]{T^{ab}}_{;b} = 0[/itex] gives some useful relations:
    [tex]
    \begin{array}{rcl}
    \theta & = & {X^m}_{;m} = \frac{-\dot{\rho}}{\rho + p} \\
    \dot{X}^a & = & \frac{-h^{am} \, p_{;m}}{\rho + p}
    \end{array}
    [/tex]
    The first equation says that the expansion scalar is positive at some event iff the density is decreasing, as measured by an observer comoving with the fluid there, and vanishes iff the density is momentarily constant there. The second equation says that the acceleration vector is the projection to the orthogonal hyperplane element of the negative of the gradient of the pressure, multiplied by a non-negative scalar.

    Summing up: the evolution of the expansion scalar of a timelike congruence is given by
    • in a vacuum region,
      [tex]
      \dot{\theta} =
      {\dot{X}^m}_{;m}
      \, + \, \omega^2
      \, - \, \frac{\theta^2}{3} \, - \, \sigma^2
      [/tex]
    • in an electrovacuum region, for some congruence which is suitably aligned with the field,
      [tex]
      \dot{\theta} =
      {\dot{X}^m}_{;m}
      \, + \, \omega^2
      \, - \, \frac{\theta^2}{3} \, - \, \sigma^2
      \, - \, 8 \pi \, \epsilon
      [/tex]
    • in a region filled with perfect fluid (and no other matter or nongravitational fields), for the congruence of world lines of the fluid elements
      [tex]
      \dot{\theta} =
      {\dot{X}^m}_{;m}
      \, + \, \omega^2
      \, - \, \frac{\theta^2}{3} \, - \, \sigma^2
      \, - \, 4 \pi \, \left( \rho + 3 p \right)
      [/tex]

    Let us consider what the last formula says about the timelike congruence defined by the world lines of the matter in a fluid ball. We see that nonzero vorticity tensor--- which we will see is associated with "swirling"--- opposes gravitational collapse (via "centrifugal force"), while nonzero expansion scalar--- of either sign--- and nonzero shear tensor both tend to promote gravitational collapse. Furthermore, both density and pressure tend to promote gravitational collapse, unless this is halted by hydrostatic forces.

    In a static spherically symmetric perfect fluid ball at equilibrium, Raychaudhuri's formula reduces to
    [tex]
    {\dot{X}^m}_{;m}
    = 4 \pi \, \left( \rho + 3 p \right)
    [/tex]
    In a static spherically symmetric elastic body at equilibrium, we might have something like
    [tex]
    {\dot{X}^m}_{;m}
    = 4 \pi \, \left( \rho + p_{\hbox{rad}} + 2 p_{\perp} \right)
    [/tex]
    In a stationary rotating perfect fluid ball, both the vorticity and shear tensors are generally nonzero (nonzero shear arises from angular velocities varying with position), and Raychaudhuri's formula becomes
    [tex]
    {\dot{X}^m}_{;m}
    = 4 \pi \, \left( \rho + 3 p \right)
    \, + \, \sigma^2 \, - \, \omega^2
    [/tex]

    In the special case of the congruence of world lines of the dust particles in a dust solution (a perfect fluid solution with vanishing pressure), the acceleration vanishes (no forces act on the dust particles), and we may write
    [tex]
    \dot{\theta} =
    \omega^2
    \, - \, \frac{\theta^2}{3} \, - \, \sigma^2
    \, - \, 4 \pi \, \rho
    [/tex]
    This shows clearly that only nonzero vorticity can prevent a dust cloud from collapsing. In the case of swirling dust clouds, we'll see that nonzero shear is associated with "nonrigid swirling", and this promotes collapse whenever it occurs. Furthermore, for a dust solution, the expansion scalar of the congruence of world lines of the dust particles is the logarithmic derivative of the density wrt proper time
    [tex]
    \theta = -\dot{\rho}/\rho
    [/tex]

    (Warning: the assumption of zero pressure is only good until it fails; exact dust solution models of collapse tend to develop caustics in the congruence of world lines of the dust particles, i.e. curves or sheets where the density diverges to plus infinity and expansion scalar diverges to minus infinity, and physically we should expect nonzero pressure to occur at these places! At present there is quite a bit of interesting work on finding exact solutions of the Einstein-Vlasov equation which models dust "probabilistically"; this turns out to neatly evade many of the problems with dust solutions.)

    Returning to our expression for [itex]\dot{J}_{ab}[/itex], we can also take the symmetric and antisymmetric parts, which gives us formulas for the proper time evolution of the expansion tensor and vorticity tensor, and we then obtain an expression for the proper time evolution of the shear tensor. Then it turns out that the time evolution of the vorticity tensor does not depend upon the curvature tensor, but the time evolution of the shear tensor depends upon the tidal tensor.

    To find the evolution of the shear and vorticity tensors, it helps to rewrite everything in terms of linear operators. In
    [tex]
    \dot{J^a}_b = {\dot{X}^a}_b - {J^a}_m \, {J^m}_b - {E^a}_b
    [/tex]
    for simplicity let us first consider the case where [itex]\vec{X}[/itex] is a timelike geodesic congruence, so that we may write
    [tex]
    {J^a}_b = \frac{\theta}{3} \, {h^a}_b + {\sigma^a}_b
    + {\omega^a}_b
    [/tex]
    Multiplying out and collecting terms as before we obtain
    [tex]
    {\dot{J}^a}_{ \; b}
    = - \, \frac{\theta^2}{9} \, {h^a}_b
    \, - \, \frac{2 \, \theta}{3}
    \; \left( {\sigma^a}_b + {\omega^a}_b \right)
    \, - \, {\sigma^a}_m \, {\sigma^m}_b
    \, - \, {\omega^a}_m \, {\omega^m}_b
    \, - \, {\sigma^a}_m \, {\omega^m}_b
    \, - \, {\omega^a}_m \, {\sigma^m}_b
    \, - \, {E^a}_b
    [/tex]
    In terms of matrix algebra, we may write
    [tex]
    \Sigma = {\sigma^a}_b, \; \;
    \Omega = {\omega^a}_b, \; \;
    {\cal E} = {E^a}_b, \; \;
    {\cal J} = {J^a}_b = {\dot{X}^a}_{\; ;b}
    [/tex]
    which are respectively symmetric traceless, antisymmetric, symmetric (but not traceless, in general), and "none of the above". Then
    [tex]
    {\cal \dot{J}} = - \, \frac{\theta^2}{9} \, I
    \, - \, \frac{2\theta}{3} \; \left( \Sigma + \Omega \right)
    \, - \, \left( \Sigma^2 + \Omega^2 \right)
    \, - \, \left( \Sigma \, \Omega + \Omega \, \Sigma \right)
    \, - \, {\cal E}
    [/tex]
    Now you can check that for any 3x3 symmetric and antisymmetric matrices A,B, it is true that [itex]A^2+B^2[/itex] is symmetric while [itex]AB + BA[/itex] is antisymmetric. Therefore the symmetric part of our matrix expression is
    [tex]
    - \, \frac{\theta^2}{9} \, I
    \, - \, \frac{2\theta}{3} \, \Sigma
    \, - \, \left( \Sigma^2 + \Omega^2 + {\cal E} \right)
    [/tex]
    while the antisymmetric part is
    [tex]
    - \, \frac{2\theta}{3} \; \Omega
    \, - \, \left( \Sigma \, \Omega + \Omega \, \Sigma \right)
    [/tex]
    From the first we see again that the scalar part is
    [tex]
    - \, \left( \frac{\theta^2}{9} + \operatorname{tr}
    \left( \Sigma^2 + \Omega^2 + {\cal E} \right) \right) \; I
    [/tex]
    And the traceless symmetric part is
    [tex]
    - \, \frac{2\theta}{3} \, \Sigma
    \, - \, \left( \Sigma^2 + \Omega^2 + {\cal E} \right)
    \, + \, \frac{ \operatorname{tr}
    \left( \Sigma^2 + \Omega^2 + {\cal E} \right)}{3} \, I
    [/tex]
    Thus, for timelike geodesic congruences,
    [tex]
    \begin{array}{rcl}
    {\dot{\sigma}^a}_{\; b} & = &
    - \, \frac{2 \theta^2}{3} \; {\sigma^a}_b
    \, - \, {\sigma^a}_m \, {\sigma^m}_b
    \, - \, {\omega^a}_m \, {\omega^m}_b
    \, - \, {E^a}_b
    \, + \, \frac{\sigma^2-\omega^2 + {E^m}_m}{3} \; {h^a}_b
    \\
    {\dot{\omega}^a}_{\; b} & = &
    - \, \frac{2 \theta^2}{3} \; {\omega^a}_b
    \, - \, {\sigma^a}_m \, {\omega^m}_b
    \, - \, {\omega^a}_m \, {\sigma^m}_b
    \end{array}
    [/tex]
    Gathering results, the evolution equations for the kinematic decomposition are:
    [tex]
    \begin{array}{rcl}
    \dot{\theta} & = &
    - \, \frac{\theta^2}{3}
    \, - \, \sigma^2
    \, + \, \omega^2
    \, - \, {E_m}^{m} \\
    \dot{\sigma}_{ab} & = &
    - \, \frac{2 \theta^2}{3} \; \sigma_{ab}
    \, - \, \sigma_{am} \, {\sigma^m}_b
    \, - \, \omega_{am} \, {\omega^m}_b
    \, - \, E_{ab}
    \, + \, \frac{\sigma^2-\omega^2 + {E^m}_m}{3} \; h_{ab}
    \\
    \dot{\omega}_{ab} & = &
    - \, \frac{2 \theta^2}{3} \; \omega_{ab}
    \, - \, \sigma_{am} \, {\omega^m}_b
    \, - \, \omega_{am} \, {\sigma^m}_b
    \end{array}
    [/tex]
    The last equation implies that if the vorticity tensor vanishes at any event on a timelike geodesic curve, it vanishes all along that geodesic. Recalling the theorem of Frobenius, this means that if some "tubular bundle" of curves belonging to a timelike geodesic congruence has even one orthogonal hyperslice, it continues to have orthogonal hyperslices along the curves in the bundle. We'll see later that this means that a small body which is initially nonrotating (in the sense of zero vorticity) cannot later start to rotate (in the sense of nonzero vorticity) unless nongravitational forces act.

    In a dust region, using [itex]{E^m}_m = 4 \pi \, \rho[/itex] we can write the evolution equations for the kinematic decomposition of the congruence of world lines of the dust particles as:
    [tex]
    \begin{array}{rcl}
    \dot{\theta} & = &
    - \, \frac{\theta^2}{3}
    \, - \, \sigma^2
    \, + \, \omega^2
    \, - \, 4 \pi \, \rho \\
    \dot{\sigma}_{ab} & = &
    - \, \frac{2 \theta^2}{3} \; \sigma_{ab}
    \, - \, \sigma_{am} \, {\sigma^m}_b
    \, - \, \omega_{am} \, {\omega^m}_b
    \, - \, E_{ab}
    \, + \, \frac{\sigma^2-\omega^2 + 4 \pi \, \rho}{3} \; h_{ab}
    \\
    \dot{\omega}_{ab} & = &
    - \, \frac{2 \theta^2}{3} \; \omega_{ab}
    \, - \, \sigma_{am} \, {\omega^m}_b
    \, - \, \omega_{am} \, {\sigma^m}_b
    \end{array}
    [/tex]
    Suppose the dust cloud is initially stationary, so that the terms appearing on the right hand sides of all three equations sum to zero. In particular, the dust must have just the right vorticity to balance the effects of nonzero shear on the expansion scalar. Bye and bye, along comes a gravitational plane wave, which adds a traceless perturbation to the tidal tensor. This causes [itex]\dot{\sigma}_{ab}[/itex] to become nonzero, which causes [itex]\sigma^2[/itex] to become positive, which causes the proper time derivative of the expansion scalar to become negative, which initiates a collapse scenario. As [itex]\theta = -\dot{\rho}/\rho[/itex] decreases from whatever value it had initially (constant along the initial segment of each world line), eventually it becomes negative. Since [itex]\theta = -\dot{\rho}/\rho[/itex] implies
    [tex]
    \rho = \rho_0 \, \exp \left( -\int \theta ds \right)
    [/tex]
    where the integral is taken wrt proper time along a world line, this implies that the density starts to increase superexponentially, which in turn cases superexponential decrease in the proper time derivative of the expansion scalar. Also, [itex]\theta^2[/itex] increases, which only makes things worse: the density increases faster and faster as the expansion scalar continues to decrease faster and faster. Only if the [itex]\sigma_{am} \, {\omega^m}_b + \omega_{am} \, {\sigma^m}_b[/itex] term quickly creates just enough vorticity to rebalance the equation for proper time derivative of the expansion scalar can the cloud avoid gravitational collapse. This suggests why cosmologists think gravitational waves may have played a part in structure formation in the early universe.

    Ellis showed that a shear-free dust must have either vanishing expansion scalar or vanishing vorticity. In the first case, we have constant vorticity tensor ("rigidly rotating dust"); in the second case, we have isotropic expansion tensor (e.g. FRW dust or Milne congruence in Minkowski vacuum). Thus, shear-free dusts are rare.

    Returning to the general case, when we include the acceleration terms, we find:
    [tex]
    \dot{J}_{ab} =
    \dot{X}_{a;b}
    \, + \, \frac{2 \theta}{3} \, \dot{X}_a \, X_b
    \, + \, \hbox{old} \; \hbox{stuff}
    [/tex]
    We can compute the scalar part, traceless symmetric part, and antisymmetric part of the new terms independently of the others (since this is just linear algebra). Carrying out this procedure, we find that the evolution equations for the decomposition of a general timelike congruence are:
    [tex]
    \begin{array}{rcl}
    \dot{\theta} & = &
    - \, \frac{\theta^2}{3}
    \, - \, \sigma^2
    \, + \, \omega^2
    \, - \, {E^m}_{m}
    \, + \, {\dot{X}^m}_{ \; ;m} \\
    \dot{\sigma}_{ab} & = &
    - \, \frac{2 \theta^2}{3} \; \sigma_{ab}
    \, - \, \sigma_{am} \, {\sigma^m}_b
    \, - \, \omega_{am} \, {\omega^m}_b
    \, - \, E_{ab}
    \, + \, \frac{\sigma^2-\omega^2 + {E^m}_m - {\dot{X}^m}_{;m}}{3} \; h_{ab}
    \, + \,
    \frac{\theta}{3} \;
    \left( \dot{X}_a \, X_b + \dot{X}_b \, X_a \right)
    \, + \, \dot{X}_{(a;b)}
    \\
    \dot{\omega}_{ab} & = &
    - \, \frac{2 \theta^2}{3} \; \omega_{ab}
    \, - \, \sigma_{am} \, {\omega^m}_b
    \, - \, \omega_{am} \, {\sigma^m}_b
    \, + \, \frac{\theta}{3} \;
    \left( \dot{X}_a \, X_b - \dot{X}_b \, X_a \right)
    \, + \, \dot{X}_{[a ;b]}
    \end{array}
    [/tex]
    In these computations, quite a few terms drop out because for a timelike unit vector field [itex]\vec{X}[/itex], the acceleration vector field is everywhere orthogonal to tangent vector field:
    [tex]
    \dot{X}^m \, X_m = 0
    [/tex]

    You might think that only a mother could love the evolution equations I just wrote down. However, in the case of a perfect fluid, we can rewrite many of the terms in the evolution equations for the congruence of world lines of the fluid elements, in order to obtain physical insight into how the fluid evolves.

    A generalization of Ellis's theorem would state that a shearfree isentropic perfect fluid must have either vanishing vorticity (e.g., FRW fluid) or vanishing expansion scalar ("rigidly rotating" fluid). (An isentropic fluid has an equation of state giving pressure as a function of density.) AFAIK, deciding whether this is true in general is an open problem. AFAIK, all known perfect fluid solutions are consistent with this conjecture.
     
    Last edited: Sep 20, 2010
  5. Sep 23, 2010 #4

    Chris Hillman

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    BRS: Timelike Congruences. III. Evolution of the Decomposition. Correction!

    Oh noooo! :mad: Nothing worse than writing a very long post and having two hard to spot (but fatal) typos appear in the formulae which whose derivation was the whole point of the post!

    But alas, two typos crept into my latex markup and I carried them along without noticing the problems until the 24 hour limit had expired, so I can't simply go back and correct the typo. They were
    • [itex]\theta^2[/itex] should be [itex]\theta[/itex] in one place in several equations
    • missing part of the trace in the traceless scalar part
    Best I can do now is to just repeat some stuff with the typo corrected:

    In the expressions for the evolution equation including the acceleration terms, I also reordered terms to bring out the structure a bit more clearly. Some of you might like to rewrite the final result as
    [tex]
    \fbox{
    \begin{array}{rcl}
    \dot{\theta} & = &
    {\dot{X}^m}_{ \; ;m}
    - \, \frac{\theta^2}{3}
    \, - \, \sigma^2
    \, + \, \omega^2
    \, - \, {E^m}_{m} \\
    \dot{\sigma}_{ab} & = & \dot{X}_{(a;b)}
    \, + \, \frac{2 {\bf \theta}}{3} \;
    \left( \dot{X}_{(a} \, X_{b)} \, - \, \sigma_{ab} \right)
    \, - \, \sigma_{am} \, {\sigma^m}_b
    \, - \, \omega_{am} \, {\omega^m}_b
    \, - \, \frac{\dot{\theta}}{3} \; h_{ab}
    \, - \, E_{ab} \\
    \dot{\omega}_{ab} & = & \dot{X}_{[a ;b]}
    \, + \, \frac{2 {\bf \theta}}{3} \;
    \left( \dot{X}_{[a} \, X_{b]} \, - \, \omega_{ab} \right)
    \, - \, \sigma_{am} \, {\omega^m}_b
    \, - \, \omega_{am} \, {\sigma^m}_b
    \end{array}
    }
    [/tex]
    It is a good exercise to study what role each term in these formulae plays in simple but nontrivial examples and in a future part I plan to do just that.

    If we use a frame field instead of a coordinate basis, physically it makes sense to prefer a nonspinning frame, and wrt such a frame, several terms in our final result drop out.
     
    Last edited: Sep 23, 2010
  6. Sep 25, 2010 #5

    Chris Hillman

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    BRS: Timelike Congruences. Correction, ad nauseum

    Obviously, I am pushing the boundaries of how much LaTeX one can try to display in PF, and I confess I don't find the PF edit windowpane large enough for easy debugging of LaTex markup... sigh... But I have no good excuse for this, it seems:

    I realized last night that there is another error :mad: in my Post #3. I wrote
    but in fact there are two more terms which I unaccountably dropped. That means that in the boxed equations in my Post #4, the second and third formulas are missing some additional terms which you need in the case of nonzero acceleration. The first formula is correct as written.

    Sorry.

    I seem to be under the weather and interest in the topic may have waned, so I hesitate to try to be specific. If anyone wants me to continue with this thread, it might be best if I start over from the beginning, trying harder to prevent any errors from creeping in.
     
    Last edited: Sep 25, 2010
  7. Sep 26, 2010 #6

    Chronos

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    Gold Member

    I dont get the last expression. The -wab term on the right side is confusing.
     
  8. Sep 26, 2010 #7

    Chris Hillman

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    Let me just clear this infection

    Hi, Chronos,

    I'll take that as encouragement to make another attempt to get it right :wink:

    One quick pointer: in part II, I actually derived the Raychaudhuri formula (giving the evolution of the expansion scalar) twice, the second time as part of the derivation (marred by goofs) of the evolution of all three quantitites (expansion scalar, shear tensor, vorticity tensor). The two derivations are the same, just written up slightly differently. The general form of the Raychaudhuri formula is correct and agrees with e.g. Hawking & Ellis and with Poisson. No books I have seen attempt to derive the non-geodesic evolution formulas for the expansion and shear tensors--- as we saw, this requires some care (although not really hard at all, if you are careful).

    Let me try to eliminate this darned stuxnet worm from my system :wink: and get back to you, maybe later today. I'll try to fit the entire derivation into one post.

    The evolution of vorticity tensor is actually the easy bit; the tricky one is the shear tensor.
     
  9. Oct 1, 2010 #8

    George Jones

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    Re: Let me just clear this infection

    What about section 15.3 of Plebanski and Krasinski?
     
  10. Oct 1, 2010 #9

    Chris Hillman

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    Don't have it at hand but it's a great book. I'll try to find a copy and take a look at that section.
     
  11. Dec 24, 2010 #10

    Chris Hillman

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    BRS: Timelike Congruences: Abreu and Visser 1012.4806

    Under the tendicity of conflicting impulses, and in the hope that the BRS may develop into something useful and used after my departure, I add this footnote to the preceding thread:

    A commendable new eprint by Abreu and Visser 1012.4806 reviews previously known forms of the Raychaudhuri equation, and introduces some new ones.

    Visser seems to consistently have a hand in producing many of the best--- and best written--- papers in classical gtr in the past decade, so I see no need to add anything beyond a few remarks:
    • Recall that the electroriemann and magnetoriemann tensors agree with the electroweyl and magnetoweyl tensors (all defined wrt some timelike congruence) in a vacuum, when the Riemann and Weyl tensors themselves agree. For many years I called the former pair the electrogravitic and magnetogravitic tensors, a terminology which I have abandoned owing to it having resulted in seeingly endless confusion. In an electrovacuum, the tidal accelerations on neutral particles are described by the electroriemann tensor, but the research literature persists in calling the electroweyl tensor the tidal tensor, which I consider confusing. Fans of the membrane paradigm should see also 1012.4869 (from whence comes the term "tendicity"--- I hope I may claim the honor of being the first to use this term informally, to denote being pulled in several directions at once :smile: ) for a visualization of the electroweyl and magnetoweyl tensors, in the vacuum case--- I hope that in future this scheme will become a standard method for conveying the physical content of numerical simulations!
    • the authors place the term [itex]R_{ab} \, u^a \, u^b[/itex], which is the trace of the electroriemann tensor, on the left hand side of their Raychaudhuri identities, and place [itex]d\theta/ds[/itex] inside a divergence term, which leads immediately to a nice proof of their equation (28)--- compare the somewhat muddled account given in MTW,
    • the authors introduce in particular an expression for [itex]R_{ab} \, u^a \, v^b[/itex], where [itex]\vec{u}, \, \vec{v}[/itex] are the timelike unit vector fields defining two distinct congruences of integral curves; recall that introducing a pair of congruences is often a natural way to treat any problem involving a frequency shift, even when we are only interested in a particular pair of (disjoint) world lines selected from these two congruences,
    • the new expression reminds me of Green's identities; conventional wisdom says that in curved manifolds, no simple relation obtains between "integral identities" and "differential identities", but I suggest that some magical combination of terms built from the Ricci tensor might prove conventional wisdom wrong--- if so, it would be particularly interesting to try to apply such ideas to two-spheres in the membrane paradigm, and in Bondi radiation formalism,
    • similarly for the new expressions using NP formalism, especially in the related case of a pair of timelike and null geodesic congruences, and of a pair of null and null geodesic congruences (keep an eye peeled for computational formulae useful in computing Penrose limits),
    • the authors didn't examine the important case of two commuting spacelike Killing vectors, which I believe should yield some connection with the work of E. T. Newman on optical distortions by gravitational plane waves,
    • study of another obvious generalization, to four congruences, just might turn up something regarding the old problem of measuring curvature by local measurements using geometric optics approximation and ideal clocks, despite a famous old result of Synge.
    I myself am unable to follow up on these suggestions, but some of you may care to try your hand.

    Anyone intending to write research papers in classical relativity (or any field, really) could profitably study 1012.4806 for stylistic tips. To paraphrase some old advice of master mathematical expositor Paul Halmos which the authors have followed (possibly unknowingly):
    • before you begin to write, make sure you have something to say---neither too little nor too much,
    • begin by saying what you are going to say,
    • say it,
    • summarize very briefly and stop.
    The paper fits in 8 pages, which I think is a really nice length. There is an unfortunate trend towards entirely unneccessary verbosity in both math and physics papers which I think is detrimental to efficient communication and to the enjoyment of the reader. In many cases, quite frankly, authors are quite openly "padding" a weak paper by repeating every idea at least three times, which seems particularly reprehensible.

    I could say more about the virtues of the Abreu and Visser eprint, but sensing that I am not at my best, I'll follow Halmos's advice and stop here.
     
    Last edited: Dec 24, 2010
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