Kinematic decomposition: expansion + shear + vorticity

Difficulty:   ★★★☆☆   undergraduate general relativity

worldlines showing expansion, shear, and vorticity
Worldlines which collectively exhibit expansion, shear, and vorticity over time.

Suppose you know the motion of some particles / fluid / observers over time, as in the diagram. At each point the gradient of the motion can be decomposed into: expansion, shear, and vorticity. This is known as the kinematic decomposition, and is an important tool in relativity.

Write \mathbf u for the 4-velocity field, then lower its index and take the total covariant derivative: \nabla\mathbf u^\flat (that’s a “flat” symbol not the letter ‘b’), which has components \nabla_a u_b or equivalently u_{b;a}. This is the gradient of the motion, expressed as a (0,2)-tensor. Now apply the spatial projection tensor P^a_{\hphantom ab} := g^a_{\hphantom ab}+u^a u_b to get the purely spatial part of the velocity gradient, meaning the part orthogonal to \mathbf u, and which we label \mathbf B:

    \[B_{ab} := P^c_{\hphantom c a} P^d_{\hphantom d b} u_{c;d} = \nabla_a u_b + u_a\dot u_b.\]

Here \dot u_b is the (dual) 4-acceleration \nabla_{\mathbf u}\mathbf u^\flat, which has components u^a\nabla_a u_b. The projectors remove the time-time, time-space, and space-time parts of the velocity gradient. However only the time-space part is nonvanishing, being -u_a\dot u_b. This follows from substituting the vector \mathbf u into the first slot of \nabla\mathbf u^\flat, which represents the direction of differentiation. Caution: many authors define this as the second slot, so the term would be \cdots\dot u_a u_b instead.

Now the symmetric part of \mathbf B is the expansion tensor \theta_{ab} = \frac{1}{2}(B_{ab}+B_{ba}) =: B_{(ab)}, and the antisymmetric part is the vorticity tensor \omega_{ab} = \frac{1}{2}(B_{ab}-B_{ba}) =: B_{[ab]}. These quantities will be explained shortly. The expansion tensor itself splits into “trace” and “trace-free” parts: \theta_{ab} = \frac{1}{3}\theta P_{ab} + \sigma_{ab}. Here \theta = g^{ab}\theta_{ab} = u^a_{\hphantom{a};a} is the expansion scalar; it is the trace of the expansion tensor, and the divergence of the 4-velocity field. It gives the rate of proportional expansion over time. \sigma_{ab} is the shear tensor. There are alternative formulae but this approach, which is largely inspired by Ellis 1971 , seems most efficient for computer algebra. In summary, the kinematic decomposition is:

    \[\nabla_a u_b = \frac{1}{3}\theta P_{ab} + \sigma_{ab} + \omega_{ab} - u_a\dot u_b.\]

So what is the physical meaning of the quantities? Expansion means the particles move apart over time, or contract in the case of negative expansion. More precisely it is the proportional expansion per unit time. (In this article all quantities are understood as measured in the fluid’s frame, in particular “time” means the proper time along the worldline(s).) The expansion scalar gives the proportional change in volume over time: \theta = V^{-1}\,dV/d\tau. A familiar example is the Lemaître-Hubble parameter H = \theta/3, but in general expansion is both position and direction-dependent. Shear (by itself) involves expansion in some directions but contraction in others. Again, this is a proportional change over time. Shear by itself does not change the volume. The eigenvectors of the shear tensor are the principal axes of shear, and since \sigma_{ab} is real and symmetric one can find an orthogonal basis of eigenvectors. Some potentially misleading language is that the expansion tensor also includes the shear; one can emphasise the isotropic (part of) expansion to distinguish \frac{1}{3}\theta P_{ab} specifically. Finally, vorticity is microscopic rotation, known as curl in 3-dimensions. At each point, it describes the rotation within an “infinitesimal” region around that point. This is distinct from macroscopic rotation, meaning an overall rotation of some extended body, as another website nicely visualises. Vorticity by itself is rigid, so does not change lengths or volume.

Define also the shear and vorticity scalars \sigma^2 = \frac{1}{2}\sigma_{ab}\sigma^{ab} and \omega^2 = \frac{1}{2}\omega_{ab}\omega^{ab}. These are positive-definite measures: \sigma \ge 0 with \sigma = 0 if and only if \sigma_{ab} = 0, and similarly for \omega. There is also a vorticity vector \omega^a which is the axis of local rotation. Much more could be said. There are formulae giving the rates of change of relative distance and direction from a given vantage point, see Ehlers 1961  for instance. There are elegant formulae using the exterior derivative, see Jantzen, Carini & Bini 1992  §2. Note we have only described the kinematics and said nothing of its causes, in particular the Einstein field equations are not assumed.

As for literature, the best textbook presentations I am aware of are Ellis, Maartens & MacCallum 2012  §4.6; and Poisson 2004  §2.3. Two classic papers are Ehlers 1961  §2.1 and Ellis 1971  §2. Translators of Ehlers (1993 , linked earlier) described it as an “outstanding review paper” and that “[d]espite its age, it remains one of the best reviews available in this area.” Ellis was republished in 2009 , also linked previously, along with an editorial note  which reviews applications, and states “[f]ew papers in relativistic cosmology have been as influential and as frequently cited”, despite being “primarily a synthesis… of earlier results”. Newtonian fluid dynamics has a similar decomposition of the velocity gradient \partial_i v_j, see perhaps Ellis or Poisson §2.2. Wainwright & Ellis, eds., 1997  is one source which gives further applications. I have assumed timelike worldlines, but Poisson §2.4, 2.6 treats the null case, for which only some of the kinematic quantities remain unambiguous. Everything I have said here assumes the fluid / particles’ frame of reference, but Larena+ 2011  §2.1 investigate other frames; see also the Jantzen+  paper, or §2.4.8, 3.3.5 of their book draft .

[Updated May 2021 to use the “del” convention for index ordering: \nabla_a u_b.]

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