Kinematic decomposition: expansion + shear + vorticity

Difficulty level:   ★ ★ ★
worldlines showing expansion, shear, and vorticity
Worldlines which collectively exhibit expansion, shear, and vorticity.

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 a very important tool in relativity.

Write \mathbf u for the 4-velocity field, then lower its index and take the covariant derivative: \nabla\mathbf u^\flat (that’s a “flat” symbol not the letter b), which is u_{a;b} or \nabla_b u_a in coordinate notation. This (0,2)-tensor is the gradient of the motion. Now apply the spatial projection tensor h_{ab} := g_{ab}+u_a u_b to get the purely spatial part (\mathbf B say) of the velocity gradient, meaning the part orthogonal to \mathbf u:

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

Here \dot u_a is the (dual) 4-acceleration \nabla_{\mathbf u}\mathbf u^\flat, or u_{a;b}u^b in coordinate notation. The latter identity displayed above follows from substituting \mathbf u into the second slot of \nabla_{\mathbf u}\mathbf u^\flat: evaluate u_{c;d}u^d, which you should recognise. 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]}. (Note some use the opposite sign convention for \omega_{ab}.) The expansion tensor itself splits into “trace” and “trace-free” parts: \theta_{ab} = \frac{1}{3}\theta h_{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. \sigma_{ab} is the shear tensor. There are alternative formulae but this approach, which follows Ellis 1971 , seems most efficient for computer algebra. In summary, the kinematic decomposition is:

    \[u_{a;b} = \frac{1}{3}\theta h_{ab}+\sigma_{ab}+\omega_{ab}-\dot u_a u_b.\]

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 an arbitrary context 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 (component of) expansion to distinguish \frac{1}{3}\theta h_{ab} specifically. Finally, vorticity is microscopic rotation, known as curl in 3-dimensions. These can be distinct from macroscopic rotation, 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  for instance. There are elegant formulae using the exterior derivative, see e.g. Jantzen, Carini & Bini  2013 draft, §2.2.3. 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 ) 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 , 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 & coauthors 2011  §2.1 investigate other frames; see also Jantzen+ §2.4.8, 3.3.5.