There is a discussion of expansion, shear, and vorticity in Wald (p. 217) and in Hawking and Ellis (p. 82). My motivation for comparing them was that although Wald's treatment is more concise, Wald doesn't define the expansion tensor, only the volume expansion.(adsbygoogle = window.adsbygoogle || []).push({});

Wald starts off by restricting to a geodesic congruence rather than any old congruence. Hawking does not.

I've put everything in consistent notation where the velocity field is u (corresponding to Wald's [itex]\xi[/itex] and Hawking's V).

The definitions are:

spatial metric: [itex] h_{ab}=g_{ab} + u_a u_b [/itex]

expansion tensor: [itex]\theta_{ab}=h_a^c h_b^d u_{(c;d)}[/itex] (Hawking)

volume expansion: [itex]\theta=\theta_{ab}h^{a b}=u^a_{;a}[/itex] (Hawking gives both, Wald only gives the first form)

shear:

[itex]\sigma_{ab}=u_{(a;b)}-\frac{1}{3}\theta h_{ab}[/itex] (Wald)

[itex]\sigma_{ab}=\theta_{ab}-\frac{1}{3}\theta h_{ab}[/itex] (Hawking)

vorticity:

[itex]\omega_{ab}=u_{[a;b]}[/itex] (Wald)

[itex]\omega_{ab}=h_a^c h_b^d u_{[c;d]}[/itex] (Hawking)

decomposition

[itex]u_{a;b}=\frac{1}{3}\theta h_{ab}+\sigma_{ab}+\omega_{ab}[/itex] (Wald)

[itex]u_{a;b}=\frac{1}{3}\theta h_{ab}+\sigma_{ab}+\omega_{ab}-\dot{u}_a u_b[/itex] (Hawking)

Am I right in thinking that Wald's reason for restricting to geodesic congruences is that under these circumstances he gets the simpler expressions shown above, rather than the more complex ones that Hawking gives?

The definition of the spatial metric would clearly have to have the + sign flipped if you were using the +--- signature (since the purpose of the term is to punch the time-time component out of the metric). Would any other signs have to be changed for +---, like the sign in the definition of the shear?

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# Differing definitions of expansion, shear, and vorticity

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