Differing definitions of expansion, shear, and vorticity

[bcrowell]

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.

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 $\xi$ and Hawking's V).

The definitions are:

spatial metric: $h_{ab}=g_{ab} + u_a u_b$
expansion tensor: $\theta_{ab}=h_a^c h_b^d u_{(c;d)}$ (Hawking)
volume expansion: $\theta=\theta_{ab}h^{a b}=u^a_{;a}$ (Hawking gives both, Wald only gives the first form)
shear:
$\sigma_{ab}=u_{(a;b)}-\frac{1}{3}\theta h_{ab}$ (Wald)
$\sigma_{ab}=\theta_{ab}-\frac{1}{3}\theta h_{ab}$ (Hawking)
vorticity:
$\omega_{ab}=u_{[a;b]}$ (Wald)
$\omega_{ab}=h_a^c h_b^d u_{[c;d]}$ (Hawking)
decomposition
$u_{a;b}=\frac{1}{3}\theta h_{ab}+\sigma_{ab}+\omega_{ab}$ (Wald)
$u_{a;b}=\frac{1}{3}\theta h_{ab}+\sigma_{ab}+\omega_{ab}-\dot{u}_a u_b$ (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?

 

[Mentz114]

Hi Ben,
looking at Stephani, I see he has the same expressions as Hawking. He regards u as a velocity field but doesn't mention if the the congruence is always geodesic so I presume it isn't.

In answer to your question - I don't know. Presumably some terms in the general expressions will be zero for a geodesic congruence, but I don't know which ones ( although I feel I ought to ).

PS: There's a wiki page on this here

http://en.wikipedia.org/wiki/Congruence_(general_relativity)

 

[George Jones]

Right now, my 4-year-old daughter is neither letting me think nor calculate as much as I would like, but to see the equivalence for geodesics of

vorticity:
$\omega_{ab}=u_{[a;b]}$ (Wald)
$\omega_{ab}=h_a^c h_b^d u_{[c;d]}$ (Hawking)

1) expand each h;
2) expand the [] barackets;
3) multiply together all factors in Hawking's definition.

In 3) use

a) $0 = u^a u_{b;a}$ (from the geodesic property);
b) $0 = u^a u_{a;b}$ (from 4-velocity normalization, $1 = u^a u_a$.

 

[Fredrik]

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?

As far as I can tell, the only result on page 217 that only holds for geodesics is $B_{ab}\xi^b=0$, and it isn't used for anything later.

Edit: I wrote that before reading George's post. I stand by my reply as far as the calculations Wald did before the definitions of $\theta$, $\sigma$ and $\omega$ are concerned. I still haven't understood why the definitions look the way they do.