Differing definitions of expansion, shear, and vorticity

In summary, there is a discussion comparing Wald and Hawking's treatments of expansion, shear, and vorticity, with a focus on their different approaches to defining these quantities. Wald restricts to a geodesic congruence while Hawking does not, resulting in simpler expressions for Wald's definitions. However, it is unclear why the definitions are different and what the implications are for using a geodesic congruence. Further exploration is needed to fully understand this topic.
  • #1
bcrowell
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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 [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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  • #2
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)
 
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  • #3
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
bcrowell said:
vorticity:
[itex]\omega_{ab}=u_{[a;b]}[/itex] (Wald)
[itex]\omega_{ab}=h_a^c h_b^d u_{[c;d]}[/itex] (Hawking)

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

In 3) use

a) [itex]0 = u^a u_{b;a}[/itex] (from the geodesic property);
b) [itex]0 = u^a u_{a;b}[/itex] (from 4-velocity normalization, [itex]1 = u^a u_a[/itex].
 
  • #4
bcrowell said:
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 [itex]B_{ab}\xi^b=0[/itex], 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 [itex]\theta[/itex], [itex]\sigma[/itex] and [itex]\omega[/itex] are concerned. I still haven't understood why the definitions look the way they do.
 
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FAQ: Differing definitions of expansion, shear, and vorticity

What is the definition of expansion in the context of fluid mechanics?

In fluid mechanics, expansion refers to the increase in volume of a fluid element due to the outward movement of its particles. This can occur in one, two, or three dimensions and is often measured by the change in density of the fluid.

How is shear defined in fluid mechanics?

Shear in fluid mechanics refers to the difference in velocity between adjacent layers of a fluid. It is caused by the frictional forces between these layers and is a measure of the deformation or strain of the fluid.

What is the significance of vorticity in fluid dynamics?

Vorticity is a measure of the local spinning or rotation of a fluid element. It plays a crucial role in the study of turbulence and is used to calculate forces such as lift and drag on objects moving through a fluid.

How do the definitions of expansion, shear, and vorticity differ in fluid mechanics?

While all three terms refer to the movement and deformation of a fluid, they have different underlying causes and are measured in different ways. Expansion is a measure of volume change, shear is a measure of velocity difference, and vorticity is a measure of rotation.

Can expansion, shear, and vorticity occur simultaneously in a fluid?

Yes, in most cases, all three phenomena can occur simultaneously in a fluid. For example, in a rotating fluid, there is both shear due to the difference in velocity between layers and vorticity due to the rotation of the fluid element. However, the relative magnitudes of these three factors can vary depending on the specific flow conditions.

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