'Expansion' of fluid world lines

[zn5252]

hello
In MTW exercise 22.6, given a fluid 4-velocity u, why the expression :
∇.u is called an expansion of the fluid world lines ?

Is the following reasoning correct ?

We know that the commutator : ∇_BA - ∇_AB is (see MTW box 9.2) is the failure of the quadrilateral formed by the vectors A and B to close.

Now If we apply this to the expression of the fluid world lines I would get :

∇_eσu - ∇_ue = ∇_eσu since a freely falling observer Fermi-Walker transports its own spatial basis (see MTW page 218) thus one can conclude that the quadrilateral formed by the time segment and the velocity segment does not close which means that the fluid expands 'or contracts'.

Regards,

 

[Bill_K]

In MTW exercise 22.6, given a fluid 4-velocity u, why the expression : ∇·u is called an expansion of the fluid world lines ?

This quote from Ex. 22.6 explains why:

Exercise 22.1 showed that the expansion θ = ∇·u describes the rate of increase of the volume of a fluid element.

 

[WannabeNewton]

One can show that $\nabla_a u^a = \frac{1}{V}u^a \nabla_a V$ where $V$ is an infinitesimal space-time volume carried along the worldline of some fluid element. So $\nabla_a u^a$ represents the rate of change of said volume per unit volume along the worldline of some fluid element.

 

[zn5252]

This quote from Ex. 22.6 explains why:

Oh I see Bill. I did not get to part b) yet .

 

[zn5252]

One can show that $\nabla_a u^a = \frac{1}{V}u^a \nabla_a V$ where $V$ is an infinitesimal space-time volume carried along the worldline of some fluid element. So $\nabla_a u^a$ represents the rate of change of said volume per unit volume along the worldline of some fluid element.

Indeed this is what part b) mentions. Thanks !

 

[WannabeNewton]

Cool! Have fun with the exercise :) MTW has the solution to 22.1 right below, if you're interested as to why the above is true.

 

[zn5252]

Cool! Have fun with the exercise :) MTW has the solution to 22.1 right below, if you're interested as to why the above is true.

Indeed I saw it and also attempted to derive my own which yielded the correct result based on the continuity equation and on the assumption that the divergence of the density is negligible...

 

[WannabeNewton]

Nice! We had a similar thread a while back that you might be interested in, where everything was done in a coordinate-free manner: https://www.physicsforums.com/showthread.php?t=702266&highlight=physical+description+concepts

Also, if you have access to Wheeler and Ciufolini's "Gravitation and Inertia", they give a very nice geometric description of hydrodynamical quantities in GR starting in section 4.5 (p.234).

 

[zn5252]

Nice! We had a similar thread a while back that you might be interested in, where everything was done in a coordinate-free manner: https://www.physicsforums.com/showthread.php?t=702266&highlight=physical+description+concepts

Also, if you have access to Wheeler and Ciufolini's "Gravitation and Inertia", they give a very nice geometric description of hydrodynamical quantities in GR starting in section 4.5 (p.234).

Great . Thanks ! I have Wheeler and Ciufolini's "Gravitation and Inertia". I will check that out. Thanks for mentioning that.

 

[WannabeNewton]

Anytime broski! :)