# 'Expansion' of fluid world lines

1. Aug 9, 2013

### zn5252

hello
In MTW excercise 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,

2. Aug 9, 2013

### Bill_K

This quote from Ex. 22.6 explains why:

3. Aug 9, 2013

### 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.

4. Aug 9, 2013

### zn5252

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

5. Aug 9, 2013

### zn5252

Indeed this is what part b) mentions. Thanks !

6. Aug 9, 2013

### 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.

7. Aug 9, 2013

### zn5252

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...

8. Aug 9, 2013

9. Aug 9, 2013

### zn5252

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

10. Aug 9, 2013

### WannabeNewton

Anytime broski! :)