Expansion tensor on rotating disk

[Dale]

Hi Everyone,

Suppose that we have cylindrical coordinates on flat spacetime (in units where c=1): $ds^2 = -dt^2 + dr^2 + r^2 d\theta^2 + dz^2$

I would like to explicitly calculate the expansion tensor for a disk of constant radius R<1 and non-constant angular velocity $\omega(t)<1$. I don't know how to do this. All I know that it starts with defining a congruence that represents the material of the disk.

Each point on the disk would have a four-velocity of
$$v=\left( \frac{1}{\sqrt{1-r^2 \omega^2}},0, \frac{r\omega}{\sqrt{1-r^2 \omega^2}},0\right) = \gamma \partial_t + r\omega\gamma\partial_{\theta}$$ where $\gamma=(1-r^2 \omega^2)^{-1/2}$

I think that is the congruence, but where do I go from there?

 

[PeterDonis]

Wikipedia has a brief definition of the expansion tensor (as well as the other elements of the kinematic decomposition):

http://en.wikipedia.org/wiki/Congru...atical_decomposition_of_a_timelike_congruence

The computations are tedious (at least, I find them so ) but straightforward.

For this particular case, it looks to me like there should be zero shear, so all the information about expansion (if any) should be in the expansion scalar, which is easier to calculate; it's just $\nabla_a u^a$, i.e., the (covariant) divergence of the 4-velocity.

 

[WannabeNewton]

Hi Dale, see http://arxiv.org/pdf/gr-qc/0312087v2.pdf

The authors do not explicitly write out the details of the calculations, rather they simply explicitly write down the expansion and rotation tensors from the definitions, but this does let you check the result of your own calculations.

As Peter remarked, the shear tensor vanishes identically as we would expect given the rotational symmetry of the entire problem. The expansion tensor is non-vanishing of course due to the fact that the 4-velocity field of the time-like congruence is not parallel to a time-like Killing field of flat space-time, which by the Herglotz-Noether theorem implies the non-Born rigidity of the congruence since the 4-velocity field has non-vanishing vorticity.

 

[Dale]

Thanks Peter and WBN, I will get back to you if I get stuck.

 

[sardar]

Hello,

The expansion tensor is a measure of how a congruence of worldlines diverges or converges. In this case, the congruence is represented by the material points on the rotating disk, and the expansion tensor can be calculated by considering the change in the four-velocity vector as we move along the worldlines of these points.

As you have correctly identified, the four-velocity vector for each point on the disk is given by v=(1/√(1-r^2ω^2),0,rω/√(1-r^2ω^2),0). To calculate the expansion tensor, we need to use the formula:

θ_ab=∇_a v_b + ∇_b v_a - (1/3)h_ab (∇_c v^c)

where θ_ab is the expansion tensor, ∇ is the covariant derivative, and h_ab is the projection tensor onto the plane of the disk. This projection tensor is given by:

h_ab=η_ab - u_a u_b

where η_ab is the Minkowski metric and u_a is the unit normal vector to the disk, which is given by u_a=(-rω/√(1-r^2ω^2),0,0,0).

Using these expressions, we can calculate the expansion tensor for the rotating disk. However, since the angular velocity is non-constant, the calculation becomes quite complicated and may not have a closed-form solution. It may be easier to use numerical methods to approximate the expansion tensor at different points on the disk.

I hope this helps. Let me know if you have any further questions.