Rotating rigid sphere stress-energy tensor

[aatw]

Hy.

Can somebody please show me the way, how to transform stress-energy tensor for sphere in rest frame to stress-energy tensor in rotating frame using Lorentz transformations?

 

[tiny-tim]

Welcome to PF!

Hy.

Can somebody please show me the way, how to transform stress-energy tensor for sphere in rest frame to stress-energy tensor in rotating frame using Lorentz transformations?

Hy aatw! Welcome to PF!

Whyever would you want to use a rotating frame?

Anyway, the rotations in the group of Lorentz transformations are exactly the same as in ordinary Euclidean space.

 

[aatw]

Thanks for your quick reply.
Maybe I didn't express myself so well.
Let's start from stress-energy tensor for sphere, which can be written as T^{ab} = (\rho + p) u^a u^b + p g^{ab}, but in rest frame where sphere doesn't rotate, it's only T^{00} element that's nonzero (I think so!).
But now sphere starts to rotate with angular speed \omega around z axis and I want to use Lorentz transformation to determine T^{ab} for rotating sphere.

 

[Mentz114]

Have you tried plugging in this velocity 4-vector ?
<br /> <br /> <br /> \left[ \begin{array}{c}<br /> -\sqrt{\omega^2r^2-c^2}\\\<br /> -\omega r\sin(\omega t) \\\<br /> \omega r\cos(\omega t) \\\<br /> 0 \end{array} \right]<br />
where r^2=x^2+y^2 into the EMT ? This seems to be in cartesian coords but easy enough to transform to polar.

I suppose another approach would be to boost in the x and y directions with velocities as given above. I'll try that later if I have time.

 

[Mmmm]

Thanks for your quick reply.
Maybe I didn't express myself so well.
Let's start from stress-energy tensor for sphere, which can be written as T^{ab} = (\rho + p) u^a u^b + p g^{ab}, but in rest frame where sphere doesn't rotate, it's only T^{00} element that's nonzero (I think so!).

No, if g = \eta then T^{ij} = P \delta^i_j where P is the pressure between the elements (just use the formula you quoted with u^a = (1,0,0,0) -which is the 4-velocity in the rest frame).

But now sphere starts to rotate with angular speed \omega around z axis and I want to use Lorentz transformation to determine T^{ab} for rotating sphere.

ok, so just use an ordinary boost in all 3 spatial directions with v=\omega r