Interval ds^2 on Rotating Disk - Explained

[exmarine]

What is ds^2 on a rotating disk?

 

[WannabeNewton]

Is this a homework problem? Anyways, it's a really easy calculation so try it yourself.

Take the Minkowski metric in cylindrical coordinates relative to a global inertial frame centered on the origin of the cylindrical coordinates and perform a coordinate transformation to a frame that's rotating with some angular velocity $\omega$ relative to this global inertial frame (centered on the same origin).

 

[exmarine]

I must be over-complicating it then. Am trying to learn GRT from Dirac's book "General Theory of Relativity". Do you mean a Lorentz transform like in SRT?

$ds^{2}=(cdt)^{2}-(dr)^{2}-(rd\theta)^{2}-dz^{2}$

All the homework links look like simpler engineering type problems? Anyway, I am stumped here.

Homework? Ha. I wish I could get to a good grad school. But I am 70 and just can't get there from here.

Thanks for any help.

 

[WannabeNewton]

Calculate the form of $ds^2$ under the following change of coordinates: $t' = t', r' = r, z' = z,$ and $\theta' = \theta - \omega t$.

 

[bcrowell]

This is not a trivial problem. In fact, it has no solution that is completely satisfactory in every way. WannabeNewton's #4 gives one possibility, but it has the undesirable properties that the t' coordinate isn't properly synchronized in terms of local Einstein synchronization, and because of this the spatial part of the metric doesn't represent distances that would be measured by a rotating observer. It's also possible to correct the t' coordinate to fix these problems, but then t' can't be extended to a global coordinate chart. I've written a discussion of this here http://www.lightandmatter.com/sr/ (section 8.1).

 

[WannabeNewton]

Wow, the entirety of chapter 8 is really awesome Ben. Thanks for the link!

In an older post on the subject of the rotating disk I posted some references regarding the issue(s) brought up by Ben so check them out after reading chapter 8 of Ben's book, if you're interested: https://www.physicsforums.com/showpost.php?p=4582800&postcount=6

 

[exmarine]

Yes, thanks Ben and Wannabe. I thought maybe I was going nuts. I look forward to studying your material!