Interval ds^2 on Rotating Disk - Explained

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Discussion Overview

The discussion centers on the concept of the interval ds² on a rotating disk, exploring the mathematical formulation and implications within the context of General Relativity Theory (GRT). Participants engage with the transformation of the Minkowski metric in cylindrical coordinates to a rotating frame and the challenges associated with this transformation.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Homework-related

Main Points Raised

  • One participant questions the nature of the problem, suggesting it may be a homework problem and encourages others to attempt the calculation themselves.
  • Another participant presents the Minkowski metric in cylindrical coordinates and suggests performing a coordinate transformation to a rotating frame characterized by angular velocity ω.
  • A participant expresses confusion about the problem, referencing their study of GRT from Dirac's book and inquiring whether a Lorentz transformation is applicable.
  • There is a suggestion to calculate the form of ds² under a specific change of coordinates involving a transformation of the angular coordinate.
  • A participant argues that the problem is not trivial and notes that the proposed solution has limitations, particularly regarding synchronization and the representation of distances for a rotating observer.
  • Another participant appreciates the resources shared and expresses interest in further studying the material related to the rotating disk issue.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the nature of the problem or the adequacy of the proposed solutions. Multiple competing views and uncertainties about the transformations and their implications remain evident throughout the discussion.

Contextual Notes

Participants highlight limitations in the proposed solutions, including issues with synchronization and the inability to extend certain coordinates to a global chart. These unresolved aspects contribute to the complexity of the discussion.

Who May Find This Useful

This discussion may be of interest to those studying General Relativity, particularly in the context of rotating frames and the mathematical challenges associated with them.

exmarine
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What is ds^2 on a rotating disk?
 
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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).
 
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?

[itex]ds^{2}=(cdt)^{2}-(dr)^{2}-(rd\theta)^{2}-dz^{2}[/itex]

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.
 
Calculate the form of ##ds^2## under the following change of coordinates: ##t' = t', r' = r, z' = z,## and ##\theta' = \theta - \omega t##.
 
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).
 
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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
 
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Yes, thanks Ben and Wannabe. I thought maybe I was going nuts. I look forward to studying your material!
 

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