Acceleration Tensor - Rotating Frame

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Discussion Overview

The discussion revolves around the transformation properties of acceleration in a rotating coordinate system, particularly whether it behaves as a rank 1 tensor. Participants explore the implications of time dependence in rotating frames and the transition from special relativity (SR) to general relativity (GR) in this context.

Discussion Character

  • Debate/contested
  • Conceptual clarification
  • Technical explanation

Main Points Raised

  • One participant questions whether acceleration transforms as a rank 1 tensor in a rotating frame, suggesting that the time dependence complicates the transformation.
  • Another participant proposes that if one understands how acceleration behaves under restricted Lorentz transformations, then the transformation should be straightforward.
  • A different viewpoint asserts that 4-acceleration is a geometric object that transforms as a tensor, implying that the nature of the metric in a rotating system is crucial.
  • Some participants note that a rotating coordinate system leads to a non-Minkowskian metric, indicating a shift towards general relativity rather than remaining within special relativity.
  • There is a contention regarding the implications of a rotating frame on the behavior of the metric, with one participant suggesting that it involves differential geometry rather than general relativity.

Areas of Agreement / Disagreement

Participants express differing views on whether acceleration transforms as a rank 1 tensor in rotating frames, with some asserting it does and others arguing against it. The discussion remains unresolved with multiple competing perspectives on the nature of acceleration in this context.

Contextual Notes

Participants highlight the complexity introduced by time dependence in rotating frames and the potential transition from special to general relativity, but do not resolve the implications of these factors on the transformation properties of acceleration.

PrinceOfDarkness
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If a coordinate system is rotating, that is time 't' is not independent, then does the acceleration transform as rank 1 tensor?

I thought that it wouldn't because when time is changing, so acceleration will change in a more complicated way than a rank 1 tensor. Perhaps as a rank 2 tensor.

This Q is really troubling me. There are two groups in my class, one saying it still transforms as a rank 1 tensor, the other saying it doesn't transform as a rank 1 tensor. Some even say that acceleration never transforms like a rank 1 tensor. I wonder how! I think it transforms like a rank 1 tensor if it goes fixed rotation, but 'rotating' coordinate system will mean that transformation is more complicated.
 
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Rotations (with presumably fixed angular velocity) are examples of (restricted) Lorentz transformations. If you know how the acceleration behaves when being subject to a (restricted) Lorentz transformation, then everything would be fine, wouldn't you say...?

Daniel.
 
PrinceOfDarkness said:
If a coordinate system is rotating, that is time 't' is not independent, then does the acceleration transform as rank 1 tensor?

You need to be a bit more specific here. The 4-acceleration is a geometric object, so it transforms as a tensor.

A rotating coordinate system will require a metric that is not Minkowskian, so you start getting into GR rather than SR.

The rotating coordinate system will be ill-behaved when r*w = c, some of the metric coefficients go to zero (or was it infinity? I'd have to double check - but I know they are not well-behaved).
 
A rotating coordinate system will require a metric that is not Minkowskian, so you start getting into GR rather than SR.
No, you're merely getting into differential geometry.
 

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