Clarification of concepts: Tetrads (Vielbeins), etc

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Discussion Overview

The discussion revolves around various concepts in gravitation, particularly focusing on tetrads, dual bases, frames, affine connections, and the Einstein-Rosen metric. Participants seek clarification on these topics, which encompass theoretical and conceptual aspects of differential geometry and general relativity.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant expresses confusion about the definitions and relationships between various bases, such as standard, normal, local, and orthogonal bases, and questions whether a tangent space or cotangent space should be referenced in relation to dual bases.
  • Another participant suggests that a good understanding of tangent spaces and dual spaces is necessary before tackling more complex concepts like affine connections.
  • There is a discussion about the nature of FIDO and FREFO observers, with some participants proposing that a FIDO is an observer using rigid rods and clocks, while others question the implications of these definitions.
  • Participants discuss the role of diffeomorphisms in Riemannian geometry and their importance for consistency in physical theories, though there is uncertainty about their physical interpretation.
  • One participant mentions the Einstein-Rosen metric and questions the presence of a common factor of beta^4 in the metric formula, indicating a lack of understanding of its derivation.
  • There is a suggestion that the exercise regarding writing a dreibein might contain a typo, as it references the tangent space when it may be intended to refer to the cotangent space.
  • Several participants recommend reviewing foundational concepts in linear algebra and vector spaces to better understand the advanced topics being discussed.

Areas of Agreement / Disagreement

Participants generally agree on the need for a solid understanding of foundational concepts before progressing to more complex topics. However, there are multiple competing views regarding the definitions and relationships between different types of bases and observers, and the discussion remains unresolved on several specific points, including the nature of the exercise mentioned.

Contextual Notes

Some participants note that the complexity of the topics may lead to confusion, particularly regarding the definitions of various bases and the implications of different types of observers. There is also mention of the potential for typos in exercises, which may contribute to misunderstandings.

Who May Find This Useful

This discussion may be useful for students and enthusiasts of general relativity and differential geometry, particularly those seeking clarification on foundational concepts related to frames, bases, and metrics.

  • #31
Breo said:
Can you give me an example of when we can "use" frefos and when fidos?

For instance, if I have two particles interacting gravitationally how a FREFO and how FIDO observiers would measure their acceleration magnitude and direction?
I don't know what you are getting at. Any coordinates or any observer can be used for any problem. The only difference is mathematical complexity. Is that what you are getting at?
 
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  • #32
Yes, any difference. Until now, I only have seen definitions so I want to see an example in practice to get a the mechanical picture in my mind.

So, how a FIDO, and then how a FREFO, would measure the particle problem? I want to see the differences
 
  • #33
Is FIDO seated on a stationary metric and FREFO on tetrads?
 
  • #34
Breo said:
Is FIDO seated on a stationary metric and FREFO on tetrads?
Yes to the second. Concpetually, I don't see FIDO as limited to a stationary metric. I've seen the term fiducial observer used in cosmological solutions, which are certainly not stationary. FYI, I've never seen your acronyms used.
 
  • #35
Free falling observer measures proper time while Fiducial observer measures natural time?
 
  • #36
Breo said:
Free falling observer measures proper time while Fiducial observer measures natural time?

What is natural time? All observers measure their own proper time.
 
  • #37
Breo said:
Free falling observer measures proper time while Fiducial observer measures natural time?
Per my post #29, fiducial observers define coordinate time for some chosen coordinates. There is nothing more 'natural' about this time than the proper time for any other observers. If you have coordinates based on a class of fiducial observers, coordinate time is proper time for those particular observers. Free fall observers measure their own proper time as well. All observers measure proper time.

I'm having trouble understanding what question you really want answered. Another observation, that may or may not be helpful, is that in GR there is no such thing as global free fall coordinates. Even if you can extend a free fall observers tetrad to cover a substantial section of spacetime via Fermi-Normal coordinates, only the origin observer is a free fall observer (in general). Fiducial observers, on the other hand, are used to build global coordinates that have some desired properties. Thus, you pick a family of observers that capture some symmetry of the manifold (static observers at infinity for SC geometry; comoving observers for a cosmological solution), and build coordinates such that coordinate time is the proper time measured by the fiducial observers. At least, that is the only sense I've seen the term used. If you have particular reference with a different sense of the term, it would be helpful to refer to that source to frame further questions.

[Edit: Let me add a clarification to the above: In some cases you can build global coordinates based on a family free fall observers (effectively using these as your fiducial observers). However, except for flat Minkowski space, you have to give up on orthogonality of your coordinates to achieve this - the metric will have off diagonal terms. ]
 
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  • #38
Coordinate time, that it was what I meant.

Thank you very much for the answers. Yours helped me a lot to understand what FIDO and FREFO are.
 
  • #39
After thinking a while about proper time. Can you give me an example with equations or not about how a frefo measures his proper time?
 
  • #40
Breo said:
After thinking a while about proper time. Can you give me an example with equations or not about how a frefo measures his proper time?
Integral of the line element along the world line of any observer is proper time. Can you clarify your question, as this fact is like the 1+1=2 of relativity, so I am having a hard time believing this is what you are really asking.
 
  • #41
Breo said:
After thinking a while about proper time. Can you give me an example with equations or not about how a frefo measures his proper time?

There's no equations necessary. Any observer measures his own proper time. That is the definition of proper time.

EDIT: Looks like PAllen is quicker to the draw than I.
 
  • #42
Yes, sure for a fiducial we take the time from integrating. But for a free falling observer... what formula gives the proper time.

Imagine you have ## e^0 = Adt -Bdx^i \\ e^i= Cdx^i ##

how to measure the proper time? in terms of the tetrad maybe?
 
  • #43
Breo said:
Yes, sure for a fiducial we take the time from integrating. But for a free falling observer... what formula gives the proper time.

Imagine you have ## e^0 = Adt -Bdx^i \\ e^i= Cdx^i ##

how to measure the proper time? in terms of the tetrad maybe?
I'm glad you clarified, because that gives a more interesting question.

First, note that a coordinate chart can represent all observer world lines that pass through the chart's coverage, and they all give the same proper time for any observer (pretty much by construction of how the metric transforms between charts). So, what I said before is trivially true: represent a free fall world line in any coordinate chart and you compute their proper time by integration; same as any observer. Further, computing which are the free fall observers in any coordinates is (conceptually - can be messy in practice) straightforward: solve the geodesic equation.

However, it seems you are asking how it is done in tetrad formalism. I admit I've never used tetrad formalism for this purpose, and am not very familiar with it, in general (I use coordinates or pure geometric formalisms; never studied tetrads much). I've only looked at tetrad formalism for local interaction, not for any integrated quantity like proper time. Hopefully someone else can explain how this might be done.
 
  • #44
Breo said:
Yes, sure for a fiducial we take the time from integrating. But for a free falling observer... what formula gives the proper time.

Imagine you have ## e^0 = Adt -Bdx^i \\ e^i= Cdx^i ##

how to measure the proper time? in terms of the tetrad maybe?

For ANY observer, as PAllen pointed out, you get the proper time by integrating along its world line. This is not just for "fiducial observers". As I stated in my last post, the free-fall observer, or any observer for that matter, does not need to use ANY equations to figure out his own proper time, he just needs to look at his watch.

I think you have to be more specific with your question.
 

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