SR textbooks discussing accelerated reference frames without delving into GR

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Logic314
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I am well aware that the theory of special relativity, like Newtonian mechanics, need not be limited to observations made within inertial frames, provided one makes a few postulates concerning acceleration (such as the fact that at any point along the worldline of an accelerating observer, certain local space-time information in the frame of the observer can be carried over from the instantaneous inertial rest frame). Unfortunately, all the common standard textbooks on special relativity that I have come across thus far either have dismissed the applicability of SR to non-inertial frames (e.g., Helliwell), or have acknowledged the applicability to non-inertial frames but have not done justice to the subject in the text (to the same extent as with inertial frames) without also bringing in general relativity and the equivalence principle. I am honestly quite tired of seeing the same bogus argument again and again stating that non-inertial frames cannot be discussed without the equivalence principle and general relativity.

I want to know whether there exist any good comprehensive SR texts out there that focus solely on special relativity (no discussion of GR) and yet devote an entire portion of the book diving deep into the theory of non-inertial frames, but entirely within the framework of special relativity. I have yet to come across such a book, but I am quite open to recommendations.
 
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I'm not sure any such beast exists. Once you've written the general methodology for curvilinear coordinate systems in flat spacetime it's not all that much extra to do GR. Covariant differentiation really only cares about whether or not you have a global inertial coordinate system, not why you don't have one.

Carroll's lecture notes on GR (obviously) cover GR, but chapters 1 and 2 cover SR (with a view to GR in the future).
 
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  • #3
Well there is one that does what you ask for:

Gourgoulhon, E. (2013). Special Relativity in General Frames: From Particles to Astrophysics. Springer. ISBN 978-3642372766.

and some hints on this topic on Wikipedia as well
https://en.wikipedia.org/wiki/Proper_reference_frame_(flat_spacetime)
 
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  • #4
There's a little of this in Taylor and Wheeler in the context of Thomas Precession,. But why? There are two reason:

1) To solve a problem or two that doesn't require the full mathematical treatment
2) To lead into GR, in which case "why stop?"

i.e. it will either be the end or the start of a book - not the whole book.
 
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  • #5
Logic314 said:
I want to know whether there exist any good comprehensive SR texts out there that focus solely on special relativity (no discussion of GR) and yet devote an entire portion of the book diving deep into the theory of non-inertial frames, but entirely within the framework of special relativity.
One reason why such a thing is so hard to find is that the scope of application of such a theoretical method would be very limited. While it is of course true that SR only requires flat spacetime, not inertial frames, in practice it's very hard to find a situation in flat spacetime which requires SR (i.e., Newtonian mechanics is not good enough) and is easier to analyze in a non-inertial frame. In practice, situations that require relativity and are easier to analyze in non-inertial frames are that way because of the presence of gravity, i.e., of spacetime curvature.
 
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  • #6
Logic314 said:
I am well aware that the theory of special relativity, like Newtonian mechanics, need not be limited to observations made within inertial frames, provided one makes a few postulates concerning acceleration (such as the fact that at any point along the worldline of an accelerating observer, certain local space-time information in the frame of the observer can be carried over from the instantaneous inertial rest frame). Unfortunately, all the common standard textbooks on special relativity that I have come across thus far either have dismissed the applicability of SR to non-inertial frames (e.g., Helliwell), or have acknowledged the applicability to non-inertial frames but have not done justice to the subject in the text (to the same extent as with inertial frames) without also bringing in general relativity and the equivalence principle. I am honestly quite tired of seeing the same bogus argument again and again stating that non-inertial frames cannot be discussed without the equivalence principle and general relativity.

I want to know whether there exist any good comprehensive SR texts out there that focus solely on special relativity (no discussion of GR) and yet devote an entire portion of the book diving deep into the theory of non-inertial frames, but entirely within the framework of special relativity. I have yet to come across such a book, but I am quite open to recommendations.

While you might find some treatments of special relativity that do what you say you want, (Rindler comes to mind, though I didn't care much for it after a brief look at his SR book), I would suggest a GR text for learning the necessary mathematics and skipping over any of the mathematics specifically related to curved space-time and/or the equivalence principle.

I would say that it's quite possible that you don't actually want exactly what you asked for. For instance, I've seen cases in the past where people said they wanted to avoid GR, but what they really wanted to avoid were graduate level tensor methods.

While it does use graduate-level tensor methods, my personal favorite is Misner, Thorne, Wheeler's treatment in "Gravitation", henceforth MTW, which has a good chapter on accelerated frames of reference (not all GR books have this, in fact I'd say most lack it). The basic approach would be to ignore all the extra discussion in the book about GR, and focus only on the elements that apply to SR.

I don't think it's really feasible to learn about accelerated frames of reference without tensor methods. One might get some insight by considering Bell's spaceship paradox, but this won't really result in a full understanding of accelerated frames, though it might shed some insight on the problem.
 
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  • #7
pervect said:
Misner, Thorne, Wheeler's treatment in "Gravitation", henceforth MTW, which has a good chapter on accelerated frames of reference (not all GR books have this, in fact I'd say most lack it). The basic approach would be to ignore all the extra discussion in the book about GR, and focus only on the elements that apply to SR.
Note that the chapter you refer to in MTW is part of the section of the book titled "Physics in Flat Spacetime", which comes before any discussion of spacetime curvature. Section 6.6, in particular, discusses coordinates centered on the worldline of an accelerated observer, in a flat spacetime context.
 
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  • #8
Histspec said:
Well there is one that does what you ask for:

Gourgoulhon, E. (2013). Special Relativity in General Frames: From Particles to Astrophysics. Springer. ISBN 978-3642372766.

and some hints on this topic on Wikipedia as well
https://en.wikipedia.org/wiki/Proper_reference_frame_(flat_spacetime)
I will definitely look into this text. Thanks for providing the reference!
 
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  • #9
In "Bryce DeWitt's Lectures on Gravitation" his chapter 2 is called "accelerated motion in special relativity" in which he spends about 20 or so pages going into the theory, and has 10 or so problems/solutions.

He talks about an accelerated meter stick, rigidity, fermi-walker transport, proper flat geometry, constant rotation around a fixed axis, and irrotational flow.

After this, he dives into general relativity concepts, so the first two chapters might suit your taste (chapter one is "Review of the Uses of Invariants in Special Relativity"). But, if it's only special relativity you're after, you're better off just spending a few afternoons if your library has it in stock rather than buying it. It is a tough book though, so be warned if you haven't studied much DeWitt.
 
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  • #10
PeterDonis said:
Note that the chapter you refer to in MTW is part of the section of the book titled "Physics in Flat Spacetime", which comes before any discussion of spacetime curvature. Section 6.6, in particular, discusses coordinates centered on the worldline of an accelerated observer, in a flat spacetime context.

Thanks for pointing that out.

One point about required background - it'd be good to read Taylor & Wheeler's SR text "Space-time physics" and become familiar with their treatment of special relativity before trying to tackle MTW. Of particular importance in my opinion is being able to interpreting the abstract idea of the invariant Lorentz interval in a way that has some intuitive physical meaning.
 
  • #11
pervect said:
(Rindler comes to mind, though I didn't care much for it after a brief look at his SR book)
In his book "Introduction to Special Relativity, 2nd Edition", Rindler has on page 33 a chapter "14. Acceleration transformation. The uniformly accelerated rod". If you solve exercise 15 on page 37, then you get also the Rindler-transformation.

But he didn't explain the spacetime-metric expressed in Rindler coordinates. I think, he should have added an additional chapter for it.

Rindler-transformation:
##ct = X \sinh(\frac{\alpha}{c}T)##
##x = X \cosh(\frac{\alpha}{c}T)##
##y = Y##
##z = Z##.

Differentiation for ##T##, using product rule, chain rule, and multiplying with ##dT##:
##cdt = dX \sinh(\frac{\alpha}{c}T) + X \frac{\alpha}{c}\cosh(\frac{\alpha}{c}T)dT##
##dx = dX \cosh(\frac{\alpha}{c}T) + X \frac{\alpha}{c}\sinh(\frac{\alpha}{c}T)dT##

Minkowski-spacetime metric:
##ds^2 = c^2dt^2 - dx^2 -dy^2 -dz^2##

Spacetime-metric expressed in Rindler coordinates, using ##\cosh^2(\varphi)-\sinh^2(\varphi) =1##:$$ds^2 = (\frac{\alpha }{c} X dT)^2 - dX^2 -dY^2 -dZ^2$$
Time dilation in Rindler frame (the observer with proper acceleration ##\alpha## is located at ##X_0=\frac{c^2}{\alpha}##):$$\frac{d\tau}{dT} = \sqrt{ (\frac{\alpha }{c^2} X)^2-V^2/c^2}$$
 
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  • #12
I wrote some notes myself on accelerating observers in SRT because I recognize your "problem". Maybe they can help (it's been a while, so there could be typos). See attachment!
 

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  • #14
Logic314 said:
such as the fact that at any point along the worldline of an accelerating observer, certain local space-time information in the frame of the observer can be carried over from the instantaneous inertial rest frame
That can be done to understand Rindler-coordinates.

rindler-coordiates.png
Source:​

The diagram shows two instantaneous co-moving inertial reference-frames ##S## and ##S'##.
A worldline of constant spacetime-distance from the origin of a Minkowski diagram is the following hyperbola:
$$x(t)^2 - c^2t^2 = x(0)^2 - 0^2= X^2\ \ \ \ \ (1)$$Differentiating every term for ##t##:
##2x\frac{dx}{dt} - 2c^2t = 0##
##\Rightarrow##
##v= \frac{dx}{dt} = \frac{c^2t}{x}##
##\Rightarrow##
##a= \frac{d^2x}{dt^2} = \frac{c^2}{x}##
##\Rightarrow##
##\alpha = a(0)= \frac{c^2}{X}##

With equation (1) follows:
$$x^2 - c^2t^2 =(\frac{c^2}{\alpha})^2$$The right side must be invariant (especially the same in every instantaneous co-moving inertial reference-frames), because the left side is invariant. Therefore, the proper acceleration ##\alpha## must be constant.

The hyperbolic angle ##\phi## between the ##x##-axis and the ##x'##-axis is ##\frac{\alpha}{c}## times the elapsed proper time of the Rindler observer along the hyperbola between crossing these two axes (events D and D' in the diagram).

From this follow the transformation formulas in posting #11.
 
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  • #15
PeterDonis said:
One reason why such a thing is so hard to find is that the scope of application of such a theoretical method would be very limited. While it is of course true that SR only requires flat spacetime, not inertial frames, in practice it's very hard to find a situation in flat spacetime which requires SR (i.e., Newtonian mechanics is not good enough) and is easier to analyze in a non-inertial frame. In practice, situations that require relativity and are easier to analyze in non-inertial frames are that way because of the presence of gravity, i.e., of spacetime curvature.
The presence of gravity will result in curvature of spacetime. and the effect of spacetime curvature is that the parallel transportation of a vector is path dependent.
Then can I ask such a question: In a flat spacetime is it possible to define an action on a vector which guarantee the movement of the vector is also path dependent?
 
  • #16
Jianbing_Shao said:
In a flat spacetime is it possible to define an action on a vector which guarantee the movement of the vector is also path dependent?
Not if "movement of the vector" means parallel transport. By definition flat spacetime has zero curvature, and nonzero curvature is required to make parallel transport path dependent.
 
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  • #17
PeterDonis said:
Not if "movement of the vector" means parallel transport. By definition flat spacetime has zero curvature, and nonzero curvature is required to make parallel transport path dependent.
If we define an infinitesimal action at each point in flat spacetime. then a vector moves from a point to another along different paths, then the movement is possibly path dependent.
If we observe the movement of a vector in a space and we are not sure if there exists a gravity field, then can we assume that the property of path dependent is just result from the infinitesimal action we defined in a flat spacetime?
 
  • #18
Jianbing_Shao said:
If we define an infinitesimal action at each point in flat spacetime. then a vector moves from a point to another along different paths, then the movement is possibly path dependent.
You can make up any "infinitesimal action" you like, but it won't have anything to do with the spacetime geometry.

Jianbing_Shao said:
If we observe the movement of a vector in a space and we are not sure if there exists a gravity field, then can we assume that the property of path dependent is just result from the infinitesimal action we defined in a flat spacetime?
No, because an arbitary "infinitesimal action" has nothing to do with the spacetime geometry, as above. In order to investigate the spacetime geometry, you need to use things that are related to the spacetime geometry, like geodesics and parallel transport.
 
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  • #19
PeterDonis said:
You can make up any "infinitesimal action" you like, but it won't have anything to do with the spacetime geometry.No, because an arbitary "infinitesimal action" has nothing to do with the spacetime geometry, as above. In order to investigate the spacetime geometry, you need to use things that are related to the spacetime geometry, like geodesics and parallel transport.
Then the structure of spacetime geometry can't be described using the language of 'infinitesimal action'? But the connection field can describe the local infinitesimal change of a components of a vector when it parallel transport along a particular path? So can it be regarded as a definition of a local 'infinitesimal action' in the spacetime?
If the property of path dependence of a vector's movement is not necessarily a result of spacetime geometry? then how to distinguish between the two.
 
  • #20
Jianbing_Shao said:
Then the structure of spacetime geometry can't be described using the language of 'infinitesimal action'?
"Infinitesimal action" is just a general term. (And I'm not sure exactly what you mean by it, although you give some hints--see further comments below.) The structure of spacetime geometry is described by the metric tensor and tensors like the Riemann curvature tensor that are derived from it. That doesn't mean the structure of spacetime geometry is described by "tensors" in general. A particular spacetime geometry is described by a particular metric tensor.

Jianbing_Shao said:
the connection field can describe the local infinitesimal change of a components of a vector when it parallel transport along a particular path?
The connection derived from the metric tensor does, yes. But that's not "infinitesimal action" in general. It's a particular connection derived from a particular metric for that particular spacetime geometry.

Jianbing_Shao said:
can it be regarded as a definition of a local 'infinitesimal action' in the spacetime?
I suppose so, although I don't see what the big deal is about the term "infinitesimal action". The point is that it is one particular thing for a given spacetime, not just generalities.

Jianbing_Shao said:
If the property of path dependence of a vector's movement is not necessarily a result of spacetime geometry?
Who said it wasn't? We have said precisely the opposite in this thread: if the spacetime is flat, parallel transport is not path dependent; if the spacetime is curved, it is. So path dependence is the result of spacetime geometry.
 
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  • #21
PeterDonis said:
No, because an arbitary "infinitesimal action" has nothing to do with the spacetime geometry, as above. In order to investigate the spacetime geometry, you need to use things that are related to the spacetime geometry, like geodesics and parallel transport.
Can you clarify this statement? Isn't it always possible to specify a metric (including a flat one) and define transport in terms of an arbitrary connection, at the expense of introducing torsion and/or non-metricity?
 
  • #22
renormalize said:
Isn't it always possible to specify a metric (including a flat one) and define transport in terms of an arbitrary connection, at the expense of introducing torsion and/or non-metricity?
Not if you're doing SR and GR. If you just want to make up theories that have no relation to reality, sure. But not if you want to actually do physics and make accurate predictions.
 
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  • #23
PeterDonis said:
Not if you're doing SR and GR. If you just want to make up theories that have no relation to reality, sure. But not if you want to actually do physics and make accurate predictions.
How does that statement square with the notion that gravitation in flat-spacetime with torsion can be physically equivalent to standard GR (e.g., The teleparallel equivalent of general relativity)?
 
  • #24
renormalize said:
Can you clarify this statement? Isn't it always possible to specify a metric (including a flat one) and define transport in terms of an arbitrary connection, at the expense of introducing torsion and/or non-metricity?
Yes, I agree with your view.
I think metric and connection are two different types conceptions. they can be defined independently. even in a space with a flat metric we can define a connection, then the movement of a vector is path dependent. so curvature is determined by the property of connection. and if you relate the metric and the connection with some equations, then you can say we can start from metric to get a non-zero curvature. but you can't say that the curvature is exclusively determined by the metric.
 
  • #25
renormalize said:
How does that statement square with the notion that gravitation in flat-spacetime with torsion can be physically equivalent to standard GR (e.g., The teleparallel equivalent of general relativity)?
Where did you get this from? Standard GR by definition assumes that spacetime is a pseudo-Riemannian manifold with a fundamental form being of Lorentzian signature (1,3), which implies that it is assumed to be torsion free and thus the connection being given by the then unique metric-compatible connection.

It's of course true that it's very likely that one needs some extension of GR because of the fact that we have to describe particles with spin, and this leads to the extension of GR to Einstein-Cartan theory with the spacetime manifold being one with torsion. This is, however, hard to observe since in astronomy we rather deal with macroscopic objects and electromagnetic radiation, all being described by classical continuum-mechanics and electrodynamical relativistic field theories, i.e., scalar fields and mass spin-1 gauge fields, which leads back to 0 torsion and thus to standard GR.
 
  • #26
vanhees71 said:
Where did you get this from?
I gave an example reference in post #23. But here is a more comprehensive review article published in February of this year: Teleparallel Gravity: From Theory to Cosmology. See §2.3 and §4.3 for the equivalence between teleparallel (torsion-based) and Einstein (curvature-based) gravity.
 
  • #27
renormalize said:
How does that statement square with the notion that gravitation in flat-spacetime with torsion can be physically equivalent to standard GR (e.g., The teleparallel equivalent of general relativity)?
That's not what the paper you reference says. It says that if you use the "teleparallel" formalism, you can still derive the metric tensor, the torsion-free Christoffel connection, and the Riemann curvature tensor of standard GR. So you can still formulate standard GR in the teleparallel formalism--but of course doing so discards torsion. If you want to make the torsion nonzero, you are no longer doing standard GR; you are using a more general theory formulated using the teleparallel formalism, which is now not equivalent to standard GR (and has no experimental evidence at all in its favor as compared to standard GR). This is all made clear in the first paragraph of the Introduction of the paper.
 
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  • #28
renormalize said:
the equivalence between teleparallel (torsion-based) and Einstein (curvature-based) gravity.
This claimed "equivalence" is based on shifting the meaning of the term "torsion". See Misner, Thorne & Wheeler, Box 10.2, p. 250, for the definition of "torsion-free connection" that is standardly used in GR. Briefly, that definition is that the covariant derivative ##\nabla## is said to be torsion-free if, for any two vector fields ##u## and ##v##, we have

$$
\nabla_u v - \nabla_v u = [u, v]
$$

Also note the statement there that a torsion-free connection in the sense given there is necessary for any theory of gravity that satisfies the equivalence principle.

In other words, physically, all of our evidence is that gravity is torsion-free, because all of our evidence is that the equivalence principle holds. Mathematically you can of course gerrymander the equations so that it looks like something that somebody wants to call "torsion" is nonzero, but the field equations still turn out to be the same as in standard GR. But that "torsion" is not the same as the physical sense of "torsion" that is standardly used in GR.
 
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  • #29
PeterDonis said:
That's not what the paper you reference says. It says that if you use the "teleparallel" formalism, you can still derive the metric tensor, the torsion-free Christoffel connection, and the Riemann curvature tensor of standard GR. So you can still formulate standard GR in the teleparallel formalism--but of course doing so discards torsion. If you want to make the torsion nonzero, you are no longer doing standard GR; you are using a more general theory formulated using the teleparallel formalism, which is now not equivalent to standard GR (and has no experimental evidence at all in its favor as compared to standard GR). This is all made clear in the first paragraph of the Introduction of the paper.
I think you're misinterpreting that paper. Let me quote from the introduction to another recent paper on teleparallel gravity (https://www.sciencedirect.com/science/article/pii/S0370269320302264):
"One of the most beautiful properties of General Relativity (GR) is its intimate alliance with the geometry of spacetime. Nowadays it is understood that the geometrical interpretation of gravity arises as a consistency requirement for the low energy effective theory describing an interacting massless spin-2 particle. Since Einstein first taught us how to think of gravity in terms of the curvature of spacetime, we have become acquainted with this description which has proven to be extremely useful for studying gravitational phenomena as well as exploring possible modifications of gravity.
However, the geometry of spacetime admits a much richer structure than that prescribed by GR once we unleash the affine sector. Remarkably, although rarely mentioned in standard textbooks, it is known that flat geometries with their well-defined notion of parallelism, provide alternative and fully equivalent representations of GR. On one hand, Weitzenböck spaces can host a Teleparallel Equivalent of GR (TEGR) [1]where gravity is identified with torsion. On the other hand, flat and torsion-free spacetimes only containing a non-trivial non-metricity can also accommodate a Symmetric Teleparallel Equivalent of GR (STEGR) [2,3]. In addition to the interest of these alternative formulations by themselves, they serve as different starting points to explore gravity theories beyond GR. The goal of this Letter is to extend previous studies in the literature on teleparallel geometries by allowing both torsion and non-metricity while keeping a trivial curvature..."
(emphasis added)
My takeaway is that the theory of GR can be consistently formulated in terms of a frame-field, along with only curvature, or only torsion, or even only non-metricity. All representations are physically equivalent, and the choice of one or another is purely a matter of convention and/or convenience.
 
  • #30
renormalize said:
I think you're misinterpreting that paper.
I think you're confusing two different meanings of "torsion", as I have already said. See further comments below.

renormalize said:
My takeaway is that the theory of GR can be consistently formulated in terms of a frame-field, along with only curvature, or only torsion, or even only non-metricity. All representations are physically equivalent, and the choice of one or another is purely a matter of convention and/or convenience.
And this is fine if you interpret "torsion" to mean a mathematical aspect of the formulation that has no physical consequences (because all of the different formulations you describe are physically equivalent). But that is not what I was using "torsion" to mean, or what "torsion" is standardly used to mean in the GR literature. "Torsion" in the standard meaning, as described in the MTW reference I gave, physically means that the equivalence principle does not hold. That is a physical difference, not just a different mathematical formulation of the same physical theory.
 
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  • #31
PeterDonis said:
And this is fine if you interpret "torsion" to mean a mathematical aspect of the formulation that has no physical consequences (because all of the different formulations you describe are physically equivalent). But that is not what I was using "torsion" to mean, or what "torsion" is standardly used to mean in the GR literature. "Torsion" in the standard meaning, as described in the MTW reference I gave, physically means that the equivalence principle does not hold. That is a physical difference, not just a different mathematical formulation of the same physical theory.
OK, I believe I understand your distinction. Your point is that adding torsion to a non-flat Levi-Civita connection necessarily violates the equivalence principle. In contrast, teleparallel-ism adds torsion to a pure-gauge (flat) connection and formulates a gravitational theory equivalent to GR (including the equivalence principle) using the torsion tensor in lieu of curvature.
 
  • #32
renormalize said:
Your point is that adding torsion to a non-flat Levi-Civita connection necessarily violates the equivalence principle.
No, that's not my point. It's not a matter of "adding torsion" to any particular connection. It's a matter of whether the covariant derivative that is obtained from the connection satisfies the condition I referenced from MTW or not. If it does, the connection is torsion-free and is consistent with the equivalence principle. If it doesn't, the connection has nonzero torsion and violates the equivalence principle. That is the definition of torsion I have been using and which is standard in the GR literature.

renormalize said:
teleparallel-ism adds torsion to a pure-gauge (flat) connection and formulates a gravitational theory equivalent to GR (including the equivalence principle) using the torsion tensor in lieu of curvature.
Here you are using "torsion" in a different sense, as I have already pointed out twice now. In the standard GR language I referenced from MTW, you are obtaining a torsion-free connection by taking a flat connection and adding a thingie to it that you are calling "torsion" (but not in the standard sense of "torsion") that makes the resulting connection obey the torsion-free condition I described above (because it has to in order to obey the equivalence principle). To say that you are "using torsion" to do this is confusing two different senses of the term "torsion". As noted, I have already said this twice before, and nothing you have posted has addressed it at all.
 
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  • #33
PeterDonis said:
To say that you are "using torsion" to do this is confusing two different senses of the term "torsion". As noted, I have already said this twice before, and nothing you have posted has addressed it at all.
To aid my understanding, can you cite a textbook or reference that clearly defines and contrasts these two distinct senses of torsion?
 
  • #34
renormalize said:
can you cite a textbook or reference that clearly defines and contrasts these two distinct senses of torsion?
I already cited you a textbook (MTW) that gives the standard sense of torsion. Wald, Chapter 3, has a similar discussion (property 5 of Section 3.1 is the same definition of "torsion-free" that MTW gives, though expressed in different notation).

I have no references for the other sense of torsion because I am not the one using it. You are.
 
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  • #35
PeterDonis said:
Who said it wasn't? We have said precisely the opposite in this thread: if the spacetime is flat, parallel transport is not path dependent; if the spacetime is curved, it is. So path dependence is the result of spacetime geometry.
In calculus. the path integral of a vector field with non-zero curl along different paths between two points will get different results. it just means that the movement of a scalar field is path dependent. A vector only contains more components than a scalar. So we can define a path dependent movement in a flat spacetime.
The path dependent integral (path dependent movement of a scalar ) is determined by curl of the vector field, it has nothing to do with the structure of space.
 

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