Can SR handle acccelerating frames?

[Ich]

This is the continuation of a discussion in https://www.physicsforums.com/showthread.php?t=431068".
Where does SR end and GR begin, in today's textbooks (not internet forums)? Do accelerating frames (coordinate systems) belong to also to SR, or is GR needed?
I say that they belong solely to GR, and find my view disputed with 4.5 : 4.5 people supporting each view.

it probably is not horribly important to draw the line anyway.

Granted, but I had more responses to that innocuous comment than to most of my disputed positions regarding real physics. So it seems people are interested.
I'm not an expert in history/philosophy of science or wherever this question belongs, so maybe I'm simply wrong. I'll argue nevertheless until someone convinces me.

Let me state also that in my opinion, if we want to separate theories, it makes sense to say flat spacetime->SR, curved spacetime->GR. I don't dispute that.
But I say that this is not how it is actually handled, and that the other position (which I believe is still the "official" one) also makes sense.


We had these quotes:

Misner, Thorne, and Wheeler, Gravitation, 1973, p. 163, "Accelerated motion and accelerated observers can be analyzed using special relativity."

Penrose, The Road to Reality, 2004, p. 422, "It used to be frequently argued that it would be necessary to pass to Einstein's general relativity in order to handle acceleration, but this is completely wrong. [...] We are working in special relativity provided that [the] metric is the flat metric of Minkowski Geometry M."

where I fully agree with DaleSpam:

Neither of these quotes refer to SR handling a non-inertial frame.

It is undisputed that SR can handle accelerated motion, as long as it is described in an inertial coordinate system. I need to see a quote concerning accelerated frames, not accelerated motion.

Now for the different objections:

we're left with Minkowski spacetime, which can be defined mathematically in at least three different ways: as a vector space, an affine space, or a manifold. But regardless of which of these options we choose, there's nothing that forces us to throw out non-inertial coordinate systems. In fact, it's quite unnatural to do so.

I don't think it's unnatural. An essential part of SR is what you must (or mustn't) read into its standard coordinates. You deal with inertial frames, and you use phrases like "the speed of light is constant". You teach Lorentz transformations, and you say you understood SR when you can handle them intuitively.

What a theory can handle or not can be established by experiments. It has nothing to do with human made definitions.

The "accelerated frame" is a human made definition, not something you can establish by experiment. So I think it's ok to let humans define whether a theory can handle a certain human-made definition.

Hmm, reading carefully, I see you have indeed classed the twin paradox as a general relativistic problem ... really?

If I did, that was not my intention.

General covariance (of the equations of motion) isn't the key principle of general relativity is it?

Yeah, general covariance (of all equations) the key principle of GR. Human made, agreed, but that's what defines GR. At least, that's my position.

SR classifies frames as privileged and non-privileged by exactly the same criteria as Newtonian mechanics. Exactly as in Newtonian mechanics, the laws of physics are form-invariant in different privileged frames, but have a different and more complicated form in the non-privileged frames. Exactly as in Newtonian mechanics, we can choose to use the non-privileged frames if we wish.

I'd say you almost have a point here. But there's one thing: Newton deals with accelerating systems by introducing fictitious forces, letting the meaning of coordinates unchanged.
But in relativity, fictitious forces mean geometry. You can't have fictitious forces without all sorts of weird coordinate transformations, i.e. time dilation, horizons, problems in defining large scale distance and such. You're in the same mess then whether spacetime is flat or not, you need the generally covariant formulation.
I say that SR excludes this kind of weird coordinates. You deal with it using general covariance and the mathematical apparatus of GR. This is GR. I know of no example where an accelerating frame is handled correctly in detail without using different line elements, as in GR.

There has been a historical evolution of our understanding of the best way to define the distinction. The definition in terms of curved versus flat spacetime has been widely accepted for decades, but it wasn't understood in 1905, or even in 1915.

That's your position, but I haven't seen your evidence yet.
I can certainly show that it all started with my position in 1915, and I've seen that there's a majority in internet forums for your position. But I've missed the change you proclaim in the literature, or even in Wikipedia. I still see GR and SR tied with the respective principles of covariance.

 

[starthaus]

This is the continuation of a discussion in https://www.physicsforums.com/showthread.php?t=431068".
Where does SR end and GR begin, in today's textbooks (not internet forums)? Do accelerating frames (coordinate systems) belong to also to SR, or is GR needed?
I say that they belong solely to GR, and find my view disputed with 4.5 : 4.5 people supporting each view.

Look up Rindler (chap 3.7) "Relativity, special , general and cosmological".

Not only that SR incorporates linearly accelerating frames, it also incorporates rotating frames (see https://www.physicsforums.com/blog.php?b=1893 ) as an example of non-inertial frames.

 

[JesseM]

Misner, Thorne, and Wheeler, Gravitation, 1973, p. 163, "Accelerated motion and accelerated observers can be analyzed using special relativity."

Ich, if you and DaleSpam don't think this quote refers to accelerated frames, why do you think they distinguish between "accelerated motion" and "accelerated observers"? I would think they're talking about the difference between analyzing an accelerating object in an inertial frame and analyzing it from the perspective of a non-inertial frame where the object is at rest, since in SR talking about what is experienced by a given "observer" is normally taken as shorthand for what measurements are made in a given object's rest frame.

Likewise, with Penrose's definition:

Penrose, The Road to Reality, 2004, p. 422, "It used to be frequently argued that it would be necessary to pass to Einstein's general relativity in order to handle acceleration, but this is completely wrong. [...] We are working in special relativity provided that [the] metric is the flat metric of Minkowski Geometry M."

Do you deny that whether the metric is flat or not is a coordinate-invariant issue, so that the metric would still be called "flat" if we expressed it in a non-inertial coordinate system? If you don't deny this, then Penrose's quote also implies that "we are working in special relativity" as long as spacetime is not curved, regardless of the choice of coordinate system.

The Usenet Physics FAQ, whose writers include a bunch of professional physicists and which is hosted on the site of physicist John Baez, says that "SR" is defined to include non-inertial frames in flat spacetime. See Can Special Relativity handle accelerations? which says:

Accelerating reference frames are a different matter. In GR the physical equations take the same form in any co-ordinate system. In SR they do not but it is still possible to use co-ordinate systems corresponding to accelerating or rotating frames of reference just as it is possible to solve ordinary mechanics problems in curvilinear co-ordinate systems. This is done by introducing a metric tensor. The formalism is very similar to that of many general relativity problems but it is still special relativity so long as the space-time is constrained to be flat and Minkowskian. Note that the speed of light is rarely constant in non-inertial frames and this has been known to cause confusion.

Likewise, the section on the Twin paradox has a subsection on The Equivalence Principle Analysis where they talk about how a non-inertial coordinate system will include a "uniform pseudo-gravitational field", and then they about talk what Einstein defined as the central features of "general relativity" vs. what modern physicists define GR to mean, explaining that modern physicists would not count this pseudo-gravitational field as a "real" gravitational field:

Here's one version of Einstein's 1907 list (without worrying too much about the fine points):

General Principle of Relativity
All motion is relative, not just uniform motion. You will have to include so-called pseudo forces, however (like centrifugal force or Coriolis force).

Principle of Equivalence
Gravity is not essentially different from any pseudo-force.

The General Principle of Relativity plays a key role in the Equivalence Principle analysis of the twin paradox. And this principle gave General Relativity its name. Even in 1916, Einstein continued to single out the General Principle of Relativity as a central feature of the new theory. (See for example the first three sections of his 1916 paper, "The Foundation of the General Theory of Relativity", or his popular exposition Relativity.)

Here's the modern physicist's list (again, not sweating the fine points):

Spacetime Structure
Spacetime is a 4-dimensional riemannian manifold. If you want to study it with coordinates, you may use any smooth set of local coordinate systems (also called "charts"). (This free choice is what has become of the General Principle of Relativity.)

Principle of Equivalence
The metric of spacetime induces a Minkowski metric on the tangent spaces. In other words, to a first-order approximation, a small patch of spacetime looks like a small patch of Minkowski spacetime. Freely falling bodies follow geodesics.

Gravitation = Curvature
A gravitational field due to matter exhibits itself as curvature in spacetime. In other words, once we subtract off the first-order effects by using a freely falling frame of reference, the remaining second-order effects betray the presence of a true gravitational field.

The third feature finds its precise mathematical expression in the Einstein field equations. This feature looms so large in the final formulation of GR that most physicists reserve the term "gravitational field" for the fields produced by matter. The phrases "flat portion of spacetime", and "spacetime without gravitational fields" are synonymous in modern parlance. "SR" and "flat spacetime" are also synonymous, or nearly so; one can quibble over whether flat spacetime with a non-trivial topology (for example, cylindrical spacetime) counts as SR. Incidentally, the modern usage appeared quite early. Eddington's book The Mathematical Theory of Relativity (1922) defines Special Relativity as the theory of flat spacetime.

So modern usage demotes the uniform "gravitational" field back to its old status as a pseudo-field. And the hallmark of a truly GR problem (i.e. not SR) is that spacetime is not flat. By contrast, the free choice of charts---the modern form of the General Principle of Relativity---doesn't pack much of a punch. You can use curvilinear coordinates in flat spacetime. (If you use polar coordinates in plane geometry, you certainly have not suddenly departed the kingdom of Euclid.)

 

[bcrowell]

I need to see a quote concerning accelerated frames, not accelerated motion.

The MTW quote *is* referring to accelerated frames, not just accelerated motion. That's why the sentence starts with "Accelerated motion and accelerated observers..." They further clarify on p. 164: "An accelerated observer can carry clocks and measuring rods with him, and he can use them to set up a reference frame (coordinate system) in his neighborhood."

More examples:

Taylor and Wheeler, Spacetime Physics, 1992, p. 132: "DO WE NEED GENERAL RELATIVITY? NO! [...] 'Don't you need general relativity to analyze events in accelerated reference frames?' 'Oh yes, general relativity can describe events in the accelerated frame,' we reply, 'but so can special relativity if we take it in easy steps!'"

Hobson, General Relativity: An Introduction for Physicists, 2005, sec. 1.14, discusses "Event horizons in special relativity" from the point of view of accelerated observers, using coordinates defined in their accelerated reference frames.

Schutz, A First Course in General Relativity, 2009. Schutz equivocates on pp. 3 and 141 about the status of accelerated observers in SR, but says, "[...] the real physical distinction between these two theories is that special relativity (SR) is capable of describing physics only in the absence of gravitational fields, while general relativity (GR) extends SR to describe gravitation itself."

So out of five post-1950 books that said something about it, all five clearly say that the SR/GR distinction is flat versus curved.

 

[yuiop]

As I understand it, linear and angular acceleration (including the analysis of what an observer in a accelerating reference frame measures) in a "flat background" can be handled by SR. GR is required if there is a gravitational source and the spacetime is curved.

 

[Dale]

Ich, if you and DaleSpam don't think this quote refers to accelerated frames, why do you think they distinguish between "accelerated motion" and "accelerated observers"? I would think they're talking about the difference between analyzing an accelerating object in an inertial frame and analyzing it from the perspective of a non-inertial frame where the object is at rest, since in SR talking about what is experienced by a given "observer" is normally taken as shorthand for what measurements are made in a given object's rest frame.

Excellent point. I think I didn't read carefully enough. I agree that the MTW quote does refer to a non-inertial coordinate system.

Do you deny that whether the metric is flat or not is a coordinate-invariant issue, so that the metric would still be called "flat" if we expressed it in a non-inertial coordinate system? If you don't deny this, then Penrose's quote also implies that "we are working in special relativity" as long as spacetime is not curved, regardless of the choice of coordinate system.

I am going to stick with my original assessment on this one. He not only said that it was flat (which as you point out is a coordinate independent notion) but that it was the Minkowski metric. The Minkowski metric is the metric of an inertial reference frame in flat spacetime. You could also consider spherical coordinates, or rotating coordinates, both of which would be flat but neither of which would have the Minkowski metric.

 

[Dale]

I'd say you almost have a point here. But there's one thing: Newton deals with accelerating systems by introducing fictitious forces, letting the meaning of coordinates unchanged.
But in relativity, fictitious forces mean geometry. You can't have fictitious forces without all sorts of weird coordinate transformations, i.e. time dilation, horizons, problems in defining large scale distance and such. You're in the same mess then whether spacetime is flat or not, you need the generally covariant formulation.
I say that SR excludes this kind of weird coordinates. You deal with it using general covariance and the mathematical apparatus of GR. This is GR. I know of no example where an accelerating frame is handled correctly in detail without using different line elements, as in GR.

Ehrenfest did all of his work on rotating frames without any of that AFAIK. It can be done, it is just a pain. I don't see a difference, in principle, between non-inertial frames in Newton and SR. In both cases the basic laws are stated in terms of inertial frames and you just have to be careful with the math to work in others.

 

[JesseM]

I am going to stick with my original assessment on this one. He not only said that it was flat (which as you point out is a coordinate independent notion) but that it was the Minkowski metric. The Minkowski metric is the metric of an inertial reference frame in flat spacetime. You could also consider spherical coordinates, or rotating coordinates, both of which would be flat but neither of which would have the Minkowski metric.

Isn't there some ambiguity when talking about a metric as to whether we are talking about the equations when expressed in a particular coordinate system or the underlying coordinate-invariant geometry? For example, Kruskal-Szekeres coordinates can be used to cover the same nonrotating black hole spacetime which Schwarzschild coordinates are often used on, and this book refers to "the Schwarzschild metric, and the Kruskal-coordinate description of this metric." Likewise this book refers to the "Schwarzschild metric in Kruskal coordinates", and this book says that Kruskal-Szekeres coordinates "provide the maximal analytic extension of the Schwarzschild metric". On the other hand other authors do distinguish between the "Schwarzschild metric" and the "Kruskal metric" so as I said it seems like there's some ambiguity in the terminology.

 

[Hurkyl]

You deal with it using general covariance and the mathematical apparatus of GR. This is GR. I know of no example where an accelerating frame is handled correctly in detail without using different line elements, as in GR.

This, I suppose, is where your problem is. The "mathematical apparatus of GR" is not a synonym for GR; it is merely differential geometry. And in this case, one only needs the fragment that is easily expressed in terms of elementary multi-variable calculus is enough

 

[Dale]

Isn't there some ambiguity when talking about a metric as to whether we are talking about the equations when expressed in a particular coordinate system or the underlying coordinate-invariant geometry?

Yes, I have noticed that ambiguity on occasion also. In this case I think they were referring specifically to the Minkowski coordinates in flat spacetime since they used both the words "flat" and "Minkowski", which seems specific to me.

 

[atyy]

Yeah, general covariance (of all equations) the key principle of GR. Human made, agreed, but that's what defines GR. At least, that's my position.

I see. That is almost certainly what Einstein believed, and which historically led to the invention of GR. I think he was wrong.

In my view, even the equations of SR can be written in generally covariant form (just use Christoffel symbols). It is true that it's not useful to do so, but it's just a change of coordinates, and we can always change coordinates. I agree that a distinguishing feature of SR is the existence of global inertial frames related to each other by Lorentz transformations. But these frames immediately imply the existence of non-inertial frames, so SR must handle accelerates frames (just that it's usually not useful to do so).

The distinguishing feature of GR is that apart from topology and signature, the metric is not fixed until the matter content is specified, and the matter content (or at least its stress-energy) is not specified until the metric is specified.

After all, there are relativistic theories of gravity like Nordstrom's second theory, which can be written in generally covariant form, but yet are not general relativity. Like in Eq 16 of http://arxiv.org/abs/gr-qc/0405030 .

 

[Al68]

This is the continuation of a discussion in https://www.physicsforums.com/showthread.php?t=431068".
Where does SR end and GR begin, in today's textbooks (not internet forums)? Do accelerating frames (coordinate systems) belong to also to SR, or is GR needed?
I say that they belong solely to GR, and find my view disputed with 4.5 : 4.5 people supporting each view.

I would note that Einstein wrote about accelerated reference frames in SR, gravitational time dilation in accelerated reference frames in SR, and predicted gravitational time dilation in gravitational fields as a result of the equivalence principle, all prior to 1915 (prior to GR).

I can't find all the links to all the papers quickly, but here's a link to a 1911 paper I found with a quick search: http://einstein.relativitybook.com/Einstein_gravity.html#FN_001

 

[Fredrik]

bcrowell has already given a few examples of books that draw the line between SR and GR at flat vs. curved. Wald doesn't address this issue directly, but says that spacetime is a smooth manifold (in all the theories from pre-relativistic physics to GR). Since the definition of "smooth manifold" includes all coordinate systems, I have to count that as one more for our side.

To define a version of SR that only includes inertial coordinate systems, we would have to be very explicit when we define it: Let "spacetime" M be the set $$\mathbb R^4$$ with the standard topology, the standard vector space structure, the Minkowski form, and the set of proper orthochronous Poincaré transformations. The members of that last set are also called "(inertial) coordinate systems".

I've never seen SR (or "coordinate system") defined like that. Actually, I don't think I've ever seen a textbook define any of the theories it's talking about, but I should probably change the subject quickly, so I don't spend half the day ranting about that. I'll just add that since no one ever writes down exact definitions of theories of physics, it's going to be impossible to settle this by examining statements made in books.

The point I was trying to make above is that if an author uses the term "manifold", or even if he uses the term "coordinate system" without explicitly defining it to be a proper orthochronous Poincaré transformation, he has already allowed all the "non-inertial" coordinate systems to be a part of the theory, even if he didn't understand that. We probably shouldn't even be talking about whether non-inertial coordinate systems are part of the mathematical framework or not. They clearly are. The more relevant issue is: Can a non-inertial coordinate system define a non-inertial object's "point of view"?

The standard way to associate a coordinate system with a world line relies on a synchronization procedure. Applied to a timelike geodesic, this procedure gives us a global inertial frame. (Here "global" just means that its domain is spacetime, not a proper subset). Applied to any other timelike curve, it fails to produce a global coordinate system. So it seems to me that the real issue is: Are we willing to consider a local coordinate system a "point of view"? A "yes" to that question is essentially the "general principle of relativity", and I guess that's why some people consider it GR.

Let G be the general principle of relativity and E Einstein's equation. We could certainly define SR to not include G, and GR to include both G and E. I can't argue that that definition of SR is wrong. I just find it very strange, like a definition of "hand" that doesn't include the index finger. Also, consider an accelerating rocket in flat spacetime. As long as we never consider the pilot's point of view, we're doing SR, and we can think of the rocket as a solid object held together by internal forces, but the moment we consider the pilot's point of view, we also have to start thinking of the rocket as consisting of non-interacting test particles, because we have switched to another theory where mass and interactions change the dynamics.

 

[bcrowell]

Having looked at some of the early presentations of GR and thought about this a little more, I think I have a better understanding of why so many people historically chose the wrong definition of SR versus GR, and why it took another 30 years for most relativists (with the notable exception of Eddington, who got it right in 1924) to converge on the definition that is both better and now universally accepted. It basically has to do with Einstein's Machian aspirations for his theory, which he expressed prominently in his 1915 paper, and which turned out not to be fulfilled. I've written up my take on the matter here: http://www.lightandmatter.com/html_books/genrel/ch01/ch01.html#Section1.5 See subsection 1.5.4, under the heading "Terminology."

 

[yuiop]

: Can a non-inertial coordinate system define a non-inertial object's "point of view"?

Would you classify Rindler coordinates as GR or SR? The "point of view" of accelerating observers can certainly be defined in Rindler coordinates but I would personally classify them as belonging to SR. In any linear or angular accelerating situation where the background is flat, we can always find at least one reference frame where the speed of light is constant and isotropic over extended distances and radar distances are isotropic and in agreement with ruler measurements over extended distances which is impossible in any situation where there is a gravitational source and curved space-time. Similarly, in any situation where linear or angular acceleration is not due to a gravitational source, we can always find a reference frame where geometry is Euclidean in any direction and over extended distances, which again is not true if there is a gravitational source. Similarly, where there is linear or angular acceleration in a flat background, there is always a reference frame where parallel transport of a vector in any direction over any path always results in the vector still being parallel to a non transported vector when the transported vector is returned to the non transported vector. These are all various informal ways of defining a flat background and if there is a flat background it belongs to SR.

 

[Fredrik]

Would you classify Rindler coordinates as GR or SR?

Definitely 100% SR.

 

[Ich]

Thanks for all the replies.

More examples: [...]

okay, okay, you win. Obviously, there is something like a consensus concerning the distinction that supports your view. Thanks for the references.

That is almost certainly what Einstein believed, and which historically led to the invention of GR.

Wald doesn't address this issue directly

Funny enough, my position relied on Einstein's 1915 paper and a remark in Wald, where he said that at least the names of the theories stem from the respective covariance principle. Maybe I misinterpreted what he wrote, or maybe he in fact supports a minority position. I don't remember the exact wording and have no copy at hand.
Ah, and of course I checked http://en.wikipedia.org/wiki/General_covariance" where I found (without references):

Albert Einstein proposed this principle [Ich: i.e. general covariance] for his special theory of relativity; however, that theory was limited to space-time coordinate systems related to each other by uniform relative motions only, the so-called "inertial frames." ...

This, I suppose, is where your problem is. The "mathematical apparatus of GR" is not a synonym for GR; it is merely differential geometry.

I think the case is somehow special, as GR introduced differential geometry in physics. The physical theory and the mathematical framework came hand in hand and are more strongly connected than usual: The physics of GR actually is geometry.
Today, it's almost a truism that physics can be expressed in a coordinate-independent way and that it must be independent of coordinates. That was far from obvious 1915 (at least to physicists), so I think it's understandable that this has been seen as a nontrivial part of GR. In fact, I think that even today students of GR struggle as much with differential geometry as with the actual physical content of GR, and that this is a major contributor to the feeling that GR is much more complicated than SR.

I still would like to make the distinction along with the necessary mathmatics, and leave SR with its old school (Poincaré-covariant, not generally covariant) formulation alone, as it IMHO still reflects the way the transition from SR to GR is experienced by learners. Of course, this distinction would be obselete when all of university-level physics is finally teached in a coordinate independent way.

But that's another topic, here I concede that SR can handle inertial frames.

 

[JustinLevy]

I agree with Fredrik.

SR can handle anything in flat spacetime.
GR is required as soon as you want curved spacetime (gravitational interactions).

And to make it absolutely clear, by curvature of spacetime I am referring to the intrinsic curvature. This is a purely geometric and coordinate system independent measure. If you start in flat spacetime in inertial coordinates, and apply a truly bizarre coordinate transformation to get a new coordinate system... the spacetime is still flat. SR can still be used.

I think that is where people are getting confused. Just because the metric isn't a simple -1,1,1,1 diagonal doesn't mean spacetime is curved and we need GR.

 

[Mentz114]

This is an interesting thread, especially to someone who believed ( along time ago ) that there was a 'Rindler spacetime'. The methodology for handling acceleration in SR is pure SR, a local frame is boosted to give a new chart. The physics is transparent and the results illuminating. The most interesting things in SR happen in accelerated frames, for instance the horizon for accelerated frames, and the change in 3D geometry in rotating frames.