This discussion is not about contradicting SR, GR, or 4D spacetime. We do not believe in SR or 4D spacetime, we have seen them too many times, we have seen SR work in accelerated motion, provide the basis for GR, and for quantum mechanics, in hundreds of applications, including MRI.
These are the points of discussion here: (a) Why take the 4D spacetime formulations as the preferred choice in SR? and, (b) Do the 4D spacetime formulations form a coherent body, a corpus, that we can use?
We provide some answers to these two questions, for discussion.
IINTRODUCTION
Different formulations of special relativity (SR) exist, such as by Minkowski [1], by Burgess [2], the original formulation by Einstein [3], by Mashoon [4], by Taylor and Wheeler [5], all different, as our different views of reality. Also, such as Clifford(3,1), where one can accept SR but not 4D, considering 3D +1D. Or, even Newtonian mechanics, only with 3D and time as absolute.
Here, we see all of them as right, but having different regions of validity, ordered as 4D ⊃ 3D +1D ⊃ 3D, in set theory, a fundamental theory in maths.
In other words, the other theories with less dimensions are all grandfathered in 4D spacetime, even those not mentioned, with nothing to discuss in that aspect.
Also, the question answers a common, cornucopia of questions of someone who begins to study SR, even in most US universities and colleges, for example, and is taught along the original formulation by Einstein [3], which is inconsistent and incomplete. The Lorentz transformation is introduced as a first-fiddle in [3], not as consequence, as in [1,2]. This creates most problems.
HIGHER LEARNING
Students may think, as reported many times, and in RG, that SR only applies to inertial motion. But, acceleration does not shut-off length contraction or time dilation. While students are fortunate to escape "relativistic mass" (not used by [3] but used originally by Einstein), they do not usually escape "rest mass".
SR as spacetime, applies to ARBITRARY motion, but we see that students are still, 100 years later, often learning a historical subset of it, and are poorly prepared for GR, taught now in undergratuate courses, that uses the same formulation of spacetime, found in [2], and hinted in [1].
The discussion aims to unite these aspects, showing a coherent body, a corpus, around the Minkowski formulation [1], such as we view today, in [2], of 4D spacetime (see NOTES, 1)
This should prevent questions we found even among researchers and in higher education, well-known professors in the US, Europe, and in RG, such as about the "inconsistencies in SR" -- meaning the first formulations by Einstein, Poincaré, or even 100+ years ago by Minkowski [1]. This us list in time. Physics has evolved out of this quagmire, using it as a support and motivation to escape.
Today, the formulation by Burgess [2], for example, would clarify most questions in SR and GR. For example, that SR and GR are not independent. There is value, for some students, to learn them at the same time, the spacetime is the same, the metric changes.
Or, about the source of magnetism, which is due in SR 4D spacetime today to the movement of charge, only. This, particularly prevents a student from thinking, a la Maxwell and Faraday, that magnetism is somehow independent from electrostatics, and also shows why MKS is not used here -- the units must be the same for B and E.
Other difficulties exist, such as listed in the growing list in [6], also [6-9]. Someone, for example, could also not avoid gimbal lock in 3D rotations, but that becomes easy in 4D [9].
While this discussion is more comprehensive, we list as important, but superfluous, some apparent stumbling blocks in gauge theories, and quantum mechanics -- use of the Schrödinger equation, the use of Clifford algebra in physics, such as STA and Cl(3,1), and other preventable problems. Participants are iinvited to add their favorite misconception, or other candidates for corpus.
Thus, formulations of the SR may change, and many had now comic misconceptions, such as "relativistic mass", whereas many points in the SR, as a corpus, are considered experimentally confirmed beyond reasonable doubt, described in 4D spacetime.
MINKOWSKI
It was Hermann Minkowski (Einstein's mathematics professor) who announced the new four-dimensional (spacetime) view of the world in 1908, which he deduced from experimental physics by decoding the profound message hidden in the failed experiments designed to discover absolute motion. Minkowski realized that the images coming from our senses, which seem to represent an evolving three-dimensional world, are only glimpses of a higher four-dimensional reality that is not divided into past, present, and future since space and all moments of time form an inseparable entity (spacetime). [1] ff.
Einstein's initial reaction to Minkowski's view of spacetime and the associated with it four-dimensional physics (also introduced by Minkowski) was not quite favorable: "Since the mathematicians have invaded the relativity theory, I do not understand it myself any more.“
However, later Einstein adopted not only Minkowski's spacetime physics (which was crucial for Einstein's revolutionary theory of gravity as curvature of spacetime), but also Minkowski's world view as evident from Einstein’s letter of condolences to the widow of his longtime friend Besso: "Now Besso has departed from this strange world a little ahead of me. That means nothing. People like us, who believe in physics, know that the distinction between past, present and future is only a stubbornly persistent illusion." Besso left this world on 15 March 1955; Einstein followed him on 18 April 1955.
ONTOLOGICAL QUESTION
Turning to the ontological question, what is place in the scheme of things in nature, of SR 4D spacetime: maths or physics?
Is 4D spacetime just a use of mathematics, or does it represent nature in ways that using lesser dimensions just can not do? Is 4D spacetime like a spreadsheet of calculations, not really nbecessary, for a reality that is entirely experienced as 3D +1D, or even 3D with time as a parameter?
Names seem to not matter, as expected, in physics. it would seem that it does not matter, then, if someone, for example, chooses a geodesic to extend Newton's First Law, overcome beautifully our limitation then (to define "straight line"), and designates it as an extended "inertial motion" -- the geometric basis is the same, the physics did not change. Or, like Mashhon, describe an infinte family of comoving observers, 3D, each one as a flash in time. One would seem to need less mathematics, it can all be solved in 3D +1D, even 3D, not needing 4D. But the context is just more limited, the region of validity is less, as we show next.
It is physically limiting to use names that do not distinguish curvature as stretching (as done in 4D spacetime), only considering curvature as bending. The spacetime that was being considered is flat, it can be seen in 3D like a stretched, flat pizza dough. The spacetime is flat but if the world trajectory is curved, it still fits with curvature as intrinsic, as stretching, so not just world straight lines "fit" in flat spacetime!
The 4D spacetime, in SR or GR, therefore, works for world-curves, as they may curve, in the world we see as flat spacetime, locally, as SR, or curved spacetime, as we see in GR. Only the 4D spacetime formulation of SR and GR can do that, we cannot use lesser dimensions, may be more.
The visible world, however, as viewed, is NOT 4D spacetime, it seems at first sight. The world is apppearing to us as 3D +1D, or 3D plus time as parameter. But, the world is governed in 4D spacetime, which manifests itself through laws that 3D or 3D+1D cannot represent, but we can see in everyday experience -- just like we do not see the Earth going around the Sun, we see the Sun going around the Earth, apparently, but we do have seasons, and the ancient Greeks measured the circumference of the Earth, with no rockets, quite correctly, through mathematics.
Therefore, the relatity, the world we live in, the truth-conditions, is at least 4D. Some just experience life less, as 3D+1D, or 3D, or even 2D. But the mechanism works in at least 4D, and 3D cannot represent it, as shown in SR and GR experiments. For example, with length contraction or time dilation for non-comoving observers, only.
Therefore, there is no ontological status anymore, and this is beautiful, not just right. This is also, perhaps, new to philosophers.
CONCLUSION
We suggest that a consistent picture emerges supporting Minkowski's view [1,2,] with no ontological question. This includes Einstein, who first rejected it publicly, at first intellectually stymied, as with the now-called Hubble flow in Cosmology, but confimed it, mathematically expanded on it, and used it further in physics, in his theory of general relativity. We are all working in cooperation, even those who do not seem to cooperate, do -- they promote patience, discussion, new ideas, new values.
PAUSE. If your notion of SR does not work with accelerated or arbitrary motion, OK, do not panic; spacetime is SR that is valid (works all the time) for accelerated motion, for non-inertial, ARBITRARY, reference systems.
Here, one should try and clarify any questions first with the discussion text itself, and then with the references by Burgess or Wheeler, cited next -- guessing definitions may not interwork!
Note that comoving and non-comoving, as well as length contraction and time dilation, are not "just words" or optical illusions -- what they denote can cross a barrier, can produce thermodynamic work. They define different physics, different truth-conditions, not just different truth values.
This may change according to the formulation of SR one uses, but we suggest that [1,2] can offer a consistent view. Later on, SR and GR can be complemented with Cosmology, and Hubble flow, without contradictions, even when comoving points separate in inertial motion, and galaxies move faster than light [7 ].
As a result, may also the rigid time in undergraduate courses [8] be better spent. Someone, for example, could also not avoid gimbal lock in 3D rotations, but that becomes easy in 4D [9].
The SR as a 4D spacetime so described, is more and more, seen in all sciences, fitting the original name of a "theory of invariants". Building a verified, coherent, corpus supporting [1] in 4D and (not only) modern variants such as [2] -- which we examine in this discussion, below, where other researchers are invited to add their views in the answers.
NOTES
1. In SR and GR spacetime, accelerated motion is defined absolutely, for example, with no absolute reference system or geodesics, of course, by extrinsic and intrisic geometric properties of the curvature.
So, it does not matter if someone chooses a geodesic to extend Newton's First Law, overcome beautifully our limitation then (to define "straight line"), and designates it as an extended "inertial motion" -- the geometric basis is the same, the physics did not change, the context is just more limited.
It is limited to not distinguishing curvature as bending, only considering curvature as stretching, in spacetime -- the spacetime that was being considered is flat, like a stretched, flat pizza dough. The spacetime is still flat if the world trajectory is curved, not just world straight lines "fit" in flat spacetime!
It all works for geodesics, world-curved as they may in the world we see as flat spacetime, locally, which is NOT spacetime but governed by a flat spacetime, there is no ontological status anymore, and this is beautiful, not just right. This is also, perhaps, new to philosophers.
But... it does not work in GR, under free fall in a gravitational field, because the spacetime is curved under gravity, as bending. So, inertial motion must be redefined, not as a name, but as a truth-condition. This is how GR actually exists, we are NOT reinventing the wheel, just may be going a bit further into what GR means, not just what it says.
REFERENCES
[1] H. Minkowski. Space and Time: Minkowski's Papers on Relativity. 2012. Online at: http://rgs.vniims.ru/books/spacetime.pdf
[2] C. P. Burgess. General Relativity: the Notes. 2009. Online at: http://www.physics.mcmaster.ca/~cburgess/Notes/GRnotes.pdf
[3] While the original but limited formulation of special relativity is still taught at various college-level textbooks, such as
Article Physics: For Scientists and Engineers J. .W. Jewett and R.A. Serway. Physics for Scientists and Engineers with Modern Physics. Thomson Brooks/Cole. 2012
[4] In another version of the SR, by Mashhoon, an accelerated observer is in effect replaced — on the basis of the hypothesis of locality — by a continuous infinity of hypothetical momentarily comoving inertial observers. Here, the accelerated observer and the otherwise identical instantaneously comoving inertial observer have the same velocity and position. In SR, two observers comoving are defined as having the same velocity and position. Cited by 36 references, with 14 occurrences of "comoving", Mashhoon paper at arXIv may further his treatment of accelerated motion in SR, and illuminate the use of "comoving" in SR and Newtonian :
at https://arxiv.org/pdf/0805.2926
[5] For the spacetime formulation, also this book:
Taylor and Wheeler. Spacetime Physics. . W. H. Freeman and Company. 1966.
[6] https://www.researchgate.net/post/What_are_the_worse_yet_enduring_misconceptions_about_mass_and_energy_in_special_relativity
[7] Cosmology, SR, and GR, at http://www.astro.ucla.edu/~wright/cosmo_02.htm
[8] This question, and links as well as comments, even apparently dissonant comments, may help provide a solution to free time in undergraduate courses, and today:
https://www.researchgate.net/post/Should_college_students_learn_about_electromagnetism_before_mechanics
[9] https://www.researchgate.net/post/Is_gimbal_lock_a_proof_at_any_speed_of_an_underlying_4D_spacetime
DISCLAIMER: We reserve the right to improve this text. All questions, public or not, are answered here. This will help make this discussion text morecomplete. References are also provided by self-search. This text may modify frequently, this is version 4.50.
Well, actually in some way SR can be adapted to some acceleration problems.
Say the constant acceleration problem starting from x=0 at t=0, where A is the acceleration.
The proceedure uses the Lagrangian
L=-mcc sqrt(1-bb) - mAx
where b = v/c is the usual beta.
After formulating the appropriate Euler Lagrange equation of motion, eventually solve with correct initial conditions, one can show the limit x=1/2 A tt at small t
and v tends to c for large times.
The Lagrangian is not that obvious, except that it leads to the correct equation of motion.
What is meant by "Different formulations of special relativity (SR) exist..."
It boils down to Minkowski space-time and the Poncare transformations. For sure there will be different ways you can approach this. But what do you specifically mean?
Juan Weisz and Andrew James Bruce : Thank you. The text included the most important formulations in the References, and elected Minkowski as the dominant formulation, pointing to one (of many) modern variants as in Burgess. To make it clear, they were now added explicitly in the text:
Also, it is now noted that the question answers a common, cornucopia of questions of someone who begins to study SR, in most US universities and colleges, for example, also by well-known professors, and in RG.
See please, in the head text.
It is not that they are poorly prepared for GR, at least in my time GR was not taught even at the graduate level in Physics, other than for students with a specific interest in researching in this area (a minority, possibly dealing with astronomy).
However , between basic modern physics course, an undergraduate electromagnetic and finally a graduate level electromagnetics a la Jackson, and exposure from Clasical Mechanics (Goldstein) I do not think we have missed out much which has to do with SR. Of course everything has probably gradually improved since Einstein.
To some extent Im self taught in GR, but miss being an expert.
At college level you already notice students resisting the idea of SR or the Quantum, seems they take it out on RG.
The trouble with magnetism coming from the magnetic dipole moments of elementary particles, then atom, then molecules, then solids means a whole lot of quantum theory and solid state, coming gradually in from separate courses, or research.
However there is reasonable exposure in mid level electromagnetics, ie.
Reitz and Milford, though with yet incomplete understanding.
The standard modern understanding of special relativity is in terms of pseduo-Riemannian geometry. In particular, Minkowski space-time which is the topological manifold R^4 (or whatever dimensions) equipped with the Minkowski metric. (There are some conventions here also.) Naturally the isometries of this manifold play a central role. While the inertial frames play a big role, one is free to use any frame. Typically, this will mean we have fictitious forces acting on test particles. Mathematically one can see this as non-trivial Christoffel symbols when building the Levi-Civita connection. However, the space-time is still flat.
Anyway, taking a geometric picture from the start makes SR feel more like GR rather quickly, especially when using arbitrary coordinates.
Electromagnetism is also a geometric theory - manifolds, bundles and connections are all central to the theory.
So, in my opinion, learning a little differential geometry helps with all aspects of classical physics. With that in mind, I would not take the early papers by Minkowski or Einstein as my guide.
But I know from experience that undergraduate physics can be all about formula with no or little mathematical understanding. Which in this case is a shame as the theory is quite beautiful once presented in the right language.
Andrew James Bruce : Thank you. Yes, differential geometry is key, and not only to SR and GR.
When Gauss defined the curvature of a surface, for example, he made explicit use of the way the surface sits in space. Evidently, this, how it sits in space, is an extrinsic “curvature as bending” rather than the intrinsic “curvature as stretching” that he defined.
The latter is in the basis of spacetime, the geometry of special relativity (avoiding Einstein first approach) and general relativity.
Burgess notices that this metric is not positive definite, unlike the metrics considered when thinking about the geometry of three-dimensional space. But distance is positive and agrees with our notion of distance in flat space if it is restricted to a purely spatial interval.
The possibility that "distance" or, better said, the interval, can be zero or negative is the main reason why the geometry of spacetime differs from the geometry of four-dimensional space.
Ed Gerck for sure the fact that the metrics used are really pseudo-metrics, i.e., the induced norm is not positive definite is quite important. Physically it gives the causal structure - so on Minkowski space-time the familiar cone or dimond. Mathematically it also leads to problems when you start to do analysis. But from a Riemannian geometry point of view there is little difference as all the basic constructions and theorems go through.
I will add that usually one does not think of space-time as embedded in some higher dimensional space, though there is no reason why you could not do this. Moreover, when dealing with string theory and brane cosmologies one does think about our space-time as being sat in the "bulk". But if we restrict to standard GR - which includes SR - then we don't need to think of our world sitting in a higher dimensional one.
Anyway, if one is thinking of research into the nature of space-time, then geometry, groups and representation theory are key. In fact, they are key to a lot of physics.
It is quite strange to hear about many interpretations of SR nobody heard of (try google "Burgess interpretation of SR" or "Mashoon interpretation of SR" and have fun) but the most important alternative interpretation - the Lorentz ether - not being mentioned at all. In comparison with the Lorentz ether, all what has been listed are essentially variants of the same spacetime interpretation. The exception of Einstein's original one, which is not really an interpretation at all.
That SR "can be complemented with Cosmology, and Hubble flow, without contradictions" is wrong, cosmology is GR domain, you cannot do it without gravtiy. The link given is ok itself but assumes GR.
Ed, you write
"But the mechanism works in at least 4D, and 3D cannot represent it, as shown in SR and GR experiments. For example, with length contraction or time dilation for non-comoving observers, only."
This remains wrong, as proven by the Lorentz ether, where only a 3D world exists and changes in time. Time is not a dimension in this interpretation, but only a parameter which describes the change. It nonetheless correctly describes (represents?) all observable relativistic effects.
" In other words, the other theories with less dimensions are all grandfathered in 4D spacetime, even those those not mentioned, with nothing to discuss in that aspect."
And they are all grandfathered by the theory that together with our world also invisible pink unicorns exist. With nothing to discuss about theories without pink invisible unicorns. Not?
Ed, if you would agree with me about the unicorns, you would have to agree also that claims made now that they exist are not at all supported by any empirical evidence. Ok? Despite the fact that all our theories are "grandfathered" by the theory that unicorns exist (which you have agreed with, afaiu). And it follows by a trivial analogy that your argument that all theories inclusive the Lorentz ether are "grandfathered" by some 4D theory shows nothing.
If you really accept that your grandfathering relation proves nothing, fine, let's start with physics. I was not able to see any physics yet proving that the Lorentz ether (where exist only 3D things changing in time) is somehow insufficient to describe it.
But this is what counts. Remember Ockham's razor: Don't introduce entities without necessity. So, as long as the Lorentz ether, where we have only a 3D world with entities changing in time, is sufficient to describe the world we observe, there is no necessity to introduce a full 4D spacetime as existing.
The Lorentz ether has a quite strong proving power (I don't know what "comproving" means), because in the simple SR variant classical spacetime SR is completely equivalent in all its physical predictions to the Lorentz ether, and for the generalization to relativistic gravity space time GR and the Lorentz ether have the same equations and are at least locally equivalent. So, there are some equivalence theorems, which, with some restrictions, allow to show that the Lorentz ether is sufficient as long as one assumes that SR/GR are sufficient.
Namely, whenever you present an experiment, and a description of this experiment in terms of a SR or GR solution, the equivalence theorems allow me to construct a similar description of the same experiment, giving the same numbers for everything observable in this experiment, using these theorems. This works always in SR, and in GR at least for all local experiments.
You have a fair chance to present an experiment where the equivalence theorems would fail. Experiments with wormholes or closed causal loops would do the job, these would be solutions where a fully equivalent Lorentz ether solution does not exist, only on some part of the solution this would be possible. I doubt you have such experiments done.
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