# Open problems in special relativity, and non-inertial motion in flat-spacetime, or things that have been discovered recently

What classical open problems are there in special relativity, including questions about non-inertial motion in flat-spacetime, but excluding questions about quantum theories.

Answers can include things which were open problems until recently.

New things in special relativity (and non-gravitational relativity) still seem to be discovered relatively recently:

For instance the book "A broader view of relativity" by Jong-Ping Hsu, Leonardo Hsu, contains formulas for transformations between some types of accelerated frames.

Abraham Ungar derived formulas for things like proper velocity composition, and for Lorentz transformation composition in terms of velocity and rotation parameters.

So it would be good to have a list of things that we still haven't worked out, or have only been worked out recently, to challenge the assumption that special relativity is all figured out.

(Due to linear frame-dragging it also seems that the usual saying that GR tends to SR in the limit of negligble gravity should be stated as GR tends to SR for negligble gravity and low acceleration, so any calculations concerning the half-way house of non-inertial relativity between SR and GR would only be valid for low acceleration. Perhaps this should be the definition of non-inertially extended special relativity: acceleration without linear frame-dragging.)

Edit: I +1ed the answer on self-force because it was an interesting general issue, but once we are talking about fields and particles then we are getting away from the question which is really about calculations in the theory of special relativity itself. Once we talk about fields and particles then I guess there are lots of problems and an accurate treatment requires quantum theories which this question is not about.

• To me, Ungar comes off as a bit crankish. He's just expressing known stuff about boosts and rotations in a different notation. He writes as though it's earth-shattering stuff, but I don't think anyone in the physics community even considers him as working on an open question. One open question is the possible validity of doubly special relativity (DSR) -- but there seem to be significant theoretical reasons why it isn't actually viable. The existence of tachyons is an open experimental question in SR.
– user4552
Aug 19 '11 at 3:09

## 7 Answers

I can think of the problem of a self-force on an accelerating charged particle as one that still needs a full solution.

Another important question in classical relativity is that of the motion of extended bodies in curved spacetime. Turns out that because the motion of an extended body in a curved space appears differently to different observers, such an object can actually translate ("swim") through space by periodic distortion of its shape [reference].

There are other open classical problems in relativity but again, as Piotr, said its hard to think of any in flat space.

• curved spacetime brings you into general relativity. And there are plenty of open problems in general relativity. Dec 7 '10 at 15:24
• And there's a paper by Bryce Dewitt on the electromagnetic self-force problem in SR if I remember right. Dec 7 '10 at 21:29
• +1 for self force, but the question specifically asked for flat-space and non-gravitational relativity because as Jerry Schirmer says there are plenty open problems in General relativity. From self-force I found the wikipedia article for Abraham–Lorentz force which says "self-fields" are frequently neglected in classical electrodynamics. Dec 8 '10 at 22:35
• @Roy: self-force is indeed a big problem and it actually can't reasonably be solved in a classical theory. You can construct contrived theories which explain it but the actual fact is that you are ignoring quantum effects (which are ultimately what saves the day). All classical theories are inconsistent on their own when looked at too closely (i.e. when approaching quantum scale). Dec 8 '10 at 23:40
• @Marek: There's a paper by Arnowitt, Deser and Misner from the late 60s where they look at the collapse of a charged fluid in isotropic coordinates, and they show that the divergence in the electromagnetic self-force cancels the divergence in the gravitational self-force as the fluid collapses into a black hole. Dec 11 '10 at 20:21

I can tell you how to generate open problems in special relativity: open up a geometry book and look at some geometry theorem. Add the word "Minkowski" before everything. Is the theorem still true? One problem that I got to thinking about recently is how to axiomatize Minkowski geometry in the same way that Hilbert axiomatized Euclid's geometry. It is a theorem that Euclidean geometry (as axiomatized by Hilbert, essentially completing Euclid's postulates) is a complete theory: there is an algorithm for deciding which statements are theorems and which are not. Is the same true of Minkowski geometry?

Would you consider something like finding a classical solution to the Yang-mills equation of motion a problem in special relativity? Because I think that's still open for non-$U(1)$ symmetry groups.

• Huh? Such solutions are known for quite some time. Not only that, these days even Yang-Mills gap question is quite close to being solved. Start here. Dec 7 '10 at 17:52
• Then why do they linearize them before quantizing them? I admit that this is not a problem that I'm actively working on at all, but especially given how ill-behaved the quark/gluon states are in QCD, if they had an analytic solution in the nonlinear regime, why do people tend to start with the linearized theory? Dec 7 '10 at 18:39
• I don't understand what you are talking about. Linearize what? Y-M equations? To my knowledge $SU(3)$ Y-M theory is treated in its full (non-linear, non-abelian) generality in also in quantum regime. For high-energetic case there is no problem (thanks to asymptotic freedom). Most unsolved problems in QCD come from understanding low-energy condensates. But also there some solutions exist (I again refer you to the above blog) and both perturbation analysis and lattice computations agree. (By the way, add @Marek to your comment next time so that I know you commented). Dec 8 '10 at 14:30
• @Marek: yes, the YM equations-the scheme that I remember from Peskin and Schroder was to take the classical equations of motion, linearize them so you get plane wave solutions, and then promote those plane wave solutions to operators, and then deal with everything else as an interaction through the Feynman diagram procedure. There's no big problem with this, as you say, in the high-energy regime, but in the low energy regime, the plane wave solution has little to do with the actual solution-there is no low energy gluon state. If this can be treated w/o linearization, why mess with the above? Dec 8 '10 at 16:33
• @Jerry: yes, you can do that and it's enough in certain cases. But not always. And then it's also possible to work with full theory. E.g. use BRST quantization to deal with the gauge freedom and obtain the full QFT. No problem with that. Of course, theory itself is brutally hard and perturbations don't really work in low-energy limit, but that is another matter :-) On a related note: you can find instanton solutions for full QCD and these are non-perturbative effects (and you wouldn't find them in linearized theory, I think) that play major role. Dec 8 '10 at 17:25

Special relativity is rather simple. Abraham Ungar's stuff that you are quoting seems quite trivial at first (and second) glance. There is no difficulty combining arbitrary rotations and Lorentz boosts. You only need to express those as 4x4 matrices. I derived such matrix one time simply to answer a forum post.

The linear frame dragging results from motion of mass, and as such is, too, a gravitational effect. The general relativity is rather complicated and there are various unsolved problems.

"Accelerating relativistic reference frames in Minkowski space-time" arXiv:1109.1796, 9 Sep 2011

Slava G. Turyshev, Olivier L. Minazzoli, Viktor T. Toth

Length_contraction#Experimental_verifications
Length Lorentz contraction - observer frame dependent ? or ...
Length Lorentz contraction - absolute, i.e. real ?

Since the occurrence of length contraction depends on the inertial frame chosen, it can only be measured by an observer not at rest in the same inertial frame, i.e., it exists only in a non-co-moving frame. ...
Another confirmation is the increased ionization ability of electrically charged particles in motion. According to pre-relativistic physics the ability should decrease at high speed, however, the Lorentz contraction of the Coulomb field leads to an increase of the electrical field strength normal to the line of motion, which leads to the actually observed increase of the ionization ability

How explain this "observed increase of the ionization ability" as a frame dependent measure? IMO, this points to an absolute effect.

• Lorentz contraction is frame-dependent. This is not an open problem.
– user4552
Aug 19 '11 at 14:52
• @Ben Crowell I'm expecting your explaination on Ionization be frame dependent. (An electron can not leave, and at the same time leave, the atom, or am I missing something?) Aug 19 '11 at 17:11
• @Helder: You are quoting the Wiki page about ionization ability. I would consider that quote to be wrong as it clearly violates SR. Aug 19 '11 at 18:03
• @Helder: edit: I think the quote from Wiki should be understood as that the ionization ability of the electron depends on its speed RELATIVE to the atom, which is correct and which is not an absolute effect. Aug 19 '11 at 18:47
• @Hans de Vries your book Fig 2.1: "Electrostatic potential of a charge moving at 0.8 c... shows how the field Φ propagates away from the charge spherically while decreasing in amplitude 1/r." It moves at 0.8c in relation to (irt) 'what'? Not irt the emmiter, not irt the observer. The only option left: it must be irt the medium-'the physical space'. The field propagation moves 'spherically' only irt to vacuum. IMO the field of a moving atom (irt vacuum) can not possess true spherical symmetry in a moving referential but, as we use light to measure it, it appears to be so. A congruent world. Aug 21 '11 at 23:58

The Einstein relativity is the viewpoint as seen from moving bodys. Since all objects are in motion we think that there is no need to study any other viewpoint. Is that all?

One subject that has been negleted is the one-way/two-way speed of light.
In Einstein SR c is the mean light speed in a closed path and it is constant to any moving observer, as it is.

My comments on this: One-way_speed_of_light#Preferred_reference_frame

A preferred reference frame is a reference frame in which the laws of physics take on a special form.

Why?

The ability to make measurements which show the one-way speed of light to be different from its two-way speed would, in principle, enable a preferred reference frame to be determined. This would be the reference frame in which the two-way speed of light was equal to the one-way speed.

This is about the CMB referential! It is unique and is common to all moving bodies and where the speed of light is isotropic. Since we are not able to measure the one-way light speed, we are persuaded that there are no effects.
How sure we are if we dont have the relevant studies?
IMO...
Afaik the computation of energy is frame dependent. Can I think that the neutrino-(originally a missing energy problem) do not exists in the absolute referential?
IMO, the flyby anomaly, chirality, matter/antimatter, and a lot of other unsolved issues can have an explanation if we dare to make the studies.

In Einstein's special theory of relativity, all inertial frames of reference are equivalent and there is no preferred frame. There are theories, such as Lorentz ether theory that are experimentally and mathematically equivalent to special relativity but have a preferred reference frame. In order for these theories to be compatible with experimental results the preferred frame must be undetectable. In other words it is a preferred frame in principle only, in practice all inertial frames must be equivalent, as in special relativity.

We are avoiding to aknowledge that the CMB reference is really special. IMO we are poisoned with prejudice.

In this paper Cosmological Principle and Relativity - Part I (2002, not peer reviewed, by my friend Alfredo that was fighting against a cancer in those days) is presented a study 'from above'. This work enable us to start to understand what kind of nuances can appear, namely in Lorentz length contraction formula, if we study light and motion from the CMB viewpoint. Einstein SR equations are also derived as a by-product.
In conclusion: Einstein viewpoint is correct but it is not all the story.

• Alfredo Oliveira is a crackpot. This paper shows a complete lack of understanding of relativity.
– user4552
Aug 19 '11 at 18:59