# Any research left to do in Special Relativity?

#### andytoh

Relativists seem to devote their research in GR. I was wondering whether SR has been fully exhausted already. Probably at the physics level, but also at the mathematical level too?

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#### Chris Hillman

Open questions in relativistic physics in flat spacetime?

Hi, Andy,

Relativists seem to devote their research in GR. I was wondering whether SR has been fully exhausted already. Probably at the physics level, but also at the mathematical level too?
On the one hand, str simply states the rules for relativistic kinematics, so in that sense there has been no mystery since 1905. But as we saw in 1907, when Minkowski introduced the notion of spacetime, a dramatic reformulation can cause a c change. A bit later the Thomas precession was discovered, which is a fundamentally important consequence of the kinematical laws. Later still, Penrose (and independently Terrell) discovered the beautiful connection between the optical experience of a relativistic observer and the Lorentz group acting by the Moebius action on the Riemann sphere. Various observations which many find surprising (sufficiently so that these observations are often called "paradoxes") have been made over the years concering relativistic kinematics, e.g. Bell's "paradox" about spaceships and string, the fact that all colliding plane waves meet "head on" in some frame, and so on. Thus, history suggests that surprises concerning relativistic kinematics are still in store.

A fundamental new topic of current interest concerns relativistic navigation, and even in flat spacetime a full relativistic beacon navigation system is still a sufficiently new idea that ineresting questions remain. For example the Coll canonical chart for Minkowski spacetime (wrt some choice of four "beacons") has some perhaps surprising properties, e.g. it has four (real) null coordinate covectors but four spacelike (real) coordinate vectors, and it is clear that there is much to say about this new and fundamental concept, which obviates all the difficulties of planetary calendars/coordinates (with inevitable coordinate singularities) as well as the procedure of making laborious "relativistic corrections" to Newtonian concepts of navigation.

So I at least have little doubt that surprising discoveries about kinematics in flat spacetime, or equivalently the geometry of $$E^(1,3)$$ still remain.

Interpreted still more broadly, if your question concerns not just the geometry of Minkowski spacetime but any kind of relativistic physics in flat spacetime, then clearly this is a huge area where research continues.

#### MeJennifer

A fundamental new topic of current interest concerns relativistic navigation, and even in flat spacetime a full relativistic beacon navigation system is still a sufficiently new idea that ineresting questions remain. For example the Coll canonical chart for Minkowski spacetime (wrt some choice of four "beacons") has some perhaps surprising properties, e.g. it has four (real) null coordinate covectors but four spacelike (real) coordinate vectors, and it is clear that there is much to say about this new and fundamental concept, which obviates all the difficulties of planetary calendars/coordinates (with inevitable coordinate singularities) as well as the procedure of making laborious "relativistic corrections" to Newtonian concepts of navigation.
Many of those problems derive from people's desire to think of the "plane of simultaneity" as some sense of reality while I believe that a "Doppler view" of reality is much more practical when it ever comes to space travel at relativistic speeds. Unfortunately, at least IMHO, to many the idea of what an observer actually measures is secondary compared to what an observer constructs as some "plane of simultaneity" reality. Hope that makes any sense.

Up to a large extent I see the same issues with explanations of GR. In my (perhaps ignorant) opinion too much emphasize is given to coordinate views and less to the intrinsic geometric properties and how they relate to the EFE with a result that people get more confused than helped in understanding GR.

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#### Chris Hillman

Observers required, coordinates optional

Many of those problems derive from people's desire to think of the "plane of simultaneity" as some sense of reality while I believe that a "Doppler view" of reality is much more practical when it ever comes to space travel at relativistic speeds. Unfortunately, at least IMHO, to many the idea of what an observer actually measures is secondary compared to what an observer constructs as some "plane of simultaneity" reality. Hope that makes any sense.
It does make sense, and I agree. (It sure is nice to be able to say that!) You will love the Coll chart if you haven't studied that eprint yet, since as you will see, there is a simple relationship with the Bondi k-calculus (consider one dimensional relative motion of two inertial beacons in Minkowski vacuum, and compute the transformation from the Cartesian chart to the corresponding Coll canonical chart)!

Up to a large extent I see the same issues with explanations of GR. In my (perhaps ignorant) opinion too much emphasize is given to coordinate views and less to the intrinsic geometric properties and how they relatate to the EFE with a result that people get more confused than helped in understanding GR.
I agree entirely; when I discuss frames, for example, I always try to stress that this concept is coordinate-free and has both a vivid geometric meaning and an immediate physical interpretation ("local Lorentz frames" of a coherent family of observers).

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#### robphy

Homework Helper
Gold Member
Relativists seem to devote their research in GR. I was wondering whether SR has been fully exhausted already. Probably at the physics level, but also at the mathematical level too?
I don't think research in SR is exhausted.

Among my research interests include what could be called "relativistic pedagogy"... ways to teach relativity... particularly ways to smoothen the transition from Galilean relativity to SR and from SR to general relativity. One outstanding problem in pedagogy is that the accelerated observer in SR has not been satisfactorally presented in textbooks yet.

Another aspect could be described as "relativistic foundations"... particularly, aspects of causality... which could be used to minimally characterize (along the lines of Alexandrov and Zeeman) and possibly develop (say) generalizations of the spacetime manifold..for various applications, like new computational techniques and new approaches to quantum gravity.

From a foundational point of view, recall that spacetime provides the arena for physical theories. It's not clear to me if topics like relativistic thermodynamics and relativistic (Hamiltonian and Lagrangian) mechanics have been fully worked out for SR (let alone GR)... not to mention presented in textbooks. (Certainly there has been a lot of work in relativistic quantum mechanics and field theory [including electromagnetism].)

I'm sure that there are still some interesting "effects" that haven't been uncovered or fully explored in SR. Such effects may reveal themselves when SR is applied to (say) engineering problems. Relativistic ray tracing (computational graphics where the finiteness of the speed of light is accounted for) may fall into this category.

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#### Aether

Gold Member
One outstanding problem in pedagogy is that the accelerated observer in SR has not been satisfactorily presented in textbooks yet.
You might like this: C.B. Giannoni, Special Relativity in Accelerated Systems, Philosophy of Science. Vol. 40, No. 3. (Sep., 1973), pp. 382-392.
C.B. Giannoni said:
The Special Theory of Relativity (STR) as formulated by Einstein is applicable only to inertial systems; however, it can easily be extended to accelerated systems by a mere reformulation that does not alter its empirical content.
You can temporarily download this paper "[URL [Broken] 1973.pdf"]here[/URL]; I intend to delete this sentence by tomorrow.

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#### Chris Hillman

Hi all, I'd like to elaborate slightly on something robphy said:

One outstanding problem in pedagogy is that the accelerated observer in SR has not been satisfactorally presented in textbooks yet.
It matters whether one is discussing the education of future physicists or a general audience in a broad sense (e.g. electrical engineers may not require exposure to conceptual subtleties in relativistic physics). My comments here concern the problem of educating the former group.

There is a sizeable and (amazingly enough) still-growing crank literature by dissidents who don't accept the Lorentz transformation; in particular, who disagree with what standard textbooks say about how they work. I will pass over this literature without further comment, on the grounds that any physics student who fails to grasp the mathematics of the Lorentz group has no chance of succeeding in his studies (unless, just possibly, at a stretch, he somehow contrives to stay far away from relativistic physics throughout his career in physics).

For the benefit of the nonscientists here, let me stress that implicitly suggesting that failing to learn how to compute correctly with the Lorentz group should be grounds for summary dismissal from any graduate program in physics is not at all equivalent to telling students that they must accept the (ludicrous) proposition that "str is 'true' once and for all". Science simply doesn't deal in 'eternal truth' in any mystical or dogmatic sense, nor does it deal in models which can never be overturned under any circumstances, and no one is suggesting the latter procedure! Rather, I am suggesting that physics apprentices need to be comfortable with the mathematics of relativistic physics in order to function as professional physicists in the 21st century. We simply do not need to consider here the unanswerable question of whether or not this will still be true in the 23rd century.

Passing on: I feel that the phenomenon which should really intrigue contemporary educators of potential future physicists is the existence of a considerable literature founded upon more subtle conceptual (and sometimes mathematical) errors. The authors of these papers may no longer be educable, but their lamentable history might suggest that better education of physics majors might result in fewer careers being compromised in this way in the future. Since educators must strive to prepare students to function, not to fail to function, it follows that they should be very interested in improving the education of future potential physicists in ways specifically aimed at helping them to avoid falling into these conceptual traps.

The existence of misinformed eprints dealing with various versions of Bell's paradox and Ehrenfest's paradox (among others) is both evidence that some positive fraction of prospective physicists are not learning in school some stuff they really need to know in order to function effectively as physics researchers, and a fertile source of "missing topics" which are overdue for consideration in any curriculum reform.

Those who have studied such papers and know what went wrong will, I warrant, recognize that some particular fundamental topics, which can be considered to fall within the scope of relativistic physics in flat spacetimes, have not yet been adequately covered in the textbooks. ("Adequately" in the sense which I have just explained.) Off the top of my head, I'd list the following as missing topics:

1. thinking about and computing with vector fields as first order linear partial differential operators, or equivalently as flows (local actions by the group of reals), or as uniparameter subgroups of diffeomorphisms, etc.,

2. computing and interpreting the kinematical quantites defined for any congruence (acceleration vector, expansion tensor, vorticity tensor) which are invaluable even in flat spacetime,

3. computing and interpreting frame fields and computations of tensors with respect to a frame field (this is how one models the physical experience of a family of observers; in order to study one observer it is often technically and conceptually advantageous to consider him one of a family of observers),

4. spatial hyperplanes and "space at a time" decompositions of tensor fields valid at the level of jet spaces ("higher order" infinitesimal approximations at an event; a tangent space is a first order jet space),

5. the existence of multiple competing notions of distance/speed "in the large" even in the case of (accelerating observers in) flat spacetime,

6. various algebraic/geometric phenomena regarding the Lorentz group and its analogues, including the Steenrod twist algebra, Penrose-Terrell and other optical phenomena, etc.,

7. familiarity with important alternatives to the usual Cartesian and polar spherical charts on Minkowski spacetime, such as the Coll canonical chart for flat spacetime and its relationship with the Bondi k-calculus,

8. distinguishing between visual experience and length and time measurements in the sense used in elementary discussions of str.

That's just off the top of my head; no doubt I have made some silly omissions in this list.

Note: various books like Eric Poisson, A Relativist's Toolkit, do discuss many of these topics, but their appearance in a book whose subtitle mentions black holes appears to fool casual readers into concluding (quite incorrectly) that the above mentioned techniques "belong to gtr".

It's not clear to me if topics like relativistic thermodynamics and relativistic (Hamiltonian and Lagrangian) mechanics have been fully worked out for SR (let alone GR)... not to mention presented in textbooks.
Some graduate level textbooks do discuss relativistic thermodynamics (e.g. MTW). There is a significant literature on all these topics but I'd agree that much of this hasn't made it into the textbooks, possibly because it is not generally considered sufficiently mature/useful by most would-be textbook authors.

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