A very rigorous mathematical setup of SR, based on assumptions (axioms) about events and signals ('paths') and a ternary relation of 'betweenness', is given by

J.W. Schutz :

Independent Axioms for Minkowski Space-Time

Pitman Research Notes in Mathematics Series 373, 1997, CRC Press , 241 pages

Axiomatic Systems for Minkowski Space-Time (1997)

https://www.ge.infn.it/∼zanghi/schutz.pdf

https://www.researchgate.net/publication/2733368 Axiomatic systems for Minkowski space-time

Foundations of Special Relativity: Kinematic Axioms for Minkowski Space-Time

Springer, 1973, https://www.springer.com/de/book/9783540065913

This setup of SR is similar to the setup of Euclidean geometry by means of axioms concerning points and lines etc. Minkowski space-time coordinates, indefinte metric etc., appear almost at the end of the theory, and are the result of a theorem, the "coordinatization theorem".

Dear C.Baumgarten,

"The concept of 4D SPACE-TIME is not necessary to derive/construct STR,Lorentz force and Maxwell's equations"

Even then' we may not ignore the 4D SPACE-TIME i.e .Minkowski geometry if we want to describe the gravitational force.

Regards

Ramesh Chandra

According to Ungar: “Einstein's Special Relativity: Unleashing the Power of its Hyperbolic Geometry”

https://www.sciencedirect.com/science/article/pii/S0898122105000271

we departed from the original formulation of Einstein in our interpretation of special relativity towards the reformulation based on Minkowski's four-dimensional geometry. In the transition process, the Einstein's formulation in terms of the addition law of the relativistically admissible three-velocities has been replaced by the Lorentz transformation. Based on the information provided by Ungar: “Einstein considered Minkowski's reformulation of his theory in terms of four-dimensional spacetime to be no more than superfluous erudition". He accepted the concept later because it was an essential basis for the formulation of general relativity.

It is still useful, however, to consider Einstein's original theory and its connection to the hyperbolic geometry of Bolyai and Lobachevski. The subject is elaborated by Barrett in: “The hyperbolic theory of special relativity”.

https://arxiv.org/pdf/1102.0462.pdf

Hello Christian, Karl, Ramesh, Janusz, and all,

Thank you.

This might depend on what we mean by "understading", as Christian said. It turns out it does not, thankfully, as I exemplify below -- even if we agree totally on the meaning of "understanding".

Ramesh and Janusz bring the point that the essence of general relativity is that gravity is described by the geometry of spacetime. Einstein had to study it to formulate general relativity, creating the Einstein tensor notation, and motivating the use of spacetime. It is common to not accept new things in the field one creates and works, Bohr of QM fame, denied to Maiman that the laser would work.

But some of the references cited by Karl seem to fail to mention Frenet and Cartan curves, essential to understand that special relativity does not work only for inertial frames, as Wikipedia says, but also for accelerated and curved trajectories. For example, it is reported that “Lecture Notes in Mathematics - Foundations of Special Relativity: Kinematic Axioms for Minkowski Space-Time” does NOT include Frenet curves, also called Frenet-Serret.

Recently, in RG, a researcher tried to show that (his) special relativity should not concern itself with curved or linearly accelerated trajectories, only with objects moving in a straight line at constant speed! Of course, this is not true, special relativity is used in RQM, for example as in the Dirac equation, where every trajectory counts in the Hamiltonian. Spacetime provides the accommodation that is needed.

Clearly, we could agree 100% on the meaning of "understand" with that researcher, but still we would NOT agree that special relativity is limited to inertial frames, to objects only moving at constant velocity in a straight line. How boring! And, useless.

Recent developments in theoretical computer science and mathematics bring to the forefront of physics, still untreated as it could, the theory of types, and what is to be considered part of reality. This agrees that physical laws cannot depend on our arbitrary choice of how we parameterize a surface, for example, used with a spacetime approach -- but missing to anyone who does not use it, and can only treat inertial frames in special relativity. Perhaps Christian can clarify better, but his approach, though simple, is self-limiting. It agrees with Einstein in an early opinion, as Janusz noted, but that was overcome in general relativity.

Further, as well-known, a physical statement can directly relate physical objects, like vectors. And although the components of said vectors may individually change when different parameters or bases are used, it must be true that both sides of an equality transform in precisely the same way. We similarly demand on spacetime that any reasonable physical law must have the form A = B, where both sides of the equation are (e.g., tensors) of precisely the same type. This ensures that once we know the components are equal in a particular basis, it is automatic that they will also be equal in any other basis we should choose to examine.

NOTE: That is why using a vector product, also with an operator, should be suspicious, because a vector product of two vectors is not a vector. This affects, for example, our reliance on Maxwell's equations and vector potentials, to be treated elsewhere.

Another point, gained by means of a spacetime method, is that although it may not be clear, Gauss’ Theorema Egregium states that the Gaussian curvature can be determined purely in terms of lengths and angles measured within the surface, and so this is a property intrinsic to the surface itself (as opposed to extrinsic, a property that would depend on how the surface is embedded into an external space).

Cheers ,Ed Gerck

Hello Ed,

It should be noted that one should not mix up the problem of describing an accelerated motion of an object (particle) in the context of SR with the problem of the transition from an inertial frame of reference to a non-inertial frame of reference. The latter is involved with a metric which in general differs from the diagonal (+,-,-,-), and which might become even time-dependent, a delicate feature which goes beyond the scope of ordinary Lorentz transformations between coordinate systems which refer to inertial frames of reference. Furthermore, please note (i) that the trajectory of any fixed point of an inertial reference frame A with respect to an inertial reference frame B, which moves relative to A, is never a curve, but a straight line, and (ii) that the basic topic of a setup of SR considers the relationship between space-time coordinates defined or constructed with respect to inertial reference frames (by means of clocks and light signals).

Karl

Karl G. Kreuzer

Hello Karl and all,

There was no mix up of those examples qua "cases" in my reply above, and it was not even mentioned. I could comment on them, if needed.

Cheers, Ed Gerck

Hello all: Regarding non-inertial frames, in special relativity the laws of nature are simpler as seen by inertial observers, whose rectangular positions and times are related by Lorentz transformations. However, they look different for observers who do not move at constant speeds relative to inertial observers. How can they be calculated?

Using a spacetime formulation, Galileo's law of motion (which Newton used as his first law) in a non-inertial frame can still be regarded as stating that particles move along geodesics in the absence of external forces. This extends what people assume by "straight line" instead of geodesics.

The geometry of spacetime with no curvature (i.e., flat) can then capture equally well the relativistic kinematics of particles that are not moving at constant velocity. For example, consider an ARBITRARY trajectory, including any that does not describe motion at a constant velocity.

First, this cannot be done without using a spacetime model, one is restricted to inertial frames. Second, the Christoffel symbols provide precisely the ‘fictitious forces’ that are required in order to ensure that the geodesics are representing straight lines, expressed in the non-inertial coordinates.

Similar to what is used in Newtonian physics, these 'fictitious forces' can be recognized as old friends; a term representing the centrifugal force and another term the coriolis force, for example, with a rotating reference frame. The fact that the Christoffel symbols do not transform as the components of a tensor is consistent with the fact that these fictitious forces can vanish in some frames — e.g. inertial ones — even if they do not in others.

See further info in the treatment by Cliff Burgess, who provided much of the text used above.

Cheers, Ed Gerck

Dear Ed Gerck,

"Regarding the non-inertial frames in special relativity"

The existence of non-inertial frames in special relativity, is a fundamental problem of the Einstein 's theory of relativity. So far known to me Einstein himself had not said any thing about it. Of course, we study the relativistic transformation equations of acceleration and force, the rate of change of the velocity with respect to the mass and the energy-momentum tensor of a distribution of matter in special relativity etc. , where the acceleration or the force are considered in an inertial frame. these considerations are against the basic nature of the inertial frame.

Let us consider a non-inertial frame in the special relativity i.e. we have two inertial frames moving with a constant acceleration relative to each other. The main problem in this consideration is what would be the maximum amount of acceleration which a particle ( massive or non-massive) may have ? Only then, the problem of non-inertial frames in the special relativity may be meaningful.

Regards

Ramesh Chandra

Dear Christian,

I observed that it is generally considered that the SRT is in harmony with the equations of electromagnetism.

I observed however that there is a disconnect between the SRT axiomatic concept of length contraction of macroscopic massive bodies and the first Maxwell equation (Gauss equation for the electric field).

We know that all macroscopic massive bodies are made of atoms immobilized with respect to each other, each atom being made of an escort of charged electrons located at some distance from their centrally located charged nuclei, which defines atomic volumes.

Macroscopic massive bodies length contraction thus involves by structure a shortening of these distances between the electronic escorts and related nuclei, which in turn involves an adiabatic increase in energy in all charged particles involved, according to Maxwell's first equation whenever multiple charges are interacting. Let's remember that Gauss's equation simply is the Coulomb law generalized to potentially be able to calculate the actual energy induced in charges when such charges are effectively assumed present at any distance of the test charge.

This obviously allows calculating the adiabatic energy increase in these atoms in physical reality.

I saw nothing in your paper on this issue, or maybe I missed it.

From my understanding, SRT is unable to account for the physical existence of this adiabatic increase in energy when electronic escorts are compressed towards nuclei in context of length contraction.

Best Regards

André

Hello Ramesh and all: You wrote, "The main problem in this consideration is what would be the maximum amount of acceleration which a particle ( massive or non-massive) may have ?"

This is not a problem in special relativity (SR), and not a problem in relativistic quantum mechanics (RQM), such as solved by Dirac.

Let's think of it as a question in three levels. The first level is SR, and simply ignores the question -- there is no limit to the maximum amount of acceleration, it is ARBITRARY except that it must be finite (no infinities in physics). In RG, Joachim Domsta has recently elaborated more on that.

The second level is RQM, that gives you clear rules but not all, for example, entanglement acceleration.

The third level is topology, that tells you that homotopy laws cannot be broken, for example.

The number of levels is, of course, arbitrary, and we can always go higher and lower, we humans are just beginning!

This, what you mention, can and has been solved in SR, which was used more than 100 years ago to create general relativity in terms of spacetime, not space and time a la Newton.

I prefer not to repeatedly indicate literature, you should have no problem finding it.

Cheers, Ed Gerck

Dear Christian,

I just finished reading you paper. Highly interesting material.

I note however that on page 27, you rather hastily dismiss the possibility that 9+1 dimensions could resolve to 3x3+1, that is, as you say: "3 times of the same" for the stated reason that "the algebraic features (for instance of rotations) are different from 3+1, since, as you clarified on page 15, that an essential feature of 3 dimensions is that the spatial rotations do not commute.

However on page 15, you specified that the number of rotators grew quadratically in a 9+1 space and that some rotators would systematically commute if the number of dimensions increased beyond 3 (in the same space). Not so in a 3x3+1 configuration.

So your argument was established not for a "3 times the same" in a 3x3+1 configuration, but for an unrelated 9+1 "same space" configuration.

If interested, you will find in Section 11 of this recent paper a sufficiently clear description of a 3x3+1 that will let you observe that each of the 3 spaces possess this essential feature that the spatial rotations in each cannot commute with those of the other two since all 3 spaces have in common only the origin.

http://file.scirp.org/pdf/JMP_2018042716061246.pdf

Best Regards

André

Dear Christian,

I have been interested in your work ever since I read and analyzed your 2015 version of this paper. I would have read this 2017 version before had I noticed it before.

I am interested in your deep analysis and conclusions regarding the special nature of 3+1, because they help me visualizing more clearly all aspects of the trispatial geometry.

Just to be clear about the trispatial geometry, the 9+1 that I explored, that I refer to as 3x3+1. The 3x3 does not mean (3 multiplied by 3 equals 9), but (3 + 3 + 3 = 9). That is 3 "normal" 3D spaces joined at their origin and perpendicular to each other, + time. This is why what you conclude about the 3+1 also applies to this 3x3+1 structure. Maybe it would be better named 3x3D+1.

I realize that adiabatic energy induction cannot be directly related to the Hamiltonian that you use as a founding principle, which is grounded on the classical Principle of energy conservation established from experiments carried out with macroscopic masses at our macroscopic level. Here is why:

The adiabatic variations of energy in such macroscopic masses that are made to interact here at ground level are so infinitesimal by nature as to be impossible to measure. For example, a 1 kg mass falling 1 meter to the ground releases the familiar 9.78 joules of kinetic energy in sync with the principle of conservation, but an easy to carry out calculation also shows that it is also adiabatically induced with a mass increment 13 orders of magnitude smaller than 1 gram.

Too small to be measured, and never considered as even existing.

They can be, and have been measured when sufficient axial distances are involved, for example, such as taking atomic clocks in altitude, but this adiabatic effect resulting from orbital contraction due to the weakening effect of the Coulomb interaction due to increased distance between the charged elementary particles of which the small mass of the bodies are made, that move to higher altitudes and those making up the mass of the Earth, have been mistakenly confused with so-called relativistic time-dilation effects due to beliefs in the misconceptions perpetuated by SRT and GRT.

They also have been assumed by popular belief to underlie the operation of the gps system even if it is well known in the specialized community that the gps and other localization systems do not use this effect, but rather multilateration to calculate locations, unrelated to time.

In 2003, Paul Marmet demonstrated from a derivation from the Biot-Savart equation that the magnetic field of an accelerating electron increases in synchronicity with its velocity related relativistic mass increment, and that this magnetic energy actually was what provided this velocity related "classical mass" increment.

Further analysis of Marmet's discovery and of the Kaufman experimental data demonstrates that the same Lorentz gamma factor related energy and mass growth with velocity of massive bodies leading to a theoretical infinite mass at the asymptotic speed of light limit, also applies to an identical energy and mass growth with proximity between elementary charged particles leading to a theoretical infinite mass at asymptotic zero distance proximity.

This second "Lorentz" gamma factor related energy and mass growth curve always was and still is undetectable from all experiments carried out with masses manageable at our macroscopic level because they happen to all occur at the infinitesimal end of the proximity mass growth curve between the charged elementary particles making up macroscopic masses nearing each other, just like velocities experimentally manageable in Newton's time all were at the infinitesimal end of the relativistic velocities growth curve.

We know now that even the fastest meteorites falling on the Earth reach at best about 60 km/s, which is way below the approximate 2000 km/s required to measurably be in the low relativistic range, which is why non-relativistic Newton remains fine for all velocities that we are likely to deal with at our macroscopic level.

This always prevented becoming aware of the adiabatic nature of the mass increases with proximity between elementary charged particles making up macroscopic massive bodies moving away from or towards to each other.

But now that the relation is established, the traditional Principle of conservation of energy can be located where it belongs, that is, at the infinitesimal limit of the adiabatic energy/mass growth curve with proximity between elementary charged particles, just like Newton's mechanics was at the infinitesimal limit of the relativistic energy/mass growth curve with velocity of massive bodies.

Since all material bodies are made exclusively of charged elementary particles, it is to this second adiabatic growth curve that I have been trying to draw attention to with my papers, because it puts in perspective that the concepts of Lagrangian and Hamiltonian as currently formulated grounded on the principle of kinetic-energy/potential-energy total conservation prevents completely exploring the submicroscopic level, where adiabatic energy/mass induction with proximity is well into the detectable range of the magnetic-energy/mass-increase growth curve.

I found that this growth curve with proximity of elementary particles is due to the adiabatic nature of kinetic-energy/electromagnetic-energy induction by the Coulomb interaction, that can be calculated with the natural variations of the well established Maxwell first equation, that is, Gauss's equation for the electric field.

I analyzed adiabatic energy variation in elementary charges in this paper:

https://www.omicsonline.org/open-access/on-adiabatic-processes-at-the-elementary-particle-level-2090-0902-1000177.pdf

Best Regards

André

One could formulate Special Relativity using just 3-dimensional kinematics.

It is self-consistent.

This formulation is useful to consider the measurement theory.

The understanding of special relativity measurement theory is very important part of understanding of SR.

But 3D consideration hides the symmetries of the theory.

4D consideration is necessary to understand symmetries and self-consistency of SR. It is also a very important part of understanding.

I believe, good SR course have to combine 3D and 4D formulations.

So one could build the course of SR without Minkowsky space-time geometry, but finally students left with two questions:

Where's the cheating? Why does this cookbook even work?

It seems difficult to separate time from space in SR since both time dilation and length contraction have been defined as being complementary, from their having been initially axiomatically and geometrically related by means of the sin2θ + cos2θ=1 trigonometric relation as clearly explained by R.E. Haskell. Ref. equations 5 to 8. So 4D seems to be minimally required:

http://www.cse.secs.oakland.edu/haskell/Special%20Relativity%20and%20Maxwells%20Equations.pdf

Time is not required however when deriving the gamma factor from electromagnetic equations, since the energy induced according to Maxwell's first equation applied to two interacting charges is adiabatically induced and varies strictly as a function of the distance separating them, irrespective of the time elapsed. Ref: the derivation of gamma from equations 53 to 66 in this paper:

http://www.gsjournal.net/Science-Journals/Research%20Papers-Relativity%20Theory/Download/3197

I have argued that space is strictly 3D and that Pythagoras's Theorem only holds in 3D space, and that therefore any theory which claims to be using Pythagoras's Theorem in 4D is in fact merely applying the 3D Pythagoras's theorem twice. In this article entitled "Pythagoras's Theorem and Special Relativity", I use the example of 7D to illustrate the point.

http://gsjournal.net/Science-Journals/Research%20Papers-Mathematical%20Physics/Download/7184

Hello all: RG, our pleasant and valued host system, is doing a great job in relevance value. However RG is currently hiding entire posts, and hiding again, and again; this is doubtful, and may affect this forum, and my postings.

There is no error message per se, but the note "Show previous answers", and having to repeat it multiple times. Past content in RG, though, is often more important than new, and I see people having to repeat posts that are so buried. If this continues, it is essentially a memoriless system, where close to nothing is taught or learned, only the few current messages matter. Not good for science, maybe good for twitter.

Suggested to RG, it is better for RG users if RG partially hide the answers, as done now, however, not totally hide the past answers -- but paginate them as before.

This affects currently all pages of questions that are long, exist for some time, like this one. This will affect this note as well, so we may have to repeat. Please understand that these are the rules currently, but we can resort to:

1. Avoid messages without factual content, fantasy postings have been and should be reduced, no willy-nilly postings or to get traffic ponts, this is not twitter, no comments on researchers, we are all just messengers here.

2. Use more space in the posting itself, use less number of postings, your message will live longer.

3. If you think that totally hiding older messages is detrimental to RG use, please let RG know.

If I post less, I may be helping your posting to live longer. If you have a question, please try looking into the hidden messages, the older postings.

Please don't complain, do something about it. As above suggested.

Hello all: Depending on your platform, this older answer may be hidden in RG. The original reference by Minkowsky on spacetime, began with, "Henceforth, space for itself, and time for itself shall completely reduce to a mere shadow, and only some sort of union of the two shall preserve independence." [1], 120 years ago.

Cheers, Ed Gerck

[1] http://rgs.vniims.ru/books/spacetime.pdf

On the fusion of time and space in special relativity, and its difficulties conceptually, as we see by attempts to use STA and Clifford(3,1), and in the original formulation of special relativity, that do not use this concept.

How can special relativity “fuse” different things, time and space, experimentally, measured in a different way? The discussion presents a experimental and mathematical position, and intends to clarify this common question:

https://www.researchgate.net/post/On_the_fusion_of_time_and_space_in_special_relativity

Malloy writes about curved spaces, that "For any flat space, a curved space exists by necessity given what may not be solved in flat spaces."

Gauss’ Theorema Egregium states that the Gaussian curvature can be determined purely in terms of lengths and angles measured within the surface — that is, in terms of the induced metric and its derivatives — and so is a property intrinsic to the surface itself (as opposed to an extrinsic property that depends on how the surface is embedded -- or sits -- into an external R3). This intoduces two kinds of curvature, extrinsic (modelled as bending) and intrinsic (modeled as stretching).

Although almost all those points, and others not listed, are in my RG publications, I tend to see things through QM.

Reality is observer-dependent, in QM. Starting with the Heisenberg principle, observer and experiment cannot be dissociated. There is no objectivity in QM -- but there is a coherent abstract view one can pursue in QM.

Now, from that common, coherent, abstract view in QM, one can pursue instances of a subjective, intersubjective, and objective natures, and mixed, ordering them into a scale, even contradictory to themselves. It is a cornucopia of choices! All validated by the same common, coherent, abstract view that has the "seed thought" of the subject. It is like a manifold, connecting different domains to one another, although no connection is seen directly between them.

QM and SR are necessary for EM, so its easy to have RQM -- it is a "match made in heaven". Gravity is modeled by extrinsic curvature of spacetime in GR, when QM is utterly ignored, incompatibly, but more than that, it is of a different type, in HoTT and TCS terminology.

In this abstract view, one is concerned with testable relationships, one is focusing on the essence of a hypothesis in the Scientific Method. A hypothesis is not a guess, a hypothesis is a testable relationship, no more and no less.

Accordingly, SR is not a guess, but a testable relationship.

In that abstract sense, SR passed the QM test and the EM test, and yet it MUST be observer-dependent, which seems to contradict several old "principles", but actually extends them, it does NOT deny them. When I summarize, for example:

"Comoving observers NEVER measure time dilation or length contraction, whereas non-comoving ALWAYS do. It does not matter the acceleration, nor the velocity."

then goodbye to "relativistic speed" or "inertial observer", they are limited to a platonic model, a human-made domain, not Nature.

In Nature, an "inertial observer" can see different realities, whether comoving, or non-comoving; reality can be the same also, no matter the acceleration, nor the velocity.

What really distinguishes observers, is whether they are comoving or not. Here, as normal in Physics, comoving is a technical word, and means "same position and velocity", not necessarily identical objects; in cosmology comoving observers see the same wavefront, so it is coherent with other related fields.

Depending on the SR formulation as the stance that one uses, even no SR, more or less "paradoxes" will result. The less "paradoxes" the better. The same with EM, thus we want to eliminate what is not necessary. Can the Maxwell equations be excluded in EM? Other questions follow.

All: As a reminder, it is easy to deal with fantasy and nonsense posters in this thread:

1. They talk against known physics, such as special relativity.

2. They add one or more of their own links, and call it referencing, but trying to get clicks while hiding self or fringe group advertising and false news, and repeat copying their own links under different titles, questions, etc.

3. When asked to stay on topic, they argue, instead of stopping.

4. When asked to correct their wrong citations by the authors themselves, they do not and continue to offend copyright.

5. One recognizes them, also, by talking about other posters, not about the subject (ad hominen attack). Then, they redefine terms in an effort to control the discussion. We do not do that as a recommended practice in science. So, they are already off-topic.

If this happens, you can treat these messages as they are, ads, and skip them, reducing noise with known fantasy or nonsense posters.