Special Relativity: Inertial Frame versus Universal Max Speed; Absolute Accelerometer Histories Hence Frame Speeds; Composition of Relativistic Mass

Special Relativity could come in two different versions which would need to be distinguished by appropriate experimental tests.

-- Related concepts are discussed at length on wiki.

• I’m looking at converting this argument fully to 4-vector form. Current thoughts as below.
• However, within any inertial frame, 4D space-time is time x 3D Euclidean space. The Minkowski metric is a 2-frame relation and captures coordinate transformations, in order to enable description of events in one set of reference frame coordinates, correctly in another set of reference frame coordinates. Within a single inertial frame, a metric between time and space coordinates is not useful.

[ Aside: Speculation and Suggestion for the Unreasonable Efficacy of Maths in Science

Maths works because we have collections { } and natural numbers 1, 2, 3, ... . The Universe works that way, too. We see { } and 1, 2, 3, ... everywhere. Thus we have science that works by maths. And science do-able in 'small pieces', that allows us to build engineering, technology and medicine, 'incrementally'. ]

[ An Aside: Examining the Self-Consistent Picture in Backwards-in-Time Travel

-- See the last section of my LinkedIn blog: 

https://www.dhirubhai.net/pulse/one-scientists-speculations-some-religious-notions-william-batty/

' 'Final' implicit-to-explicit assumptions, backwards-in-time travel and prognostication'.

In an earlier LinkedIn blog, I concluded that backwards-in-time travel, even transport of a single bit of information, could create a singularity at the point jumped back to, (maybe a 'harmless' singularity), and 'split' the Universe, due to the butterfly effect. That is, it would simply not be possible to create zero perturbation, under 'normal circumstances', as any backward propagated information, matter, or person, would couple to the world's chaotic weather system, via heat interaction with the atmosphere. The conclusion then following on the assumption that classical fields, e.g., temperature or velocity, cannot be multi-valued at any given point in space-time.

Backwards-in-time travel with split Universe branches, is consistent. It would appear to be no more 'problematic' than the N-multiverse of classical, extant, 4D space-times, with their respective fully deterministically-evolving mass-energy contents, (and resulting space-time curvatures), required to 'explain' 'probabilistic determinism'.

How about the more usually claimed 'you do not go back in time and kill your own grandfather, thus preventing your own birth, because had you done so, you would not be here to make the jump back in time' ?

Consider yourself at 1 Jan 2020. You travel 'the slow way' through time to 31 Dec 2020. You then jump back in time to 1 Jan 2020 and meet yourself. There are now two of you at 1 Jan 2020. You travel together, 'the slow way' to 31 Dec 2020. Then one of you jumps back. The other proceeds into the future, beyond 31 Dec 2020.

It might be tempting to argue that the self-consistent picture is symmetric. That there are always two of you between 1 Jan 2020 and 31 Dec 2020. And that each views the other in the same way.

However, what happens at 31 Dec 2020 ? Only one jumps back. They cannot occupy the same physical space. One jumps back from spot X1 and one jumps back from spot X2. And each 'you' has memory. When one 'you' proceeds past 31 Dec 2020, he sees 'his other you' jump back in time, not 'self self'.

Consider labels on T-shirts, (1) and (n), and natural number, n, can be actually infinite if required:

(1) .......... | .......... (1) ... (n) .......... | .......... X1 (1) or X2 (n)

....... 1 Jan 2020 .................... 31 Dec 2020..............................

If the T-shirts say (1) and (n), and the backwards-in-time-traveller increases his T-shirt index by +1, when he jumps, then there exist two possibilities for a single backwards-in-time-traveller beyond 31 Dec 2020:

(a) From X1, wearer of T-shirt (1) jumps back and increments his T-shirt to (2). Then the consistent solution is that n = 2, and wearer of T-shirt (2) proceeds into the future, beyond 31 Dec 2020, from X2.

(b) From X2, wearer of T-shirt (n) jumps back and increments his T-shirt to (n+1). Then the consistent solution is that n = n+1, thus n is integer actual infinity. And that actual infinity has been 'achieved' by unit increments. I'm not clear if this is, (1) and (n), or (1) ... (n), i.e., an infinity of self's, to, n, actually infinite.

In (a), wearers of T-shirts (1) and (2) have different memories, the situation is not symmetric.

The extant, 4D space-time Universe, with its mass-energy contents, is 'as it is' (at that level of emergence). So is it single and self-consistent, or branched ? Either would appear possible. Knowledge of the future, based on existing information, would appear to be different to the singularity and splitting caused by backwards-in-time propagation of information or matter.

And variants of the experiment could have two time travellers jumping backwards-in-time, Then four, then eight, ... . After 33 jumps, the population of the Earth would have doubled, half of the new total being different aged 'self's' -- given a big enough time machine. So apart from the 1 Jan 2020 singularity, there would appear to be possible straightforward energy, number, mass and volume conservation issues (?). Maybe such doubling would not be possible. And equally, in (b) above, what about all the wearers of T-shirts, (n), with n = 1, 2, 3, 4, 5, 6, ..., especially if no-one ever writes a T-shirt with 'infinity' on it; always instead, a finite, positive integer ? n actually infinite self's between 1 Jan 2020 and 31 Dec 2020 would obviously be much more of a problem !

As for the notion of Universe splitting at the jumped-back-in-time point, 1 Jan 2020, what would T-shirt pairs, (1) and (1), or, (1) and (0), mean (if we extend, n, to include zero) ?

As wearer of T-shirt (1) has not time-jumped, (1) and (1), would correspond to a first split-Universe branch which only had 'self' in it. Thus would be different to the self-consistent picture. But why should that be disallowed ? The Universe is 'as it is'. And the (0), in (1) and (0), might mean the same thing. However, it looks tidier to use, (1) and (1), with just the natural numbers, n, than to introduce, (0), and then miss out T-shirt 'pair', (1) and (1).

A related argument can be constructed, to examine time-traveller memories, by the initial traveller wearing a green T-shirt at 1 Jan 2020. Then changing to a red T-shirt at 31 Dec 2020, if not jumped already. And the time-traveller changing to a blue T-shirt at 31 Dec 2020, if has jumped already. With respective jumps from launch points X1 and X2.

'Split' Universes and 'single self-consistent' Universes could both be 'reality', with constraints under some 'reasonable assumptions'.

Backwards-in-Time-Travel by Quantum Teleportation with Post-Selection

In a previous blog, I also considered the self-consistent picture in backwards-in-time travel, in the form of an arXiv paper on quantum teleportation with post-selection. That paper is appropriate to mention here, in the context of self-consistency, as it is argued potentially to provide paradox free backwards-in-time travel.

https://www.dhirubhai.net/pulse/prof-feynman-1964-youtube-classical-physics-maths-continuum-batty/, 'An Aside: Best Guess on How to Win a Nobel Prize in a Week?: ...', based on S. Lloyd, et al., arXiv:1007.2615v2.

The idea is to use quantum teleportation by quantum entanglement, but in combination with a closed time curve. Thus instantaneous collapse of the wavefunction, but instantaneous at 'two distinct but coincident' points in time, in curved space-time.

Generally, quantum teleportation requires a classical co-channel, to allow propagation of an Alice -> Bob type signal, in order to allow decoding and actual information transfer beyond 'random 50:50'. A key notion of the arXiv paper, is that post-selection, i.e., wavefunction collapse at a point in the future, can influence 'instantaneously' wavefunction collapse at a 'time-coincident point in the past', on a closed time curve. This avoiding the need for the classical co-channel.

And I speculate whether time-ordered wave-function collapse of a labelled collection of qubit pairs, q1-q2-...-qn, might allow actual information transfer without a classical co-channel ?

-- Though I still need to check, just how wave-function collapse is detected experimentally, to know if this is viable. The classical co-channel does appear to be fundamental to information transfer. However, it is certainly not mentioned explicitly in all applications of quantum entanglement and teleportation. I don't currently understand why that is. Perhaps some initial hardware experiments, say, are happy to demonstrate quantum teleportation, without actual useful information transfer. Perhaps other reasons. I'm not sure. Still on the spare time reading list.

Rather than a single collection of qubit pairs, q1-q2-...-qn, perhaps use a collection of collections of qubit pairs, Q1-Q2-...-Qn. At time t = tc, collapse (or not) the qubit pair collections Q1, Q2, ..., Qn at 'one-end', in well-defined fashion. Then at time t < tc at the 'other-end', measure each corresponding qubit pair collection Q1, Q2, ..., Qn. Presumably, the collected measurements for each qubit pair collection, Qr, would be different, depending on whether it had been collapsed at the 'other-end' or not. In which case, information transfer would have been achieved 'instantaneously', without requirement for a classical Alice -> Bob co-channel. Perhaps collections of collections of qubit pairs, for digital binary (collapsed or non collapsed) information transmission, suffices. If so, then this looks conceptually simpler than the conventional quantum teleportation scheme, for instantaneous transfer of information over arbitrary distances.

I have not yet followed up the maths of the arXiv paper in detail. However, what is important are the concepts. Irrespective of maths models, current or future, if the Universe contains such behaviour then it is extant. If the Universe does not contain such behaviour, then it is not extant. The thought experiments are well motivated either way.

The chronology protection conjecture is a hypothesis first proposed by Stephen Hawking that the laws of physics prevent time travel on all but microscopic scales.

https://en.wikipedia.org/wiki/Chronology_protection_conjecture

]

[ Aside: Two-Slit Experiment and Low Electron Gas Density

In a previous blog, I argued that the Schrodinger Equation could be solved as a Heat-Diffusion-Equation-like pre-calculation, once-only, before inserting the particle mass, m*, and Planck's constant, hbar. I did sketch the algebra pretty carefully. It looks right. However, I still have not found time to programme this up and check eigenvalues and eigenfunctions, to confirm, or deny and correct, the assertion. On my to do list, as an important calculation.

However, even if classical pre-computation were possible, what would that imply about the probabilistic interpretation of the wavefunction ? Could it reduce to simple electron gas number density, or not ? That is not obvious and can be tackled from another angle anyway. For electron gas calculations based on semi-classical Boltzmann Transport Equation type formulation, and kinetic theory, the gas is simply assumed to exist. Then its internal workings are developed in time. The gas itself is not considered to evolve 'into and out of' its collective form. Thus for a 2-slit experiment, we could consider the reservoir of electrons as simply existing. One electron at a time through the system is then arguably a statement about electron gas density. However, electron gases are not uniform. And the resulting diffraction pattern is a whole gas effect, even if electrons, as particles, strike the screen one electron at a time. So how is it known that only one electron traverses the slits at a time ? If this is actually a time average statement, then electrons could still traverse the slit in 'bunches' and we would still have (some) interference. I do not remember from my Uni-days' reading of the 2-slit experiment, how precisely one electron through the system at a time, was guaranteed. As opposed to one electron at at time, as a time average. I will try re-reading. I do not remember this being discussed routinely. Wiki first.

-- I know that there are other key issues to consider, like non locality, hidden variables and Bells inequalities. ]

[ Probabilistic Determinism Fundamental Intelligences Might 'Specify' the Coin Flip

Classical determinism says that an identically flipped coin will come up heads 1000 times in a row, for instance. Probabilistic determinism captures small variations in initial conditions and applied forces, as well as small variations in environment and perhaps asymmetry. Non exact local time invariance means that conservation of energy is not exact; non exact local translation invariance means that conservation of momentum is not exact; and non exact local rotation invariance means that angular momentum conservation is not exact.

The fair coin flip and the fair die roll have been studied since the 1700's and it is known that the fair coin flip is not p = 1 / 2, due to asymmetry at typically the ~10% level. Immediately apparent by simple examination of head and tail asymmetry of the 'average' coin. The coin will land 'heavy side down' more often. And the fair die roll is typically not p = 1 / 6, due to asymmetry of drilled pips, at perhaps the 1 in 10^6 level. Also, detailed modelling studies show that for a die rolled onto a bouncing die table, the die lands preferentially on the side initially uppermost. And that a skilled die roller, can indeed control the outcome of the roll. The system is shown not to be chaotic below an infinite number of bounces.

The ideal p = 1 / 2 is therefore actually a probability distribution function, PDF, about p ~ 1 / 2. Similarly for the fair die roll, p ~ 1 / 6. But why should the fair die roll come out p ~ 1 / 2 even for near symmetry ? Why not p(head) ~ 1 / 4 and p(tail) ~ 3 / 4, say, even with near perfect symmetry ? The PDF captures variation in initial conditions, ... . However, those initial conditions ... , are also specified uniquely deterministically by the evolution of the system embedding the local environment. Thus classical probabilistic determinism requires an N-multiverse, of extant static 4D space-times, each sub-Universe with its own respective mass-energy distribution evolved fully deterministically with time, from some t_initial to t_final (with corresponding time evolved space-time curvature). Then a mind worldline traversing the N-multiverse, as each classical individual sub-Universe branch is selected, by personal choice or otherwise, in a converging tree structure, at each probabilistic decision, e.g., the fair coin flip.

However, what meta-rule for selecting the individual sub-Universe at each probabilistic branch point ? And how is the branching 'decision' made ? Speculative meta-physics suggests perhaps, Leibniz-monad type fundamental intelligences. For instance, maybe associated with elementary particles in Bohmian wave fashion. And in an info-theoretic type Universe, perhaps associated with broadly cellular automata type structure. In any such picture, irrespective of the distribution of N-multiverse, sub-Universes, at each probabilistic branch point, the meta-rule for individual sub-Universe selection or weight must be specified. Simplicity says colloquially 'an appropriate' PDF, with around 'p ~ 1 / 2' for the 'fair coin flip'. But that is an arbitrary assumption. If the 'decision(s)' is / are enacted by 'fundamental intelligences', then the coin flip need not be fair, and the fact that it appears approximately to be so empirically, could reflect the fact that the 'fundamental intelligences' specified that to be the case. 'Make it so'.

If the 'info-content' or 'decision making tendencies' of any Bohmian wave / cellular automata or other type 'fundamental intelligences' happened to be accessible experimentally, then such speculations could become science rather than metaphysics. ]

[ Aside: Huge Bitwise Storage and 'Tower' Exponentials

For integers up to 8 in each ‘placeholder’, ‘tower’ exponentials like 8^8^8^8 would take only 4 x 4 bits of binary storage.

Is there any ‘fairly obvious’ separate radix and mantissa type formulation, for ‘towers’, that would allow their multiplication and division, addition and substraction ?

Perhaps giving factors as a rational approximation, say, to adequate accuracy ? Then maybe even fully integer formulation and manipulation ?

Would it be possible to build ‘arbitrarily large’ bitwise storage on such a scheme ?

Thus structures something like, for instance,

n = Sum_i=0^M Sum_j=0^M c_ij base^(r_i^r^j)

forming products, n1*n2, in the same form, with all quantities in the 64-bit integer range, say, and

r_i^r_j = Sum_i’=0^M c_i’ij base^r_i’

etc.,

or of broadly related form. A lot of the manipulations fully pre-computed up-front and ‘stored’ (or hardware-implemented) for ‘effective O(1) recall’. ]

Classical Maths Continuum Equation Invariance Looks Unlikely / Impossible ?

Consider the following 1D equations. Can they be invariant under 1D Lorentz transformations ? Especially with time- and position-dependent source terms and equation-parameter heterogeneity ?

Diffusion equations:

d2T / dx2 + g(x,t) = 1/k dT / dt

d/dx K(T;x) dT / dx + g~(x,t) = 1 / Cp(T;x) dT / dt

Wave equation:

d2W / dx2 - 1/c^2(x) d2W / dt2 = s(x,t)

Newton's second law (for constant mass):

F(x,t) = m d2x / dt2

1D Lorentz transformation:

t' = gamma . (t - v.x / c^2)

x' = gamma . (x - v.t)

gamma = gamma( v ) = 1 / sqrt(1 - v^2/c^2)

I will post the results, once I have made the transformations from (x,t) to (x',t').

If results are explicitly v-dependent, then frames are not inertial. In which case, non v-dependent, invariant forms would be required.

-- Question to self: Do such exist already ? I don't know. The general relativistic approach is to take some 4-vector relation, e.g., a^mu a_mu = g^2, say, and convert to general coordinate form. I will web search (and think some more):

https://en.wikipedia.org/wiki/Relativistic_heat_conduction

https://en.wikipedia.org/wiki/Relativistic_wave_equations

and for some detailed discussion of relativistic Newton's laws,

https://galileo.phys.virginia.edu/classes/252/relativistic_mass.html

https://en.wikipedia.org/wiki/Relativistic_mechanics

General Coordinate Transformations, Space-Time Crystals & No-Boundary Universes

I once made the above, a^mu a_mu = g^2, conversion to general coordinate form, for a 'rocket' moving radially in the gravitational field of a black-hole, at constant experienced gravitational field, g, i.e., utilising the Schwarzschild metric. The result was a pair of nonlinear equations in r and t, containing exponential terms from Christoffel symbols, constructed by means of a Lagrangian variational formulation. I never solved the equations. I attempted initially, circular motion around the black-hole. However, this seemed to be complicated by Fermi-Walker transport and reference frame rotation considerations, so I stuck to the easier radial problem.

That problem in turn, was prompted by an 'infinite vantage point' notion. That is, in flat space, fly around the planet, at finite height above the circumference. At close to the speed of light, tangential speed, the twin paradox says that time 'goes slow in the rocket'. Pass radially, from Earth to rocket, a recording of Earth history over elapsed time. Then play-back that Earth history, on the rocket, at high speed. Consider N such rockets, as N gets arbitrarily large and finite, and with steady increase in corresponding radial heights. Whether or not there happens to be any maximum tangential speed, c_mech, for mechanical transport, could the 'outermost rocket' 'see' (much of) Earth history, in (an arbitrarily) 'small' time ?

As complete speculation, in a gridded space-time, with finite minimal dx and dt, if dynamical solutions happened to be eigenvalue in form, and also not 'continuous' to 'infinity', then E = E(k) = E(m . dx/dt) and the (classical) solution would be broadly energy-band-like. Like a space-time crystal. Something roughly of Bloch theorem type. With an inertial 'effective mass'. Then kinetic energy might be limited to a maximum value, within a band, irrespective of dx/dt, and inertial 'effective mass' would vary with dx/dt. Particular variation depending on specific definition of inertial 'effective mass', e.g., 1 / m* ~ d2E / dk2 or ~1/k dA(E) / dk, for area A enclosed by contour E(k) = E, say.

Capturing only (regular) 6-nearest neighbour connectivity in 3D Cartesian coordinates, a 3D Poisson matrix has simple, c_norm . sin( m . n . PI / (N+1) ), eigenvectors, and thus respective eigenvalues. And the easiest 'no-boundary Universe' would be a reciprocal lattice vector type cube, i.e., periodicity connecting each face to its opposite face. Else Dirichlet or Neumann boundary conditions, say, essentially irrelevant for any space-time (sub-)volume for which any contained matter and / or energy field, and its gradient, both went essentially to zero at cube faces, i.e., for cube faces so distant, in space and / or time, that detailed space-time structure was not 'sampled' and did not matter there.

More generally, for (some real, R, or rational, Q, not complex, C) no-boundary Universes in n-D (no imaginary perpendicular time):

f(x_1, x_2, ..., x_n+1) = a^2,

then check for boundedness and closure, like,

x^2 + y^2 + z^2 = r^2, say, as a 2D no-boundary Universe, 'throwing away' the 3D embedding space. And presumably, there are known general topological results at each n-D. To search.

Higgs in Gridded Space-Time and QFT's Without Infinities Or Non-Rational Reals

-- And as further related Q's-to-self:

1. What might the Higgs mechanism look like, in a gridded space-time, with finite minimal dx and dt ?
2. How do all the gauge field theories look, when evaluated only over finite gridded space-time, with finite minimal dx and dt ? Then working to N arbitrarily large and finite, including all useful function evaluations as power series to x^N, within a given radius of convergence, no infinite numbers at all will ever be generated (avoiding 1 / (n - n)).
3. So do all infinity and renormalisation problems go away, or not ? And subtracting, N1 - N2, with N1 and N2 both arbitrarily large but finite and well specified, could we obtain for instance, N1 - N2 = m, for small and meaningful, m, for example ?
4. And strictly, to the 'uncertainty' experimentally-accessible, all scientific computation in mat-vecs, with only rational numbers, m / n, no other real numbers, and no actually infinite numbers at all. Thus excluding all actual continua and continuum infinity, aleph-1. Not even integer infinity aleph-0. Instead, N arbitrarily large but finite.
5. Circumference = PI x diameter, as an approximation to, and idealisation of, a circle-like N-polygon. No gradients specified for vertical lines. Instead note that (x_1, ..., x_n-1) = const, in n-dimensions (and if measured, then to experimental accuracy and statistical reproducibility). Etc.
6. Analysis 'replaced by', in practice 'reduced to', discrete applicable mathematics.

Observation: science works pretty well already. So where might an approach such as that outlined above, actually make calculable and measurable, thus systematically falsifiable, differences ? That is, actually matter as science ?

And if no such, well and good. However, scientific good practice to check that out first.

Special Relativity: Inertial Frames and Maximum Universal Speed(s)

Consider Special Relativity with frame S’ moving at speed v relative to frame S, parallel to the x-axis.

If the theory is obtained on the premise that science is the same in all inertial frames, then Newton’s laws must be the same in all inertial frames, thus

F = m a

However, if the speed of light is a limiting universal speed, then m = m(u), and becomes infinite as u -> c

F = m(u) a

for a moving mass, m(u), with speed, u, moving within a given inertial frame. Directions of F and a, unspecified and arbitrary, (though in the same direction as each other), relative to direction of u. Speeds in the direction of F and a could be much lower than u -> c, so that the combined description in terms of SR and Newton's laws remains good.

Assume a form, m(v), for a moving mass m = m(v) in S, e.g., at origin of the system S’ and regarded from S. With m = m(v), SR is not inertial in the sense of science being the same in all inertial frames. That v is not dy/dt or dy’/dt’ within the frames S and S’, say. 'Constant' mass can be assumed coordinate independent within each inertial frame. Thus that m(v) captures explicitly, relative velocity between inertial frames. If the mass, m(v), is now pushed in the y-direction, for constant F and a, y-acceleration will be v-dependent in S and S’. And combination of Newton's laws with SR, in a combined thought experiment, should remain good for dy/dt, dy'/dt' << v.

Equally, consider frame S' at rest, and 'own' frame S, moving at speed, -v, relative to S', along the x-axis. Then the speed, -v, of 'own' frame, would appear to be obtainable immediately, simply by examining 'how hard to accelerate' was mass, m = m(v) = m(-v), in the y-direction.

Checking the derivation of that m(v), it can be 'obtained' from conservation of momentum arguments, say, utilising 2D velocity transformations, e.g., for dy/dt in terms of relative speed of inertial frames, v, parallel to the x-axis. For relative velocity parallel to the x-axis, no transformation of dy/dt should be expected, except in combination with m(v).

-- Conservation of momentum follows from translation invariance. Is that obvious for inertial frames in relative motion along the direction of translation ?

— Umm. And I should check in just which direction that conservation of momentum was being applied. To do.

Speeds Greater Than Speed of Light in Vacuum, c ?

Therefore, if ‘inertial’ is taken to mean that science is the same in all inertial frames, then electromagnetic wave speed in a vacuum, c, is also the same in all inertial frames. However, there is no obvious reason why a mechanical system should not accelerate indefinitely. And if not, why should its upper limit be, c, the speed of light in vacuum ?

Considering a mechanical system accelerated to 10c, we see that the Lorentz transformation, (x, t) -> (x’, t’) reverses x and t, thus x<->t’ and t<->x’. (Up to imaginary number i, speed c, and +/- depending on the branch of the square root, but cancelling between x and t, x’ and t’.) Then compare d2x/dt2 = a = const, with d2t/dx2 ~{ 0 to +/- to 0 } != const. Not surprisingly, passing through the singularity at v = c, the inertial frame definition no longer holds. The model would need extending above v = c and probably as v -> c as well.

It would therefore be worthwhile considering if a Michelson-Morley experiment could be performed up to, and perhaps beyond, the speed of light.

• For instance, in micro-machined form: 10^5 atoms x 10^5 atoms ~10^-14 kg (Avogadro's number ~10^26 and atomic weights 1 to ~300), with applied force F ~ 1000 N (about 1/10 that for a car from 0 mph to 100 mph in 10 s), reaching 10c over a lab length of 50 m.
• And obviously, even without micro-machining M-M, simply accelerating anything mechanical to above the speed of light in vacuum, would be hugely important (!). In fact, micro-machined engines have been suggested as alternative battery storage technology, due to their high energy density. Thus the energy density goes in the correct direction. So perhaps micro-machined engines, rockets, etc., maybe in materials like SU-8, might be feasible ?

[ Say, 1 mg of propellant, to 1000 ms^-1 in 1 us. Thus 1 mm^3 at water density, (10 mm)^3 at air density. Which does, perhaps, flag some length scale discrepancy at: ~10^10 atoms ~ (2 x 10^3)^3, thus ~1 um per side. Though in terms of container surfaces ~ (10^5)^2 maybe ~0.1 mm per side and not too far out, as an order of magnitude estimate. Key issues would be efficiency, given heating of propellant and exhaust gases, and speed of propellant gases.

-- Some related articles: wiki/Rocket_engine, micro-rockets, e.g., IEEE Spectrum, and self-assembled nano-rockets for bio-organic drug delivery, Radboud University. ]

• Do long-time-run accelerator experiments, like the LHC, preclude such a possibility ? Not immediately. Maybe different upper limits, c_n, for different systems, n, say. I do not know how much 'stuff' has actually been tested. What about accelerating 'all' (!) charge-neutral 'stuff' mechanically, for instance ?

Earth speed for M-M ether experiments is about c / 10,000. And obviously, SR appears to work fine. So the question would be, are there any differences to be discovered where it has not been tested ?

-- And it has not been tested, almost everywhere.

Interestingly, NASA is examining the Alcubierre drive, 1994, which requires negative mass (for instance the Casimir vacuum between parallel plates) to achieve faster-than-light traversal of the Universe, without ever exceeding the speed of light locally.

https://www.msn.com/en-gb/news/spotlight/is-nasa-actually-working-on-a-warp-drive

Inertial Frames and Relativistic Mass

If SR is good ‘as is’, then the Universal upper speed is c, the electromagnetic wave velocity in free space, and appears to take precedence over the definition of an inertial frame as one in which the laws of science are indistinguishable from other inertial frames. For F and a in the y’-direction in S’, and m = m(v), any large enough v parallel to the x-axis, should be obtainable from a = a(v) = F / m(v). In S’, the mass m = m(v) is at x’ = 0, and initially at rest at y’ = 0, until pushed in the y’-direction along the y’-axis. And combination of Newton's laws with SR, should remain good for dy'/dt' << v. Similarly for S' viewed as at rest, with S moving at, -v, with respect to S'.

Assuming arbitrarily high accuracy of measurement, over arbitrarily short times, this situation does not appear to be 'rectified' by finite signal propagation speeds, for any systems travelling at below the speed of light, c.

-- However, those assumptions will not be good, in practice.

More commonly understood ‘inertial’ would have m = const, i.e., independent of v, not m = m(v), within any given inertial frame S’, moving at speed v relative to another inertial frame S.

And for completeness in this context, consider vector (inertial) mass, mL and mT, different in perpendicular directions according to speed in those directions.

Mass in special relativity - Wikipedia, ‘Relativistic mass’, Tolman 1912.

Retaining a scalar mass, a linearity assumption would imply that successive 'addition of 3-vector component speeds', from inertial frame to inertial frame, along perpendicular axes, and in any order, for any given inertial frame relative velocity 3-vector, should always give the same result for the scalar mass. And agree with the result for the scalar mass using the magnitude of the relative velocity 3-vector in the direction of propagation.

In this case, taking ux, uy, uz ~ c- / sqrt(3), would seem to imply finite m(u), from the product of x, y, z gamma factors, thus possible continued acceleration past u -> c to u > c. Whereas using u -> c directly, in the (1,1,1)-direction, gives the expected m(u) -> infinity, as u -> c, thus always u < c. That is, for gamma = gamma(v),

gamma( ux ) . gamma( uy ) . gamma( uz ) != gamma( sqrt( ux^2 + uy^2 + uz^2) )

-- Though maybe correct application of the above, would also require use of 2D (and 3D) velocity addition formulae. I have still to try that ... .

-- And simple 1D composition, purely along the x-direction, does show an immediate and simple generalisation, below.

Tolman 1934, argues that m = m(v) holds only for particles moving at less than the speed of light. And that m = E / c^2 holds for all particles, including those moving at the speed of light. However, there is no indication of evaluation for v > c, which would appear not to be disallowed immediately, based on successive linear ‘addition of 3-vector component speeds’, i.e., successive ‘Lorentz transformations’ on mass in x-, y- and then z-directions, say. Unless the m = E / c^2 form precludes it.

Examining this, 1 / sqrt(1 - v^2/c^2) is singular and then imaginary, but 1 / sqrt(1 + p^2 / m(v=0)^2) is not immediately obviously problematic as v -> c or for v > c. For v >> c, factors of i and +/- 1 cancel in dx/dt or dt/dx, etc., though the Lorentz transformations do not maintain the ‘inertial’ definition for v >= c and would need reworking anyway.

LHC Thought Experiments

Examining plots of increasing electron mass with speed on wiki, presumably masses are not obtained directly. Perhaps instead, from increase in magnetic field, required to maintain a circular orbit at increasing speed. If so, the relativistic effect is on EM field, not on mass directly. Thus I would be interested to know, what experiments, if any, have been performed to 'push' circulating charged particles radially. And is the mass thus known to be mT = m(v) = gamma(v) m(v = 0), in the radial direction, or not, with v the tangential speed ? Clearly, for circulating charge, a radial 'push' would change radius of 'orbit'. Thus a linear accelerator might be better, for such an experiment. However, then there would still be the questions of: (a) how to supply the radial or perpendicular 'push', and (b) the collective effects of relativistic charged particle interactions.

-- Would a mechanical 'push', rather than an EM 'push', be possible, in the radial or perpendicular direction ? Perhaps fall under gravity, say ? For instance, do LHC magnets need any asymmetry up-down, versus left-right, across a diameter, because particles have mass and circulate so many times ? If not, why not, if relativistic particle masses produce 'heavy' particle weights ?

On the LHC thought experiment, F = ma, with F = mg and mv^2 / r, gives g ~ 10 ms^-2, v^2 / r ~ 10^14 ms^-2 and v / c ~ 99% thus m = m(v) increasing x10,000. Hence gravity effects would be hidden as a x10,000 variation at 1 in 10^13. Thus unlikely to be detectable above standard magnetic field non uniformity. Despite the fact that fall under gravity, s = 1/2 g t^2 ~ 5 m for t ~ 1 s. (About 11,000 orbits per second, with 10^9 collisions from an initial proton bunch of 10^11 protons, per second, i.e., perhaps < 100 s lifetimes.) Thus the routine ‘radial pushes’ to avoid walls, left-right or up-down, would presumably give no handle on the actual transverse mass values, mT = m(v).

Absolute 'Accelerometer Histories' Thus 'Absolute Speeds' (Local in Space and Time)

I had in mind a thought experiment, with a screen between me and the y-direction-pushed mass, m = m(-vx) = m(vx). Whether I am at rest, or moving in the x-direction, I see the mass only instantaneously, through a gap in the screen. I cannot see relative motion. I push that mass instantaneously in the y-direction. It either moves easily, or it does not. The mass looks exactly the same, either way. I make an instantaneous test. I am not using relative x-motion information. I can see none. The experiment is performed (almost) entirely within my own inertial frame. However, I can detect uniform x-motion. I should not be able to do that, i.e., detect uniform x-motion of my own inertial frame, from observations purely within my own inertial frame.

I guess the answer is that the y-directional push observation, is not purely within my own inertial frame ?

— However, that particular y-directional means of assessing my constant relative x-speed, still strikes me as ‘odd’.

If mT = m(v) = gamma(v) m(v = 0), then it would appear that v might be determined purely within a given reference frame, by observation of acceleration in the perpendicular or radial direction. Have those experiments been performed ? Perhaps such experiments are performed routinely ?

Certainly, radial adjustments of LHC-type circulating beams, to avoid walls are routine, (Quora, Erik Kofoed). I remember that accelerating charges radiate. And circular motion demands acceleration towards the centre. Also, systems will always be non uniform in the broadest sense, thus perhaps 'unstable'. Then the implicit assumption would be that radial pushes, to correct for losses and instability, would be consistent with relativistic masses. Is that immediately checkable ? Would it be obvious, if the usual transverse / radial mass relation, mT = gamma( v ) m(v = 0) did not hold true, for tangential speed, v ? That is, are those routine and expected pushes good support, or not, for the conventional, m = m(v), description ? Presumably, no flagged anomalies, implies that all is good. But this would still not really be an, m = m(v), effect. Interpretation instead, in terms of EM field dependence on tangential speed, v, say ? Are there any indirect means of inferring the situation in the perpendicular or radial direction ? I will web search.

Assuming m = m(v), it would appear to be 'much harder' to accelerate instantaneously, a mass, m(v) = m(-v), in the perpendicular (or radial) direction, when travelling past it (tangentially) at high speed, -v, than when at rest relative to the mass, v = 0.

-- Question to self: is there something wrong with that argument ?

Can I see it, even within SR ? Stand next to a small mass at rest, m = m(vx = 0), and give it a light push, supplying an instantaneous impulse along the y-axis. The small mass moves easily. Zoom past the same mass in a rocket, travelling at speed, vx <~ c, (c the speed of light in vacuum), in the x-direction, and give the same mass, m = m(-vx) = m(vx), the same instantaneous impulse along the y-axis. The mass barely moves and it feels like you are pushing on an immovable mountain. Is it true ? Why ?

-- Finite duration of impulse, and thus the most immediate relevance of clocks running slow, also finite signal propagation speeds, have been taken out of the problem formulation, initially. Certainly, a classical mechanics problem could be formulated in this broad fashion, in other circumstances and scenarios.

-- First 'rough' thoughts being that energy has inertia. And the possible 'complications' of 'transverse mass', mT = m(vx), with velocity components, vx >> dy'/dt' and dy/dt, in perpendicular directions. Therefore possible 'instantaneous acceleration' of 'inertia generated by y-impulse to speed, dy'/dt' and dy/dt' as that 'y-inertia' achieves immediate 'x-speed, -vx' (?). For instance, is a more general vector mass, m_vec(vx,vy,vz), required or implied ? Perhaps along with the 2D (and 3D) velocity addition formulae ? And likely then, with an appropriate, relativistic, presumably 4-vector, formulation of Newton's laws ?

Is it obvious that such a consistent formulation exists ? And is it obvious that any such, would take the 'standard form' ? For instance, considering an intuitive 'linearity and associativity and commutativity' assumption, for composition of any general, Lorentz transformation related mass variation, from the individual perpendicular components, vx, vy, vz, of some relative 3-vector velocity, v, between inertial frames. I'll think about it a bit more.

--The definition of inertial that I was adopting, was that the laws of physics were indistinguishable between inertial frames, e.g., constancy of EM wave speed in vacuum, like the Michelson Morley experiment.

What bothers me particularly, about the instantaneous y-direction test, to determine the x-speed of ‘own’ inertial frame, vx, from ‘entirely within’ (?) that frame, is that ‘I’ move fast, but the mass, m = m(-vx) = m(vx), ‘gets harder to push’.

However, speeds ‘are absolute’ in the sense of acceleration / deceleration records for a collection of inertial frames all initially at relative x-speeds, vx_i = 0, accelerated / decelerated to a range of constant relative speeds, { vx_i }, i = 1, …, N, all non zero relative to the initial inertial frame, vx_0 = 0.

If my ‘accelerometer history’ shows acceleration / deceleration, but that of the test mass, does not, then how can I find the test mass ‘harder to push’, when ‘I’ must have gotten more massive ?

I would now assert that ‘speed is absolute’ in the sense of absolute ‘accelerometer histories’ from an initial mutual inertial frame. Thus if ‘accelerometer histories’ say that the mass, m, has not accelerated along the x-axis, i.e., m = m(vx = 0) continues to hold true, but that my own ‘accelerometer history’ says that I am now moving at vx ~< c along the x-axis, then I will have gotten more massive and the mass m = m(vx = 0) has not, in the initial mutual inertial frame.

Then it would not be true to argue that m = m(-vx) = m(vx). When I apply the same impulse along the y-axis, the mass will move as easily as initially.

The contrast being between a small mass which moves easily and a v ~ c- relativistic mass that feels ‘like pushing a mountain’.

If the ‘accelerometer histories’ are ‘the other way round’, then I suggest that applying the same impulse along the y-axis, the high relativistic mass will mean that it does feel ‘like pushing a mountain’.

And I would argue that this is a variant of the ‘twin paradox’, which is not a paradox, but implies that ‘accelerometer histories’, hence final accelerated/decelerated-to-constant relative inertial frame speeds, are absolute, in the sense described above (and local in space and time).

Thus I suggest that the answer to the original question would be ‘not true in general, but true in special cases’, depending on absolute ‘accelerometer histories’ relative to the original mutual inertial frame.

Composition of Relativistic Masses

As for combining x-direction constant relative speeds, v and ux’, into a compound mass variation, m = m(ux) = m( ux’, v ), if

ux = ( ux’ + v ) / ( 1 + v . ux’ / c^2 )

then the composition rule for m(v) = gamma(v) m(v=0) appears to be not,

gamma(ux) = gamma( ux’, v ) != gamma(ux’) gamma(v)

as might be ‘expected’ from the simplest ‘linear, commutative, associative’ decomposition, but rather,

gamma(ux) = gamma( ux’, v ) = gamma(ux’) gamma(v) ( 1 + v. ux’ / c^2 )

which reduces to the previous form for v = 0 or ux’ = 0, or as a low-speed approximation for ux’ or v or both << c, and for v = -ux', i.e., ux = 0.

-- For 'addition' of relative inertial frame speeds along the x-axis, similar generalisation is obtained for composition based on individual ux,' uy', uz != 0,

gamma(u) = gamma( ux’, uy', uz', v ) = gamma(u’) gamma(v) ( 1 + v. ux’ / c^2 )

where,

u'^2 = ux'^2 + uy'^2 + uz'^2.

Further related generalisation might therefore be expected for 'addition' of relative inertial frame speed components along the x-axis, y-axis and z-axis. I will try it.

Accelerometer Histories with Attached Chronometers

I can see the standard event-based picture, (xA,tA) and (xB,tB), etc., and I will try writing it out fully explicitly, with all events and equations actually listed.

And I know that my (currently) suggested 'interpretation' is non standard. Right or wrong. Still thinking this through ... .

I am arguing that starting from any collection of initial, mutual inertial frames, vx_i = 0, i = 0, ..., N, the Lorentz transformations are not everything, I am arguing that the 'accelerometer histories' are everything, i.e., giving (non-)accelerated/decelerated 'absolute speeds' vx_i ~ c- OR 0, i = 1, ..., N, from that initial vx_0 = 0, in the example variants considered.

So I am talking SR 'hysteresis', as opposed to pure constant vx != 0 'equation of state', as the Lorentz transformation analogy when combined with the 'twin paradox'.

-- Is the 'twin paradox' General Relativity ? As it captures acceleration, not only constant, non zero, relative speeds between inertial frames ?

-- It is also interesting to speculate whether the Big Bang might have provided such an initial mutual inertial rest frame (without subsequent 'accelerometer histories' being known explicitly) so that velocities relative to that initial frame would be absolute. However, the initial singularity, inflation, the global structure of Universe space-time, space-time curvature, time-varying inhomogeneity and anisotropy of mass-energy, make that question look non trivial to me.

I guess to answer that question in part, I could ask how old am I, and how old is the test mass, in each configuration ? If I accelerate/decelerate from vx_i = 0, am I at a different age to if I do not accelerate/decelerate ? Similarly for the test mass ? So perhaps what is required, is to attach chronometers to myself and to the test mass, and then compare y-direction ease of push, i.e., mass feels light, or mass feels like pushing a mountain, in each scenario.

So the difference from the usual SR configuration, is that SR typically simply considers different inertial frames. However, I am asking the question 'How do I fill in the time evolution, from all frames at rest, vx_i = 0, i = 0, ..., N, to some frames not at rest, vx_i !=0, i = 1, ..., N ?' And that appears to be very important.

-- I could try 'integrating' the Lorentz transformations over time, which would presumably still be SR, not GR ? That is, I am not, I think, attempting to describe curvature of space-time by mass-energy (?). I am considering accelerating (between) inertial frames by mechanical or EM means, say. I derived the 'correct' 1D gamma(v) composition, yesterday. Bottom line, though, would be, do (pointwise) successive Lorentz transformations integrate to zero relative velocity, vx_i = 0, or not, e.g., vx_i ~ c- ? 

-- However, as I am now considering time evolution and histories, I am considering at least extended 2D space-time, (x, t0 <= t <= t1). And notionally extended 3D, with the y-direction push, even if I consider everything on the x-axis. Thus I guess, differently accelerated/decelerated inertial frames, corresponding in some sense to different, extant 2D (3D) extended space-times. And that I am considering different experiments, in different extant extended space-times, and at different times, tA, tB, in those different space-times. Thus I will be capturing the various different acceleration schemes and their specific time dependencies. Even if I do not capture explicitly the respective actual energy expenditures, etc.

Then I am arguing that viewed from within the initial inertial frame, vx_0 = 0, I can tell whether the y-direction ease of push of the mass feels light, or mass feels like pushing a mountain, in each scenario. And that depends on whether integrated, vx_i = 0 or vx_i ~ c-, for the mass, not the difference between integrated, vx_i and vx_i', for the mass and myself

-- viewed consistently from within the initial inertial frame, vx_0 = 0.

Though all the usual questions of the effect 'viewed' from within my own frame, or that of the test mass, still apply. In which case, how can an instantaneous push 'feel' different, i.e., 'light' or 'weight of a mountain', from within different frames ? And must I now disentangle both different frames and different frame accelerometer histories ?

-- Always working consistently within a single frame, to avoid apparent paradox. Or else, transforming correctly between frames, to 'explain' one frame 'view', (events only to start with, (xA,tA) and (xB,tB), say, not yet any additional finite signal propagation complications) from within another frame.

-- If I now have chronometers attached, can I have multi-valued chronometer ages, depending on the detailed specific 'accelerometer histories' ? Thus this would actually be 'equation of state' on the integrated vx_i for the mass, in the, vx_0 = 0, inertial frame, from the point of view of y-direction ease of push. But 'hysteresis' from the point of view of attached chronometer age, i.e., clocks only ever run slow, for vx_i, vx_i' >< 0. Clocks never run fast for any inertial frame corresponding to integrated speeds, vx_i, vx_i' ! = 0 and i = 1, ... , N, when viewed consistently from within the initial inertial frame, vx_0 = 0 ?

Maybe (probably) the issue is again, consistent view from within a single inertial frame. And I am now considering the view from initial frame(s) at rest. I should try connecting that frame with my frame and the test mass frame, event-wise, (xA,tA) and (xB,tB). 'Hysteresis' on chronometer age, appears to drop out of the 'equation of state' Lorentz transformation form, on integrated vx_i, vx_i', for ease of push of the test mass in the y-direction. Next on the to do list ... . 

-- I am still not 'seeing' the expected (?) 'symmetric', y-direction push of the mass feels 'light', or y-direction push of the mass feels 'like pushing a mountain', result.

x-Direction Mechanical Michelson Morley with Mass Acceleration in y-Direction

To determine the 'resistance' of the y-direction push, would require consideration of instantaneous acceleration of the test mass in the y-direction, d2Y/dt2 or d2Y'/dt'2, for either the initial vx_i = 0 frame(s), test mass or own accelerated/decelerated vx_i, vx_i' ~ c- frame.

Thus consider a mechanical Michelson Morley-type experiment, with motion in the x-direction of a system containing a point mass, vibrating on a spring about the origin, along the y'-axis. For vibration of point mass, m', about y' = 0, with spring constant k, from an initial speed, u',

Y'(t) = (u' / omega') sin( omega' . t' ) with omega'^2 = k / m'

Considering systems S at rest, (x, y, t), and S' moving at speed, v < c, along the x-axis in S, (x',y',t'), the 1D Lorentz transformation is:

t' = gamma . (t - v.x / c^2)

x' = gamma . (x - v.t)

gamma = gamma(v) = 1 / sqrt(1 - v^2/c^2)

If S' is initially coincident with S, the x, x'-position of the mass in S, S' is,

X'(t) = X(t) - vt

and X'(t) = 0 for all t and t' => X(t) = vt.

Then by the definition of an inertial frame,

Y'(t') = Y(t)

so that at the point mass on the spring,

(u' / omega') sin( omega' . t'(t) ) = (u / omega) sin( omega . t )

t'(t) = gamma . (t - v . X(t) / c^2) = t / gamma

hence,

(u' / omega') sin( omega' . t / gamma ) = (u / omega) sin( omega . t )

where either

u = u',

or

u = u' / gamma

which is inconsistent if omega = omega', i.e., if m = m', unless v = 0 so that gamma = 1 and u = u'. And non trivial and non standard as a relation for, m' = m'(v), if taken to define the necessary velocity dependence of the point mass vibrating about the origin, along the y'-axis in S', with initially coincident S', moving along the x-axis at speed, v, in S.

For instance,

m'(v) = m(v = 0) / gamma(v)^2 or t-dependent solution of a transcendental equation

Thus something wrong here. Reasoning, algebra, non invariant forms, other, ... ?

Or maybe spring constant, k, must scale too ? Though it is associated with y-direction force, only as a multiplier of y-displacement, not y-speed. Then definition of an inertial frame, would appear to indicate no requirement for spring constant scaling. And looking at the 4-vector form of Newton's laws appears to give rise to a nonlinear mass-on-spring equation, but one that solves as a simple quadratic in explicit time-stepping form. I will test that solution, e.g., for stability, against the 'zeroth order' non relativistic mass-on-spring approximation, some time soon. And check that the difference is small, not large.

However, maybe the solution as derived above is correct and should be expressed,

u' = u . gamma

omega' = omega . gamma

Thus amplitude of oscillation,

u / omega = u' / omega'

does not change.

But frequency in S' coordinates tends from omega to infinity as v -> c-.

For frequency in S' to go up, mass would have to go down, as omega'^2 = k / m'. Hence the observed m'(v) relation.

Perhaps this is intuitive in the form: (a) x-direction Lorentz transformation does not affect y-displacement amplitude, and (b) moving clocks run slow, thus long intervals and time periods become short intervals and time periods, and frequencies go up.

Alternative Derivation with Integrated Accelerometer Counters

My understanding of the twin paradox is that what counts is acceleration. Acceleration is absolutely detectable. The laws of physics no longer take their simplest form.

Thus along with a chronometer, we could add a counter for summed/integrated acceleration/deceleration. We could display that number alongside our chronometer. The chronometer could be arbitrarily multi-valued relative to the summed/integrated acceleration/deceleration counter. And the acceleration/deceleration counter would always show absolute speed, relative to the frame in which we started summing/integrating.

Relative to the frame in which we started summing/integrating, we can therefore always ascertain whether we are moving, and therefore by implication, more massive.

To ascertain 'ease of push in y-direction', we need instantaneous acceleration in y-direction, d2Y/dt2 or d2Y'/dt'2. I can see that gamma factors will enter in t or t'.

The question is then, who is at rest ? And relative to whom ?

Consider the original rest frame. With test mass either at rest or moving relative to that rest frame, in the x-direction. And with me either at rest or moving relative to that rest frame, in the x-direction.

We can all be point masses, so as to avoid body rotation effects, etc. And so that all 'pushes' are 'direct and central'.

In the original rest frame, if test mass is at rest, it is easy to push it in the y-direction. If the test mass moves at close to c, it is hard to push it in the y-direction. This is true whether I am moving or not. However, in the original rest frame, my mass will also vary with my speed, thus in a free-space situation, there will be 4 combinations: (i) small mass pushes small mass, (ii) small mass pushes large mass, (iii) large mass pushes small mass, (iv) large mass pushes large mass.

I think all of these situations are detectable uniquely, from the local, instantaneous push in the y-direction experiments.

I then have to consider the effect relative to my own frame. I am at rest, so my mass is small, in my own frame. However, I and the test mass carry our own chronometers, and our own integrated acceleration/deceleration counters relative to the initial rest frame. Thus we both know our absolute speeds relative to the initial rest frame.

My integrated acceleration/deceleration counter can say 0 or c-, relative to the initial rest frame. Both are consistent with my being at rest in my own frame. What about the test mass ? That can also say 0 or c-, relative to the initial rest frame. To keep things simple, consider all motions only along the positive x-direction. The paired counters can be (me, test mass) = (0,0), (0,c-), (c-,0), (c-,c-).

We have already discussed those four possibilities viewed from the initial rest frame. All good.

Now how about from my rest frame ? Consider (0,0) or (c-,c-), all speeds in same +ve x-direction. From my own rest frame, both of those should show an easy y-direction push, as relative speed is zero. No problem for (0,0). Any problem for (c-,c-) ? In the original mutual rest frame, we see a hard y-direction push, between two large masses, so how to reconcile ? Hardness of push is detected via instantaneous acceleration in my own rest frame.

Let initial mutual rest frame, (0,0), be S, (x,y,t).

Let final mutual rest frame, (c-,c-), be S', (x',y',t').

1D Lorentz transformation is:

t' = g (t - v . x / c^2)

x' = g (x - v. t)

y' = y

g = 1 / sqrt(1 - v^2 / c^2)

We need d2Y'/dt'2 in S', where F' = m' d2Y'/dt'2 ,with Y'(t') the y'-displacement of mass in S' coordinates.

By definition of inertial frame, Y(t) = Y'(t').

We will say that I apply 1 Newton in S', in the y'-direction.

What force do I apply in S ? Also 1 Newton. These are inertial frames.

Thus F = m d2Y/dt2 = 1 in S.

And F' = m' d2Y'/dt'2 = 1 in S'.

How to relate d2Y/dt2 and d2Y'/dt'2 ?

We have Y(t) = Y'(t') and will assume mass is X'(t) = X'(t') = 0.

As X'(t) = X(t) - vt, X(t) = vt.

Thus at the mass,

t'(t) = g (t - v . X(t) / c^2) = t / g.

Hence with Y = Y' and t varying from t' only by g, at the position of the mass,

d2Y/dt2 = d2Y'/dt2 = d2Y'/d(gt')^2 = (1/g^2) d2Y'/dt'2

However,

m d2Y/dt2 = m' d2Y'/dt'2 = m' g^2 d2Y/dt2

so 

m' = m / g^2 = m (1 - v^2 / c^2)

... which is simply wrong again.

Why ? Reasoning, algebra, non invariant forms (the above would appear to hold for general force, F(t) = F'(t'), thus presumably similar considerations to the earlier mass-on-spring case), spring constant scaling required (if y-direction force is imagined to applied by a compressed spring of zero length), other, ... ?

Perhaps the partial second derivative, d2Y'/dt'2, needs constructing fully, for x, t != 0, then substituting, x = X(t), later.

Intuitive Form as y-Displacement Invariant with Clocks Running Slow ?

-- Still thinking about this ... . However, perhaps again, this is intuitive in the form: (a) x-direction Lorentz transformation does not affect y-displacement amplitude, and (b) moving clocks run slow, thus long intervals and time periods become short intervals and time periods, and in oscillatory cases, frequencies go up.

Following that line of reasoning, consider y-directed acceleration, with y-coordinates unaffected by inertial frames in relative constant motion along the x-direction. Then time runs slow in any moving system, i.e., t’ = t / gamma = t sqrt(1 - v^2 / c^2) -> 0 as v -> c, at x’ = 0 or x = vt, which will give larger y-acceleration when viewed from any moving system, i.e., 'easier push'. And 'easy push', i.e., the same initial y-acceleration, when viewed from any system at rest. Thus push is 'easy' or 'easier', never 'harder'.

-- Which is 100% diametrically opposed to the 'mass increases with relative speed' argument. That sounds experimentally testable. Is the argument correct ?

In fact, the previous argument looks almost consistent with wiki, but not quite:

Mass in special relativity - Wikipedia

' ... The precise relativistic expression (which is equivalent to Lorentz's) relating force and acceleration for a particle with non-zero rest mass, m, moving in the x-direction with velocity, v, and associated Lorentz factor, gamma(v), is

fx = m . gamma^3 ax = mL . ax

fy = m . gamma ay = mT . ay

fz = m . gamma az = mT . az ... '

for forces, fx, fy, fz, and accelerations, ax, ay, az, defined in rest frame S, (x,y,z,t) -- not S', (x',y',z',t').

Then,

mT = mL / gamma^2

as deduced above.

Thus as,

mL = m / (1 - v^2 / c^2)^3/2 -> infinity for v -> c,

mT -> mL (1 - v^2 / c^2)

and becomes progressively less than mL.

Both mL, mT ->infinity as v -> c. However, this formulation does not answer directly, the original question as posed. Presumably, the wiki description is correct, thus these two latest descriptions still need reconciling. Still thinking about this ... .

And the key question then appears to be, should those 'm . gamma's', actually only be m's ? If wiki is correct, then I need to understand why. And the answer to that question presumably drops out of the invariant, four-vector Newton's equations, which is where I will look next ... .

Einstein's Suggested Experiment for mL and mT

I note that the following 1906 paper by Einstein (wiki):

https://myweb.rz.uni-augsburg.de/~eckern/adp/history/einstein-papers/1906_21_583-586.pdf

appears to be a suggested experimental examination of the ratio of longitudinal to transverse masses of accelerated electrons, comparing Einstein-Lorentz predictions against those of two competing models. As noted in wiki, Einstein has in his 1905 paper, the alternate mT form,

mL = m / (1 - v^2 / c^2)^3/2

mT = m / (1 - v^2 / c^2)

based on a different definition of force. And stating that care is required in these calculations. Wiki also points out that Einstein later moved away from use of relativistic mass, altogether.

Interestingly, Einstein's suggested experiment appears to be aimed at ascertaining only the ratio of mL and mT. Therefore that experiment would not address the key questions raised here, which is whether there is one factor of gamma too many, in the Lorentz-Einstein formulae for mL and mT, as given on wiki. And what is the correct form of the equations for a (mechanical) 'transverse oscillator'.

The difference between a transverse mass which goes to zero, and one which goes to infinity, (always choosing 'correct' reference frame, to avoid paradox and inconsistency), is clearly large (!).

Is the Lorentz transformation for electromagnetism, the same as the 'Lorentz transformation' for other system types, e.g., mechanical? If so, why?

If someone were to build a micro-machined rocket, say 10^10 atoms in SU-8, as a suggestion, is it known that after optimising the micro-machined rocket physics, powered by hydrocarbon fuel or other rocket fuel, not electromagnetic propulsion, that such a rocket could not, for instance, reach ten times the speed of light in vacuum, 10 c ?

Perhaps over the length of a long lab ?

Is the experimental evidence totally compelling to the contrary ? If so, what are some examples of such evidence ?

And perhaps think about the implications of any electromagnetic equivalent of the sonic boom.

I know of the Michelson Morley experiment, which demonstrated invariance of speed of light in vacuum. Have there been any ‘mechanical’ or ‘other’ Michelson Morley type experiments ?

Why should ‘mechanical’ or all ‘other’ systems, say, have the same maximum speed, or even have a maximum speed ? That is, why should mechanical systems be constrained by an ‘electromagnetic’ upper speed ?

How well has this upper speed been tested, for ‘all types’ of systems ?

Are there other fundamental reasons why the ‘conventional Lorentz transformation’ can be inferred to apply ‘everywhere’ ? Rather than assumed or asserted ?

Is there comprehensive experimental evidence ?

I know for instance, of related considerations like increasing LHC magnetic fields, to constrain ever faster circulating particles. However, that is again broadly electromagnetic. And I know lifetime of the muon through the atmosphere type arguments.

Is there a lot more ‘direct’ evidence, for all types of systems ?

If the speed of Earth relative to the Sun is c/10,000, do any oscillator frequencies vary relativistically as a result by 1:10^8, over the course of a year? Or simply on rotation of experimental equipment through 360 degrees? If not, why not?

Consider a mass on a spring, oscillating about the origin, O’, along the y’-axis in frame S’. S’ moves at speed v along the x-axis relative to frame S. At time t = t’ = 0, frames S and S’ are coincident. If the frequency of the oscillator viewed within S’ is omega’, with y’-amplitude, y’_amp, what is the frequency and amplitude viewed within S ? Same frequency or different frequency ? If different, higher or lower in which frame, S or S’ ? Same amplitude or different amplitude ? If different, higher or lower in which frame, S or S’ ? And viewed both from the point of view of the Lorentz transformation, and from the perspective of relativistic masses, mT(v) and mL(v), transverse and longitudinal to direction of motion along x and x’ axes ?

Consider a point on the Earth’s equator, ‘moving in two circles’. Ignore planetary tilt and non circular orbit. Relative to the North pole, this point revolves at ~Mach 1, or v / c ~10^-6. Relative to the Sun (or distant stars ?), this point orbits at roughly ~Mach 100, or v / c ~ 10^-4.

Choose as a unique inertial frame, the current frame at that equatorial point, then in ~3 months’ and ~9 months’ time, the respective inertial frame will be moving at v / c in the perpendicular direction. And in ~6 months’ time, the respective inertial frame will be moving at roughly -2 v / c.

Thus why not expect / predict ‘seasonal’ Lorentz transformation differences due to orbit around the Sun ? Nothing to do with ‘radial’ gravitational red-shift.

I am considering the notion that we have ‘mechanical’ and ‘other’ Michelson Morley type experiments effectively running 24/7 all year round. Simply described, a harmonic oscillator, e.g., mass on a spring, oscillating about the origin, along the y-axis, perpendicular to high-speed motion along the x-axis. Or any other oscillator with well defined direction.

Making some order of magnitude estimates:

Earth rotation looks like ~ Earth circumference ~ Pi diameter / 24 hours ~ 3 x 8000 miles / 24 hours ~ 1000 miles / hour (!) ~ 1.6 x 10^6 m / 4000 s ~ 400 m / s ~ speed of sound ~ 300 m / s. Thus v / c ~ 10^-6. Hence v^2 / c^2 ~ 10^-12. Thus small effect.

Transistor gate length ~ 10 nm at 1 GHz, v ~ 10 x 10^-9 m x 10^9 s^-1 ~ 10 ms^-1, thus small.

I seem to remember that electron diffusion speed in a conductor is typically about 1 mm / s, even with near speed of light EM wave propagation. Thus also low.

However, power transistor finger width (perhaps end effects) ~ 10 microns at 1 GHz ~ 10^-5 m x 10^9 s^-1 ~ 10^4 ms^-1. Thus v / c ~ 10^-4 *if* charge ‘relaxation’ might occur on these time and length scales.

For thermal effects, contributing analytical series summation terms with k t / L^2 ~ 1, thus v ~ L / t ~ k / L with thermal diffusivity k ~ 1 - 100 x 10^-6 m^2 / s and L ~ 1 Angstrom ~ 10^-10 m (atomic) to ~1 m (macro) -> v ~ 10^6 - 10^-6 m / s thus v / c ~ 1% to 10^-12 %. Say, v / c ~ 1%, for polycrystalline graphite at atomic dimensions.

Where might effects at v^2 / c^2 ~ 10^-8 be relevant ? For instance, WCDMA communications signals can have -120 dB noise floors. And ‘logarithmic high accuracy’ is required for, say, spectral regrowth and intermodulation distortion products. Thus 1 in ~6 orders of magnitude can be relevant.

Thus even if, v / c ~10^-4, Earth speed relative to the Sun (or distant stars ?), might not (?) show Lorentz transformation frequency scaling effects, other than relative to a rest frame in the Sun (or distant stars ?), could v / c ~ 1% to 10^-4 be visible in any electron or phonon oscillations, say, in ‘typical’ or ‘experimentally accessible’ (electronic type) systems ?

And could any effects coming in at roughly the ~10^-8 level, say, ever impact device or system performance, say, if ‘logarithmic sensitivities’ are relevant at the ~6 order of magnitude level, or more sensitive ?

The question must always be answered with respect to any given single reference frame. Or else correct transformations used to relate results expressed in different coordinate systems. I can see 1 in 200 million with, say, a factor, 1/2, from a gamma-factor, 1/sqrt(1 - v^2/c^2), if that factor is correct and not from say, a gamma^2 term.

The science always looks the same from entirely within a given inertial frame. So my question becomes about ‘relativistic masses’, along longitudinal and transverse directions, relative to direction of Earth motion. Considering the case of a mass on a spring, oscillating about the origin, along the y-axis, with Earth motion at ‘large’ speed along the x-axis, are you asserting that the frequency of oscillation would never be observed to change ? Either over the course of a year, or with ‘immediate and local’ rotation of experimental equipment, at intervals, around 360 degrees ? Thus capturing velocities both parallel to, and perpendicular to, the velocity of the Earth ? (Relative to the Michelson Morley experiment reference frame, I had remembered speed of Earth as c/10,000. I had thought that speed was relative to the Sun, but it would likely have been relative to some notional ether. I should and will check whether that was therefore relative to distant stars, or some other reference frame.)

• Speed relative to the Sun might therefore not be physically correct. However, the question is still good as posed. Ultimate application might be different.

Or, as ‘relativistic mass’ is a contentious topic, consider the classical harmonic oscillator equation in the ‘perpendicular’ direction, with ‘time running slow’ in ‘moving systems’ to capture the Lorentz transformation colloquially. Is it ‘obvious’ that oscillator frequency does not ever vary ? And for instance as gamma^2, say, (my own calculation) ? And the same question for any and all oscillators, which oscillate along a well defined direction.