linear stellar aberration

(Romer ≈ Rømer ≈ Römer ≈ Rœmer ≈ Roemer)

[See also ppt-videos "no simultaneity for Einstein's Special Theory of Relativity," and unlisted priors, Correcting Einstein: The Other End Is Running Late (trailer) and the Einstein-Minkowski Acausality Paradox (trailer)]

[See also articles Space-Time-and-Light Relativity, and angular stellar aberration]

PREFACE AND SETUP:

Newton was onto something: He had calculus which his predecessor mathematician Galileo had lacked, and he could thereby easily formulate the processes of varying velocities and accelerations... So, let us work-up an arithmetic-think-about electric train all cars all identical accelerating equally by identical motors, placed in outer space away from Earth gravity (simpler) and initially at rest (such region exists or such relation can be constructed by adjusting velocity to v=0 and waiting 4 years for ³¹P low-energy-excited-state to thermalize from 'deceleration-shock', and the test-criterion might simply satisfy this-theory) and we'll use spherical photon-clocks (which Newton did-not-have but his corpuscular theory of light could have a sufficiently short pulse that'd suffice) and allow regulation of photon speed by space itself (our advised-assumption over emission; and already having electromagnetic impedance) and monitor all the function profiles from both outside and inside, of 'what we know and see'... n.b. all observers see the exact-same events, flags, data, each-others' clocks, in the exact-same places, but, not-exactly-the-same velocities, Doppler-shifted-colors, stellar-aberrations, remote-delays, (unless proven otherwise)...

(We'll also avoid, 'tricky' experiments such as changing acceleration mid-photon, or capturing or reflecting photons broadsideways, which can be fun postulation-and-calculation and may-even explain anomalies in the cosmic-big-bang, But, for the intent of finding 'Time' we'll assume experimental condensation of 'subparticle-like' laser-pulse-ruler clock-photon-travel-distance-lengths to 'spontaneous-infinitesimal'-time maintaining continuous interaction with its mirrors, to minimize 'pause, between mirrors,' and affirm a true calculation of time not-involving clock-behaviorals, as such is the nature of calculus such as Newton invented, so-that we may extract the 'best sense' thereof—and meanwhile-then watch for any cumulative effects in our equations—emphatically, assuming clock-photon-pulses are by-design retaining perfect sequential-coherence...)

The following is a ‘pure’ Newtonian analysis of the same-situation as Einstein's Space-Time [and-Light] Relativity:

SPECIAL CASE #1: stepped-constant velocity 'inertial frame' - OUTSIDE FRAME

The train jump-starts at t₀ fully self-synchronized everywhere along its length such that the velocity is constant thereafter and moves smoothly-evenly every car at the exact-same velocity straight-line (no gravity-bend) and exact-same-increasing distance as timed and measured in the outside frame—such that differential-distances and lengths remain constant the-same simultaneous-measurement-differences in the outside frame (not by inside-photon-semiroundtrips), and there-is-no grid-apparent 'length-contraction' (molecular forces and points-of-perspective to-be-determined in fine-finding kinetic and potential mass-energies, developing deBroglie wavelengths and resolving Faraday's transformer paradox etc.)...

So for t₀₊ (instantly settled) acceleration a=0 for constant velocity v=v and constantly increasing distance every car from its starting position, d = d₀ + vt, and photon-clock-race-length L=L and clock-photon-transit-event times t_n and cosmic-space-frame track and station clocks are fully grid-synchronized for considering both what we know and how it looks and conveniently located along the way (specially-engineered gray-scale-strip-clocks readable continually-instantly-exactly at every point along the track, or, cheaper tick-phase-locked-cumulator kind) and there may be adjunct observers in both frames placed at their windows so-as to confirm each such photon-pulse-arrival-passing/bouncing-event and corresponding data, on-the-spot simultaneously...

Simultaneously by design when the train starts the inside clocks each emit a photon-pulse toward their opposite end-mirrors (considering only inline traverse photons here, as transverse photons 'fall' rearward during acceleration phase, scattering as all sides are mirrored no-longer straight-perpendicular-orthogonal to the traverse; And we're allowing that photons be detected without interfering with their isochronous pulse), and each clock-photon-pulse travels at the speed-of-light c its clock-length L between its front and rear mirrors which direct-distance-length in the outside frame is shortened or lengthened by the train-velocity distance-increment to ct₁ = L − vt₁ arriving at event time t₁ = L/(c+v) sooner-c-rearward or later-c-chasing-forward than when both were =L/c at station-stop, and then promptly counter-likewise each photon-pulse bounces to return and travels t₂ = L/(c−v) all at the same speed-of-light on the grid frame and so all arrive back at their original mirrors outside-frame-simultaneously in the clock-photon-pulse-roundtrip-time t₁₊₂ = 2L c/(c²−v²)...

(Sidebar note: the entrapment of photons between mirrors gives rise to rest-mass as surely as capturing photons in atomic electron orbitals; and their release reduces rest-mass; And foreward-to and rearward-from Dopplers exactly balance end-vs-end adding no secular first-order momentum but second-order energy in the Doppler-difference; But, timing may exchange that mass-energy with the mirrors... See prior discussion.)

And thereafter coasting, for subsequent photon-pulse bounces at the front and back mirrors the roundtrip-times are unchanged at constant velocity and the train-car photon-pulse-clocks all appear to be running slow, pulsing by dilation factor γ² = 1/(1−v_c²) in the right-outside-frame-view where similar-but-standstill still-fully-synchronized roundtrip-photon-pulse-clocks count their-own-gear-time as still t_1+2,v=0 = 2L/c relatively-faster than the traveler-inside-clocks, (recall that Newtonian analysis of Einstein [co]relativity also observed clock-tripping but only by factor γ due to assumed accelerated-length-[self]contraction by γ).

And therefor-also train-frame-time-itself, as 'photon-clocks-miniaturized-to-infinitesimal', is likewise dilated...

(Sidebar note on time and energy-motion in 'infinitesimal' particles: Einstein seems to have intuited, in addition to equationed, that mass-energy-equivalence is actual action, that particles are energy-convolved-on-itself and internally scattering—subnote the contemporarily-infamous multivalued neutron decay rate—in contrast to modern zero-entropy-nondecaying-particle-electrodynamics theory, but did-not-carry-out such a concept of internal-energy-motion in particles, nor discover what slows-and-mutually-captures photons to themselves in particle-creation, nor define particle-internal-speed of energy, nor what binds photons linearly in free space,—and physicists pronounce their notion that e.g. electrons are point-particles to great energies but avoid saying electrons have a Schwarzschild radius around that point or that the speed-of-light is slower near that point, or, what-it-is-that photons contain that cooperates only with its-own-anti-equal in another photon to enforce paired-or-quad-[re]paired quantum-number-rules to lock energy into multiple spatially-extended created-particle-anti-pairs...)

But n.b. single-observer outside view-think of the whole-train is 'warped' from the whole-frame-view because cars farther are seen later by lightspeed at greater distances and so appear to be starting late, running late, relatively, and the train will be visibly 'crushing-ahead' and 'extruding-behind'—an expanding, renormalizing, linear-stellar-aberration effect.

But now immediately the question arises as to think-versus-view train times and lengths—because—while grid measurements indicate equal lengths preserved unchanged by fully-synchronized-acceleration, the sense of what the travelers must be experiencing is, changed, by directness of lengths aboard assumed measurable as by laser-pulse-semiroundtrip-ruler no longer sync'ed end-to-end—the clocks afore show time-now both grid and travel advanced, delayed less, to rearward observers, than when the train was at standstill, and counter-likewise clocks abaft show time-now retarded, delayed more, to foreward observers, and yet the many indirect, mirror-images each observer sees of its-own clock by photons fore-and-aft-roundtrip are timing-out equal-forward-versus-rearward once each roundtrip is accomplished after startup, and, there's no sense of room-time-or-length dilation adjudged against equally-dilated-clocks-and-rulers therein—and these effects are cumulative and further compounded in combinatorially-direct-and-indirect-mirror images, and, ultimately, mathematically, not,-un-noticeable, (Galilean naval-speed specious correlativity assumptions notwithstanding)—which means, onboard, thinking-and-feeling-of room-local time and length and speed of light is mostly unchanged by velocity except as it pushes direct observation paths to linear-stellar-aberration (observable both of inside and of outside 'stars')...

(N.B. this, linear aberration, is additional-to the inline-angular-refocusing effect in foreward and rearward telescopy, neither-of-which is as-easily measured as angular-stellar-aberration, as inline-dispersion is self-compensating.)

However, that said, the view is 'warped' for travelers too—fore and aft clocks and consequent traverse-distances and lengths are observed later farther, equationally repositioning-and-catching-up their apparent operations, and, any simplistic-briefish view is segmented into compound-schedule regimes—

0. t₀₋ the experiment pre-startup, phase, all-stop regime, undilated/uncontracted 'til the t₀ startup-velocity-step event
1. t₀₊ immediately-post-startup constant-velocity all-other times and traverse-distances are contracted in dilated-view
   (i.e. traveler time and traverse distance are dilated in the outside view right-outside and fully-sync'ed outside-frame)
2. t₀₊₊ thence direct/roundtrip-return partial-dilations update, catching-up by expanding photon-distance fore and aft
3. t₀₊₊₊ then started-roundtrip,-return partial-dilations update, catching-up by expanding photon-distance aft and fore
   (n.b. cotraveler clock views update sooner foreward and rearward later to equally-dilated apparent normalcy again)
4. t₀₊₊₊₊ multibounce-views cumulate to roundtrips-equalized plus the emitter-position/indice/sum-of-direct-leftovers
   (i.e. photon-distance is pathlength which in a mirrored-closed-system is twice-or-longer-than its objective diameter)

But also the catch-up process results in momentary scattering of a linear, sort: Consider a clock apart and two paths of view, one path 2 units direct and 2 units folded-down to roundtrip, the other the alternate order roundtrip first then direct—the clock-views of the already partway-mid-half photons go out-of-sync out-of-simultaneity the instant the train steps to its constant velocity—soon seen but eventually self-correcting once the alternate-path-portions are traversed, (this linear-multipath might be experimentally measurable but due to its logistical-complications we'll do just direct and semiroundtrip portions herein)...

(Note here that an Einsteinian-Minkowskian will suggest the notion of 'proper-time' as 'belonging-to' calculation of timelike dimensionality, along with time-biasing assuming the other-end of a roundtrip is, the-halfway, to simplify the equations, a linear transformation process; however, the momentary-loss-of-sync makes this piecework challenging...)

(Not that it's crucial here but Zeno paradox #? implicates velocity derivation requires infinitesimally-distinct time spots.)

It's a thinkers' paradise—

INSIDE FRAME: (insider view)

Now notice the inside observers watching locally passing track grid clocks, see their-own clock-time-photon-pulses return full-cycle in a grid-roundtrip-time at the same-instant fully-grid-synchronized right-outside grid-clocks read t₁₊₂ = γ² 2L/c—but back-when while the train had been sitting waiting to start, those roundtrip-grid-times had been =2L/c, so, the observers all, inside and outside, and at all scale-lengths big-or-small, observe the right-outside grid-time is faster than right-inside train-time (between shared events) when the train is moving, (no topological-Einsteinian-gedanken-rubbernecking-watching-prior-passed-now-far-past clocks, these proper-local-mutual-point observations are instantaneous undelayed like mutual-contact), so by-comparison the inside traveler-observers' own gear-time by their-own-clocks is running-slower (unfelt shorter times) than local-outside grid-time (larger times), and one-unit of inside time t_1+2,v = t_1+2,grid/γ² (slower by γ⁻² smaller)—but—the insider view of time t_v, length L_v, velocity v_v = d_v/t_v, lightspeed c_v too, depends, mathematically, on train-time-itself which is dilated (technically ?Einstein's assumption of a 'space-time-basis' ruling rulers and timing clocks, real even without, those rulers and clocks, i.e. mathematical consistency across experimental gaps); and particularly traveler-velocity calculates dγ⁻²/tγ⁻² = v the same, both distance and time equally γ²-contracted, and lightspeed is measured as-assumed-the-same-constant c...

But now firstly how could that be—how and-when did the inside clocks slow down—how could the inside observers see the outside-grid-time station clocks speed up when/while nothing unusual happened outside but that the train-itself and-its observers-inside had sped-up (and instantaneously-so, as by 'high school senior physics exam' rules)—

Well—firstly—the inside observers felt a 'serious-jerk' at t₀ but they're measuring time on photon-clocks which don't-particularly-suffer 'jerk' in the traverse direction, but pumping and acceleration and especially velocity and cumulatively position, in Newtonian calculus—so the only useful consideration is 'the other-end started late', and-apparently-so both fore-and-aft, which calculates to direct t = L/(c∓v) and return (±) in the right-outside-view... and, for the 'infinitesimal local reality', response-time-and-length γ²-dilations/contractions are instantaneous (for this Special Case #1)...

(n.b. 'serious-jerk', means it yields instantaneously 'serious-pumping' which yields 'serious-acceleration' for that instant, which yields 'serious-velocity-difference' which was given as constant-thereafter and so which yields consequently-subsequently 'serious-continual-position-change'—uniformly throughout... see also nomenclature, the derivative of acceleration is 'pump' not 'jerk'—it's all 'serious'.)

Now, this odd 'warp' should be understood for its modus operandi: the abruptly-accelerated traveler perceives clock time as instantly-unchanged (sic: imperceivably dilated relative-to-itself) such that fore and aft views, and the outside frame, are abruptly time-and-length-contracted and, rushing time-and-distance-length offset-differences relative to the insider which further affects their apparent time-lapse rate Δt_seen,v/Δt_v = (1+v_c)γ² = 1/(1−v_c) faster-or-slower (albeit a 'tough-chew' for photon-travelers at v_c = −1) toward their new apparent-positions fore-advanced-nearer aft-retarded-farther 'til the nearer traveler clocks are again seen normalized at their newly dilated clock rate and new direct-positions d_1,v = L/(1+v_c)γ² = (1−v_c)L at t_1,v = L/(c+v)γ² = (1−v_c)L/c linearly-stellarly-aberrated cross-compared to their semiroundtrip-measure ruler-length (implicit 'til settled) (d_1,v+d_2,v)/2 = L their original length, measured at t_1+2,v = L/c (and clocks further-farther further-later)—and then only outside-clocks continue time-contracted rushing rearward in their γ²-contracted frame also-offset by that same (1−v_c) linear stellar aberration—And therefor roundtrip-simultaneity is retained, at velocity, along-with re-dilating-effects updating catching-up across space, but direct-simultaneity is-not...

(So apparently Einstein's assumption of a lightspeed-problem-to-solve never existed in the traverse direction, but, had Newton figured this thus-far he'd likely have put it away in a 'Secret Principia' to avoid an Inquisition in his-own-church, as physicists were then gaining ground in mathematics but so-also were the churches before them—cf AD 1582 Pope Gregory XIII had realigned the calendar by 10 days and 3 months to correct slippage in the Julian Calendar over two millennia, And, soon thereafter but separately as an issue, with the advent of the AD-160-corrected-arithmetic the new 1611-KJV Bible several Genesis birth-centuries were fixed in Newtons' English so-changing that story from grotesque-myth to discernable reality)....

(Notice that emission Doppler Shift is adjusted by the traveler's clock-rate inverse, γ² for the traveler looking at the non-traveler, γ⁻² for the non-traveler looking at the traveler; and, reflection Doppler Shift is not-adjusted, already γ²⁻²-auto-compensated for the traveler, and not-involved for the non-traveler, except for surface-electron mobility phase effects.)

So the accelerated traveler clock is instantly linearly dilated (roundtrip slower) and aberrated (time-biased along its infinitesimal length, to be integrated) and the traveler roundtrip-photon-ruler is instantly linearly dilated (roundtrip longer) and aberrated (directionally position-biased along its finite, length, to be integrated) as the images of clocks and rulers fore and aft are old-views, delayed at distances, seen in the traveler's new-view, directionally, foreward, time advanced, distance foreshortened, and rearward, time retarded distance aft-lengthened, including everything, including references, a ruler held forward is directly shorter, and behind is directly longer, and, roundtrip is constant, dilated, such that linear comparisons of-the-traveler by-the-traveler (by integrals adding-up the infinitesimal and finite biases) detect no observable aberration even as such detected no observable dilation, except, its quick-change-old-to-new a savory-glimpse of parallaxic-FTL sprinkled on pre-synchronized 'fully-cooked' past-history 'settling-cooling', and v_c still =v_c and c still =c, meanwhile the unaccelerated, non-traveler clocks and rulers, are unaffected-by/unfeeling-of, the travelers', movements, transitions, antics, and so appear faster-shorter in the traveler view...

BY EQUATIONS: (t ≜ time, L ≜ objective length, d ≜ distance)

t, t_grid (fully synchronized everywhere, same rate, all directions)
L, L_grid (where L=ct semiroundtrip, equally all directions)
d, d_grid (where d=ct direct, all directions)

Pre-startup, t₀₋ (v=0):

t_v=0 = t_grid (relative to t=0, right inside right outside, everywhere)
L_v=0 = L_grid (semiroundtrip, equally both [all] directions)
d_v=0 = d_grid (as v=0; relative to d=0, direct = semiroundtrip)

t_by(L) = t_grid − |L_grid/c| (delayed t=L/c, equally both directions)
t_v=0,by(L) = t_by(L) (as v=0; delayed t=L/c, equally both directions)
t_by(d) = t_grid − |d_grid/c| (delayed t=d/c, equally both directions)
t_v=0,by(d) = t_by(d) (as v=0; delayed t=d/c, equally both directions)

Post-startup, t₀₊ (v>0):

γ² ≜ 1/(1−v_c²) ≡ 1/(1+v_c)(1−v_c) ≡ ((1/1−v_c) + (1/1+v_c)) / 2
d_trav,v = vt_grid (as v≥0; relative to d=0, t=0)

t_v = t_grid/γ² (relative to t=0, right inside right outside, everywhere)
L_v = L_grid/γ² (equally both directions, grid synchronized simultaneous)

t_d,v,c = |d_grid/(c±v)| = d_grid/c(1±v_c) (±foreward, relative to d=0, t=0)
t_v,c,d = t_d,v,c/γ² = d_v/c(1±v_c) (±foreward but only photons pertain)
⇔ d_v = d_grid/γ² (direct d linear stellar aberration, for c unchanged)
(whereas grid-direct and grid-semiroundtrip are equal, as v_grid=0)
(and photon speed is unaffected by its emitter-travel thereafter)
(redacted convincingly in Special Case #2 'stepped acceleration')

t_L,v,c = |L_grid/(c±v)| = L_grid/c(1±v_c) (likewise, ±foreward)
t_v,c,L = t_L,v,c/γ² = L_v/c(1±v_c) (±foreward but only photons pertain)
(and linear stellar time-aberration 1/(1±v_c) is the same as piecewise)

so—cross compared—

L_L,v,c = L_grid/(1±v_c) (direct L vs semiroundtrip L; ±foreward)
L_v,c,L = L_v/(1±v_c) (direct L vs semiroundtrip ruler; ±foreward)
d_v,c,d = d_v/(1±v_c) (apparent linear stellar aberration; ±foreward)

and viewing clocks afar—

t_by(L),v = t_grid − |L_grid/c| (delayed t=L/c, equally both directions)
t_v,by(L) = t_by(L),v/γ² (as v≥0; delayed t=L/c, equally both directions)
= t_v−|L_v/c| (as v≥0; delayed t=L/c, equally both directions)

d_by(d),v = d_target − vt_grid (remaining, unequally by direction)
t_by(d),v = t_grid − |(d_target−vt_grid)/c| (unequally by direction)
= t_grid(1±v_c) ∓ (d_target/c) (±foreward, linear stellar aberration)

d_v,by(d) = d_by(d),v/γ² = d_v,target − vt_v (unequally by direction)
t_v,by(d) = t_by(d),v/γ² = t_v − |(d_v,target−vt_v)/c| (unequally by direction)
= t_v(1±v_c) ∓ (d_v,target/c) (±foreward, linear stellar aberration)

e.g. ∂t_by(d),v = ∂t_v/(1∓v_c) (±foreward, receding clocks appear slow)

So—the traveler actually observes linear stellar aberration of co-moving lengths advancing/retarding clocks fore/aft...

And, for co-moving lengths, at the instant such photon arrives, its source-point finally-apparently there steps velocity to the same (in the traveler's expanding-new-view of the cosmos) and thereafter the relative distance remains constant apparently-and-understood... meanwhile-however, the traveler's-own inside reality-clocks (general common finite approximations of timekeeping, assuming 'time-speed' is the same inside subnanoscopic atoms, crystals, molecules, femtoscopic radionuclei, electrons, eigen-quarks, gluons, Planck-length-waves, etc.) were already measuring at the roundtrip-second-order-slower clock-speed, since T₀... which means the self-seen velocity of the train-inside-looking-afar is boosted, fore and aft, not-equivalenced as Galileo, Newton, Einstein, had guessed-and-or-supposed, (cf Einstein SR contraction was essentially Lorentz-Fitzgerald Contraction which had already appeared in Maxwell E-M equations but which is a topic with troubles hors d'oeuvre, see ppt-video the Faraday-Maxwell Tease, trailer, centuries-old transformer equivalence paradox)...

And, the traveler actually-does gain on photons coming-back (but if seen to act-upon or communicate it's no-sooner than any nearer observer can learn of it or do-so: it's not FTL-communication), and for the moment this is novel, that the contraction-frontward will-be-the-same velocity-dependent-proportion for all things, including in-line photons foreward (indeed all, photons traveling-with the clocked-pulse from the front since its emission-instant), but also, each measurable contraction is delayed by its distance, at-the-speed-of-light, and therefor the photons cannot-be-met by frontward-shrinkage of the clock that cannot shove photons already in flight back, any sooner, but it will-be sooner because those photons are already nearer, sooner... and so the combined-apparent-velocity afar, is, FTL, (Einstein's assumptions notwithstanding)...

And so we'll have both, times and distances direct, and times and lengths semiroundtrip, corresponding, and, compared in measuring, direct-trip against ruler-direct-trips measured by time-delays contracted/expanded equally but differently-fore-versus-aft by linear-stellar-aberration, and fore/aft-direct-insensitive semiroundtrip against ruler-semiroundtrips contracted/expanded equally, and, cross-compared, exposing, that linear-stellar-aberration, (but worse both compounded), yet each is compensated compared to its own-kind rulers; And, the traveler may compute, these, and know the changes-occurred in its-own perception-frame due to acceleration felt, resulting in apparent differences in the grid frame—(the universe shrinks because the traveler perception-process stretches)...

And all being used to cross-compute an A-student's 'gedanken' of arithmetic...

(The mathematics, might not be so easily done as in Einsteinian-Minkowskian style—as contraction and expansion are origin-dependent, not-simply a matrix-transformation, and, like angular stellar aberration of a distant star, which itself might be moving, did not include the local object of the telescope, co-moving, so angular aberration was apparent, and the time-asimultaneity may be apparent if objective lengths have molecular restructuring... tbd)

So, for the extended startup-phase t₀₊ and closing-out at the speed of light, 'the other end is starting and running late', so, the apparent-and-understood-velocity of the other, ends, is, for that extended phase instant, behind by the time light takes traveling the distance-apparent-and-understood, minus the distance traveled by its end mirror that same, but which is outside-measure while inside-measure is soon renormalized by their laser-pulse-rulers their roundtrip-γ²-expanded distance and time—but without its mirror-travel time in the photons' path, it sees distance shortened for shortened time while the speed of light appears the same constant for all—but all can, think-about, what's happening, and know, because they felt it, inside, or-not outside...

So, on longterm self-consistency then, all distances shrink/expand in cross-comparison only, by second-order-only, due to measuring both, direct and, roundtrip, times, while presuming light-speed is constant... (cf geodesic acceleration)

And a last note: There will be further considerations on the expansion of space and whether local, velocity, changes, as objects cross into new space, or remains constant as light-speed presumably does even at extremely large distances...

So, next, we'll try a smoother start, as should-do-okay for maximum train-velocity not-too-near-lightlike, and, the results are crudely the same as jumping,-the-velocity-forward to meet-or-race oncoming photon-pulses; meanwhile the distant will appear-and-be-understood to be stationary-on-the-grid, approaching-or-receding-relatively for observers, inside, at the complement of the traveler's velocity, but at the 'penalty-of-believing' that distance is aberrated by first-order velocity proportion, understood to shrink-afore and stretch-abaft by that instant-stretch-of-velocity-starting-delay,—

SPECIAL CASE #2: stepped-constant acceleration 'forced frame' - OUTSIDE FRAME

Very similar to Case #1 but with the acceleration phase expanded-out, nonzero time, (more-realistic jerk-start and jerk-to-coast with constant acceleration meanwhile 'the other ends are running late'), the train gets up to speed still fully self-synchronized, differential-distances and lengths still equally constant on the grid—just taking more time than instantaneous, (and travelers can watch inline time-lag as relative-velocity cumulating position and simultaneity difference)—and, the linear stellar aberration calculations must yield exactly the same results as in Case #1, being dependent only on velocity, for time and distance-covered-per-time, both in grid view and onboard view—and, we will still watch for cumulations as-that-may-persist into the coasting phase especially now with more accelerationings...

The train as-before starts at t₀ from station stop, fully self-synchronized everywhere along its length but, in this case such that the acceleration is constant thereafter, the velocity constant-increasing, and moves smoothly-evenly every car at the exact-same acceleration, straight-line (no gravity-bend) and exact-same-increasing velocity and exact-same-parabolically-increasing distance compounding as timed and measured in the outside frame—such that differential-distances and lengths remain constant the-same simultaneous-measurement-differences in the outside frame (not by inside-photon-semiroundtrips)... 'til it achieves maximum velocity and switches-abruptly-back to zero-acceleration, constant-coasting,-velocity, thereafter, and there-is-no grid-apparent 'length-contraction' (molecular contractive forces 'tbd' ibid)...

So for t₀₊ (instantly settled), acceleration a=a for constantly-increasing velocity v=at and constantly second-order-increasing distance every car from its starting position, d = d₀ + ½at², and, in addition to its photon-clock-race-length L=L, we'll also include a special car-photon-race-length L_car=L_car between mirrors mounted in adjacent cars for a cumulative-drift experiment (simplified so that it's unaffected by mechanical-length-compression-or-stretch under the force of acceleration pushing-or-pulling depending on mounts and stiffness), and, cosmic-space-frame track and station clocks are fully synchronized as-usual...

And again, simultaneously by design when the train starts the inside clocks each emit a photon-pulse toward their opposite end-mirrors (still considering only inline traverse photons, as transverse photons still 'fall' rearward during acceleration phase but now more slowly, still scattering), and each clock-photon-pulse travels at the speed-of-light c its clock-length L between its front and rear mirrors which direct-distance-length in the outside frame is shortened or lengthened by the train-acceleration-velocity-profile distance-increment to ct₁ = L − (v₀t₁+½at₁²) but which we'll calculate from its roundtrip midway initial return velocity v = v₀ + at₁ and simplify to čt₁ = (c+v)t₁ = L + ½at₁² arriving at event time t₁ = (č−√(č²−2aL))/a sooner-c-rearward or later-c-chasing-forward, and then promptly counter-likewise each photon-pulse bounces to return and travels ĉt₂ = (c−v)t₂ = L + ½at₂² all at the same speed-of-light on the grid frame and so all arrive back at their original mirrors outside-frame-simultaneously in the clock-photon-pulse-roundtrip-time t₁₊₂ = (2c−√(č²−2aL)−√(ĉ²−2aL))/a which expanded ≈ (L/č)(1−½(aL/č²)+½(aL/č²)²−⅝(aL/č²)³+...) + (L/ĉ)(1−½(aL/ĉ²)+½(aL/ĉ²)²−⅝(aL/ĉ²)³+...) accurate for small acceleration-length-product aL≪č²,ĉ² and which ≈

[NEW CONSTRUCTION MARKER 2018]

NEW-NEWTON V. OLD-NEWTON: (space-time-light relativity)

Einstein's solution had been to compromise, his arithmetic, for settling an argument between two camps of physicists: By assumptively setting the speed of light to both, emission-velocity c, and, space-velocity c, and reception-velocity c for good measure, he produced his Special Relativity, But, with Newton there's none-of-that, and the new-Newtonian values, relative to the old-Newtonian values, are thus calculable.... QEI QEF QED

EXTRAS: Conserved Newtonian spacetime-mass-moments: [2007]

• m ≡ mass ≡ mass-energy;
• m d ≡ mass × position ≡ center-of-mass;
• m d/t ≡ mass × velocity ≡ momentum;
• m d/t² ≡ mass × acceleration ≡ force;
• m d²/t² ≡ mass × velocity-squared ≡ kinetic energy (×2);
• m d³/t² ≡ mass × gravity ≡ mass-squared ⇒ Schwarzschild radius;
• m t²/d³ ≡ mass (inertial) / gravitational mass ⇒ a coefficient-constant
• m / d ≡ potential energy (but we've noted an envelope correction)
• m / d² ≡ force of potential energy (but we've noted an envelope correction)

* [P.S. In the college textbook, Spacetime Physics, Taylor and Wheeler, 1969, it was understood that Special Relativity had been tested as many-times-or-more than Euclidean-Newtonian theory; But the review and development herein is nontrivial—the true test was probably never done before, and whatever is left to Einstein’s assumptions must be found-out-first before attributing them to any sort-of-object-relativity… and ‘til-then, ‘You just can’t get there from here’ (Equations don’t dispute equations), It’s all part, of the calculation]

* [We note also that the Biblical Book of Daniel by several storied analogies discusses the perceptual relations of space, time, and light, counting steps on two horizontal dimensions, weighing on the vertical third, dividing by time for all rates of travail and travel (or dividing the past and future), that when exposed in the perfect fire that is light, on three dimensional actors and accoutrements, reveals the fourth like another dimension timely lord over all]

A premise discovery under the title,