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About Celeritas

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  1. I meant only that twin B maps the twin A clock in his own spacetime system, as he goes. If he had a tracking console, it would be populated on the display because it's current location is being updated and calculated. You say B measured the current A clock location and readout as he goes. But that's impossible. He's remote from A, and must consider the light transit time, the speed of light, and determine the A velocity and A location (as he goes) per whatever model he uses. One may be able to prove B wrong, depending on how he is figuring it all. Maybe you can, maybe you can't ... We might imagine other clocks crossing paths during B's roundtrip. Does B predict the correct readout of those clocks when they intersect, or not? The owners of those clocks could verify that. Which theory predicts that accurately? If both do, then I'd say there is no way to know (yet) if one theory matches reality and the other does not. For example, assume for a moment that the 1-way = 2-way in nature. If conventions using 1-way <> 2-way produce the same solns, then both theories are valid far as spacetime solns are concerned, however only 1 theory matches what is real. We cannot measure the 1-way speed of light, and so I figure we may never know the answer to that. LET theory could be correct, or SR could be correct. There may well be various reasons though, for presuming one theory superior to the other, but there may be no known way to ever prove that. Then, Occam's Razor says use the simplest theory, and that theory is the most likely to match mother nature (but not guaranteed to). I think Einstein picked 1-way = 2-way because it produces the simplest derivation and the simplest transforms. Also, since he was not commencing his derivation from a master frame, it seems to me that 1-way = 2-way would be the natural first choice. It would always be the natural first choice, if starting from a master frame or not. I figure he would not say that his convention was the most likely to match reality, as he could not prove it. Best regards, Celeritas
  2. But I was speaking strictly of how the A clock exists "in the B spacetime system" over the round trip, to determine A's aging. Not "seeing" the hands of the A clock as B goes. Again, using everyone's proper time alone (per each themselves), produces the correct relative aging, and is the easiest approach. No dispute there. My point has been that B should be able to accurately predict the extent of twin A aging, using only A-coordinate values (per B), using the MCCIRF method. And, we could imagine a well preplanned flight test. Twin B cannot ignore how A exists in B's own spacetime system during his turnabout. If he does ignore it, B will predict A to age only 3.2+3.2=6.4y (not 10y) on reunion. That's wrong, only because it's incomplete. When I say sense-of-simultaneity, or sense-of-now across space, I mean line-of-simultaneity. If "sense" is a problem, just assume I said B's "simultaneous space", instead of B's sense-of-simultaneity. That's fine md. Nobody has a problem with the Time Handoff scenario, and it's certainly the simplest case. However, I've only been trying to point out what I mentioned above, in the 3rd para of my first response above. All good points, and I've been trying to do so. I've never worked on this before. There's a learning curve, as I go. For any two clocks momentarily intersecting in spacetime, they have a specific set-of-readouts. SR makes its prediction. Any valid theory must predict the correct readouts, and the owners of those 2 clocks know what their clocks then read. All in the cosmos must agree, if the transformations they use are valid. I'm primarily looking to see what changes in convention cause a change in that set-of-readouts (wrt SR), versus what changes do not. Best regards, Celeritas
  3. It was late when I posted last night, and I suppose "and components of them" was vague. The 1-way speed of light has never been verified by test, and may never be. So until it is, it's certainly fair to say that a convention used is a personal preference. It cannot be known that it represents something true in nature. I should add, while (say) Lorentz Ether Theory (LET) uses the same time sync procedure as SR, the meaning of "in sync" means something different for LET vs SR. In SR, you are assumed in sync after running the procedure, however in LET your assumption of such is mistaken while yet unawares. Philosophically, one theory (eg Lorentz Ether Theory, LET) may assume a master frame, while the other (eg SR) does not. One theory (LET) possesses a much more convoluted meaning than does the other (SR). So while they use the very same LTs, they mean something different. Light signals cannot prove which theory is right. So, it's a philosophical choice as to which theory one prefers. A block universe and a master ether frame are two such philosophical choices. Eternalism and Presentism are philosophies of spacetime. So, I agree in all that. The important thing, is that any valid theory (of space and time) must accurately predict the time readout of moving clocks upon flyby, anywhere and everywhere in spacetime. And, we may imagine virtual clocks of all frames at all points in spacetime, with virtual clocks replacable by real clocks. What is real, are the measurements and the predicatability of theories per measurements. With enough consistency of predictability, mathematically sound theories are then accepted. I'm still of the opinion that the accrual of twin A coordinate time must match the accrual of twin A proper time. If twin B does NOT consider the accrual of twin A coordinate time during his turnabout, then twin A ages only 3.2+3.2 = 6.4y over the interval (which is wrong, because it's incomplete). If twin B DOES consider it, then twin A ages the required 3.2+3.6+3.2 = 10y over the interval, which matches the proper time accrual of twin A. This is only to say that the present coordinate time of a remote moving clock is the proper time of that clock at that spacetime location, and while the use of coordinate time is less convenient it is no less correct. During turnabout, the fact that B's added consideration of twin A coordinate-time produces the total required aging of twin A (10y), lends weight that the rapid advancement of the A-clock readout (in the B system) is valid and representative of something real, just as proper time is assumed representative of something real. Just for a moment ... let's assume that all moments in time coexist in a fused spacetime continuum. IOW, let's assume that particular philosophy matches reality. As such, the wild advancement of the A clock in the B system during turnabout, is the result of a real mechanism, not (say) a mere mathematical artifact. The current readout of a remote moving clock then depends upon where your sense-of-simultaneity then slices the twin A worldline, and said intersection dynamically changes while transitioning inertial frames of reference (eg B's turnabout). Your sense-of-simultaneity is governed by the angular orientation of your spacetime system in spacetime which is dependent on your current velocity, and is dynamic during proper acceleration. So, it would then model what is real. Also, the rapid advancement in A-clock coordinate time has no impact on twin A, just as turning your head has no impact on the stars in the night sky. It's only a change in POV of the observer. Twin A always experiences the passage of time as completely normal, ans everyone else does. Lastly, I'm still studying the impact of differing time sync conventions, and I'll need more time before saying more in that regard. And as you (md) had mentioned, I'm not even sure (as yet) that these differing conventions make sense, or whether all the assumptions used in their derivation even make sense. It's been somewhat a challenge. One thing is for certain, if no convention can be proven true to nature, then you'd have to be a mascocist to use a non-midpoint reflection event. Best regards, Celeritas
  4. Agreed md, the LT derivation was based on a kinematic model. My point, was only to say that an instant acceleration is unrealistic, and anything less than instant "allows for continuity of" the A clock in the B spacetime system during his turnabout. It's not so easy to ignore it, if not instantaneous. You've argued that only proper time should be considered, because its absolute, while coordinate time is relative per POV. My point has been that coordinate time of a remote clock at a point in spacetime, is the proper time of that clock at that location in spacetime. Upon B/C flyby, each predict a different time of the A clock, but this does not mean they are wrong. They are both correct, and the eventual receipt of light signals will confirm that. As such, there is no reason (in my thinking) that the accrual of clock A coordinate time (per twin B) should not match the accrual of proper time (per A), ie 10y-A. And it does, and makes complete sense, using (for example) the MCCIRF method. It's certainly not the convenient method, but just as correct. If B ignores his turnabout, he must predict that twin A ages only 3.2+3.2 = 6.4y-A. However, we know he must age 10y-A. Therefore, the twin A existence in the B spacetime system during turnabout, cannot be ignored (instant turn, or not). The X clock reads 5y-A during twin B's complete turnabout, because no time can pass per anyone in the cosmos in an instantaneous turnabout (by definition). Well, if B had no idea of relativity theory, he'd calculate the Newtonian predictions as he went. He'd be wondering how he got to planet X as fast as he did, with no good explanation. He'd be confused until he figured out how to derive the LTs In thought experiment, the sweep at A may be logically deduced. The eventual receipt of light signals verifies LT solns for any and all spacetime events in all-inertial scenarios. Now, imagine a clock D colocated and comoving with B. Twin B makes his turn at planet X, but D keeps on coasting. The eventual receipt of light signals will prove to D that when B turned, the A clock was 2.4 ly downrange and read 3.2y-A. That's a valid test. The B clock made that same prediction at the turn, so he was right, as confirmed by D's eventual verification. Next, B instantly lands on planet X. The eventual receipt of light signals will confirm that upon landing, twin A was 3 ly downrange with a readout of 5y-A. That's a test. So either the A clock truly advanced from 3.2y-A to 5y-A in no time at all (due to POV chnage), or it jumped in readout by magic, or B's SR prediction of the A clock readout while inertial is flawed (and Newton was right all along). The current A clock readout depends only on the current POV. When B transitions frames of reference at turnabout, his POV dynamically changes wrt those who remain inertial. His POV changes because his spacetime system rotates in 4-space. His sense-of-simultaneity rotates as his spacetime system rotates. The current A clock readout (and location) depends only on where B's sense-of-simultaneity then intersects the A worldline. Since his spacetime system dynamically rotates during turn, the remote A clock wildly swings in readout (and location). This has no affect on twin A or his clock, who always experiences time to pass steadily as normal. Turning your head at night, has no impact on the stars themselves. It's a POV change. What's required, is that all moments in time coexist, as inches on the ruler do. Ie, a fused spacetime continuum. That all worldlines are laid out in the continuum in their entirety, eternal in a sense. That we for reasons unknown, we do not experience that directly, however the measured relativistic effects (to date) are the proof of it. The required time desynchronization of moving bodies, is a clear example of why it should exist as such, IMO. But ... All that said, Lorentz's version of relativity (as re-interpretted after 1905) possesses the very same LTs, but means something completely different. It also uses the same time synchronisation procedure. Also, coordinate time means something different in Einstein's theory, versus Lorentz's theory. Therefore, I'll have to agree that the meaning of theories, and components of them, is subject to philosophical choice. The measurements, and noted predictability, are not. Does that sound fair md? Best regards, Celeritas
  5. md63356, BTW, wrt my derived transformations for the case (2), where ... ..... 0 < e < 1 ..... cout ≠ cback ..... and where e ≠ ½ ........ ... (where e = ½ would be the SR convention) Considering the A and B worldlines from A/B flyby to BC flyby (Time Handoff scenario), while the gamma factor remains at γ = 1/√(1-v²/c²) for the relation between the A & B clocks, it also turns out that moving contracted lengths at rest in the B frame (as noted by A) must contract to values different from 1/γ. No doubt, because cout ≠ cback ≠ c, and spatial-offsets from the B worldline produce different coordinate values (of B per A) than does SR. All this, makes it very clear why Occam's Razor should win, and Einstein's convention be assumed as "the most likely" true-to-nature. Alas, it seems impossible to ever prove such, or so it would seem today. Best regards, Celeritas
  6. You had mentioned (prior) the momentarily co-moving and co-located inertial reference frames (MCCIRFs) that coincided with the twin B during his own own proper accelerations. At any moment during B turnabout, the local speed of light from A should be speed-c per twin B and the corresponding MCCIRF, and they should also agree on the doppler shift of the light from twin A. And, we may assume that we knew the frequency of the light transmitted by A per A in a pre-planned test event. Although, given superfast unrealistic proper accelerations by B, twin B may be required to coast momentarily to verify and validate that doppler effect. Consider this md ... We might imagine the twin's executing a pre-planned flight test precisely as we've been discussing here, with instant (or virtually so) B turnabout at a planet X. Planet X is at rest with earth, separated from earth by 3 ly proper, and it's clocks are synchronized with the earth (and twin A) clocks. Everytime the flight test is executed, the relative aging result is the very same, because the flight test is conducted identically every time. Next, imagine that at any point in time during the B turnabout at planet X, we have twin B "go to coast" and hence becomes inertial. Here, I am making a point as to "the validity of" the A clock sweeping continguously from 3.2y-A thru 5y-A to 6.8y-A (in the B system) during his turnabout (which I say is consistent with SR). Further, let's imagine that B is told to "go coast" precisely when he is momentarily co-moving and co-located with Planet X. The B clock then reads 4y-B, and twin B is now inertial. What do you think the twin B navigation system will declare of the current range to twin A (on earth), and for the current time on the clocks of planet X, earth, and twin A? Per B, I submit those clocks must then all read 4y, and with twin A 3 ly downrange. For that to happen, which I submit would be the only outcome consistent with SR (and reality), it must be true that the A clock contiguously swept from 3.2y-A to 5y-A in the B spacetime system during B's instant (or virtually instant) turnabout. Therefore, it must be true that the A clock contiguously swept from 3.2y-A thru 5y-A to 6.8y-A in the B spacetime system during B's instant (or virtually instant) turnabout, in the complete twins scenario we've been discussing here. Yes? Best regards, Celeritas
  7. Well, I did not post the complete derivations, so that's understandable. I'm having some difficulty understanding exactly what they mean too. I derived transforms for 2 cases ... ... (1) 0 < e < 1 ... where 1-way = 2-way (speed-of-light) ... (2) 0 < e < 1 ... where 1-way <> 2-way (speed-of-light) ... unless e= ½ Case (1) seems improper IMO. The LTs are derived exactly as done in OEMB, with an invariant c, and closing rates of c-v (outbound) and c+v (inbound). Yet, to select a non-midpoint reflection event requires cout <> cback. So an invariant c and non-midpoint reflection event seem mutually exclusive. This is why I derived transforms for case (2). Case (2) determines the cout and cback for the selected e. My derivation seems proper, however I am not yet certain the 3 aformentioned assumptions that I made (during derivation) were proper. I must admit, the solns for a non-midpoint reflection event seem suspicious, but I'm still trying to understand it and verify it is actually right. I've been following your advice, in assuming that unexpected results may not necessarily be wrong. In fact, I'm still trying to convince myself that the derivation is logically sound, so this may take awhile. Fair enough md. You quoted "doesn't cause", and wonder regarding the precise meaning of that. Your prior point regarding "proper-time accrual" being the important (and easiest) thing, is certainly a cause for the relative aging. That speaks to the comparison of the 2 worldline lengths. However ... It is equally true that the proper accelerations (ie frame transitioning) by twin B allows that worldlines geometry to become attained. That's about it, IMO. It really comes down to relative simultaneity. I like to say the same thing just a little differently ... It's about the relative angular orientation between the spacetime systems, which in the twins scenario is dynamic ... which also allows for re-location for an absolute age comparison. One twin must depart the originating common frame, or there be no relative aging differential. He who properly accelerates to transitions frames, always ages the least. So the geometry defines the relation of the worldlines and their lengths, the worldline length(s) define the proper time(s) accrued, and B's proper acceleration creates the worldlines geometry with re-colocation. Very true, causality is dictated by locality, and hence by the transfer of light signals. I would only add that ... This does not then lead that when twin B executes his (instant, or virtually instant) turnabout, that twin A cannot sweep contiguously from 3.2y-A to 5y-A to 6.8y-A within the B spacetime system. Twin A must do so, if SR is correct. And IMO, that's the more interesting part. It matters not, that B does not "see this" via light signals. However, for the light received by twin B from twin A, the corresponding doppler shifts notable by twin B (during his transition of frames) provide a validation of this, per SR. The meaning of the theory seems incomplete IMO, if this point is not considered or understood. This is to say that when twin B arrives back on earth, before he even asks A to show his clock, B has already predicted that A must have aged 10 yr ... 3.2-A inertially out, 3.6y-A during instant (or virtually instant) turnabout, and 3.2y-A inertially back (=10y-A total). Each twin has each their own proper time accrual, but each twin also has their prediction of the other's aging over the common interval, and everyone's correct. Best regards, Celeritas
  8. accidental post ... thanx, Celeritas
  9. Update to my prior post, which is the 7th post back from here. My apologies. My previous derivation of the time transformation for a non-midpoint reflection event (with 1-way = 2-way speed-of-light) was missing a 1/c2 term, which produced the correct numeric result (given c=1) but with incorrect units. The space transformation (for x→x') was correct. .... (Eqn 1, old) ...... t' = γ[ (1/γ2 - (2ec-v-c)v)t - (v+(1-2e)c)x)/c2 ] ....... ← incorrect time transform (posted prior) .... (Eqn 1, fixed) ... t' = γ[ (1/γ2 - (2ec-v-c)v/c2)t - (v+(1-2e)c)x/c2 ] ....... ← correct time transform (posted here) .... (Eqn 1) ............. t' = γ[ (1/γ2+(v+(1-2e)c)v/c2)t - (v+(1-2e)c)x/c2 ] ....... ← I reorganized the corrected time t coefficients this way ) The correct transforms are as follows ... .... (Eqn 1) ... t' = γ[ (1/γ2+mv/c2)t - mx/c2 ] .... (Eqn 2) ... x' = 2eγ(x-vt) where … m = v+(1-2e)c where … 0 < e < 1 where … γ = 1/√(1-v²/c²) For e = ½ , the above transformations reduce to the LT's of Special Relativity. Best regards, Celeritas
  10. md65536, This supercedes my prior related post. So it turned out to be a very silly misapplication of a parentheses in 1 spot. Couldn't believe it, but on the other hand, yes I can. My eyes were buggy from looking at pages full of parens. So with the correction (which applies to EQN 3 below), the spreadsheet shows it matches SR precisely for reflection event e=½, be it proper or coordinate values produced. It varies from SR the moment e <> 1/2, and it should. (1) e ... is a variable that specifies the point-of-reflection as percentage of the roundtrip, so 0 < e < 1. (2) I assume the 2-way speed of light is invariant c, but the outbound light speed (along +x) and inbound light speed (along -x) is determined based on the selected reflection event e. (3) Assumption I ... the linear coefficent alpha (that arises from integrating the partial derivatives eqn) is equal to a = 1/gamma, just as per SR. Einstein was able to deduce that a = 1/gamma per phi(v) = 1, but I could not do that because the transforms did not reduce much. Therefore, I just assumed what Einstein deduced, that a = 1/gamma. (4) Assumption II ... there's that point in the derivation where you plug the interim eqn for time Tau (T) into X = cT. However, what speed-of-light to use? The 1-way outbound, the 1-way inbound, or the averaged 2-way speed-of-light? The options are cout = c/(2e), cback = c/(2(1-e)), or c (=cround_trip) where c is the average 2-way light speed (invariant c). Well, I used cout = c/(2e) for the same reason Einstein substituted x'/(c-v) for t instead of t = x'/(c+v) ... so I used t = x'/(cout -v), and for X = cT I used X = coutT. I'm sure this derivation is correct, given the assumptions I made. How much use this is I do not know, but it is interesting to compare what happens for time sync conventions that differ from SR, and again, this matches SR for SR's e=0.5. Here's a repost of my most prior, with corrected space transform for x→X ... *************************************************************************************************** As derived from the 3 TAUs Eqn in OEMB Section 3, but where e specifies the reflection event ... ... 0 < e < 1 ............← Use e=0.5 to produce SR transformation solns ... e(T0 + T2 ) = T1 ... e [ T0(0,0,0,t0) + T2 { (0,0,0,t0 + x’/(cout -v)+x’/(cback+v) } ] = T1[ x’,0,0,t0 + x’/(cout -v) ] My interim substitutions were these ... ... where Einstein uses X=cT ... I used X = coutT ... where Einstein uses t = x'/(c-v) ... I used t = t = x'/(cout -v) ... and of course ... x' = x-vt The transforms attained were ... ... (Eqn 3) ... X = cout [ 1/(cout -v) - m/n ] (x-vt) / γ ... (Eqn 4) ... T = [ ( n + mv) t - mx ] / (γn) where … ... 0 < e < 1 ... cout = c/(2e) ... cback = c/(2(1-e)) ... m = [ (1-e)cback - ecout + v ] ... n = (de -v2) ... de = (cout*cback) + (cout-cback)v ... γ = 1/√(1-v2/c2) *************************************************************************************************** Thank you for the help md. Much appreciated! Best regards, Celeritas
  11. md, Yup, I see exactly what you mean now. It's beat the crap outa me here. In my most prior derivation here, while I believe I have the time transform correct (t -> T), the space transform x -> X seems to be mistaken. Been looking at it, but I don't see the mistake yet. It may have to do with my 2nd assumption I had mentioned prior, not sure? The spreadsheet that processes the transforms produces the correct B values per A, but only for events ON the B worldline, and no matter what the reflection event (e) selected. However, when I set e=0.5 in the derived eqns, and reduce it by hand, it does not seem to reduce to the SR LT for X (it should). However, for any coordinate inputs that are off the B worldline, they certainly don't match the SR soln. But while I would figure them not to, I doubt my solns differ "properly". That said, I posted thinking it was right, but clearly I posted too quickly. If I figure it out, and am 100% certain, I'll repost it. I do see what you mean though, about too complicated, and whether it can even work. Thanx, Celeritas
  12. md65536, So I did put a derivation together for this. It allowed a reflection event at 0 < e < 1, versus SR's midpoint reflection event of e = 1/2. It also assumes the 2-way speed of light is always c (ie averaged), but does calculate the required 1-way light speeds (outbound vs return) based on the selected e. I derive the transforms in the same manner as done in OEMB. However, I had to make 2 assumptions ... (1) the linear coefficent alpha (that arises from integrating the partial derivatives eqn) is equal to a = 1/gamma, just as per SR. Einstein was able to deduce that a = 1/gamma per phi(v) = 1, but I could not do that because the transforms did not reduce much. Therefore, I just assumed what Einstein deduced, that a = 1/gamma. (2) there's that point in the derivation where you plug the interim eqn for time Tau (T) into X = cT. However, what speed-of-light to use? The 1-way outbound, the 1-way inbound, or the averaged 2-way speed-of-light? I had cout = c/(2e) and cback = c/(2(1-e)), and c (=cround_trip) being the averaged 2-way light speed (invariant c). When I got to that part, X = cT, I derived the x->X transform for all 3 light speeds options (cout = c/(2e) and cback = c/(2(1-e)) and cRoundTrip=c). Not sure if any of those are definitively proper, but in doing so I found "the results did in fact support what you've stated prior in this thread", given I understood you correctly ... The LT solns "for a clock that intersects you at some point", are the very same as per SR. So in relation to the OP's Time Handoff scenario, observer A obtains the same transform solns for the B POV (who passed him by prior), no matter if SR's LTs are used, or these modified transforms are used. For any clock that does NOT intersect you, such as a clock comoving with B (that's separated from B) never intersects A, the transform solns then DIFFER from SR for that clock. The transforms are ugly, compared to the nicely reduced LTs of SR. And I mean, ugly. Here's the transforms I got ... As derived from the 3 TAUs Eqn in OEMB Section 3, but where 0 < e < 1 ... ... e(T0 + T2 ) = T1 ... e [ T0(0,0,0,t0) + T2 { (0,0,0,t0 + x’/(cout-v)+x’/(cback+v) } ] = T1[ x’,0,0,t0 + x’/(cout-v) ] The transforms attained were ... ... (Eqn 3) ... X = c[ ( (1/(cout -v) - m )/ (γn) ) (x-vt) ] ........... ← for the leading c, I used each of ... cRoundTrip=c ...... cout = c/(2e) ...... cback = c/(2(1-e)) ... (Eqn 4) ... T = [ ( n + mv) t - mx ] / (γn) where … ... 0 < e < 1 ... cout = c/(2e) ... cback = c/(2(1-e)) ... m = [ (1-e)cback - ecout + v ] ... n = (de -v2) ... de = (cout*cback) + (cout-cback)v ... γ = 1/√(1-v2/c2) Thanx, Celeritas
  13. Temporarily under reconstrunction Thanx, Celeritas
  14. Sorry, the EDIT feature locked me out before I could make this EDIT ... In my prior, under my response to the 2nd quote (last sentence), when I stated the above I meant this ... What I want to do next, given the derivation whereby 1-way = 2-way and e <> 0.5, is verify that per A's POV ... that Eqn 1 solns match SR prediction for intersecting clocks NOT on the A or B worldlines. I'm assuming they will. Whereby Eqn 1 was ... .... ((Eqn 1) ... t' = γ[ (1/γ2 - (2ec-v-c)v)t - (v+(1-2e)c)x)/c2 ] Best regards, Celeritas
  15. Well, they are events. I wouldn't say it's easier, but rather that clocks intersecting are a best confirmation of the LT solns, because those clock carriers can tell everyone what their own clocks read on flyby. Of course, the eventual receipt of light signals are a confirmation of the LT prediction as well, if the owners of intersecting clocks refuse to tell us . Events in spacetime are what relativity is all about. Two postulates, a few assumptions, and events. The train arrives at the station when the small hand points to 7 and the little hand to 12. But no better event though than 2 clocks at flyby, because their readouts are then absolute. It does verify the prediction of the LTs. My derivation of the LTs for a non-midpoint reflection event (but with 1-way = 2-way speed-of-light), resulted in this generalized transform eqns (for variable time sync convention) ... .... ((Eqn 1) ... t' = γ[ (1/γ2 - (2ec-v-c)v)t - (v+(1-2e)c)x)/c2 ] .... ((Eqn 2) ... x' = 2e(γ)(x-vt) whereby e defines where the reflection event occurs in the transceiver/reflector frame ... ie reflection at t'1 = e( t'0+t'2 ) where e is in the range of ... 0 > e < 1. So for e = 0.5 (the SR convention), Eqn 1 reduces to SR's transforms t' = γ( t - vx/c2 ) and x' = γ(x-vt) as follows ... .... ((Eqn 1) ... t' = γ[ (1/γ2 - (2ec-v-c)v)t - (v+(1-2e)c)x)/c2 ] .... (............ ... t' = γ[ (1/γ2 - (2½c-v-c)v)t - (v+(1-2½)c)x)/c2 ] .... (............ ... t' = γ[ (1/γ2 - (c-v-c)v)t - (v+(1-1)c)x)/c2 ] .... (............ ... t' = γ[ (1/γ2 + v2)t - vx/c2 )] whereby the terms (1/γ2 + v2) in the time transform always equals unity. So for SR's e = ½ ... then Eqn 1 reduces to SR's ... .... (................ t' = γ[ t - vx/c2 ] However ... for e <> 0.5, then the Eqn 1 solns say this ... Wrt the time handoff scenario (for easy reference), if the A observer runs Eqn 1 for the outbound B clock, it produces the same time solns (as SR) for the B clock readout anywhere on B's own worldline. However, for spacetime events NOT on the B worldline, the time soln differs from that of SR. However since the 1-way = 2-way speed of light, all it really means is this. The transforms for events "upon" the A and B worldlines, are as per SR. Event's off the worldlines, do not match SR. But since the 1-way = 2-way is maintained "while changing the reflection event" (e <> 0.5), all this really means is this ... Whereby SR says the reflector's clock should be set to t'1 = ½( t'0+t'2 ) upon photon reflection eg say 3y-X (X is a clock at rest with A where B turns around), it may be instead set to (say) 3.8y-X per the value of e selected (where e<> 0.5). Nothing else changes, so it's just a clock set to a different time readout than SR would. The relative velocity, relative rate of tick, and geometry of worldlines remain the very same. So it's just a clock that is set to a different readout than SR would dictate, nothing more, nothing less. It's NOT as though the 1-way light speeds were altered (wrt SR) to match the non-midpoint reflection event. That's what I want to derive next. What I want to do next, is verify (per A's POV) that ... intersecting clocks are predicted as per SR, given neither of those clock's worldlines are the A or B worldlines. I'm assuming they will. Well, I'm in the process of deriving the transforms for 1-way <> 2-way, as a function of the selected e, although I must say it's somewhat a mess because nothing reduces much. I'm not going to concern myself with the reductions though, I'm only interested in making the required substitutions as in OEMB. That's all you should need. A spreadsheet can handle anything you throw at it, ugly or not, assuming it's (of course) correct. So, I'll keep plugging along here, although I'm not moving fast because of other life related matters. I'll let you know what it reveals md. And, thank you for your time on this. Much appreciated. Best regards, Celeritas
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