Celeritas

Tim88, I see I never got back to you in relation to one of my early posts in this thread ... If I had actually said what you paraphrased here, I’d agree with your assessment … It’s not about something being done to a far away clock. It’s about how the remotely located (and inertial) twin A clock exists within the non-inertial twin B spacetime system … Here's my understanding ... During B’s virtually-instant turnabout, the twin A worldline shifts wildly (even superluminally) within the B spacetime system, and twin A also advances wildly along his own worldline as it exists within the B spacetime system. This, the result of twin B’s line-of-simultaneity (sense-of-now across B-space) rapidly rotating within spacetime wrt inertial spacetime systems (eg twin A’s). Within the B spacetime system, the A clock time-readout must tick wildly forward, relative to B’s normal steady-ticking clock rate (per B). Twin B never sees this happen by the receipt of light signals, but can deduce this (1) per the measured doppler shift of light from A, and per the LTs ... for example, I consider the collection of momentarily co-located and co-moving inertial observers wrt twin B. Twin A never experiences anything unusual, time passing steadily per he as normal. B's clock ticks normal per himself. I'm talking only about how the twin A clock exists within in the B spacetime system. This must happen in the B system, for otherwise B could not predict (or agree) that A must age more than himself over the round-trip, as twin A is in motion per B over the entire interval. However, geometrically, a rotation of B's spacetime system by his own proper acceleration cannot ever cause moving-lengths to become non-length-contracted, within B's own system. What I say here is not about optical effects. It's about how remotely located clocks exist within the non-inertial spacetime system, during a virtually-instant proper acceleraton (B's turnabout). Now, does what I say here change your stated position at all? Best regards, Celeritas

In regards to my prior post 223 ... which addressed proper length, contracted length, proper time, and coordinate time (partly shown below) and as in relation to the muon scenario, I had a blunder on 1 word there. Correction shown below ... wrt the word MUON (in brown highlight) in the text referenced above, it should have said EARTH. Very sorry for any confusion that may have caused. Best regards, Celeritas

Well, I do realize there would be a sync problem globally, for clocks scattered at differing x' values. Also, for (say) a thick ruler, where the ruler has a non-ignorable thickness wrt x'. Primed clocks would exist at different values of x'. But as you just mentioned in your prior post, there should be no sync problem for various clocks at rest on the y’-axis that reside only on a specific single x' value x'=0 at t'=t=0 (they remain in sync per themselves during the instant proper acceleration). That's what I intended when I showed the time LT for x'=0 at t'=t=0. For simplicity, my ruler may be taken as a string of wonder point-clocks along the y'-axis at x'=0, the proper acceleration being instantaneous at t'=t=0. Does this sound reasonable, or no? At any rate, I had just re-posted that diagram, which now essentially says ... the systems are purely inertial, and coincidentally their origins align at t=t'=0. When trying to keep a figure and its verbiage "non-lengthy" and "busy-minimal" (yet complete enough for sake of point), its does get tricky at times trying to balance those, ie detail is traded off for simplicity I do understand that (in the primed system) aftward clocks would need advanced to become synchronous with forward clocks of the primed system, after proper acceleration completes, and inertial motion commences. I have a nice Rindler diagram that depicts this, for clocks at each end of a Born-rigid body of single point acceleration. All very good points, thanx Tim88. Best regards, Celeritas

This is a re-post of my original related post, necessary because the prior post's IMAGE LINK was deleted, and the prior post is no longer EDIT'able. For reference, my prior post ... post 225 robinpike, So the LTs explain the relative measure of time for any 2 observers moving relatively. Per your study of that math, you do not yet understand WHY the LTs do what they do. A derivation of the LTs from scratch (on your own) would help you considerably in that regard. Usually, if one understands the derivation then one envisions the mechanism, and your answer lies therein. Also, the fine spacetime diagrams presented thus far in this thread, have apparently not gotten you there. I would recommend you spend a good week studying spacetime figures, if you haven't already. Often, they are the ticket for a more expedient and complete understanding of WHY the relativistic effects exist, as they present the abstract math per the LTs both geometrically and visually. Ok, that said, I will try and attempt answering your question using only a 2-spacetime figure (with time implied), which most folks are plenty familiar with. Read this carefully, think about it before responding, and let me know if it helps any ... ************************************************************************************* Note: scenario description was modified to include statement for clock synchronization, and to delete the instantaneous proper acceleration of the primed system at co-located origins. The systems are now defined as purely inertial, to simplify the description for sake of point (ie. why relative time exists). Corrections per Tim88 review. Best regards, Celeritas

michel123456, I'm not sure what you mean here, as written? The moving object is the atmosphere. The earth frame does not hold the atmosphere in motion, and so a direct measurement by the earthbound observer records only the proper length. The earth frame observer cannot directly record the moving contracted-length that an observer moving wrt the earth (eg the muon) would measure. The earthbound observer must divide that proper length by gamma to determine how observers who move wrt the earth measure the moving atmosphere's thickness. If I may make a point here. This is off topic, but it might relate to one interpretation of your statement above (not sure), but I'll mention it anyway ... The muon holds the atmosphere in motion, and so the atmosphere directly measured by the muon is length-contracted (to L=2km) wrt its proper length (L0 = 10km), given the velocity v = 0.98c. Let's say the muon holds up his wonder ruler, and takes an instantaneous measurement of the moving earth atmosphere. His ruler measures it at 2km. Let's say the muon used a ruler precisely 2km long to make that measurement, ie its proper length. The muon might then consider what an observer (in a system x",y",z",t") moving at u=-0.98c (in the opposite direction wrt himself) would measure for his ruler's length. In that case, the muon can simply divide his ruler's own proper length by gamma, so L" = 2km / 5 = 0.4 km. That's the muon's moving-ruler-length per the dbl-primed system. However, this does not lead that the earth's atmosphere is 0.4 km thick per that other dbl-primed system. One must apply the Composition of Velocities formula first to determine the velocity between the earth's atmosphere and the dbl-primed system, then determine the gamma factor for that velocity. That calculation is ... u = (w+v)/(1+wv/c²) u = (.98+.98)/(1+.98(.98)/1²) u = 0.99979 ... velocity of earth's atmosphere wrt the dbl-primed frame Then one determines gamma for that velocity u ... γ = 1/√(1-u²/c²) γ = 1/√(1-.99979²/1²) γ = 49.5 The earth's atmosphere is contracted per the dbl-primed system since it moves, and so its moving contracted-length is given by ... L" = L0 / γ L" = 10km / 49.5 L" = .202km ... thick So while the muon may hold his own ruler (proper length = 2km) as the same length as the moving earth atmosphere, he divides his ruler's proper length by γ = 5 to determine the length of his own ruler per the other dbl-primed system ... 2 km / 5 = .4 km. However, he must divide the earth-atmosphere's proper length (10km) by γ = 49.5 to determine the length of the earth's atmosphere per the other dbl-primed system ... 10 km / 49.5 = .202 km. I mention this, only because many have difficulty in keeping the frames of consideration straight. Also, as to what is invariant and what is not. The proper-length (L0) of the earth's atmosphere is invariant. Any frame that moves relatively wrt earth can directly measure only what exists per they, ie a moving length-contracted earth atmosphere, which in relation to the direct measurement per earth observers is L = L0 / γ. Best regards, Celeritas

Thanx, I understand your points, and agree. I did forget to make a statement regarding clock sync. I'll enhance the post for future readers with an EDIT per your comments here. I'm thinking I'll state it this way ... no setting into motion, and the 2 inertial systems coincidentally read t=t'=0 at the momentary co-location of their spatial origins. The purpose of the figure was regarding how relative time (per LTs) arises, not what impacts from proper acceleration eg in the twins scenario. However, if I were to keep the scenario as stated (I won't), I would (for simplicity) just state that the proper acceleration of the primed system (that puts it into inertial motion) is "virtually instant". Then, there would be no need to re-synchronize the clocks upon completion of the acceleration, since x' in t = γ(t'+vx'/c²) is essentially zero, and the time is essentially t=t'=0. It's a trade-off of detail for simplicity for sake of point. Nonetheless, for any proper acceleration that is "not so instant", the primed system would indeed have to re-synchronize their clocks immediately upon completion of their own proper acceleration, and what happens during the non-inertial period of relative motion would need accounted for. Thanx. Best regards, Celeritas ************************************ EDIT ... Well, I updated the figure for my prior post, I deleted the old image, and went to insert the new image's link. However, I then found that my EDIT period expired apparently, as the EDIT button no longer exists for that post. So, I could not edit it, AND my original post lost its image. So, I will repost that prior post with updated figure. Sorry about that ! How long does the EDIT opportunity last? 24 hr or so, maybe?

robinpike, So the LTs explain the relative measure of time for any 2 observers moving relatively. Per your study of that math, you do not yet understand WHY the LTs do what they do. A derivation of the LTs from scratch (on your own) would help you considerably in that regard. Usually, if one understands the derivation then one envisions the mechanism, and your answer lies therein. Also, the fine spacetime diagrams presented thus far in this thread, have apparently not gotten you there. I would recommend you spend a good week studying spacetime figures, if you haven't already. Often, they are the ticket for a more expedient and complete understanding of WHY the relativistic effects exist, as they present the abstract math per the LTs both geometrically and visually. Ok, that said, I will try and attempt answering your question using only a 2-spacetime figure (with time implied), which most folks are plenty familiar with. Read this carefully, think about it before responding, and let me know if it helps any ... Best regards, Celeritas

In supplement to swansont's post ... Note 1 ... any length is measured between 2 simultaneous events per the measurer, one event co-located with one end-point of the considered length, the other event co-located with the other end-point of the considered length. Note 2 ... a stationary length need not be considered by 2 simultaneous events at each end, although it may be and usually is. If the 2 events located at each end of the considered length exist at different moments in time (per he who co-moves with this length), their spatial separation is the very same no matter the duration between those 2 events ... because every point of said stationary length (including its endpoints) possess spatial locations that are independent of time. 1) wrt SPATIAL LENGTHs ... PROPER LENGTH ... If a length measured by an observer does not move over duration, it is a stationary length. This is referred to as a proper length, and is the longest length measurable for the said length. He divides his measured stationary-length by gamma if he wishes to know what an observer in relative motion would measure that length as (ie its contracted-length, per POV). Since the atmosphere is stationary wrt the earth, earth-frame rulers directly measure the atmosphere's thickness at its PROPER LENGTH (10km). Earth-frame rulers cannot measure the atmosphere's contracted-length directly, because the atmosphere does not move wrt the muon. The contracted-length measurable directly by the muon's ruler must be calculated by earth-frame observers ... contracted-length = proper-length / gamma = 10km / 5 = 2km. CONTRACTED LENGTH ... If a length measured by an observer moves over duration, it is a moving length. A moving length is referred to as a contracted-length, and is always shorter than the measure of its proper-length. He mulitplies his (directly measured) moving-contracted-length by gamma if he wishes to know what an observer who co-moves with the moving length would measure that length as (ie its proper length). Since the atmosphere moves wrt the muon, the muon's ruler directly measures the atmosphere's thickness at its contracted-length (2km). The muon cannot measure the atmosphere's proper length directly, because the muon is not at rest with the atmosphere. The proper-length measurable directly by earth-frame rulers must be calculated by muon-frame observers ... proper-length = contracted-length x gamma = 2km x 5 = 10km. 2) wrt TIME INTERVALs ... PROPER TIME ... If the clock exists at both events, that clock measures the proper time for the time interval between the 2 events . This is the shortest duration measurable for the defined time interval. The carrier of that clock multiplies his measured duration by gamma if he wishes to know what an observer in relative motion would measure for the duration between those same 2 events (ie its coordinate time, per POV). Since the muon's clock is AT both events (EVENT 1 ... muon enters atmosphere here, EVENT 2 ... muon exits atmosphere there), it directly measures the PROPER TIME for the interval (6.8 μs). The muon cannot directly measure the coordinate-time measured directly by earth-frame clocks, because the muon is AT both events (it does not move wrt itself). The coordinate-time measurable directly by earth-frame clocks must be calculated by the muon as follows ... coordinate-time = proper-time x gamma ... 34 μs = 6.8 μs x 5. COORDINATE TIME ... If the clock (even virtual) cannot exist at both events, the frame co-moving with that clock measures coordinate time between the 2 events. Coordinate time varies per v, and is always a greater duration than the proper time. The carrier of that clock divides his measured duration by gamma if he wishes to know what an observer AT both events would measure for the duration between those events (ie its proper time). Since earth-frame clocks are not AT both events (EVENT 1 ... muon enters atmosphere here, EVENT 2 ... muon exits atmosphere there), they directly measure the COORDINATE TIME for the interval (34 μs). Earth-frame clocks cannot directly measure the proper-time measured directly by the muon's clock, because earth-frame clocks are not AT both events (they move wrt the muon). The proper-time measurable directly by the muon's clock must be calculated by earth-frame observers ... proper-time = coordinate-time / gamma ... 6.8 μs = 34 μs / 5. Best regards, Celeritas

robinpike, All relativistic effects are measurable. If measurable, then real. For otherwise, Newton mechanics would not model reality either. Your concern seems to be this ... Q) does the disagreement between observers of relative motion produce a conflict in the description of reality? While POVs can differ on the relative measure of time, they never disagree on the time readout of 2 clocks when momentarily co-located in 4-space. Any observer, using his own clock and ruler (and/or light signals), can accurately predict what those 2 clocks will read at their flyby event. That, is all that is important. It matters not, that I say their flyby event occurred at x,t = 2,7 while you say it occurred at x',t' = 45,111 ... and so long as others can predict our clocks' time readouts for any event we are located at. The LTs accurately predict intersections in space and time, and the readouts of clocks at intersections, and no matter how we define our units of measure. Add, the LTs apply to the case of 2 hypothetical clocks executing a flyby anywhere in space and time, not just real clocks having done so, doing so, or soon to do so. The LTs map each point (event) of one inertial spacetime system, to a unique point of another inertial spacetime system, moving relatively or not. *************************************************************************** 3-space systems (with time implied) are defined by cartesian coordinate systems, using the euclidean (space) metric. It defines a distance between any 2 points per the Pythagorean theorem ... d² = x² + y² + z² d = √(x² + y² + z²) Herman Minkowski (Einstein's math teacher who thought him lazy), defined a new metric that relates any 2 classical cartesian systems in relativistic motion, given the 2 relativity postulates true. It's called the Minkowski (spacetime) metric, necessary to make the 2 postulates of relativity compatible (they were not under Newton's model). It defines the distance between 2 points in 4-space. For time-like intervals (where causality is observed), the Minkowski metric is defined as ... s² = (ct)² - x² - y² - z² s = √( (ct)² - (x² + y² + z²) ) so if considering only the location of a moving body of velocity v traveling along only x (x=vt), it may be modeled as ... s = √( (ct)² - (vt)² ) s = t√(c²-v²) s = ct√(1-v²/c²) s, is the length of a SPACETIME INTERVAL. It is an invariant (constant) in relativity, because no one can say a clock did not tick the precise amount its own hands indicated as it traveled thru spacetime from one event at its own location to another event at its own location. So observers of differing frames all agree on the value of s, and hence the value of tau (t') for said clock between those events ... s = ct' The reason is this ... Minkowski converted the time tau of the moving observer (t') into a spatial length, by multiplying it by c (where c=1), so it then has units of distance instead of time. This simply converts time to a 4th spatial axis, hence the term 4-space. And why not? You and I are in relative inertial motion. You (as a stationary inertial observer) hold me moving thru both space and time. Yet, I hold myself as stationary (as you do) passing only thru time. Who is right? We are both right. It may then be said that a passage thru time, is a passage thru space. If you and I are passing thru space in parallel, at the same rate, we both declare ourselves stationary, co-moving, and passing only thru time. Yet, others in the cosmos moving relatively, will disagree. Those moving at a virtual c wrt us, will claim we travel virtually only thru space, and not thru time (our clocks having slowed to a virtual zero tick rate, per they). The key words here ... RELATIVE and POV ... folks can disagree yet all be correct. No preferred POV, no paradox ... s = ct√(1-v²/c²) ct' = ct√(1-v²/c²) t' = t√(1-v²/c²) t' = t/γ Wrt the spacetime system origins ... Time t' represents the aging of the moving clock for the defined spacetime interval. Time t represents the aging of the stationary POV's own clock wrt the defined spacetime interval, per its own POV. Those in relative motion hold themselves the stationary, and the clock traversing the spacetime interval as ticking slower. Hence they record a longer time-interval for the clock (that exists at both events) to complete its journey. That's called a COORDINATE TIME, and differing POVs will disagree on the length of said time interval (its variable per POV, not an invariant). Bottom line, the clock that exists at BOTH events marks the PROPER TIME between them in 4-space, and is the SHORTEST recordable duration between those events. All in the cosmos agree on the duration said clock records, ie it's an invariant. So disagreements wrt simultaneity and the measure of space and time, do not prevent accurate spacetime predictions. Truely, confusion only stems from a lack of total understanding of the overall mechanism of relativity in collective. One other point ... For me personally, it was not until I realized that ... all moments in time co-exist as the inches on a ruler do (a POV we are not fully privied to), that the LT solns made complete sense to me. The only difficulty arises in explaining WHY we only perceive an ever changing NOW, that progresses only toward the future, given all moments in time co-exist. Regardless, the model works, has been long validated, and still stands. Best regards, Celeritas

michel123456, The extent of a clock’s own aging is marked by the time accrued by its own hands. The less time accrued by a clock over a defined spacetime interval, the shorter its traversal thru the 4-space respectively. You asked … Does the moving contracted observer always travel the shorter path thru spacetime? During segments of all-inertial relative motion, yes, but only per the stationary POV. However here, a comparison of the 4-space traversal lengths do not produce a valid relative aging comparison in any absolute sense, because the 2 observers can disagree with each other as to who aged more, and both are correct (because they each hold themself stationary and the other moving). That relative aging comparison is not apple-to-apple. The twins scenario allows for an absolute relative aging comparison, because all observers agree on the outcome. No disagreement exists, because the same 2 events define the spacetime interval upon both worldlines. Both spacetime intervals are defined by a common pair of events (only 2 events). Both clocks are co-moving and co-located at both those events, and so any disagreement vanishes (x=0, and v=0 so no relativistic effects). So when the interval is completed, they can have no disagreement as to who aged less (over the entire interval in collective). All in the cosmos agree. Everyone, and no matter what the frame, must agree on the readout of 2 co-moving clocks in-the-same-place-at-the-same-time. The relative aging comparison is hence, absolute, given total consensus. One note here ... in all-inertial scenarios, it requires 3 or 4 events to define the 2 spacetime intervals for comparison ... which introduces the relative disagreement for the measure of space and time (and hence for relative aging as well), caused by the relative motion. One more point wrt your question ... during periods of non-inertial relative motion, the twin A clock ticks faster than B's own clock within the B spacetime system. For that particular segment, A travels a longer distance thru 4-space than B (per B), even though A is always moving and length-contracted per B. This is a case where it can no longer hold true that the moving length-contracted POV travels the shorter distance thru spacetime, per the stationary POV. But again, the comparison of 4-space traversal lengths does not constitute a valid relative aging comparison between the clocks in an absolute sense, unless both worldlines possess the very same 2 events that define the interval of consideration. Best regards, Celeritas

robinpike, Nice to make your acquaintance. If I may try and contribute here ... I’ll likely repeat much of what others have already said. First, while it may seem as such to you, it is certainly not the case. There are no preferred frames. The principle of relativity, and first postulate, states that … mechanics hold equally good in any inertial frame. Since everyone uses the same formulae from their own POV, no frame can be preferred, for otherwise they would have to start calculations from the preferred frame to attain correct solns. Not required by relativity. While inertial frames are more convenient to work with than non-inertial frames, they are not preferred. Accurate spacetime solns are attainable from any POV. wrt physically real ... A moving contracted ruler is measured as such, just as the proper length of a ruler is measured as such. If real is to be taken as “exists as per measurement”, then moving contracted lengths are as real as everything else that is measured. Its only that per relativity, the moving contracted length and stationary proper length co-exist, as per POV as a function of relative velocity between the measuring apparatus and that being measured. wrt like tick rate ... While I prefer to stick to interval lengths, one interpretation of relativity is this ... The rate at which time passes by me per me, is the same rate at which time passes by you per you, and no matter what our relative motion. This is commonly referred to as the rate of PROPER TIME. We do not know what that rate is, but it is assumed invariant whatever it is. For lack of any better definition, we refer to it as (for example) 1 sec per sec, which is somewhat pointless since there is no absolute reference for motion, let alone time's rate. However, for identical clocks made in a common inertial frame by the perfect vendor, they must tick at the same rate whenever they happen to arrive in any common inertial frame, since the rate of proper time is the same for all inertial frames. This is why the twins’ clocks tick at the same rate when within a common inertial frame, eg before B’s departure and upon B’s return. While they tick at differing rates relatively when in relative motion (per relativity), it remains true that the rate of proper time is the same, always. The proper rate of tick may be viewed similarly as the proper length of a ruler. A ruler’s proper length never changes, ever, even though relatively moving observers record it length-contracted. Similarly, the proper rate of tick never changes, ever, even though relatively moving observers record your tick rate as slower. A body never changes in and of itself by the gazing of a passer-by. wrt differing aging ... Given the rate of proper time is the same for all (no matter what), then the only way in which B can age less than A is if he travels a shorter path-length than twin A thru 4-space over the defined spacetime interval. He who travels the shorter distance, must accrue the least seconds by his own clock. He ages the least (eg twin B). Remember, each twin holds himself the zero reference for all motion, and simply awaits the 2 events to come to him. The SPACETIME INTERVAL is defined as Event 1 → Event 2 ... Event 1 … twins depart Event 2 … twins re-unite Each twin holds himself as passing only thru time, and this makes his own clock his measuring device for distance traversal thru spacetime (4-space). One’s clock, is his own odometer for such. The twins start at a same-point-in-spacetime, and end at a different same-point-in-spacetime. B ages less because he travels the shorter path between the 2 events! This description did not consider the frame-to-frame relations as per the LTs (that follows below), but is completely consistent with it ... During relative motion the relativistic effects exist between the frames, with differentials in relative tick rate governed by the LTs (and by the relative geometry of their reference frames, given acceleration). Per each, their clock rates differ during their relative motion, and they age per the seconds accrued by their own clock's hands. The only difference in this description, is that we consider the relative rate of time (per the LTs) induced by the Lorentz symmetry of spacetime, versus only the invariant proper rate of time in 4-space (which we are not fully privied to by casual observation). The result is just the same. So tick rate is one thing, and accrued seconds (aging) is another thing. The former being a function of v (and geometry), the latter being a function of the former and the world-line length over the spacetime interval. There are no preferred frames, and no paradoxes exist, wrt relativity theory. Great minds have not toppled the theory in 116 years, and countless have tried. Hope this helps. Best regards, Celeritas