Exploring the Intertwining of Space and Time in a Galilean Universe"

[grav-universe]

This is for anybody who might wish to cling to the idea that there is space and there is time, each of which are measured individually as sole and separate characteristics of the universe by either a ruler or a clock, but that they do not otherwise intertwine as within the context of Relativity into such a concept as space-time. Some might even further claim that objects in motion won't be measured to contract in length, whereas we live in a purely Galilean universe, so objects will be measured at the same length as they would when they are stationary.

Let's imagine for now that we do live in such a Galilean universe. Observers A1, B1 and C1 are stationary to each other in the same place with identical rulers and clocks. Each observes the others' clocks to be ticking at the same rate as their own and to have the same reading as well as rulers being the same length. Observer C1 now accelerates to a speed v. Within a Galilean universe, observers A1 and B1 will still measure the same length of the ruler C1 carries as they did before. Now let's say we repeat the experiment, but this time we will set B1's clock 5 seconds forward, so while the clock of A1 reads TA = 0 when C1 departs, B1's clock will read TB = 5 seconds. Is it possible that B1 will now measure C1's ruler to be contracted after C1 accelerates? A purely classical thinker might say no, that the reading on B1's clock should have no bearing upon the length B1 now measures of C1's ruler, that rulers measure only length and clocks measure only time, so since neither B1's ruler nor C1's ruler has changed in any way, then B1 should still measure C1's ruler in the same way.

Okay, now let's add a second set of observers A2, B2, and C2, that are also initially stationary to each other in the same place with identical rulers and clocks, but they are a distance d from the first set of observers. Repeating our last experiment, we will set the clock of B1 forward 5 seconds. According to the observers A1 and A2 in frame A, C1 and C2 depart at the same time at TA=0, along the line between each set of observers with C1 in front. Both C1 and C2 accelerate at the same time in the same way to a speed of v, so A will still measure a distance of d between them. With a Galilean universe, according to A, the rulers of the observers C1 and C2 in frame C will not have contracted, so the C observers also measure the same distance d between themselves and will radio that back, that the distance they measure between themselves using their own rulers is still d.

Now let's look at what frame B measures using frame B's own clocks and rulers. C2 departs at TA=0 when B2's clock also reads TB2 = 0. Since B1 has set his clock 5 seconds forward, however, then according to the clock of B1, C1 departs at TB1 = 5 seconds. From the perspective of frame B, then, according to frame B's clocks, C2 departs first and C1 departs 5 seconds later. B1 and B2 still measure a distance of d between themselves using their own rulers. Since according to frame B, however, C2 departs 5 seconds earlier than C1, then C2 has already traveled for 5 seconds toward C1 before C1 then departs as well, so the distance between C1 and C2 as frame B measure them after C1 departs also is now less than d. As before, C1 and C2 radio back that they still measure a distance of d between themselves, so from the perspective of frame B, C1 and C2's rulers must have contracted to the same proportion as the lesser distance frame B measures between them in order for the observers in frame C to still measure the same distance d between themselves.

This simple exercise demonstrates that what we measure on rulers and clocks, of space and time, are closely intertwined. If we were to now drop observers A2, B2, and C2 altogether again from our experiment, then we could say that even within a Galilean universe, if B1 simply sets his clock forward somewhat, it is possible that he could now measure the ruler of C1 to be contracted when C1 is in motion while A1 does not. Of course, one can also see from this that it is really not enough for B1 to measure using a single clock and a ruler, one which measures only distance and the other which measures only time. As we have seen, in order to gain a proper measure, one must place a clock at either end of some distance traveled and they must be properly synchronized. For a single observer, then, that would mean that he must carry not one but two clocks, placed at either end of his ruler, producing a sort of "clock-ruler". We have also seen that the way we synchronize clocks directly affects the distances we measure, showing a direct relationship between clocks and rulers, between space and time. No longer do we measure just space or just time as purely independent properties of the universe, but rather space-time.

 

[Q-reeus]

grav-universe, could I first suggest that sometimes a picture can be worth a thousand words! As regards your thought experiment, you are using a double accounting procedure in that B1 is simply not making correct allowance for having changed it's clock reading. Galilean Relativity is internally self-consistent and I think it's fair to say if it weren't for observational evidence (Michelson-Morley etc.) no-one including Einstein would have dreamed of coming up with SR. It's a fact of nature that only intrudes in a classical way at high energies, and that doesn't apply to your scenario.

 

[PeterDonis]

Galilean Relativity is internally self-consistent and I think it's fair to say if it weren't for observational evidence (Michelson-Morley etc.) no-one including Einstein would have dreamed of coming up with SR. It's a fact of nature that only intrudes in a classical way at high energies, and that doesn't apply to your scenario.

This isn't quite true, since Maxwell's Equations are Lorentz invariant, not Galilean invariant, so basically all of electrodynamics constitutes "observational evidence" against Galilean Relativity, even at low energies. I don't have Einstein's papers in front of me, but as I remember it the fact that electrodynamics was Lorentz invariant was one of the chief clues that pointed him towards SR.

 

[Q-reeus]

This isn't quite true, since Maxwell's Equations are Lorentz invariant, not Galilean invariant, so basically all of electrodynamics constitutes "observational evidence" against Galilean Relativity, even at low energies. I don't have Einstein's papers in front of me, but as I remember it the fact that electrodynamics was Lorentz invariant was one of the chief clues that pointed him towards SR.

I take your point that EM is not Galilean Invariant. What constitutes low velocity departures though? Agreed that say the magnetic field of a straight current-carrying wire can only really be explained in an SR context, to a good approximation magnetostatics and electrostatics can be considered separate phenomena within a Galilean context. But yes, Trouton-Noble etc. showed there was something wrong that needed a fix - SR.

 

[JesseM]

This isn't quite true, since Maxwell's Equations are Lorentz invariant, not Galilean invariant, so basically all of electrodynamics constitutes "observational evidence" against Galilean Relativity, even at low energies.

Is that really true? The old idea was that Maxwell's equations would only hold exactly in the rest frame of the Ether, and that observers in motion relative to the ether would see slightly different laws (just a Galilei transform applied to the equations, assuming the laws governing rulers and clocks were Galilei-symmetric). If the Earth had a relatively small velocity (small fraction of c) relative to the ether, then what low energy experiments of the type that might have been performed in the 19th century (apart from measurements of the speed of light like the Michelson-Morley experiment) would be expected to give noticeably different results if this theory were correct?

 

[grav-universe]

grav-universe, could I first suggest that sometimes a picture can be worth a thousand words!

As regards your thought experiment, you are using a double accounting procedure in that B1 is simply not making correct allowance for having changed it's clock reading. Galilean Relativity is internally self-consistent and I think it's fair to say if it weren't for observational evidence (Michelson-Morley etc.) no-one including Einstein would have dreamed of coming up with SR. It's a fact of nature that only intrudes in a classical way at high energies, and that doesn't apply to your scenario.

Right, I am not stating anything for or against Galilean kinematics. We all know that SR includes length contraction, so it would not do much good to use SR for the experiment, as a true classical thinker would just say of course SR involves length contraction but Galilean kinematics does not, and rulers are used to measure only distance and clocks are used to measure only time. But we have seen that even observers in a Galilean universe will measure length contraction if the clocks are not synchronized properly. The proper synchronization of clocks within a Galilean universe is not the point either, however, which is why I didn't go into detail about that, but the main point is only that what we measure for distances and length contraction of objects in motion depends strongly upon how we synchronize our clocks, showing a direct relationship between clocks and rulers, between space and time, since a proper measure of distance requires more than just our ruler.

For an example that is independent of the kinematic theory involved, let's say we want to measure the length of a ship in motion along the length of our ruler, so we could mark off the places that the front and back of the ship happen to be at the same time. We will place clocks all along the length of the ruler. Let's say that according to our frame, the back of the ship is at 20 meters along our ruler and the front is at 50 meters when clocks at each of those places read the same time T. So we would say that the ship is 30 meters long, but that would depend upon how we have synchronized our clocks. If we synchronize them somewhat differently, perhaps adding one second per ten meters along the length of the ruler, then if the back of the ship passes the 20 meter mark at T, the front of the ship will now pass the 50 meter mark at T + 3 seconds according to the synchronization of our frame, so we would say that since the back of the ship passes the 20 meter mark and the front of the ship passes the 50 meter mark 3 seconds later, then the front of the ship was at something less than 50 meters at the same time that the back passed 20 meters, so the length of the ship is now measured as less than 30 meters long, all depending upon how we set our clocks in order to measure the lengths of objects in motion.

 

[grav-universe]

Let me put it this way. Let's say that clocks in each frame are meant to be synchronized in a particular way according to the convention of the kinematic theory involved, but that a clock at one end of a distance traveled by an object has become misaligned. Observers using that clock in their measurements will now measure different distances and length contractions for objects in motion than they normally would when the clocks are properly synchronized according to the convention, indicating that it requires more than just a ruler to measure distance and length contraction, that it requires clocks as well, so that we no longer have just space and just time, measured independently by a clock or ruler, but that it requires both, and these are two components that are tightly interwoven into the overall concept of space-time.

 

[Q-reeus]

Right, I am not stating anything for or against Galilean kinematics. We all know that SR includes length contraction, so it would not do much good to use SR for the experiment, as a true classical thinker would just say of course SR involves length contraction but Galilean kinematics does not, and rulers are used to measure only distance and clocks are used to measure only time. But we have seen that even observers in a Galilean universe will measure length contraction if the clocks are not synchronized properly. The proper synchronization of clocks within a Galilean universe is not the point either, however, which is why I didn't go into detail about that, but the main point is only that what we measure for distances and length contraction of objects in motion depends strongly upon how we synchronize our clocks, showing a direct relationship between clocks and rulers, between space and time, since a proper measure of distance requires more than just our ruler.

For an example that is independent of the kinematic theory involved, let's say we want to measure the length of a ship in motion along the length of our ruler, so we could mark off the places that the front and back of the ship happen to be at the same time. We will place clocks all along the length of the ruler. Let's say that according to our frame, the back of the ship is at 20 meters along our ruler and the front is at 50 meters when clocks at each of those places read the same time T. So we would say that the ship is 30 meters long, but that would depend upon how we have synchronized our clocks. If we synchronize them somewhat differently, perhaps adding one second per ten meters along the length of the ruler, then if the back of the ship passes the 20 meter mark at T, the front of the ship will now pass the 50 meter mark at T + 3 seconds according to the synchronization of our frame, so we would say that since the back of the ship passes the 20 meter mark and the front of the ship passes the 50 meter mark 3 seconds later, then the front of the ship was at something less than 50 meters at the same time that the back passed 20 meters, so the length of the ship is now measured as less than 30 meters long, all depending upon how we set our clocks in order to measure the lengths of objects in motion.

OK - think I get what you are driving at now. If you DON'T synch clocks right then even in Galilean Relativity distance will be out too, so in that sense they are linked and not separate. And in SR the 'unsynching' is there from the start so to speak so neither can be treated properly without recourse to the other. Hope that's about it. It was just an exhausting exercise reading through your first post, and your intention was lost on me!
I should take this opportunity to make some corrections to my last post:
Remarks there about magnetostatics/electrostatics are somewhat skew of the mark as reading from the opening paragraph of "On The Electrodynamics Of Moving Bodies" (see http://www.fourmilab.ch/etexts/einstein/specrel/www/), Einstein's opening address dealt with basic electrodynamic phenomena (induction). His first postulate (relativity of motion) explained things better but as JesseM says did not overthrow low velocity Galilean/Maxwell EM. The real death blow was the second postulate (constancy of c for all observers). Reference to Trouton-Noble was also a bit skew as that specifically related to attempts to find the 'aether' and by then Lorentz-Fitzgerald contraction was well in use.

 

[grav-universe]

OK - think I get what you are driving at now. If you DON'T synch clocks right then even in Galilean Relativity distance will be out too, so in that sense they are linked and not separate. And in SR the 'unsynching' is there from the start so to speak so neither can be treated properly without recourse to the other. Hope that's about it. It was just an exhausting exercise reading through your first post, and your intention was lost on me!

Right, that's it exactly. Perhaps that last example with marking off distances on a ruler would have been better starting out with the OP, I'm thinking now in hindsight.

 

[PeterDonis]

If the Earth had a relatively small velocity (small fraction of c) relative to the ether, then what low energy experiments of the type that might have been performed in the 19th century (apart from measurements of the speed of light like the Michelson-Morley experiment) would be expected to give noticeably different results if this theory were correct?

I'm not sure; there may not have been any. But the theoretical difference was clear: the inherent symmetry properties of electrodynamics and Newtonian mechanics were different, and the question was, which of the following possibilities holds in nature?

(1) Galilean invariance holds for everything at the fundamental level: the apparent Lorentz invariance of electrodynamics is a low energy "illusion";

(2) Lorentz invariance holds for everything at the fundamental level; the apparent Galilean invariance of Newtonian mechanics is a low energy "illusion";

(3) Electrodynamics and mechanics fundamentally have different invariance properties.

Nobody liked (3), basically because of Occam's razor, so the choice was between (1) and (2), and the main reason everybody was kind of assuming (1) was simply because there was no obvious way (2) could work, since it would require a Lorentz invariant theory of mechanics and nobody had one.

The statement I was responding to was basically, if the MMX hadn't taken place, nobody, including Einstein, would have come up with SR. I don't think that's the case because of the obvious theoretical issue I just described; what Einstein did was to make the idea of a Lorentz invariant theory of mechanics seem "obvious" once you look at things the right way. This made (2) a viable possibility even though no experiment except the MMX had yet cast any doubt on (1); and I think it's quite plausible that Einstein would have followed basically the same line of reasoning even if the MMX hadn't (or hadn't yet) been performed.