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grav-universe

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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.