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Accelerating Measurements |
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Suppose that, in terms of inertial coordinate system S, two particles separated by a spatial distance L are moving in the negative x direction along the x axis at constant speed v = –√3 / 2, and at some instant they both begin to undergo constant proper acceleration in the positive x direction. As a result of this acceleration the particles are gradually brought to rest in terms of S, and the acceleration continues until the two particles are both moving at speed v = +√3 / 2 in the positive x direction, at which time their accelerations stop. Since the particles have the same initial velocity and undergo identical histories of acceleration, their spatial separation in terms of S remains a constant L. The world lines of the two particles are shown in the space-time diagram below. (We’ve arbitrarily scaled the separation L for this example.) |
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The proper accelerations begin at events B and C, and the particles come to rest (in S) at events D and E, and the proper accelerations stop at events G and H. Both of these world lines represent hyperbolic motion, with the light-like asymptotes of BDG passing through the origin. The “center” and asymptotes of the hyperbolic world line CEH are shifted by L to the right. |
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Now, letting S′ denote the inertial coordinate system in terms of which the particles were initially at rest, we see that the spatial distance between the particles in terms of S′ is initially 2L and gradually reduces to 2L/7. Similarly, letting S″ denote the inertial coordinate system in terms of which the particles are ultimately at rest, we see that the spatial distance between the particles in terms of S″ is initially 2L/7 and gradually increases to 2L. The relative speed between S′ and S″ is 4√3 / 7. In both of these coordinate systems the particles experience the same history of acceleration, but staggered in time, consistent with the secular and monotonic changes in spatial distance. |
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We can also consider an accelerating system of coordinates, Σ, consisting of the time slices of the sequence of inertial coordinate systems in which the left hand particle is instantaneously at rest at the origin. In terms of Σ the left hand particle is always at rest, and the right hand particle is initially at a spatial distance of 2L, then approaches to a distance of L, and then recedes again to a distance 2L. Naturally this does not conflict with the fact that both particles are undergoing the same proper acceleration, because the coordinate accelerations are not the same as proper accelerations when dealing with accelerating coordinates. |
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To illustrate the palpable significance of these effects, suppose a solid rod is carried along with the particles, in such a way that its left end always coincides with the left particle. If the acceleration is slow enough, we can regard this rod as being always in equilibrium in terms of its momentarily co-moving system of inertial coordinates. (It doesn’t matter if we sometimes re-orient the rod, since it will always be in equilibrium when aligned with the x axis.) Hence the rod is essentially in Born rigid motion, which implies that the right hand end of the rod must be in hyperbolic motion but with a lesser proper acceleration than the left-hand end (and the particles). The is shown by the world line AFI. |
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It’s important to recognize the constraints that are placed on the motions of the two particles and the various parts of the rod by the stipulated conditions. The two particles were stipulated to begin their proper accelerations simultaneously in terms of S, so those events are not simultaneous in terms of S′. Also, we stipulated that the left end of the rod always coincides with the left particle, and that the rod is always in Born rigid motion. It follows that the right end of the rod must begin to accelerate at event A, which is simultaneous with event B in system S′ (and Σ). If the beginning of the acceleration of the right end of the rod was delays until event C, the rod would not be in Born rigid motion, i.e., it would not be in equilibrium, its proper length would be distorted, so it would no longer be 2L in length. |
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Overall, in terms of S, the two particles begin separated by a distance 2L, equal to the length of the solid rod, and then the right particle’s world line deviates from the right end of the rod, and arrives at the center of the rod, a distance L from the left particle, and then the right particle moves back to the right end of the rod. Needless to say, the derivation of this description from the previous description is purely mathematical, once we have fully (and self-consistently) specified the sequence of events in terms of S, and of course once we have determined that inertial coordinate systems are related by Lorentz transformations. We don’t introduce any new physics by describing the same sequence of events in terms of different systems of coordinates. We also note that, in terms of any of the inertial coordinates systems (S, S′, S″), the usual equations of dynamics relating force to accelerations apply, according to the principle of relativity. In terms of the accelerating system Σ the same laws of physics still apply, but in that case we must include the acceleration terms (analogous to centrifugal and Coriolis accelerations) to compensate for the acceleration of the coordinates, and those terms are simply defined so that the results are consistent with the behavior as described in terms of inertial coordinates. |
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