Some Comments on Acceleration

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    So far in these essays I have looked almost exclusively at observers who occupy and mark inertial frames of reference, observers who exist in a state of uniform motion relative to each other and to their surroundings. Now I want to extend my considerations into the realm of non-inertial motion, in which at least some observers occupy and mark frames of reference that accelerate.

    I want to begin with the fact that uniform velocity comes to us as a purely relativistic motion while acceleration represents an absolute state of motion. Thus, two observers with an unchanging velocity between them cannot, by any means, determine which of them is moving, but if that velocity between them were to change, they would have no doubt as to which of them is accelerating.

    I can recall a good example of that fact from my own experience. One summer morning in the late 1950's, when Southern Pacific still ran passenger trains, I rode one from Los Angeles to Tulare, California. The train made a stop in Mojave and to my left I could see only the rusty-colored, reddish-brown side of a boxcar in a freight train parked on the siding. After a time I noticed the boxcar moving, heading southward as, I assumed, the freight train left Mojave on its way to Los Angeles. I not only saw the boxcar moving, but I also felt it moving in much the same way in which our far ancestors felt the celestial bodies moving as they revolved about an Earth standing stationary at the center of the cosmos. And, like my far ancestors, I was wrong, as I saw when I caught a glimpse of the landscape through the gap between the boxcar and its neighbor and felt when the wheels of my car bumped over the joints in the rails on which they rolled. I immediately understood that the engineer of my train had left the Mojave station so smoothly that I had not noticed the slight forward acceleration. I had only discerned my train's motion when I felt the mini-accelerations of the wheels bumping on the track.

    Does that fact have anything to do with relativistic physics? Indeed, it does. For starters, it gives us the only possible resolution of the infamous twin paradox.

    Originally called the clock paradox, the paradox of the twins originated in a comment Einstein made in Section 4 of his original paper, "On the Electrodynamics of Moving Bodies". He noted that if we have two clocks, at points A and B respectively, if we have synchronized the two clocks with each other, and if we then move Clock A to Point B, then we will see that Clock A will run slow relative to Clock B. It didn't come out explicitly as a paradox, but it caused Einstein enough worry that he revisited the idea in 1907. In 1911 Paul Langevin made the paradox explicit by introduced the story of traveling twins in place of the wayfaring clocks. That gave us the paradox in the form we know today.

    We have a rocketship that travels to one of the planets of Alpha Centauri at about 87% of lightspeed (Lorentz factor = 2) and then returns to Earth. When the ship returns to Earth the crew compares their clocks with clocks on Earth and finds that the ship's clocks have ticked off a different amount of time than Earth's clocks have done. We want to determine by how much the ship's clocks and the Earth's clocks differ. To accomplish that task we need only consider the first leg of the journey.

    We assume that the ship takes a negligible amount of time to accelerate and to decelerate, so we concentrate our attention on the behavior of the various clocks while the ship moves at a uniform speed. At 87% of lightspeed, as observed from Earth, the ship takes about five years to cross the 4.3 lightyears to Alpha Centauri. Occupying an inertial frame that differs from Earth's (approximately) inertial frame, the ship's clocks tick off time half as fast as Earth's clocks do, so they show an elapse of 2.5 years when the ship reaches Alpha Centauri.

    In the ship's frame the Lorentz-Fitzgerald contraction has shortened the distance to Alpha Centauri to 2.15 lightyears, so the star, moving toward the ship at 87% of lightspeed, reaches the ship in 2.5 years. But also in the ship's frame the clocks near Alpha Centauri tick half as fast as the ship's clocks do, so they must show an elapse of 1.25 years when the ship reaches its destination.

    So what do the ship's crew see on Alpha Centauri's clocks when they arrive at one of that star's planets? Do they see an elapse of 5 years or 1.25 years over what they saw on Earth's clocks when they left Sol system?

    We can resolve that question by looking at one of our tacit assumptions. We have assumed that the clocks of Alpha Centauri remain synchronized with Earth's clocks in all inertial frames. But when the ship goes into its moving frame it goes into a frame in which Alpha Centauri's clocks differ from Earth's clocks by 3.75 years. That fact, that Alpha Centauri's clocks run "fast" relative to Earth's clocks, accounts for the Lorentz-Fitzgerald contraction that makes the distance from Sol to Alpha Centauri shorter for the ship than it is for observers on Earth. That temporal offset, when added to the time that elapses on Alpha Centauri's clocks in the ship's moving frame, gives an elapsed time of 5 years, thereby resolving the paradox. And, of course, the same analysis applies to the return leg of the ship's voyage.

    Thus we see that the symmetry of Special Relativity gave us a paradox and that the acceleration of one set of observers provided the asymmetry necessary to resolve the paradox.

    So what happens in that accelerating frame of reference? How does the Universe look to the people occupying that accelerating frame? How can they and the people occupying inertial frames reconcile their measurements of events? Can we properly incorporate a state of absolute motion into the theory of Relativity? And does it make a difference whether the acceleration comes from rocketpower or from gravity? I will address those questions and others in the essays that follow.


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