BurtJordaan
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Why is relativity so hard to learn?[10X]
May 5th, 2017, 12:13 am

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After wrapping my head around the biggest issues of special relativity theory, I obviously had to move on to general relativity, where gravity takes the center stage. Einstein used uniformly accelerated frames to 'build a bridge' between special relativity and gravity. He established that a uniformly accelerated frame is precisely equivalent to a hypothetical 'uniform gravitational field' - a thing that does not really exist. However, reasonably small stretches of space and time (in a 'real gravitational field') will very, very closely resemble a uniformly accelerated frame.

Engineers normally understand gravity pretty well in terms of how to calculate it and use it in their designs, be it a tall building or the trajectory of a missile or a rocket. They understand that gravity is 'stronger' at ground level than at altitude, and they understand the equivalence between acceleration and gravity in broad terms. But, tell them that the tail end of a uniformly accelerated frame in free space must necessarily accelerate a bit harder than the front end, and they usually turn (frowning) to a black or white board, draw a sketch and try to convince you that the equivalence principle does not imply that sort of thing.

True, the 'Galilean equivalence principle'[1] does not imply it, but Einstein's version surely does. The worst way to defend Einstein's position is to cite Lorentz contraction as the culprit. It will open up a can of worms on the observability (or not) of Lorentz contraction, believe me. However, it is true that if the frame does not Lorentz contract as viewed by any inertial frame, it will be physically stressed and potentially be stretched and eventually break.[2] The secret lies in the fact that the propagation speed of light is not isotropic in accelerating frames.

Einstein used a light signal transmitted from the 'floor' of a (free-space) accelerating lab to its 'ceiling' to show that in order to be compatible with the principle of special relativity, the light must take longer to reach the ceiling than it will take in an inertially moving frame. According to an inertial frame observer, the ceiling has moved while the photon was in flight and it hence explains why the light has taken longer than it 'should have', while still moving at speed c. For an observer in the lab, the ceiling has not moved away from the floor (as measured by her lab rulers) and either the light traveled slower than c, or the ceiling clock ran faster than it should have, or both happened. So, which is it?

The only indisputable point of view is that the ceiling clock has gained some time on the floor clock, because the first option (light slower than c), is observer dependent. An observer at the ceiling would actually have reckoned that the light traveled faster than c, while an inertial observer would have reckoned that the light's speed was exactly c. However, the floor and ceiling clocks could later be slowly brought together and compared - the result will be that the ceiling clock has gained time on the floor clock during the acceleration period.

How does this relate to "the tail end of a uniformly accelerated frame in free space must necessarily accelerate a bit harder than the front end", as stated above? The simplest way to look at it is that acceleration is measured in meters per second per second, or m/s2, meaning that the observed acceleration will be quite sensitive to the elapsed time measurement. A clock that gains time will measure a slightly larger elapsed time and hence a slightly lower acceleration than the other one. When compared to the 'floor clock', this is what happens to the 'ceiling clock', both in a uniformly accelerated lab in free space and in a static lab on Earth.

I hope this removes some of the mystery of accelerating frames of reference and helps readers to better understand Einstein's 'bridge' between special and general relativity. We will cross the bridge to general relativity properly in the next installment.

--
Regards
Jorrie

[1] Loosely defined, the 'Galilean principle of equivalence' stemmed from his observation that the acceleration of a test mass due to gravitation is independent of the amount of mass being accelerated. This boils down to the fact that the inertial mass of a body is precisely equal to its gravitational mass (wikipedia.org/wiki/Equivalence_principle).

[2] The stretching effect of acceleration on rigid systems first became well known in the 1980s as "Bell's spaceship paradox", where a light-weight string between two spaceships undergoing identical accelerations will be stretched and eventually break. It was originally called a "paradox", because the physics was not properly understood when introduced as a thought experiment in the late 1960s (wikipedia.org/wiki/Bell's_spaceship_paradox).