Hello and thank you for your time. I've been wondering about Einstein's Equivalence Principle core idea. That is, from experiment alone, one cannot differentiate between a gravitational field and a non-inertial (accelerating) frame of reference. So, Einstein thought about an elevator (I guess) out in space that is getting tugged on, pulled in an accelerated manner, and then found/made a geometrical hyperspace that fit this idea. My question is - wouldn't the "tugging force" have to keep accelerating? Gravitational fields keep "pulling" as long as their is enough mass-energy present, so wouldn't the analogue of the "tugger" have to keep accelerating? What about the light speed limit? If it has to keep accelerating, would it not also have to keep introducing energy to keep accelerating? If we adhere to Special Relativity, should this not equate to requiring more and more and eventually infinite energy?
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1$\begingroup$ Earth can accelerate you to approx. 11km/s before you hit the surface, the sun to approx. 42km/s. No infinities here. $\endgroup$– CuriousOneCommented Mar 15, 2016 at 21:43
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2$\begingroup$ 1) "wouldn't the "tugging force" have to keep accelerating?" yes, it would (so what?). 2) "What about the light speed limit?" constant acceleration doesn't make bodies exceed the speed of light (see en.wikipedia.org/wiki/Space_travel_using_constant_acceleration) . 3) "would it not also have to keep introducing energy to keep accelerating?" yes, it would (so what?) $\endgroup$– AccidentalFourierTransformCommented Mar 15, 2016 at 21:44
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$\begingroup$ thank you for your responses. CuriousOne - Yes, I realize these are constant accelerations but for the analog case (tugged elevator in free space), one would have to keep this acceleration going, akin to a gravitational field, which as compared to any large mass (earth, sun, moon, etc) this equates to tugging for a very long time! Eventually one would be approaching the speed of light. $\endgroup$– Curious CoderCommented Mar 15, 2016 at 22:04
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$\begingroup$ Accidental: Thank you also for your thoughts. You mentioned, "so what?" That is the point - are we to assume then that the Lorentz-Contraction just keeps applying, eventually making the elevator a traveling plane, contracted to virtually no width at all? Which then implies more and more energy to the tugged-system. How can this scenario really be equivalent to a gravitational field? Gravitational bodies have been "tugging" for eons upon eons. I don't see how these two situations are truly "equivalent" (at least when adhering to SR). $\endgroup$– Curious CoderCommented Mar 15, 2016 at 22:05
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2$\begingroup$ The equivalence principle doesn't say anything about "this has to be so for infinity", it's a local principle in space and time. If you want to keep the motion going forever, though, you can do that, too... that's what happens in rotating systems. "Approaching the speed of light" is observer dependent and therefor meaningless. $\endgroup$– CuriousOneCommented Mar 15, 2016 at 22:18
3 Answers
There are a couple of misconceptions in your question.
The first is that the limiting speed of light, and the increasing inertia of the object being accelerated are relevant only in other reference frames. From the perspective of the object, it can keep accelerating indefinitely in accordance with f=ma where f is the applied force in the object's (instantaneous) rest frame and m is the object's rest mass. To an outside observer the cumulative effect of the acceleration is an asymptotic approach to c, but that is beside the point- it is the acceleration in the object's own frame that mimics the effect of gravity.
The second is that you imply that gravitational fields keep accelerating objects indefinitely, which is not true- they will accelerate an object until the object moves to a position in which there some form of local minimum of gravitational PE. If you found yourself at the centre of the Earth you would be weightless.
The point of the principle of equivalence is that the object that is subject to the acceleration experiences the acceleration as if the acceleration were gravity- it doesn't mean that bringing about acceleration through some kind of mechanical process is the same as gravity: the effects are the same, but the causes are different.
In relativity a constant force doesn't mean a constant velocity and linear growth of velocoty). It means a linear growth of momentum, according to the law $$ F = \frac{dp}{dt}$$ Under a constant force, the momentum will grow indefinitely, but the velocity will only approach assymptotically the speed of light.
As for the source of energy for that, it comes from the gravitational field. What is important to consider is that the energy/momentum of a gravitational field is not a tensor, which means it doesn't obey the usual laws of transformation of energy and momentum between frames of reference, at least not if the trnasformation of the coordinates is not localized. That is the case of the accelerating frame of reference = the transformation from inertial to accelerating frame of reference is not localized, but affect the whole spacetime, and because of that it happens that, even if in a inertial frame the enrgy and momentum of gravitational field of Minkowski space is 0, then in an accelerating frame the energy and momentum of the same space becomes infinite. The gravitational field seen from the accelerating frame becomes an endless source of energy and momentum.
In analogy to 'fictitious forces' you can also actualy the energy of gravitational field a 'fictitious energy'. If you want to consider the law of of energy conservation it is necessary, but it can appear from no physical source if you change the coordinates.
As you approach speed of light in any (assumingly possible) manner, it becomes increasingly hard to push/pull further; it truly requires an infinite amount of energy to accelerate a massive particle (no matter how small) to a velocity exactly equal to the speed of light. You can never break this limit. So, yes, adhering to special relativity you would require infinite amount of energy to make this happen.
In other words, as you constantly apply your force on the accelerating system, it becomes harder and harder to keep that same amount of force for longer time periods because it is not going to suffice to overcome a constantly increasing inertia.
