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I am a bit confused about the particulars of inertial reference frames. For example, I was just thinking about the following thought experiment.

Thought experiment:

Imagine the only things in the universe are a pair of spaceships (and their fuel, occupants and so on). At $t=0$, the first spaceship is travelling at $v_1(0)=1$ms$^{-1}$ and the second spaceship is travelling at $v_2(0)=2$ms$^{-1}$. Both spaceships are accelerating at a rate of $a=5$ms$^{-2}$.

Two seconds later, their velocities are $v_1=11$ms$^{-1}$ and $v_1=12$ms$^{-1}$, respectively. They have both accelerated, but crucially they are traveling with constant velocity with respect to one another and, thus, constitute a pair of inertial reference frames.

With the exception of these two spaceships there is literally nothing else in the universe, so what have they accelerated with respect to, if not one another?

The simplest explanation seems to be that they haven't accelerated at all, but surely the occupants of both ships would experience that feeling in their stomach which suggests that one is accelerating. Also, the speed dial in the cockpit would be increasing. This suggests that both ships are accelerating with respect to something, and that something doesn't exist because they are travelling at a constant velocity with respect to one another and there is nothing else. This suggests that there is a fundamental reference frame of the universe, violating the cosmological principle.

Question: can somebody please explain the flaw in my above argument, which I am quite certain is wrong?

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    $\begingroup$ There are no extended reference frames when you are accelerating. This may give you some intuition en.wikipedia.org/wiki/Bell%27s_spaceship_paradox $\endgroup$ – ClassicStyle Sep 20 '16 at 14:41
  • $\begingroup$ "but crucially they are traveling with constant velocity with respect to one another and, thus, constitute a pair of inertial reference frames." That's not correct. Each frame is evaluated on its own, and each is accelerating, and each is not inertial. $\endgroup$ – garyp Sep 20 '16 at 15:08
  • $\begingroup$ By measuring the speed of light one can conclude whether or not he/she/it is in non-inertial frame of reference. Using special/general relativistic effects, the coordinates of an accelerating object is described by Rindler metric , in which the speed of light as observed by an accelerating object is not $c$. (Which is a result of formulating acceleration in such way that it is Lorentz invariant, i.e. including special relativistic effects, as it has been asserted in several answers here) $\endgroup$ – Gigi Butbaia Sep 20 '16 at 16:30
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Acceleration is not as relative as you might think. Acceleration of an object always is the result of a force acting on the object. Classically we have $F=ma$, and in special relativity the expression becomes a little more complicated, but the idea is essentially the same. When you you are in a frame and you want to find out if your frame is accelerating, then you can go look for the force that accompanies it, and in principle you should be able to find out.

For instance, imagine that you're in a spaceship. If you are accelerating, then you will be pushed to the back of the spaceship, which is an easy thing to check. In that case you will know that you are not in an inertial frame! Acceleration is measurable.

The difference with movement with constant velocity is that someone in an inertial frame cannot possibly find out if his frame his moving at all; it would not even make sense to ask that question, for as you say, it all depends on the reference point.

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I like the answer by Sjors Heefer, but I have a few points to add.

Your model of the world is overly simplified. Sure, if the two spaceships are the only things out there, acceleration is relative. But the world is not like that.

According to General Relativity, there is a kind of medium filling the entire Universe: the gravitational field which provides the notion of local absolute acceleration. So one might say for example that he is accelerating with 1 $m/c^2$ with respect to this gravitational field, or more technically, with respect to the local Lorentz frame.

In Special Relativity, a special case is considered which is the Minkowski solution of Einstein's equations, or simply flat spacetime. This spacetime also provides a notion of (global) absolute acceleration, though no notion of absolute velocity. So it makes sense to say that e.g. I am accelerating and you are moving freely, and not the other way around.

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surely the occupants of both ships would experience that feeling in their stomach which suggests that one is accelerating.

This certainly suggests that a force is acting on the ships and their occupants, and that means that they are accelerating.

This suggests that both ships are accelerating with respect to something

Correct. But what is the something they are accelerating relative to? If there is a force acting on you, then you must also be experiencing a 4-acceleration. This means that you are not moving along the geodesics of 4D space-time. The 4-acceleration can then intuitively be interpreted as the acceleration of an object relative to 4D space-time itself.

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