Does the twin paradox work in an almost empty universe?

Question

Background

The twin paradox is the popular thought experiment involving twins, one of whom makes a journey to a nearby star in a high-speed rocket (travelling at a velocity close to the speed of light) and when he reaches the star, changes direction and returns home at the same speed. When he gets home, he finds that time has slowed down during his journey and that his twin who has remained on Earth has aged more. (The paradox being that because both twins have travelled at the same speed with respect to each other, why has one aged faster than the other?)

My understanding is that the solution is as follows. When he arrives at the star and changes direction, the travelling twin switches from a reference frame travelling away from Earth at a velocity near the speed of light to another reference frame travelling towards Earth with a velocity of the same magnitude but in the opposite direction. In contrast the non-travelling twin stays in the same reference frame. It is often pointed out that the travelling twin experiences an acceleration when he switches from one reference frame to another, whereas the non-travelling twin does not experience an acceleration. (This acceleration could be a Dirac delta function i.e. an instantaneous switch of direction)

Questions

My first question is what is the travelling twin’s acceleration with respect to? Since in special relativity there are no absolute reference frames. I assume it is the Earth??

Secondly, and this is the question which is intriguing me the most, suppose hypothetically there were only two objects in the Universe namely the two twins. If, in this hypothetical universe, one twin travelled on a rocket for a distance of say ten light years and then returned back to meet the original twin. Would the travelling twin’s clock still run slow with respect to the other twin? since there would be no distant background stars/galaxies in this hypothetical universe to measure the acceleration against? Is it just as valid to say that the “stationary twin” is the one in motion? Additional comment

In a various replies to this question, it has been mentioned that the moving twin feels an acceleration and the stationary one does not, but in this hypothetical Universe containing only the two twins I cannot see how this can be the case

Answer 1

Velocity is relative, but acceleration is not: you can feel it (and it feels just like gravity). But as I said in the question comments, acceleration is a red herring.

The real issue is that the travelling twin occupies two (or more) distinct reference frames. That means their worldline (their trajectory through spacetime) is not a straight line, and any inertial observer measuring that worldline will agree on that, although different observers will disagree on the lengths of the segments of that trajectory, depending on their speed relative to the traveller.

So the important factor is that the worldline has a bend in it, and so it cannot be transformed into an unbent line. It doesn't make much difference whether it's a perfectly sharp bend (your Dirac delta function), or a smooth gradual curve.

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There's a variant of the twin paradox that requires no acceleration, but it has 3 participants, A, B, and C. A is based on Earth, C is based on Alpha Centauri (which we can assume is at rest relative to Earth).

B's rocket takes off. It's moving at constant speed $v$ (relative to Earth) as B passes A, and at that moment they use radio to synchronise their clocks to zero. B then travels towards Alpha Centauri.

One year later (according to B's shipboard clock), C sees B approaching. C starts their rocket and heads towards Earth. C's rocket has a speed of $v$ relative to Alpha Centauri. (According to the relativistic law of composition of velocities, B & C have velocity relative to each other of $2v/(1+v^2)$, using natural units, where the speed of light $c=1$). Just as C passes B, C synchronises their clock to B's clock. That is, C "clones" the value of "1 year" from B's clock the instant they pass one another.

One year later, according to that clock, C arrives at Earth, and as C passes A, they compare clocks. C's clock says "2 years". A's clock says "9 years".

Answer 2

As Richard Feynman said about a similar situation, we can't do the experiment of removing all of the matter from the universe to see what would happen, so we don't really know.

But from what we do know, it seems like there would be spacetime even if there was hardly any matter in it. The geometry of spacetime defines the meaning of acceleration and elapsed time. The twins move "relative to spacetime" and that determines who's really accelerating and what their ages are at the end. There's nothing suggesting it's possible to have some sort of pre-spacetime that the twins could move in, but that would leave it ambiguous which one was accelerating.