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Suppose I have a universe with just two objects, say two astronauts, that are placed opposite one another (or alternatively our universe in a very deep and empty region of space with nothing nearby for many many Mpc so that no stars etc. are visible). Then I apply a force on the first astronaut directly parallel to the line between the two astronauts, sending him on a collision course with the second astronaut. The first astronaut feels this force as an acceleration. All well and good so far.

From the second astronaut's perspective the first astronaut started accelerating towards her, so by the laws of motion a force must have been applied on the first astronaut. However the first astronaut can equally reason the exact same way and come to the conclusion that a force must have been applied on the second astronaut to make her accelerate towards him. However we applied no force on the second astronaut and she doesn't feel any acceleration at all, which breaks the expected symmetry of this situation (namely that there is no way to tell whether 1 is accelerating towards 2 or 2 is accelerating towards 1 in a universe with 1 and 2 as the only objects).

I don't understand how to resolve this seeming paradox. Any help is appreciated!

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You are expecting the situation to be symmetric between the two astronauts. This would be true if both astronauts were inertial frames, i.e., not experiencing acceleration or gravity. Galilean relativity (and its generalization, Einstein's relativity), indeed guarantees that if you have two inertial frames, neither is "more correct", there is no experiment which you can do to tell that one of them is moving and the other one isn't or vice versa.

But this is not true if one of the frames is not inertial, i.e., it accelerates. While the acceleration (i.e., a force) is being applied, the accelerating astronaut can perform experiments that can conclusively demonstrate that he is accelerating - while if the second astronaut performs the same experiments he will discover that he's not accelerating. For example, imagine the following experiment: Each astronaut picks up a ball and puts it in space 10cm in front of his face. Now,

  1. For the non-accelerating astronaut, the ball will remain 10cm in front of his face - if the astronaut was moving (in constant speed) or rotating around some axis, both he and the ball will continue the same movement, and the ball will forever appear to be 10cm in front of his face.
  2. For the accelerating astronaut, the ball will be moving in the old velocity, but the astronaut's velocity changes, so the distance of the ball to the astronaut's face will change. The astronaut notices that and deduces that he is being accelerated (while the ball is not).
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  • $\begingroup$ Right, I understand that. Does this still hold if we disallow adding extra markers like the balls and seeing how the astronaut moves relative to them? e.g. if we replaced the astronauts with electrons, would it be possible for the accelerating electron to be able to tell that it is accelerating when the only other thing in the universe is another electron it starts moving relative to (rather than the other electron accelerating)? $\endgroup$
    – Hadi Khan
    Commented Jul 10, 2023 at 14:54
  • $\begingroup$ An electron obviously would not be "aware" of its acceleration (it has no self-awareness :-)), but an outside viewer can see which of the two electrons is being accelerated, because the accelerated electron would produce distinctive photons (see physics.stackexchange.com/questions/46946/…). $\endgroup$
    – Nadav Har'El
    Commented Jul 10, 2023 at 16:24
  • $\begingroup$ Moreover, I don't think it is fair to exclude additional "markers" for doing the experiments. You are asking whether the astronauts can figure out who's being accelerated. They usually need instruments to do experiments. But even without instruments, they have their own bodies. They can feel accelerated by movements of different body parts or their gut contents relative to the rest of the body. Would you exclude that too? $\endgroup$
    – Nadav Har'El
    Commented Jul 10, 2023 at 16:25

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