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There are many questions exploring variations upon the Twin Paradox, but none I have seen seem to resolve this thought experiment.

It is often said that the 1g force we feel on earth due to the surface pushing us in the opposite direction of gravity's pull is indistinguishable from a 1g force applied by a spaceship, and that no experiment can tell a difference. With that in mind, consider the following.

Twins Alice and Bob synchronize clocks. They then accelerate away from Earth with a constant perceived 2g force for the same amount of time, side by side. At this point, both twins agree to go on separate journeys and meet back at earth with carefully timed paths (adjusted for relativistic effects as needed, given that simultaneity is relative).

Both twins will undergo the same forces at the same "times", but from different sources, as follows:

  • while Alice reverses course back towards earth at 2g and then decelerates at 2g to land on the surface, Bob likewise deflects from his course with a force of 2g followed by an opposite force of 2g
  • while Alice then remains on Earth feeling 1g for "6 months" (as shown by her local clock), Bob reduces acceleration to 1g for the same "6 months" relative to his local clock. As we know, accelerating 1g for 1 year should bring Bob to "near" the speed of light relative to something, so 6 months (0.5c) tries to keep both twins still alive when Bob returns
  • after the 6 months, Alice takes another round trip away from earth and back to feel the same acceleration as Bob who is now deflecting course again (carefully changing acceleration twice just as Alice would) to start the journey home
  • this process is repeated one more time so that Bob's deceleration back to Earth is still matched in magnitude by Alice.

This scenario seems to be equivalent to the original Twin Paradox, except we went to great lengths to ensure both twins experienced the same measurable acceleration at all times. And yet, if Bob is younger than Alice when he arrives home, it would seem we just did an experiment where we did in fact distinguish between a normal force from Earth and the accelerating forces by the spaceship, albeit not in the same directions or with any absolute sense of simultaneity.

What detail am I missing that reconciles this or renders it an impossible experiment? Please ELI5 even if a deep mathematical answer is also provided.

Stev0
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  • "we went to great lengths to ensure both twins experienced the same measurable acceleration at all times" - no we didn't, because acceleration has a direction as well as a magnitude. If at step 1 Alice changes the direction of her acceleration by 180 degrees and Bob changes the direction of his acceleration by some other angle, then this introduces an asymmetry between Alice and Bob. – gandalf61 Mar 30 '24 at 17:17
  • So is this to say it is only because of the direction of acceleration that such an experiment cannot be crafted (where if they matched directions they would necessarily be together the entire journey)? And thus a typical spacetime diagram of this experiment would not collapse the accelerations from all 3 spatial dimensions onto the x-axis but rather only describe the projection of their journeys on the x-axis? – Stev0 Mar 30 '24 at 17:27
  • To explore this further, suppose instead Alice and Bob orchestrated this whole experiment with a fixed space offset that was accounted for, but it would allow Bob to fly by Earth without crashing into it if needed in order to match both direction and magnitude during the journey. – Stev0 Mar 30 '24 at 17:36
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    Acceleration itself does not cause time dilation, only the velocity which arises due to that acceleration causes time dilation. This is known as the clock postulate. See https://physics.stackexchange.com/q/704658/123208 & https://physics.stackexchange.com/q/498799/123208 – PM 2Ring Mar 30 '24 at 17:41
  • Thanks @PM 2Ring. So does this mean it is in fact possible to do experiments to distinguish between forces applied by spacetime deformation from gravity, and those applied by a spaceship to perfectly simulate the forces due to gravity? – Stev0 Mar 30 '24 at 17:46
  • "and that no experiment can tell a difference" The Equivalence Principle says that a local experiment cannot distinguish between uniform acceleration and gravity, where "local" means that the gravitational field appears uniform in that spacetime region. You can detect a difference if the region is large enough for you to detect that the field isn't actually uniform. – PM 2Ring Mar 30 '24 at 17:47
  • So if you take off from Earth in a ship accelerating vertically at 1 g you will initially feel a weight equivalent to 2 g. But after an hour, the acceleration due to gravity will drop to ~ g /120, so you'll feel lighter. – PM 2Ring Mar 30 '24 at 18:00
  • In this experiment, I am specifically requiring that Alice's and Bob's spaceships are designed to simulate forces regardless of how much of an influence gravity is playing on that net force – Stev0 Mar 30 '24 at 18:03
  • So it seems, in conclusion, that gravity has a lesser effect on time dilation than the "velocity and position" (both of which are relative) that is caused by actual movement through interstellar space. Either that or there is still something I'm missing to explain why it matters whether a gravitational field is present. – Stev0 Mar 30 '24 at 18:09
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    Recommended resource: 2004 article by Andrew J. S. Hamilton and Jason P. Lisle The river model of black holes (The article is about spacetime curvature in general, it's not specific to black holes.) The river model is not a new theory. The river model is a heuristic; it provides a way to think of velocity time dilation and gravitational time dilation in relation to each other. I expect the heuristic will provide you with the means to assess the merits of the version of the twin scenario that you propose in your question. – Cleonis Mar 30 '24 at 19:23

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it would seem we just did an experiment where we did in fact distinguish between a normal force from Earth and the accelerating forces by the spaceship

Indeed, you have proposed such an experiment. The equivalence principle is frequently misunderstood and simply doesn’t apply to the scenario you describe.

The equivalence principle is usually formulated in terms of “local” measurements where “local” is a specific technical term meaning that the measurements are done over a small enough region of space and time that tidal effects are negligible. Another formulation is that it applies to a uniform gravitational field.

Your thought experiment is designed explicitly such that neither condition applies. Because this is not a local experiment and tidal gravity is significant, the equivalence principle is not directly applicable.

Instead, this would simply be calculated using the standard approach of integrating the spacetime interval over the two separate worldlines. No simplification via the equivalence principle can be used.

Dale
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  • Thanks! It seems some of my weak areas are showing through on my journey to understand SR/GR. Also, I believe some of the confusion came from oversimplifications like "the ground is accelerating toward the inertial object rather than the other way around.". Taken literally, one might conclude this acceleration causes the same kind of 4-velocity changes as navigating through the ether. – Stev0 Mar 31 '24 at 01:38