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I'm aware that the twin 'paradox' is generally just a mistake of (mis)treating the traveler's frame as an inertial one, whereas it isn't, because he accelerates to make the turnaround. There just on detail I can't get. I apologize for the length, and thanks to anyone who bears with me.

I've been looking at a great website for relativity simulations, https://www.refsmmat.com/jsphys/relativity/relativity.html. It shows multiple observers, and all clocks from each one's frame of reference.

For the twin 'paradox', it shows the duration of the trip from two inertial frames: 1) the twin who we call the one 'at rest' 2) an observer that passes #1 as the traveler takes off and 'moves' in sync with him, but does not make the turnaround with him. The simulation assumes instantaneous acceleration, and the traveler makes the turnaround at t=50 on his own clock - we can say that this was agreed upon from the beginning.

From #1's perspective, the traveler accelerates to .866c and makes the turnaround at t=50, which is t=100 for #1. The traveler continues at .866c until he comes home. His clock shows t=100, and #1's shows t=200.

From #2's perspective, the 'home' is travelling away at .866c, the traveler decelerates to 0, waits t=50 on his clock, then accelerates to .99c. He arrives home at t=400 on #2's clock. From #2's perspective, the turnaround happened at t=25 on #1's clock. The math works out approximately: according to #2, the 'home' was travelling at .866c * 400 units of time, giving a distance travelled of 346.4, and the traveler raced towards #1 at .99c * 350 units of time, giving 346.5.

What bothers me is like this:

Presumably, all parties can know how much fuel was taken on board and what amounts were expended. From the home perspective, 33% of the fuel was used accelerating to .866c, 33% reversing the acceleration, and 33% accelerating home. However, from #2's frame, 33% was used decelerating from .866c to 0, and after a wait of 50, 66% was used to accelerate to .99c. For arguments sake, let's give the spacecraft a mass of 1kg. At .866c its kinematic energy is 8.986×10^16J. So, 33% of the fuel was expended to transfer that much energy on the first leg. At .99c is kinematic energy is 5.472×10^17J - about 6x the energy of .866c. How could #2 explain how just 2x the fuel (the 66%) imparts 6x the energy?

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  • $\begingroup$ Why does one frame see the traveler accelerate initially and another frame see the traveler already at speed? $\endgroup$
    – Asher
    Commented Nov 2, 2017 at 20:08
  • $\begingroup$ @Asher, because the other frame is moving at .866c in frame #1, but is inertial in his own frame, watching #1 move away at .866c. If you take a look at the simulation, it will be easier to see. $\endgroup$
    – Reinstate Monica
    Commented Nov 2, 2017 at 20:10
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    $\begingroup$ acceleration is not relative. All inertial observers will see Twin 2 accelerate, even if the acceleration happens to bring him into the same frame as a particular observer. $\endgroup$
    – Asher
    Commented Nov 2, 2017 at 20:17
  • $\begingroup$ @Asher - sorry I misread your comment; either way, I wrote that frame #2 sees him decelerate to 0 - decelerating is a form of acceleration. $\endgroup$
    – Reinstate Monica
    Commented Nov 2, 2017 at 20:23
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    $\begingroup$ That values for energy and energy-transfer is are frame-dependent is a feature even of Galilean/Newtonian mechanics. See, for instance, physics.stackexchange.com/questions/230054/…. $\endgroup$
    – dmckee --- ex-moderator kitten
    Commented Nov 2, 2017 at 21:43

2 Answers 2

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The answer is that increases in kinetic energy are frame dependent. You don't need to venture into relativity to find apparent contradictions of this sort, as the following example shows.

Imagine that you expend a certain amount of fuel to accelerate your 1000kg car from rest to a speed of 10m/s. The KE of your car increases by 50,000 Joules. From the perspective of a helicopter passing overhead at 100m/s in the opposite direction, your car appears to have accelerated from 100m/s to 110m/s. If you work out the corresponding increase in KE from the perspective of the helicopter you will find it is 1,050,000 Joules, a vastly greater increase for the same expenditure of fuel.

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When a rocket motor accelerates a reaction mass, the kinetic energy of the reaction mass increases.

When a rocket motor decelerates a reaction mass, the kinetic energy of the reaction mass decreases.

Now let us use heat pump terminology: A motor burns fuel to produce energy, which is used to pump kinetic energy from a reaction mass to a spacecraft. One joule of energy can pump many joules of energy, if there is a reservoir from where kinetic energy can be taken.

In the OP there was no mention about reaction mass, so the scenario is very unphysical. I recommend using electric trains on the surface of a planet instead of rockets.

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