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Let's suppose we have two particles that move together (parallel), and they have a certain acceleration, their acceleration is identical, so both are at relative rest. My doubt is, as they approach the speed of light, will the relative rest remain always? or at some point they will begin to observe a relative speed or acceleration?

Thank you

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As @Luke says, they will always be at relative rest.... but only in the LAB frame.
The distinction is necessary because the two simultaneous events that are used for comparison are spatially separated, and thus depend on the frame of reference.

In the problem stated, the worldlines of each traveler are congruent hyperbolas (corresponding to the same proper-acceleration).

In the LAB frame, the situation can be described as one worldline is a translation in space of the other.

So, at any time T in the LAB frame,
the two simultaneous events on the traveller worldlines have parallel tangent lines..
and this distance between corresponding events doesn't change with T.

But in the frame of the LEFT-TRAVELER (ALICE),
those LAB-simultaneous events aren't simultaneous.
Instead, in the LEFT-TRAVELER (ALICE) uses a different pair of simultaneous events on the traveler worldlines... and their tangent lines are not parallel.... they don't have the same velocity in this frame.

Accel-BellRocket-robphy-pSE-updated

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  • $\begingroup$ So... Could we say that each one of them is watching how his partner moves slowly away at higher and higher speed? (to the limit the speed of light) $\endgroup$ – Pedro Nov 8 '18 at 12:20
  • $\begingroup$ At this time, I can't give a definite answer... since "watching" involves sending light-signals... and I'm still trying to sort out how uniformly-accelerated observers make and interpret measurements (an ongoing study of mine). [Note that while Alice says these events are simultaneous, Bob doesn't... the spacelike line orthogonal to Bob's tangent line isn't shown (a radius vector of Bob's hyperbola).] $\endgroup$ – robphy Nov 8 '18 at 12:31
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No, they will always be in relative rest.

This must be so because being "close" to the speed of light also depends on the frame of reference. For example, if you have two particles at rest in your lab system, then a spaceship flying by with 99% of the speed of light will see them move by at a very large speed close to the speed of light. However, the real physical facts cannot depend on the frame which we use to describe them (which is one of Einstein's basic principles for the formulation of relativity). Hence the particles cannot observe a relativ speed or acceleration.

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