Objective velocity (Relativity theory)

Question

Suppose that I have a velocity of (approximately) $c$. Then I decelerate with $c$ per hour for exactly one hour.

Then I would know that I would be standing still in the end, because the decelleration from $c$ to zero in one hour is exactly $c$ per hour. I will have deduced that my velocity is zero without referring to any inertial reference frame. Not relative to any specific reference frame is what could be called 'objective'. I have thereby given rise to the concept of objective velocity, which is contrary to relativity theory.

Note that the above method is not totally correct, since my clock would be going slower in the first half hour of the deceleration (from the inertial frame of reference at the end point). It should be possible to compensate for this and so the decelleration period would be less than an hour on my own clock. The casus would then be altered a bit, but it would still give rise to an objective velocity.

Is there such a thing as objective velocity, or is there something wrong with the thought experiment?

Answer 1

You are assuming that if you accelerate at $c$ m/sec/hour for one hour your your velocity will increase to $c$ and thereafter remain unchanged, so if you decelerate at $c$ m/sec/hour for one hour your your velocity will return to zero again.

Firstly we need to point out that your velocity, measured in your rest frame, is always zero. So I assume you're asking about your velocity relative to a non-accelerating observer who is stationary with respect to you when you start accelerating. In that case the time dependence of your velocity is given by the relativistic rocket equation:

$$ v = c \tanh \left( \frac{aT}{c} \right ) $$

where $T$ is the time measured by you on the rocket. I discuss this in more detail in my answer to Does the pilot of a rocket ship experience an asymptotic approach to the speed of light?.

Relative to the non-accelerating observer your acceleration tends asymptotically towards $c$ but never reaches it. If you accelerated for some time greater than one hour then decelerated for just one hour you would not return to zero velocity relative to the non-accelerating observer.