bahamagreen said:
The relativistic velocity addition equation would be used by the "stationary" observer.
The observer in the accelerating frame would not use that equation. He would measure a constant acceleration (equivalence principle) and have no reason not to integrate that to find a net relative increasing velocity (with respect to prior velocity). He can make this measure without information from outside his frame that would present results of externally viewed length contraction of time dilation (he may want to look outside and confirm the acceleration is not a local gravity source).
Only an inertial observer can integrate their acceleration over time to get their velocity in the manner you suggest. The reason for this is that the basis vectors of the accelerating observer are constantly changing.
One way to describe the correct way of getting the velocity is to say that one needs to use the "covariant derivative", not the ordinary partial derivative, but I suspect that that's not the level that we want to approach the problem at.
There's a couple of approaches we can take that may give more insight without as much sophisticated math.
Using four-vectors, as one previous poster did, you can always say that a = du/dtau, where a is the four-acceleration and u is the four-velocity, dx/dtau. We take the magnitude of the four-acceleration, we know it's direction.
You won't get the correct answer if you try to differentiate the ordinary velocity to get the four-acceleration, i.e the proper acceleration a is not dv/dtau, it is du/dtau.
The difference between u and v is that u = dx/dtau, not dx/dt, tau being the proper time of the accelerated observer, and t being the coordinate time of the inertial frame.
Less formally, we can see that if we drop off "comoving" objects at various times in our journey, at any particular instant the velocity that the comoving object measures for it's relative veocity to the origin, O, is the same velocityh that an observer at the origin, O, measures for the comoving object.
We can then use this fact to find the velocity our accelerating observer measures relative to the origin, O, at any time T, by replacing it with the velocity that the observer at the origin measures for the co-moving observer.
We can also see that the velocity measured by the comoving object will be the same as the velocity measured by the instantaneously co-located accelefrating observer.
This is why we can't integrate to get v, an it's also the reason that the ordinary velocity v does not increase without bound.
Note that the four velocity can be integrated, and does increase without bound.
