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So here's the setup: I'm in a spaceship, without windows as always, and the ship is accelerating upwards at a constant rate of $1\,\text{g}$. So inside the spaceship it feels like I'm being pulled down by gravity as normal. But this constant acceleration can't last forever: I have to approach $c$ asymptotically. I believe the equation for my speed is:

$$v/c = \tanh(at/c).$$

That's what an outsider would observe. Correct? Not really my question though. What I want to know is, as my speed approaches $c$ and the outsider observes my acceleration slowing down, do I feel that? Or does time dilation and length contraction mean that I would feel the exact same acceleration/gravity inside, forever?

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From the accelerating frame, you're always instantaneously at rest, so there is no problem in experiencing the 1g force indefinitely.

The reference frame will observe the rocket to be accelerating, but as you point out it will approach $c$ asymptotically. Intuitively, the force causing the acceleration is doing work some on the rocket, increasing its energy. The energy is free to increase indefinitely (up to practical considerations, of course). This is consistent with the relation $E = \gamma m c^2 = \frac{mc^2}{\sqrt{1-v^2/c^2}}$, where $E \rightarrow \infty$ as $v \rightarrow c$.

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Would length contraction also take effect, making it appear to me (in the ship) that I am also approaching my destination faster and faster, indefinitely? Or would my apparent unlimited acceleration be solely due to time dilation? –  doug65536 Oct 26 '13 at 4:18
    
Length contraction and time dilation are essentially the same thing. Usually length contraction is defined by counting the time between the front and back of an object at a given velocity. –  chase Oct 27 '13 at 6:06
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Short answer is what you feel about your acceleration is not directly related to the observations of any "outsider".

Consider if there were more than one observer, for example, one of whom was at your launch pad, and another was in a second rocket accelerating towards your starting position. They would both observe different relative speed for you. You obviously could not feel different accelerations depending on which observer you were thinking of (or which was currently measuring your position).

The observer, if they knew relativity theory, could in fact figure out that you were experiencing 1g acceleration, from observations of your relative speed over time, using the inverse equation, and knowing their own acceleration.

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