Perception at relatavistic speeds If one were to be traveling at near the speed of light, their mass would be $m_{rel}= m_0 / \sqrt{1 - \frac{v^2}{c^2}}$.  For the mass to double the speed would have to be $86.6\%c$
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To better phrase the question, perhaps I should use relativistic momentum: $p_{rel}=m_0v/\sqrt{1 - \frac{v^2}{c^2}}$
In a space ship at this speed in zero gravity & acceleration, the mass would be weightless anyway.  But to swing ones arm left and right, would it seem to have more inertia to move in the direction of travel ?  Or does relativistic time compensate this and although the arm moves slower the perception of time corrects this and it appears to move as it should with the same effort?
 A: From the reference frame of the space ship, your body is stationary. When you swing your arm back and forth, it has a non-zero speed in this frame, and thus its mass, or rather momentum, increases. The space ship's velocity of $v_\mathrm{ship} = 0.866c$ doesn't add to this. That means that the increase in momentum is exactly the same as if you swing your arm right now, on Earth. Thus, you can perform the experiment yourself, and verify that the momentum-increase is negligible. It is there, nonetheless.
From the reference frame of an observer that sees the space ship passing by at $0.866c$, the ship, your body, and your arm has increased their momenta by a factor of 2. When you swing your arm, its velocity will alternate between being slightly below $v_\mathrm{ship}$, and slightly above $v_\mathrm{ship}$. Hence, the observer will measure your arm's increase in momentum $\Delta p_\mathrm{arm}$ to be slightly below 2, and slightly above 2.
Note that velocities at this speed don't add like this
$$v_\mathrm{tot} = v_\mathrm{ship} + v_\mathrm{arm},\qquad\mathrm{(wrong!)}$$
but like this
$$v_\mathrm{tot} = \frac{v_\mathrm{ship} + v_\mathrm{arm}}{1+(v_\mathrm{ship} v_\mathrm{arm}/c^2)}.$$
EDIT: Also, note that what increases is not really mass, but rather momentum, as pointed out by @Jimnosperm. The argument is the same, though.
