I'm trying to analyse non-inertial motion in special relativity.
First I'll start off with inertial motion. In my reference frame, my particle has coordinates   

$x^\mu = (t, x)$  

And in the particle's reference frame, it has coordinates  

$x'^\mu = (\tau, 0)$  

My understanding is that since we are in flat (affine) space, the particle's position transforms as a vector, and I can boost it by its velocity $v$ to give  

$\begin{pmatrix}\gamma & v \gamma\\ v \gamma & \gamma\end{pmatrix} \begin{pmatrix} \tau \\ 0 \end{pmatrix} = \begin{pmatrix} t \\ x \end{pmatrix}$  

Which tells me that $x = v\gamma\tau$ and $t = \gamma\tau$, as usual. It's clear that in both frames we have $\dot{x}^\mu\dot{x}\_\mu = -1$, from explicit calculation and from the fact that $\dot{x}^\mu\dot{x}\_\mu$ is a Lorentz scalar, and our two frames are related by Lorentz boost.

I start to have problems when I move to non-intertial motion. If I take my reference frame to be the same, and the particle's reference frame to now be its instantaneous reference frame, from what I can see nothing should change, apart from the particle's velocity is now some (as yet unspecified) function of time, and the Lorentz boost becomes time dependent. In my new non-intertial scenario my particle still has $x = v\gamma\tau$ and $t = \gamma\tau$ from the same analysis above, and it's clear that it still has $\dot{x}^\mu\dot{x}\_\mu = -1$ in its own instantaneous reference frame. By the fact that this is a Lorentz scalar, I expect this quantity to be the same in my frame.

By explicit calculation in my reference frame, I find that   

$\dot{x}^\mu = (\gamma^3 v \dot{v}\tau + \gamma, \gamma^3 v^2 \dot{v} \tau + \gamma \dot{v}\tau + \gamma v) $  

Which gives   

$\dot{x}^\mu\dot{x}\_\mu = -1 + \gamma^4\dot{v}^2\tau^2$  

Clearly this not acceptable.

My general relativistic intuition tells me that my problem lies in the fact that I've treated the manifold as a vector space, and that I should only use the Lorentz boost on elements of the tangent space. So maybe I should start with the particle's velocity 2-vector instead of its position. But I can't see why this approach should work for inertial motion and not for non-intertial motion, and my understanding is that because Minkowski spacetime is flat, it should be OK to treat elements of the manifold as vectors.

Any ideas?