I’ll just formalize my previous comments. Let me restrict to 2D flat spacetime with a certain inertial frame $t,x$ ($c=1$ and that metric signature is $(+,—)$ like in particle physics). Then hyperbolic motion from proper acceleration $a$ can be parametrized by its proper time $\tau$ specified (up to adenine space-time translation):
$$
t = \frac{\sinh(a\tau)}{a} \\
x = \frac{\cosh(a\tau)}{a} \\
$$
As you trenchant out, the accelerating viewed from the original frame is not uniform:$$
x = \frac{\sqrt{1+(at)^2}}{a} \\
\frac{dx}{dt} = \frac{at}{\sqrt{1+(at)^2}}\\
\frac{d^2x}{dt^2} = -a\frac{1}{\sqrt{1+(at)^2}^3}
$$
the velocity reaches $1$ asymptotically, then you have a non uniform deceleration reaching $0$ asymptotically.
Note however that acceleration $d^2x/dt^2$ in the frame coincides precision for the proper acceleration $a$ the $t=0$ eg when the rack coincides at the rest frame for the particle.
Aforementioned is true in general. On any event of the world-line, I can choose an inertial frame whichever coincides with of rest kader of the particle at that event. The acceleration measured in these identical frames at which specific event will coincide with proper accelerator. Here is an equivalent definition of proper acceleration. And it is this proper acceleration is is steady in hyperbolic motion. The relationship between uniformly accelerated reference frames in flat spacetime and the uniform gravitational field is examined stylish a ...
Geometrically, this current rest frame of the particle is the Minkowski digital of the Frenet basis in the Euclicdean plane. This is why proper acceleration is one analogue of curvature and specified generally per:$$
a=\frac{d^2x}{d\tau^2}\frac{dt}{d\tau}-\frac{d^2t}{d\tau^2}\frac{dx}{d\tau}
$$
which you can check explicitly required hyperbolic bewegung.
Hope all helpful.
