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In $d{s}^2=(\alpha x{)}^{2}d{t}^2-d{x}^2$, a constant-$x_0>0$ observer has $t=c\tau /(\alpha x_0)$. The nonzero Christoffel symbols are ${\Gamma }^t{}_{tx}={\Gamma }^t{}_{xt}=1/x$ and ${\Gamma }^x{}_{tt}={\alpha }^{2}x$. Its covariant acceleration components are $a^t=0$, $a^x=c^2/x_0$, with invariant magnitude $c^2/x_0$. This is constant proper acceleration, not a covariantly constant four-vector. In inertial coordinates the worldline is the hyperbola $X^2-c^2{T}^2=x_0^2$.

A static observer at Schwarzschild radius $r>2M$ must maintain the outward proper accelerationIt approaches the Newtonian value $M/r^2$ at large radius and diverges at the horizon.

For rectilinear motion write the four-velocity in an inertial frame as $U=(c\cosh \eta ,c\sinh \eta )$, where $\eta$ is the rapidity and $\tau$ is proper time. Differentiation gives $A=c\dot{\eta }(\sinh \eta ,\cosh \eta )$, with invariant $A\cdot A=-c^2{\dot{\eta }}^2$ in the $+-$ metric signature. The signed proper acceleration is therefore $a=c\dot{\eta }$. Constant signed acceleration gives $\eta ={\eta }_0+a\tau /c$; specifying initial position alone leaves the initial rapidity ${\eta }_0$ undetermined.

A particle released from rest in a constant electric field parallel to its motion follows a spacetime hyperbola and has rapidity proportional to proper time.

This force law realizes hyperbolic motion with constant proper acceleration.

An instantaneous reversal is an idealization. A real traveller changes velocity over a finite time, follows a smooth world line, and has finite proper acceleration. Special relativity still calculates the elapsed proper time using the home inertial frame and the same integral; general relativity is not needed merely because the traveller accelerates in otherwise flat spacetime. The clock hypothesis identifies an ideal clock's reading with this geometric proper time.

One explicit smooth trip between the same $O$ and $R$ starts and ends at rest in the home inertial frame. ChooseIts velocity reverses continuously at $T/2$, and its proper acceleration is bounded because $v_{\max }<c$ and $dv/dt$ is finite. With $v_{\max }=0.6c$ and $T=10\mathrm{yr}$, numerical integration givesThis comparison fixes the reunion spacetime events and maximum speed, not the turnaround distance: the smooth trip reaches $v_{\max }T/\pi \simeq 1.910$ light-years, whereas the ideal trip reaches $3$ light-years. The smooth traveller spends less time at high speed and therefore accumulates more proper time than the ideal traveller, while still accumulating less than the home clock.

For a sufficiently short smooth reversal replacing the corner, the proper time approaches the piecewise inertial result. A reversal interval of home duration $\delta t$ contributes between zero and $\delta t$ to the clock reading; shrinking that interval cannot create a finite extra loss concentrated at the corner. The finite difference survives in the long coasting legs. Keeping the same reversal manoeuvre while lengthening those legs changes the age difference. Conversely, two travellers following mirrored paths with the same speed history accumulate equal proper times, even though both accelerate. Thus acceleration distinguishes paths and enables reunion here, but an amount of acceleration alone does not determine the age difference.

The changing simultaneity assignment also becomes continuous. At a traveller spacetime event $(t,x(t))$ with instantaneous velocity $v(t)$, the momentarily comoving inertial frame assigns home time $t_H=t-v(t)x(t)/c^2$. Differentiating givesDuring the reversal at positive $x$, the last term advances the assigned home time rapidly. It describes the traveller's changing relativity of simultaneity, not a locally measured alteration of the home clock's ticking.

An inertial observer of a fixed-$y$ flat brane in metric $d{y}^2+e^{-2y/L}{\eta }_{\mu \nu }d{x}^{\mu }d{x}^{\nu }$ is accelerated relative to the higher-dimensional bulk. Normalization and ${\Gamma }_{\mu \nu }^y=e^{-2y/L}{\eta }_{\mu \nu }/L$ give $a^y=-1/L$. This is its invariant bulk proper acceleration, not a nonzero intrinsic four-dimensional acceleration on the flat induced brane metric.