Four-acceleration ($A^{\mu }=D{U}^{\mu }/d\tau$)

The four-acceleration is the covariant derivative of four-velocity along a massive observer's worldline: $A^{\mu }=d{U}^{\mu }/d\tau +{\Gamma }_{\nu \rho }^{\mu }{U}^{\nu }{U}^{\rho }$. In Cartesian Minkowski spacetime coordinates the Christoffel symbols vanish, leaving the ordinary proper time derivative. Differentiating the constant velocity norm gives $U\cdot A=0$; in the instantaneous rest frame, $A=(0,\mathbf{a})$, so $A\cdot A=-a^2$ with $+---$ metric signature. The magnitude $a$ is the proper acceleration.

For uniform circular motion in an inertial laboratory, the Lorentz factor is constant. With metric signature $+---$, four-acceleration is $A^{\mu }=(0,{\gamma }^2\mathbf{a})$ and proper acceleration has magnitude ${\gamma }^2{v}^2/r$. The instantaneous spatial acceleration is perpendicular to velocity, even though the particle has nonzero proper acceleration.

Proper acceleration is the acceleration measured in the particle’s instantaneous rest frame.

In a Cartesian inertial frame, let $\mathbf{v}$ and $\mathbf{a}=d\mathbf{v}/dt$ be the coordinate velocity and acceleration of a massive particle. With $\gamma =(1-v^2/c^2{)}^{-1/2}$, differentiating its four-velocity givesIn the $+---$ metric signature, its invariant Minkowski norm is $A^{\mu }{A}_{\mu }=-{\alpha }^2$. Expanding that contraction gives the displayed proper acceleration. Resolving the coordinate acceleration into components parallel and perpendicular to $\mathbf{v}$ yields the equivalent formula ${\alpha }^2={\gamma }^6{a}_{\parallel }^2+{\gamma }^4{a}_{\perp }^2$. In particular, constant coordinate acceleration and constant proper acceleration are different prescriptions.

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 $X^2-c^2{T}^2=x_0^2$.

For motion on a fixed axis, the signed spatial four-acceleration in the instantaneous rest inertial frame is $a={\gamma }^3\dot{u}$. Its magnitude is the scalar proper acceleration. Constant signed $a$ from rest gives $\gamma u=at$ and the hyperbolic motion worldline $x=(c^2/a)(\sqrt{1+a^2{t}^2/c^2}-1)$.

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 uniformly accelerated observer has a null boundary beyond which light signals can never reach the observer.

For the standard right-wedge hyperbola and a stationary source at $x=L$ emitting rightmoving photons, the retarded null coordinate is $p=t-x=-e^{-a\tau }/a$. Equating it to the emission value $t_e-L$ gives the displayed reception relation. Differentiating yields the Doppler frequency ratio $\nu /{\nu }_0=d{t}_e/d\tau =e^{-a\tau }$. The limiting emission time $L$ is a supremum: emission exactly at that time is never received at finite proper time. These expressions use $c=1$ and acceleration continued indefinitely.

Let an observer start at rest at $T=0$, $X=R=c^2/a$ and accelerate towards increasing $X$ with constant proper acceleration $a>0$. A signal emitted simultaneously from $X=R-d$ travels along $X=R-d+cT$. The observer follows hyperbolic motion, $X=R\cosh (a\tau /c)$ and $cT=R\sinh (a\tau /c)$. Their intersection obeysA finite future reception exists only for $0\le d<c^2/a$. At equality the ray lies on the limiting null horizon and never reaches the observer; for larger separation it remains outside the observer's receivable past. The proper reception time tends to $d/c$ as $a\to 0$. The simultaneous emission and initial-rest conditions are essential; changing either changes the timing problem.

In units $c=1$, a receiver on the hyperbolic motion worldline $t=a^{-1}\sinh (a\tau )$, $x=a^{-1}\cosh (a\tau )$ has retarded null coordinate $p=t-x=-a^{-1}{e}^{-a\tau }$. A right-moving light ray emitted at $(t_0,L)$ has $p=t_0-L$, so it meets the receiver only if $t_0<L$. The photon wave covector is proportional to $(1,-1)$, and contraction with the receiver's four-velocity gives its measured frequency as ${\nu }_0(\dot{t}-\dot{x})={\nu }_0{e}^{-a\tau }$. The intersection condition gives the second displayed equality. Emission at $t_0=L$ lies on the Rindler horizon and never reaches the receiver at finite proper time; $L$ is the supremum of receivable emission times, not an attained latest time.