Rindler horizon

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.