The Rindler chart: causal horizons of a uniformly accelerated observer and the restrictions of radar time and distance

(Chart domain, wedge and boundary.) From [L3], $\mathrm{\Phi }$ is a smooth bijection onto $W$ with the stated smooth inverse defined for $x^1>\mathrm{\mid }{x}^0\mathrm{\mid }$, so the coordinate domain is exactly $W$ and $\rho =0$ is excluded; on $W$ the coordinates satisfy $U=\rho e^{\theta }>0$ and $V=-\rho e^{-\theta }<0$, and conversely $U>0$ and $V<0$ define $W$, since $x^1=(U-V)\mathrm{/}2>\mathrm{\mid }{x}^0\mathrm{\mid }=\mathrm{\mid }U+V\mathrm{\mid }\mathrm{/}2$ is equivalent to $U>0,V<0$. The boundary is $x^1=\mathrm{\mid }{x}^0\mathrm{\mid }$, split into the two half-planes $x^1=x^0\ge 0$ (where $V=0$, $U\ge 0$) and $x^1=-x^0\ge 0$ (where $U=0$, $V\le 0$); these are the parts of the full null planes $V=0$ and $U=0$ that bound $W$, while the full planes extend beyond.

(The worldline $O$ and its parametrization.) By [L3] the curve $\theta \mapsto \mathrm{\Phi }(\theta ,{\rho }_0,y_0,z_0)$ is a flow line of $n={\rho }^{-1}{\mathrm{\partial }}_{\theta }$ restricted to $\rho ={\rho }_0$; its tangent with respect to $\theta$ is ${\mathrm{\partial }}_{\theta }={\rho }_{0}n$, so by [L4] its four-velocity is $cn$, its proper time satisfies $d\tau ={\rho }_{0}d\theta \mathrm{/}c$, i.e. $\theta =c\tau \mathrm{/}{\rho }_0$ up to an additive constant, and its proper acceleration is $c^2\mathrm{/}{\rho }_0$. Along it, $U={\rho }_0{e}^{\theta }$ and $V=-{\rho }_0{e}^{-\theta }$.

(The future null lines of the plane through $O$.) Let $p,q$ be events in the plane $y=y_0$, $z=z_0$ and $w:=q-p$. Since $w^2=w^3=0$, one has $\mathrm{\Delta }U:=U(q)-U(p)=w^0+w^1$, $\mathrm{\Delta }V:=V(q)-V(p)=w^0-w^1$, and $\mathrm{\Delta }U\mathrm{\Delta }V=(w^0{)}^2-(w^1{)}^2$; hence $w\in {\bar{C}}^{+}$ iff $\mathrm{\Delta }U\ge 0$, $\mathrm{\Delta }V\ge 0$ and $\mathrm{\Delta }U\mathrm{\Delta }V\ge 0$, and $w$ is future null iff in addition $\mathrm{\Delta }U\mathrm{\Delta }V=(w^2{)}^2+(w^3{)}^2=0$, i.e. iff either $\mathrm{\Delta }V=0<\mathrm{\Delta }U$ or $\mathrm{\Delta }U=0<\mathrm{\Delta }V$. Differentiating $U=\rho e^{\theta }$ and $V=-\rho e^{-\theta }$ along a curve in $W$ gives $dU=e^{\theta }(d\rho +\rho d\theta )$ and $dV=-e^{-\theta }(d\rho -\rho d\theta )$, so the future null lines through a point of $O$ in the plane are exactly the constant-$U$ lines ($d\theta =-d\rho \mathrm{/}\rho$) and the constant-$V$ lines ($d\theta =+d\rho \mathrm{/}\rho$).

(The logarithmic inequality.) For $r>0$ with $r\mathrm{\ne }1$, the mean value theorem applied to $\log$ on the interval between $1$ and $r$ gives $\log r-\log 1=\frac{1}{\zeta }(r-1)$ for some $\zeta$ strictly between $1$ and $r$; since $\zeta >1$ when $r>1$ this gives $\log r<r-1$, and since $0<\zeta <1$ when $0<r<1$ it gives $-\log r>1-r$; equality holds only at $r=1$. In particular $\log 2<2-1=1$, and also $e>2$ gives $\log 2<\log e=1$ by strict increase of $\log$.

(Causal criteria for the worldline $O$.) For an arbitrary displacement $w$, $w\in {\bar{C}}^{+}$ is equivalent to $\mathrm{\Delta }U\ge 0$, $\mathrm{\Delta }V\ge 0$ and $\mathrm{\Delta }U\mathrm{\Delta }V\ge (w^2{)}^2+(w^3{)}^2$: this follows from $w^0=(\mathrm{\Delta }U+\mathrm{\Delta }V)\mathrm{/}2$ and $\eta (w,w)=-\mathrm{\Delta }U\mathrm{\Delta }V+(w^2{)}^2+(w^3{)}^2$ and the inertial cone criterion [L5]. (Forward.) Let $q\le p$ with $p$ on $O$, so $w=p-q\in {\bar{C}}^{+}$; then $\mathrm{\Delta }V=V(p)-V(q)\ge 0$ and $V(p)=-{\rho }_0{e}^{-{\theta }_p}<0$, so $V(q)\le V(p)<0$. Similarly $\mathrm{\Delta }U\ge 0$ with $U(p)>0$ gives nothing yet, but the same argument with the roles of $U$ and $V$ reversed and the sign of ${\rho }_0{e}^{{\theta }_p}$ shows $U(q)\le U(p)$; to obtain $U(q)>0$ from $p\le q$ one uses $w=q-p\in {\bar{C}}^{+}$: then $U(q)-U(p)\ge 0$ and $U(p)={\rho }_0{e}^{{\theta }_p}>0$, so $U(q)\ge U(p)>0$; and $V(q)\ge V(p)$ gives no upper bound. (Reverse.) Suppose $V(q)<0$; choose ${\theta }_p$ with $-{\rho }_0{e}^{-{\theta }_p}>V(q)$, i.e. $V(p)>V(q)$, which is possible because $-{\rho }_0{e}^{-{\theta }_p}\to 0^{-}$ as ${\theta }_p\to +\mathrm{\infty }$; then $\mathrm{\Delta }V>0$, and $\mathrm{\Delta }U={\rho }_0{e}^{{\theta }_p}-U(q)$ and $\mathrm{\Delta }U\mathrm{\Delta }V$ tend to $+\mathrm{\infty }$ as ${\theta }_p\to +\mathrm{\infty }$ while the transverse displacement $(y_0-q^2,z_0-q^3)$ is fixed, so for all sufficiently large ${\theta }_p$ the three general inequalities just established hold and $q\le p$. Similarly, if $U(q)>0$, choose ${\theta }_p$ very negative so that $U(p)={\rho }_0{e}^{{\theta }_p}<U(q)$ and $\mathrm{\Delta }U>0$, while $\mathrm{\Delta }V=V(q)+{\rho }_0{e}^{-{\theta }_p}\to +\mathrm{\infty }$; then $\mathrm{\Delta }U\mathrm{\Delta }V\to +\mathrm{\infty }$ and $p\le q$ for all sufficiently negative ${\theta }_p$. Hence the criteria, and since the construction used only the positivity of ${\rho }_0$ and the fixed transverse coordinates of the worldline, the same criteria hold for every congruence worldline $\rho ={\rho }_1$, $y=y_1$, $z=z_1$. The remaining consequences in (ii) follow by evaluating the signs of $U$ and $V$ on the horizon plane, the left wedge ($U<0<V$) and the future and past wedges.

(Limits along $O$ and along the reflector family.) By [L9], ${\rho }_0{e}^{-\theta }\to 0$ and ${\rho }_0{e}^{\theta }\to +\mathrm{\infty }$ as $\theta \to +\mathrm{\infty }$, while ${\rho }_0{e}^{\theta }\to 0$ and ${\rho }_0{e}^{-\theta }\to +\mathrm{\infty }$ as $\theta \to -\mathrm{\infty }$; hence along $O$, $V=-{\rho }_0{e}^{-\theta }\to 0^{-}$ and $U\to +\mathrm{\infty }$ as $\theta \to +\mathrm{\infty }$, and $U\to 0^{+}$ with $V\to -\mathrm{\infty }$ as $\theta \to -\mathrm{\infty }$.

(Clause (iii): the radar geometry.) Suppose first ${\rho }_m>{\rho }_0$. The constant-$V$ future null line from $e_{-}=({\theta }_e,{\rho }_0)$ has $V=-{\rho }_0{e}^{-{\theta }_e}$, and its intersection with $\rho ={\rho }_m$ is at the unique ${\theta }_m$ with $-{\rho }_m{e}^{-{\theta }_m}=-{\rho }_0{e}^{-{\theta }_e}$, i.e. ${\theta }_m={\theta }_e+\log ({\rho }_m\mathrm{/}{\rho }_0)={\theta }_e+\mathrm{\mid }\log ({\rho }_m\mathrm{/}{\rho }_0)\mathrm{\mid }$ since ${\rho }_m>{\rho }_0$; the return leg must be the constant-$U$ line through $q_m=({\theta }_m,{\rho }_m)$ (the constant-$V$ line leaves $\rho$ increasing), and it meets $\rho ={\rho }_0$ at the unique $\theta (e_{+})$ with ${\rho }_0{e}^{\theta (e_{+})}={\rho }_m{e}^{{\theta }_m}$, i.e. $\theta (e_{+})={\theta }_m+\log ({\rho }_m\mathrm{/}{\rho }_0)={\theta }_e+2\log ({\rho }_m\mathrm{/}{\rho }_0)$. If instead $0<{\rho }_m<{\rho }_0$, the same computation with the roles of $U$ and $V$ exchanged, using the two formulas $U=\rho e^{\theta }$ and $V=-\rho e^{-\theta }$, gives ${\theta }_m={\theta }_e+\log ({\rho }_0\mathrm{/}{\rho }_m)$ and $\theta (e_{+})={\theta }_m+\log ({\rho }_0\mathrm{/}{\rho }_m)$, i.e. in both cases ${\theta }_m={\theta }_e+\mathrm{\mid }\log ({\rho }_m\mathrm{/}{\rho }_0)\mathrm{\mid }$ and $\theta (e_{+})={\theta }_e+2\mathrm{\mid }\log ({\rho }_m\mathrm{/}{\rho }_0)\mathrm{\mid }$ with midpoint ${\theta }_m$. Uniqueness of $q_m$ and $e_{+}$ holds because each constant-$U$ or constant-$V$ line meets each of the lines $\rho =\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}>0$ exactly once, and $e_{-}\le q_m$ and $q_m\le e_{+}$ are future null by step 1.3.

(Clause (i).) Step 1.1 gives the bijection, its domain, the boundary description and the identification of the horizon planes with $V=0$ and $U=0$ on the wedge closure; step 1.3 identifies the two families of future null lines in the plane through $O$ with the constant-$V$ and constant-$U$ lines and their differential equations; step 2.2 gives the asymptotic limits of $U$ and $V$ along $O$.

(Clause (ii).) The two criteria are step 2.1; the stated consequences for the horizon planes, the left wedge and the future and past wedges follow by computing $U$ and $V$ there, using that $x^1<-\mathrm{\mid }{x}^0\mathrm{\mid }$ gives $U<0$ and $V>0$, $x^0>\mathrm{\mid }{x}^1\mathrm{\mid }$ gives $U>0,V>0$, and $x^0<-\mathrm{\mid }{x}^1\mathrm{\mid }$ gives $U<0,V<0$; the horizon interpretations as boundary of the causal past and future of $O$ follow from the two criteria together with the limits of step 2.2, and the invariance in ${\rho }_1,y_1,z_1$ is the last sentence of step 2.1.

(Clause (iii): the radar assignments and the distance comparison.) By [L6] and step 2.3, the radar time is ${\tau }_{\mathrm{r}\mathrm{a}\mathrm{d}}(q_m)=(\tau (e_{-})+\tau (e_{+}))\mathrm{/}2=({\rho }_0\mathrm{/}c){\theta }_m+{\tau }_0$, since $\theta =c\tau \mathrm{/}{\rho }_0+\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}$ along $O$ by step 1.2, and the radar distance is ${\rho }_{\mathrm{r}\mathrm{a}\mathrm{d}}(q_m)=c(\tau (e_{+})-\tau (e_{-}))\mathrm{/}2={\rho }_0\mathrm{\mid }\log ({\rho }_m\mathrm{/}{\rho }_0)\mathrm{\mid }$; the radar-time level sets in the coaxial family are the sets ${\theta }_m=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}$, which are precisely the restrictions of the Rindler rest slices to $y=y_0$, $z=z_0$, $\rho \mathrm{\ne }{\rho }_0$; indeed every such $q_m$ has emission parameter ${\theta }_e={\theta }_m-\mathrm{\mid }\log ({\rho }_m\mathrm{/}{\rho }_0)\mathrm{\mid }$, and [L3] identifies the unrestricted $\theta$-level slices as rest slices. The rest distance in the slice $\theta ={\theta }_m$ is $\mathrm{\mid }{\rho }_m-{\rho }_0\mathrm{\mid }$ by [L7] and the induced metric $d{\rho }^2+d{y}^2+d{z}^2$ of [L3]. With $r={\rho }_m\mathrm{/}{\rho }_0\mathrm{\ne }1$, the radar distance is ${\rho }_0\mathrm{\mid }\log r\mathrm{\mid }$ and the rest distance is ${\rho }_0\mathrm{\mid }r-1\mathrm{\mid }$, so by step 1.4 the radar distance is strictly smaller than the rest distance for $r>1$ and strictly larger for $0<r<1$, with equality only in the degenerate case $r=1$ in which both are zero; in particular at ${\rho }_m=2{\rho }_0$, ${\rho }_0\log 2<{\rho }_0$.

(Clause (iv): the restrictions, and the example.) Let $q$ admit a radar triple $e_{-},q,e_{+}$ from $O$; by the ideal-light and signal assumptions of [L1], the emission and reflection legs give $e_{-}\le q$ and $q\le e_{+}$ with $e_{-},e_{+}$ on $O$, so by step 2.1's criteria $U(q)>0$ and $V(q)<0$, i.e. $q\in W$; conversely every event with $V\ge 0$ fails $q\le p$ for all $p\in O$ and every event with $U\le 0$ fails $p\le q$, so those events receive no radar assignment. For the example $q=x^{-1}(2{\rho }_0,0,y_0,z_0)$: $U(q)=V(q)=2{\rho }_0$, and its displacement from the event of $O$ at $\theta =0$ is $(2{\rho }_0,-{\rho }_0,0,0)$, with $\eta =-4{\rho }_0^2+{\rho }_0^2=-3{\rho }_0^2<0$ and positive time component, hence future timelike, so $q$ is causally after that event by [L5]; the displacement from the event of $O$ at $\theta =\log 2$, namely $({\rho }_0\sinh \log 2,{\rho }_0\cosh \log 2,0,0)=(\frac{3}{4}{\rho }_0,\frac{5}{4}{\rho }_0,0,0)$ by [L9], is $(5{\rho }_0\mathrm{/}4,-5{\rho }_0\mathrm{/}4,0,0)$, future null, so $q$ lies on a future null ray from $O$; but $V(q)=2{\rho }_0\ge 0$, so by the criterion $q$ cannot reach any event of $O$ and receives no radar assignment, showing that the protocol does not extend even to events causally after $O$.

(Clause (iii): the limiting cases.) As ${\rho }_m\to 0^{+}$ we have $\mathrm{\mid }\log ({\rho }_m\mathrm{/}{\rho }_0)\mathrm{\mid }\to +\mathrm{\infty }$ by strict increase and unboundedness of $\log$ (equivalently $\log ({\rho }_0\mathrm{/}{\rho }_m)\to +\mathrm{\infty }$), so the round-trip proper time $(2{\rho }_0\mathrm{/}c)\mathrm{\mid }\log ({\rho }_m\mathrm{/}{\rho }_0)\mathrm{\mid }$ and the radar distance ${\rho }_0\mathrm{\mid }\log ({\rho }_m\mathrm{/}{\rho }_0)\mathrm{\mid }$ grow without bound, while the reflector's proper acceleration $c^2\mathrm{/}{\rho }_m\to +\mathrm{\infty }$ by [L4]; as ${\rho }_m\to +\mathrm{\infty }$ the same divergence of $\mathrm{\mid }\log ({\rho }_m\mathrm{/}{\rho }_0)\mathrm{\mid }$ holds while $c^2\mathrm{/}{\rho }_m\to 0^{+}$.

(Clause (v): scope.) Every clause was computed from the adopted model and the cited mathematical suppliers; the several limiting cases (equal radii excluded, the degenerate $r=1$ value, the two null half-planes versus the full horizon planes, the two limiting reflector radii) were treated explicitly. No apparatus, measurement, statistical or empirical input enters, no choice principle was used, and the statements are conditional model computations.