Summary

Added notes

This animation also highlights an unavoidable property of far-away events in space-time, since the direction of your world-line matters: When on your clock a far-away event happens is not set in stone until such time as light-rays from that event have the chance to reach you. As a result the readings on the far-away clocks above (on either end of the animation) depend on the assumption that the voyage will continue as planned.

Equation appendix

The figure was drawn using Mathematica. At some point we may add code here to construct a roundtrip to any destination that you like. First, however, some notes on the relationships used are provided here.

the trajectory

Let's start by imagining that our traveler starts from rest at x_o=c²/α, t_o=0, and the trip is divided into quarters. The first quarter involves acceleration rightward, the second two quarters involve acceleration leftward before and after a destination event at {2x_c, 2t_c}, while the fourth involves acceleration rightward again to bring the traveler to rest back home.

First take a look the velocity-measure most simply connected to acceleration, namely hyperbolic velocity angle or rapidity η, as a function of traveler-time τ and the quarter round-trip turn-around time τ_c:

${\displaystyle \eta [\tau ]={\begin{cases}0&{\mbox{ if }}\tau \leq 0\\{\frac {\alpha \tau }{c}}&{\mbox{ if }}0<\tau \leq \tau _{c}\\{\frac {\alpha (\tau -2\tau _{c})}{c}}&{\mbox{ if }}\tau _{c}<\tau \leq 3\tau _{c}\\{\frac {\alpha (\tau -4\tau _{c})}{c}}&{\mbox{ if }}3\tau _{c}<\tau \leq 4\tau _{c}\\0&{\mbox{ if }}\tau _{c}<\tau \end{cases}}}$.

This is useful because rapidity in turn relates simply to other speed measures in (1+1)D, including proper-velocity w ≡ dx/dτ = c sinh[η], coordinate-velocity v ≡ dx/dt = c tanh[η], and Lorentz-factor γ ≡ dt/dτ = cosh[η]. Hence we can integrate them to determine map-time elapsed and distance traveled. In perhaps simplest form, the resulting integrals for each constant proper-acceleration segment may be written as:

${\displaystyle \alpha ={\frac {\Delta w}{\Delta t}}=c{\frac {\Delta \eta }{\Delta \tau }}=c^{2}{\frac {\Delta \gamma }{\Delta x}}{\text{ }}{\stackrel {v\ll c}{\rightarrow }}{\text{ }}a={\frac {\Delta v}{\Delta t}}={\frac {1}{2}}{\frac {\Delta (v^{2})}{\Delta x}}}$.

The map-trajectory for galactic-coordinates {x,t}, parameterized using traveler time τ and the quarter round-trip turn-around time τ_c, looks something like:

${\displaystyle ct[\tau ]={\begin{cases}c\tau &{\mbox{ if }}\tau \leq 0\\{\frac {c^{2}}{\alpha }}\sinh[\eta [\tau ]]&{\mbox{ if }}0<\tau \leq \tau _{c}\\2ct_{c}+{\frac {c^{2}}{\alpha }}\sinh[\eta [\tau ]]&{\mbox{ if }}\tau _{c}<\tau \leq 3\tau _{c}\\4ct_{c}+{\frac {c^{2}}{\alpha }}\sinh[\eta [\tau ]]&{\mbox{ if }}3\tau _{c}<\tau \leq 4\tau _{c}\\4c(t_{c}-\tau _{c})+c\tau &{\mbox{ if }}\tau _{c}<\tau \end{cases}}}$,

and

${\displaystyle x[\tau ]={\begin{cases}{\frac {c^{2}}{\alpha }}&{\mbox{ if }}\tau \leq 0\\{\frac {c^{2}}{\alpha }}\cosh[\eta [\tau ]]&{\mbox{ if }}0<\tau \leq \tau _{c}\\2x_{c}+{\frac {c^{2}}{\alpha }}\cosh[\eta [\tau ]]&{\mbox{ if }}\tau _{c}<\tau \leq 3\tau _{c}\\{\frac {c^{2}}{\alpha }}\cosh[\eta [\tau ]]&{\mbox{ if }}3\tau _{c}<\tau \leq 4\tau _{c}\\{\frac {c^{2}}{\alpha }}&{\mbox{ if }}\tau _{c}<\tau \end{cases}}}$.

Here t_c ≡ (c/α)sinh[ατ_c/c] and x_c ≡ (c²/α)(cosh[ατ_c/c]-1) are galactic map-coordinates for the first turn-around event at traveler-clock time τ_c. In terms of the destination distance x_d = 2x_c on the galactic map, this second equation suggests that the total roundtrip time on traveler-clocks is Δτ_round ≡ 4τ_c = 4(c/α)acosh[1+(α/c²)x_d/2]. Does that look right?

causality-gap

For the A and B destinations at the left and right ends (respectively) of the shuttle's oscillation, the causality limits look something like:

${\displaystyle ct_{\text{A}}[\tau ]=ct[\tau ]\pm (x[\tau ]-x[0])}$, and

${\displaystyle ct_{\text{B}}[\tau ]=ct[\tau ]\pm (x[2\tau _{c}]-x[\tau ])}$.

Of course centered in this causality-gap is the local map-time t[τ].

tangent-fff equations

The tangent free-float-frame time of events for a star along our trajectory at the A and B positions may look something like:

${\displaystyle ct_{\text{A}}[\tau ]=ct[\tau ]-\tanh[\eta [\tau ]](x[\tau ]-x[0])\,}$, and

${\displaystyle ct_{\text{B}}[\tau ]=ct[\tau ]-\tanh[\eta [\tau ]](x[\tau ]-x[2\tau _{c}])\,}$.

This equation arises because -1 ≤ tanh[η] ≤ +1 is dt/dx for fixed time-isocontours associated with an extended frame of yardsticks and synchronized clocks which is moving relative to the fixed axes of an x-ct plot in flat spacetime.

radar-separation equations

We discuss these with c=1 and α=1 to minimize sprawl. In all for a constant proper-acceleration roundtrip there are four function changes, 5 intervals, and thus 5×5=25 zones involved. The plan for each of these 25 zones is to solve radar time τ[t,x] ≡ ½(τ⁺[t,x]+τ^-[t,x]) = τ_o where τ⁺[t,x] solves u=u_B[τ⁺] and τ^-[t,x] solves v=v_B[τ^-]. These in turn have been used (e.g. here) to plot radar isochrons and radar-distance grid lines for proper time/distance intervals of 0.2c²/α for all 25 zones is an x-ct diagram's field of view.

Using the linked example figure, for example working our way up from the magenta-shaded 00 zone at the bottom center of the traveler world line, we get for the radar isochrons:

${\displaystyle t_{\tau _{o}}[x]={\begin{cases}\tau _{o}&{\mbox{ in zone 00 (magenta)}}\\x\tanh[\tau _{o}]&{\mbox{ in zone 11 (blue)}}\\2\sinh[1]-(x-2\cosh[1])\tanh[\tau _{o}-2]&{\mbox{ in zone 22 (cyan)}}\\4\sinh[1]+x\tanh[\tau _{o}-4]&{\mbox{ in zone 33 (green)}}\\4\sinh[1]+\left(\tau _{o}-4\right)&{\mbox{ in zone 44 (yellow)}}\end{cases}}}$,

and for the radar-distance contours in the same zones:

${\displaystyle x_{\rho _{o}}[t]={\begin{cases}1+\rho _{o}&{\mbox{ in zone 00 (magenta)}}\\{\sqrt {e^{2\rho _{o}}+t^{2}}}&{\mbox{ in zone 11 (blue)}}\\2\cosh[1]-{\sqrt {e^{-2(\rho _{o}+2)}+(t-2\sinh[1])^{2}}}&{\mbox{ in zone 22 (cyan)}}\\{\sqrt {e^{2\rho _{o}}+\left(t-4\sinh[1]\right)^{2}}}&{\mbox{ in zone 33 (green)}}\\1+\rho _{o}&{\mbox{ in zone 44 (yellow)}}\end{cases}}}$.

To create the plot above, similar functions are needed for all 25 hk zones, where h={0,1,2,3,4} and k={0,1,2,3,4}.

The twelve zones 01, 02, 03, 10, 14, 20, 24, 30, 34, 41, 42 and 43 may require the principal value (0^th branch) of the Lambert W or product log function defined implicitly by z = We^W, namely

${\displaystyle W_{0}[x]=\sum _{n=1}^{\infty }{\frac {(-n)^{n-1}}{n!}}\ x^{n}=x-x^{2}+{\frac {3}{2}}x^{3}-{\frac {8}{3}}x^{4}+{\frac {125}{24}}x^{5}-\cdots }$

The remaining eight zones, namely 04, 12, 13, 21, 23, 31, 32, and 40, can be written out explicitly.

Footnotes

1. ↑ Mary Doria Russell (2008) The Sparrow (Random House, NY).
2. ↑ Carl E. Dolby and Stephen F. Gull (2001) "On radar time and the twin paradox", Amer. J. Phys. 69 (12) 1257-1261 abstract.

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