File:RoundTripToVega.gif
RoundTripToVega.gif (425 × 165 pixels, file size: 520 KB, MIME type: image/gif, looped, 51 frames, 51 s)
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Summary
DescriptionRoundTripToVega.gif |
English: The local nature of time: Four clocks on a 1-gee constant proper-acceleration round-trip shuttle to and from Vega 25 lightyears away, as seen from the ship's point of view. The one-way trip takes about 6.6 traveler years and about 27 map years, with turn-around points for the ship's proper-acceleration/deceleration halfway between earth and Vega. Although the earth-clock (whose concurrent value on the left clock-face is obtained unambiguously from a radar-time calculation) and the local map-clocks (right clock-face) are synchronized from the earth's point of view, they are only synchronized from the ship's point of view when our ship is docked, either at home (blue dot) or at Vega (green dot). Inspired by clocks on the spaceship wall in a sci-fi novel[1], this animation also illustrates the relativity of simultaneity in a less abstract way than is usually done with help from constant-speed (Lorentz-transform) models that allow neither acceleration nor curved-spacetime. In this case the extended-simultaneity model used for the two "distant-location clocks" on each end is the much more robust radar-time model discussed by Dolby and Gull[2]. The faded red clock hands on the Sol and Vega clocks show "tangent free-float-frame" time rates of change, which require a moving frame of synchronized clocks embedded in flat spacetime. The green dashed lines correspond to past events on the corresponding "far-away" clock which we've not yet detected, while the green dotted lines correspond to future events there on which our subsequent actions can have no effect. |
Date | |
Source | Own work |
Author | P. Fraundorf |
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 xo=c2/α, to=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 {2xc, 2tc}, 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 τc:
- .
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:
- .
The map-trajectory for galactic-coordinates {x,t}, parameterized using traveler time τ and the quarter round-trip turn-around time τc, looks something like:
- ,
and
- .
Here tc ≡ (c/α)sinh[ατc/c] and xc ≡ (c2/α)(cosh[ατc/c]-1) are galactic map-coordinates for the first turn-around event at traveler-clock time τc. In terms of the destination distance xd = 2xc on the galactic map, this second equation suggests that the total roundtrip time on traveler-clocks is Δτround ≡ 4τc = 4(c/α)acosh[1+(α/c2)xd/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:
- , and
- .
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:
- , and
- .
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]) = τo where τ+[t,x] solves u=uB[τ+] and τ-[t,x] solves v=vB[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:
- ,
and for the radar-distance contours in the same zones:
- .
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 (0th branch) of the Lambert W or product log function defined implicitly by z = WeW, namely
The remaining eight zones, namely 04, 12, 13, 21, 23, 31, 32, and 40, can be written out explicitly.
Footnotes
- ↑ Mary Doria Russell (2008) The Sparrow (Random House, NY).
- ↑ 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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Date/Time | Thumbnail | Dimensions | User | Comment | |
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current | 20:20, 19 January 2016 | 425 × 165 (520 KB) | Unitsphere | Add thrust-reversal dots on the ship clock, and correct the tangent-fff arrow dynamics. | |
15:25, 18 January 2016 | 425 × 165 (519 KB) | Unitsphere | Added "causality windows" to the Sol and Vega clocks. | ||
12:21, 28 August 2014 | 425 × 165 (477 KB) | Unitsphere | Add a fourth clock to show symmetry in origin and destination "far-away" times. | ||
00:28, 17 May 2014 | 360 × 183 (493 KB) | Unitsphere | User created page with UploadWizard |
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