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1005:= asinh. Here the first equation tells us how proper-velocity w changes with elapsed traveler-time τ and map-time Δt, while the second tells us how map-position x changes with traveler-time τ as well as with state-of-motion (the work-energy equation) since work is mαΔx and change-in kinetic energy is ΔK = mcΔγ (given that K = (γ-1)mc).
202:
a relativistic world arises because proper velocity change is proportional to acceleration times the change in map rather than traveler time. As those two begin to differ, the advantage (over the dotted straight line above) emerges. Of course, building a spaceship that can do this is another story.
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This plot illustrates how a spaceship capable of 1 g acceleration for 100 years can power a round trip to most anywhere in the visible universe, and back in a lifetime or less. Additional time will have elapsed on earth by the time that you return. This advantage of constant proper acceleration in
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For equations that work at any speed, we begin by treating the "proper" time τ on the clocks of a traveler as a local-variable, whose value we'd like to figure out relative to the local value of the traveler's position x and time t on the yardsticks and synchronized clocks of a reference map-frame.
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The bronze thrust profile inset above for a 5 traveler-year one-way trip to
Proxima Centauri shows how the thrust profile (proportional to cross-sectional area) would vary during the trip. The orange mesh line around the profile corresponds to the thruster turnaround point in the trip. Even longer
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The dotted line shows how far the same round trip would take a traveler if
Newtonian physics applied instead. As you can see, the lightspeed limit improves travel prospects for the accelerated traveler even though it makes things worse for the couch-potato who waits at home for the traveler to
567:
The space-time
Pythagorean theorem or "metric-equation" for flat space-time, namely (cdτ) = (cdt) - (dx) with "lightspeed" constant c, requires that we define "proper" (in addition to "coordinate") values for the velocity and acceleration, as well as for the time, experienced by our traveler.
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Furthermore, it should be noted that, when discussing large distances, the expansion of the universe is important. For example, the edge of the universe (~46 billion light years) can not be reached no matter how fast you accelerate because the cosmic horizon is 16 billion light years.
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535:. Here the first equation tells us how coordinate-velocity v changes with elapsed map-time t, while the second tells us how map-position x changes with map-time t as well as with state-of-motion (the work-energy equation) since work is mαΔx and kinetic energy is K = ½mv.
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At low speeds (w<<c) of course, map & traveler clock times go at the same rate i.e. dt ≈ dτ, the velocity-parameters (coordinate, proper & angle) are essentially the same i.e. v ≈ w ≈ cη, coordinate and proper acceleration are about equal i.e.
1824:{{Information |Description=This plot illustrates how a spaceship capable of 1-gee acceleration for 100 years can power a round trip to most anywhere in the visible universe in a lifetime or less. Of course, additional time may have elapsed on earth by th
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to derive the equations of constant acceleration. For equations that work at any speed, we'll also give you some practice treating time as a local instead of as a global variable i.e. as a value connected to readings on a specific clock.
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632:γ ≡ dt/dτ = cosh = Sqrt = 1/Sqrt, where coordinate-velocity v ≡ w/γ = c tanh. For incremental changes when proper-acceleration is constant and all motion is along that direction we can also write α =Δw/Δt = cΔη/Δτ = cΔγ/Δx = γ
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as the independent variable so as to avoid thinking of time as a global variable. Note that unlike coordinate-velocity v≡dx/dt, proper-velocity w≡dx/dτ always equals momentum per unit mass and has no upper limit. Can
1220:– You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.
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586:α felt by a traveler equals the "length-contracted" proper-velocity derivative c dη/dτ i.e. constant c times the traveler-time τ derivative of rapidity η. Holding α fixed thus allows one to derive
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of yardsticks and synchronized clocks. If we use those synchronized clocks to define simulaneity between separated events, the above also tells us about
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trips would exponentially decrease traveler time for the trip, but only at the price of a comparably-exponential increase in fuel mass on launch.
647:. Hence these equations are spectacular for exploring constant-acceleration round-trips between stars, as illustrated in the figure at right.
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The figure at right illustrates how one might apply the equations of constant proper-acceleration to a specific (if fanciful) roundtrip.
1121:; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled
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Thus the equations, analogous to the
Newtonian ones, for unidirectional constant proper-acceleration at any speed might be written:
982:{\displaystyle x=x_{o}+{\tfrac {c^{2}}{\alpha }}\left(\cosh \left-\cosh \left\right)=x_{o}+{\frac {c^{2}}{\alpha }}\Delta \gamma }
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FullSimplify[ DSolve[ { c Sinh] == x', alpha == c eta', x == xo, eta == etao }, {x, eta}, {tau} ] ]
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In terms of increments instead of differentials for constant unidirectional acceleration, we can therefore also write:
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figure how map-position x and proper-velocity w = c sinh depend on traveler-time τ, given this information?
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return. Relativity also can significantly reduce the (still probably exhorbitant) fuel-costs for such trips.
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The following pages on the
English Knowledge (XXG) use this file (pages on other projects are not listed):
1226:– If you remix, transform, or build upon the material, you must distribute your contributions under the
314:(which is not the acceleration felt by our traveler at high speeds) allows one to derive the v<<c
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as initial coordinate-velocity to eliminate the two constants of integration. Mathematica's output is:
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805:{\displaystyle w=c\sinh \left=w_{o}+\alpha \int _{0}^{\tau }\gamma d\tau '=w_{o}+\alpha \Delta t\,}
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FullSimplify[ DSolve[ { v == x', a == v', x == xo, v == vo }, {x, v}, t ] ]
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familiar from intro-physics texts for coordinate-velocity v and map-position x. Can you do it?
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Permission is granted to copy, distribute and/or modify this document under the terms of the
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622:{ { eta -> etao + (alpha tau)/c, x -> (alpha xo + c^2 (-Cosh + Cosh))/alpha } }
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Thus the equations for constant coordinate-acceleration in one direction might be written:
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As above we specify two initial (τ=0) conditions, in this case for initial map-position x
516:{\displaystyle x=x_{o}+v_{o}t+{\frac {1}{2}}at^{2}=x_{o}+{\frac {v^{2}-v_{o}^{2}}{2a}}}
578:τ) is just w ≡ dx/dτ ≡ c sinh, where η is referred to as hyperbolic velocity-angle or
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I, the copyright holder of this work, hereby publish it under the following licenses:
73:
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Added a short-trip thrust profile to suggest vehicular structures required as well.
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traveler motion from the perspective of stationary observers anywhere on the map
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For a change of pace from most texts, lets discuss the assumptions needed for
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Round-trip times and distances for space trips of constant acceleration.
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Traveler-speed on the map can be expressed in several ways, including:
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Here we've added the intial (t=0) boundary-conditions by defining x
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Edwin F. Taylor & John
Archibald Wheeler (1966 1st ed. only)
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assertions are local to the traveler's position in the map-frame
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P. Fraundorf (2012) "A traveler-centered intro to kinematics",
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constant proper-acceleration equations that work at any speed
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Creative
Commons Attribution-ShareAlike 4.0 International
1068:, Chapter 1 Exercise 51 page 97-98: "Clock paradox III" (
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The following is what
Mathematica needs to pull it off:
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The following is what
Mathematica needs to pull it off:
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Creative
Commons Attribution-ShareAlike 3.0 Unported
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GNU Free Documentation License, version 1.2 or later
1117:, Version 1.2 or any later version published by the
338:{{v -> a t + vo, x -> (a t^2)/2 + t vo + xo}}
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low-speed constant coordinate-acceleration equations
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Commons is a freely licensed media file repository.
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Creative Commons Attribution-ShareAlike 2.0 Generic
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Creative Commons Attribution-ShareAlike 1.0 Generic
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Creative Commons Attribution-ShareAlike 2.5 Generic
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and initial rapidity or hyperbolic velocity-angle η
593:In the above discussion we are using proper-time τ
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626:Note that we also get these bonus relationships:
51:(548 × 573 pixels, file size: 40 KB, MIME type:
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1902:Патување низ вселената со постојано забрзување
1237:https://creativecommons.org/licenses/by-sa/4.0
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1243:Creative Commons Attribution-Share Alike 4.0
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266:Deriving unidirectional constant acceleration
137:Deriving unidirectional constant acceleration
8:
1203:– to copy, distribute and transmit the work
590:for map-position x and proper-velocity w.
1932:Космічні подорожі з постійним прискоренням
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1255:You may select the license of your choice.
1864:The following other wikis use this file:
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1135:GNU Free Documentation License
1124:GNU Free Documentation License
1114:GNU Free Documentation License
1056:(W.H. Freeman, San Francisco)
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562:proper-acceleration in (3+1)D.
554:proper-acceleration in (1+1)D.
246:A hi-speed round-trip puzzler.
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1515:original creation by uploader
331:as initial map-position and v
302:v as dx/dt where x and t are
279:Low-speed (Newtonian) version
147:Low-speed (Newtonian) version
1888:Usage on en.wikiversity.org
1286:Items portrayed in this file
381:{\displaystyle v=v_{o}+at\,}
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574:w (map-distance x per unit
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1943:Usage on www.wikidata.org
1928:Usage on uk.wikipedia.org
1918:Usage on ru.wikipedia.org
1908:Usage on pt.wikipedia.org
1898:Usage on mk.wikipedia.org
1878:Usage on cs.wikipedia.org
1868:Usage on ar.wikipedia.org
1228:same or compatible license
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582:. The frame-invariant
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76:. Information from its
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1598:15:21, 10 January 2010
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542:= Δv/Δt = ½Δ(v)/Δx.
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1844:Proper acceleration
1839:Interstellar travel
1209:– to adapt the work
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300:coordinate-velocity
1937:Власне прискорення
1912:Aceleração própria
993:where again Δf ≡ f
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197:Roundtriptimes.png
47:Roundtriptimes.png
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1174:4.0 International
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1054:Spacetime Physics
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1178:3.0 Unported
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1074:archive copy
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1015:P. Fraundorf
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230:P. Fraundorf
87:You can help
77:
45:
22:File history
1324:copyrighted
1224:share alike
1218:attribution
1190:1.0 Generic
1186:2.0 Generic
1182:2.5 Generic
576:proper-time
238:Added notes
195:Description
127:Added notes
1830:File usage
1819:Unitsphere
1811:377 × 389
1790:Unitsphere
1782:371 × 388
1761:Unitsphere
1753:377 × 391
1732:Unitsphere
1724:489 × 498
1703:Unitsphere
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1645:Unitsphere
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1615:Unitsphere
1607:548 × 573
1583:Dimensions
1535:media type
272:a computer
27:File usage
1580:Thumbnail
1577:Date/Time
1550:image/png
1465:inception
1091:Licensing
1027:Footnotes
179:Licensing
169:Footnotes
53:image/png
1947:Q7572639
1267:Captions
1207:to remix
1201:to share
1192:license.
580:rapidity
308:map-time
221:Own work
103:Contents
1813:(10 KB)
1784:(10 KB)
1755:(11 KB)
1726:(14 KB)
1697:(16 KB)
1668:(19 KB)
1639:(21 KB)
1609:(40 KB)
1594:current
1589:Comment
1291:depicts
1271:English
1076:at the
1001:, and η
999:initial
533:initial
187:Summary
117:Summary
49:
1064:
227:Author
217:Source
995:final
529:final
1586:User
1249:true
1246:true
1188:and
1141:true
1138:true
1132:GFDL
1060:ISBN
1020:talk
912:cosh
865:cosh
670:sinh
306:and
207:Date
17:File
1070:pdf
997:- f
636:.
600:you
531:- f
154:3.2
144:3.1
1184:,
1180:,
1176:,
1085:).
977:γ
974:Δ
969:α
923:η
915:
909:−
895:η
882:τ
879:α
868:
854:α
795:Δ
792:α
769:τ
755:τ
748:γ
743:τ
734:∫
730:α
700:η
687:τ
684:α
673:
485:−
1127:.
1048:↑
1042:.
1034:↑
1017:(
1011:a
1003:o
989:,
964:2
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919:[
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899:o
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872:[
861:(
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825:=
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355:=
352:v
333:o
329:o
312:a
296:a
176:5
166:4
134:3
124:2
114:1
89:.
55:)
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