in this class we will investigate how accelerated motion ...cblake/class4_acceleratedmotion.pdf ·...
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Class4:AcceleratedMotion
InthisclasswewillinvestigatehowacceleratedmotioncanbestudiedwithinSpecialRelativity;
inthenextclasswe’lldiscusshowthisisequivalenttomotioninagravitationalfield
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Class4:AcceleratedMotion
Attheendofthissessionyoushouldbeableto…
• … determinetheworldlineofanacceleratedobjectinspecialrelativity
• … representthismotiononaspace-timediagram
• … describehowtheclockratevarieswithinalongrocketshipundergoingacceleratedmotion
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AnalyzingaccelerationinSR
• Oneofthecentre-piecesofclassicalphysicsisNewton’s2nd Law:�⃗� = 𝑚�⃗�
• WhatdoesaccelerationlooklikeinSpecialRelativity?
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AnalyzingaccelerationinSR
• Whatdoesuniformaccelerationlooklikeinframe𝑆?
• Initialguess:'(')= constant→ 𝑣 = 𝑣, + 𝑎𝑡
• Doesthismakesense??
𝑥
𝑦
𝑣𝑡
Frame𝑆
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AnalyzingaccelerationinSR
• No– velocitywilleventuallyexceed𝑐!
• No– “dv”involvessubtractingvelocitiesatnearbypoints,butweknowvelocitiesaddinstrangewaysinrelativity!
• No– “dt”involvestimeintervals,butthesevarybetweenframes
• '(')= constantisunphysicalandcan’tholdinallframes
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AnalyzingaccelerationinSR
• Let’sdefineuniformaccelerationinadifferentway–as“feelingconstanttotheobjectbeingaccelerated”
• (Theacceleratedobservercanalwaysmeasurethisratebygentlydroppingarockoutofthewindow)
https://www.decodedscience.org/jet-airplanes-take-off-speeds-and-aircraft-performance/5453
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AnalyzingaccelerationinSR
• Howcanweanalyze thisintermsofinertialframes?
• Setupalargenumberofinertialframeswitharangeofrelativevelocities,suchthattheacceleratingobjectistemporarilyatrestineach– analyze themotionbypatchingalltheframestogether
𝑆′ 𝑆′′𝑆
𝑣 𝑣 + ∆𝑣
𝑥
𝑦
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AnalyzingaccelerationinSR
• Wedefineconstantproperaccelerationby
𝛼 = '(5
'6= constant
𝑑𝑣8 =momentaryincreaseinspeedfromrestin𝑆′(non-relativistic,sinceinitially𝑣8 = 0)
𝑑𝜏 = propertimeelapsedontherocket-shipclockinasmallinterval(= 𝑑𝑡8 in𝑆′)
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Worldlineofacceleratingobject
• Wecandeterminethespace-timeco-ordinates𝑥(𝑡) oftherocket-shipintheoriginalframe𝑆,intermsofthepropertimeco-ordinate𝜏 intheship’srestframe
• Thevelocityin𝑆 is𝒗 = 𝒄 𝐭𝐚𝐧𝐡 𝜶𝝉𝒄
– atlargetimes𝑣 → 𝑐,buttheship’sspeedneverexceeds𝑐
• The𝑡 co-ordinatein𝑆 is𝒕 = 𝒄𝜶𝐬𝐢𝐧𝐡 𝜶𝝉
𝒄
• The𝑥 co-ordinatein𝑆 is𝒙 = 𝒄𝟐
𝜶𝐜𝐨𝐬𝐡 𝜶𝝉
𝒄
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Acceleratingspace-timediagram
• Uniformly-acceleratedobjectsaremovingalongahyperbolainaspace-timediagram,𝒙𝟐 − 𝒄𝒕 𝟐 = 𝒄𝟐/𝜶 𝟐
𝑐N
𝛼
• Theobjectremainsaconstantproperdistance fromtheorigin(soundsconfusing,butthisisadistanceinspace-time,notspace!)
• Theobject’sspace-timediagramisthesameineveryframe (wheneveritentersmyframeat𝑣 = 0,Ifinditsaccelerationtobe𝛼)
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Acceleratingspace-timediagram
• Considerthisrocketsendingandreceivinglightsignals
• Justbyundergoinguniformacceleration,therockethascutitselfofffromlargesectionsofspace-time!
Therocketcanneverreceivesignalsfromhere
Therocketcanneversendsignalstohere
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Clockratesinarocketship
• Nowconsidera“longrocket”,suchthatitsfrontandbackhavedifferentworldlines,butareseparatedbyafixedproperlength𝑑𝐿 atanyinstant(i.e.,𝑑𝑥 = 𝑑𝐿 when𝑑𝑡 = 0)
𝑑𝐿
𝑐N/𝛼
Back
Front
• Theproperaccelerationsofthefrontandbackmustbedifferent!
• PQ
RS− PQ
RT= 𝑑𝐿
• 𝛼U < 𝛼W
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Clockratesinarocketship
• Synchronizethefrontandbackobserverclocksat𝜏 = 0,whentherocketisinstantaneouslyatrestin𝑆
• Atsomelaterinstantwhentherocketisatrestin𝑆′,𝑣 =𝑐 tanh 𝛼𝜏/𝑐 forbothends.But𝛼’saredifferent!→ ∆6S
∆6T= RT
RS
• Morepropertimehaselapsedatthefrontoftherocket!Thefrontclocksarerunning“faster”!
• Thisphenomenonoccursin theacceleratingframeofthelongrocket– it’snotaninertialframe,sodoesnotcontradictSR