Class4:AcceleratedMotion

InthisclasswewillinvestigatehowacceleratedmotioncanbestudiedwithinSpecialRelativity;

inthenextclasswe’lldiscusshowthisisequivalenttomotioninagravitationalfield

Class4:AcceleratedMotion

Attheendofthissessionyoushouldbeableto…

• … determinetheworldlineofanacceleratedobjectinspecialrelativity

• … representthismotiononaspace-timediagram

• … describehowtheclockratevarieswithinalongrocketshipundergoingacceleratedmotion

AnalyzingaccelerationinSR

• Oneofthecentre-piecesofclassicalphysicsisNewton’s2nd Law:�⃗� = 𝑚�⃗�

• WhatdoesaccelerationlooklikeinSpecialRelativity?

AnalyzingaccelerationinSR

• Whatdoesuniformaccelerationlooklikeinframe𝑆?

• Initialguess:'(')= constant→ 𝑣 = 𝑣, + 𝑎𝑡

• Doesthismakesense??

𝑥

𝑦

𝑣𝑡

Frame𝑆

AnalyzingaccelerationinSR

• No– velocitywilleventuallyexceed𝑐!

• No– “dv”involvessubtractingvelocitiesatnearbypoints,butweknowvelocitiesaddinstrangewaysinrelativity!

• No– “dt”involvestimeintervals,butthesevarybetweenframes

• '(')= constantisunphysicalandcan’tholdinallframes

AnalyzingaccelerationinSR

• Let’sdefineuniformaccelerationinadifferentway–as“feelingconstanttotheobjectbeingaccelerated”

• (Theacceleratedobservercanalwaysmeasurethisratebygentlydroppingarockoutofthewindow)

https://www.decodedscience.org/jet-airplanes-take-off-speeds-and-aircraft-performance/5453

AnalyzingaccelerationinSR

• Howcanweanalyze thisintermsofinertialframes?

• Setupalargenumberofinertialframeswitharangeofrelativevelocities,suchthattheacceleratingobjectistemporarilyatrestineach– analyze themotionbypatchingalltheframestogether

𝑆′ 𝑆′′𝑆

𝑣 𝑣 + ∆𝑣

𝑥

𝑦

AnalyzingaccelerationinSR

• Wedefineconstantproperaccelerationby

𝛼 = '(5

'6= constant

𝑑𝑣8 =momentaryincreaseinspeedfromrestin𝑆′(non-relativistic,sinceinitially𝑣8 = 0)

𝑑𝜏 = propertimeelapsedontherocket-shipclockinasmallinterval(= 𝑑𝑡8 in𝑆′)

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

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)

Acceleratingspace-timediagram

• Considerthisrocketsendingandreceivinglightsignals

• Justbyundergoinguniformacceleration,therockethascutitselfofffromlargesectionsofspace-time!

Therocketcanneverreceivesignalsfromhere

Therocketcanneversendsignalstohere

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

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