Class 2: Index Notationcblake/Class2_IndexNotation.pdf · Invariants in index notation •We have seen that a useful quantity in Special Relativity is the space-time interval @TU=−5U@6U+@3U+@RU+@SU

Class2:IndexNotation

Inthisclasswewillstartdevelopingindexnotation,thekeymathematicalbasisof

Relativity.Wewillalsolearnhowtodescribeflowsofenergyandmomentum.

Attheendofthissessionyoushould…

• … knowsomeexamplesof4-vectorsandtensors,objectswhosecomponentstransformbetweendifferentinertialreferenceframesusingtheLorentztransformations

• … bedevelopingsomefamiliaritywithindexnotation:thedifferencebetweenup- anddown-indices,howonemaybeconvertedintotheother,andsummationrules

• … understandhowthedensity/flowofenergy/momentummaybedescribedbythematter-energytensor𝑇"#

Class2:IndexNotation

Whatisa4-vector?

• A4-vector isanarrayof4physicalquantitieswhosevaluesindifferentinertialframesarerelatedbytheLorentztransformations

• Theprototypical4-vectorishence𝒙𝝁 = (𝒄𝒕, 𝒙, 𝒚, 𝒛)

• Notethattheindex𝜇 isasuperscript,andcantakefourvalues𝜇 = 0,1,2,3 ,oneforeachelement(e.g.,𝑥4 = 𝑐𝑡).Itdoesn’tmean“tothepowerof”.

• Wewillmeetsubscriptindicesshortly!

Indexnotation

• Whenwriting𝑥" todescribeanarrayofquantities,weareusing“indexnotation”– theconvenientmathematicalapproachforcalculationsinRelativity

• Forexample,bytheendoftheunitwewillbeencounteringequationslike…

• Aaargh!𝑅8"#9 = 𝜕"Γ8#

9 − 𝜕#Γ8"9 + Γ">9 Γ8#

> − Γ#>9 Γ8">

Indexnotation

• Goodnotationisalwaysveryimportant…

• Wewillspendtimepractisingusingindexnotation

https://archive.org/details/methodoffluxions00newt

Producing4-vectors

• 4-vectorsareuseful,becauseweknowhowtheircomponentstransformbetweeninertialframes

• SincetheLorentztransformationsarelinear,thesum/differenceof4-vectorsisalsoa4-vector

• Inparticular,thedifferenceinspace-timeco-ordinatesisa4-vector,𝒅𝒙𝝁 = (𝒄𝒅𝒕, 𝒅𝒙, 𝒅𝒚, 𝒅𝒛)

• New4-vectorsmayalsobeobtainedbymultiplying/dividingbyaninvariant,suchasthepropertimeinterval𝑑𝜏 ortherestmass𝑚4

4-velocityand4-momentum

• The4-vector𝑣" = DEF

DG= (𝛾𝑐, 𝛾𝑢E, 𝛾𝑢J, 𝛾𝑢K) isknownas

the4-velocityofaparticlewith3Dvelocity𝑢 = (𝑢E, 𝑢J, 𝑢K)

• The4-vector𝑝" = 𝑚4𝑣" =𝛾𝑚4𝑐, 𝛾𝑚4𝑢E, 𝛾𝑚4𝑢J, 𝛾𝑚4𝑢K = (M

N, 𝑝E, 𝑝J, 𝑝K) isknown

asthe4-momentumofaparticle

• Thisimmediatelytellsushowtheenergyandmomentumofaparticletransformbetweenframes:

𝐸P = 𝛾(𝐸 −𝑣𝑝E𝑐 )𝑝EP = 𝛾(𝑝E −

𝑣𝐸𝑐 )

“Down”4-vectors

• A“down”4-vector inSpecialRelativityisobtainedsimplybyreversingthesignofthefirstcomponentofa4-vector

• Forexample,adown4-vectoris𝑥" = (−𝑐𝑡, 𝑥, 𝑦, 𝑧)

• Thiswillbeaveryusefuldeviceincalculations,aswewillnowexplore!

Invariantsinindexnotation

• WehaveseenthatausefulquantityinSpecialRelativityisthespace-timeinterval𝑑𝑠U = −𝑐U𝑑𝑡U + 𝑑𝑥U + 𝑑𝑦U + 𝑑𝑧U

• Inindexnotation,thiscanbewrittenas𝑑𝑠U = 𝑑𝑥4𝑑𝑥4 +𝑑𝑥V𝑑𝑥V + 𝑑𝑥U𝑑𝑥U + 𝑑𝑥W𝑑𝑥W = ∑ 𝑑𝑥"𝑑𝑥"W

"Y4

• Inindexnotation,thisisabbreviatedas𝑑𝑠U = 𝑑𝑥"𝑑𝑥"

• Greekindiceswhichrepeatonthetopandbottomofanexpressionarealwayssummedfrom0to3

• Notethatwecanuseanylettertoindicateasummedindex–𝑑𝑥#𝑑𝑥# and𝑑𝑥>𝑑𝑥> areexactlythesame!

Lorentztransformations

• TheLorentztransformationsarewritteninindexnotationas𝒙′𝝁 = 𝑳𝝁𝝂𝒙𝝂

• Thisis“fourequationsinone”,since𝜇 = 0,1,2,3

• Why??Theindex𝜈 appearsonthetopandbottomoftheR.H.S.soissummed,leavingasingleup-index𝜇

Howcanwemakesenseof“𝐿"#𝑥#”??

“Ifindoubt,writeitout…”

Lorentztransformations

• Let’swriteitoutexplicitly:

• Itisanalogoustoamatrixmultiplication:

𝑥′" = 𝐿"#𝑥# = _𝐿"#𝑥#W

#Y4

𝜇 = 0 → 𝑥′4 = 𝐿44𝑥4 + 𝐿4V𝑥V + 𝐿4U𝑥U + 𝐿4W𝑥W

𝜇 = 1 → 𝑥′V = 𝐿V4𝑥4 + 𝐿VV𝑥V + 𝐿VU𝑥U + 𝐿VW𝑥Wetc.

𝑐𝑡′𝑥′𝑦′𝑧′

=

𝛾 −𝑣𝛾/𝑐−𝑣𝛾/𝑐 𝛾

0 00 0

0 00 0

1 00 1

𝑐𝑡𝑥𝑦𝑧

𝑥′" 𝑥#𝐿"#

Raisingandloweringanindex

• Thetransformationfroman“up”toa“down”4-vectorcanbewrittenas𝒙𝝁 = 𝜼𝝁𝝂𝒙𝝂.Again,thisis“fourequationsinone”.

• 𝜂"# isamatrix−1 00 1

0 00 0

0 00 0

1 00 1

thatreversesthe1st sign

• Thisisknownasloweringanindex (𝑥" → 𝑥")

• Similarly,toraiseanindexwecanwrite𝑥" = 𝜂"#𝑥#,where𝜂"# isthesamematrixasabove

• Thesamegoesfor2Dquantities,e.g.𝐿"# = 𝜂"8𝐿#8

Gradienttransformations

• Considerafunctionofspace-timeco-ordinates𝑓(𝑐𝑡, 𝑥, 𝑦, 𝑧),whichhasgradientsatapoint V

Nefeg, efeE, efeJ, efeK

.Whatareitsgradientswithrespecttoco-ordinatesin𝑆′, 𝑐𝑡P, 𝑥P, 𝑦P, 𝑧P ?

• Bythechainrule: efeEPF

= eEi

eEPFefeEi

• Since𝑥# = 𝐿"#𝑥′",wehaveeEi

eEPF= 𝐿"# (“ifindoubt,writeit

out”)so efeEPF

= 𝐿"#efeEi

• ThegradientofafunctiontransformsusingtheLorentztransformations:𝝏𝝁𝒇 =

𝟏𝒄𝝏𝒇𝝏𝒕, 𝝏𝒇𝝏𝒙, 𝝏𝒇𝝏𝒚, 𝝏𝒇𝝏𝒛

isadown4-vector

Matterandenergy

• TodevelopGeneralRelativityweneedtodescribehowmatter-energyisdistributed,andwhereit’sgoing

• Thisisachievedbyanobjectknownastheenergy-momentumtensor𝑻𝝁𝝂 (ateachpointofspace-time)

• Fornow,wecanthinkofa“tensor”asa2Dmatrix

• 𝑇"# hastwoindicesbecausemomentumhasadirection,butcanalsobetransportedindifferentdirections(e.g.,afluxof𝑥-momentuminthe𝑦-direction,if𝑥-movingparticlesaredriftingin𝑦)

Matterandenergy

• Itraisesanimmediatequestion:howdoesaquantitywith2indicestransformbetweendifferentinertialreferenceframesSandS’?

• TheLorentztransformationofa4-vector𝑥":

• TheLorentztransformation ofa2Dtensor𝑇"#:

𝑥′" = 𝐿"#𝑥#

𝑇′"# = 𝐿"9𝐿#8𝑇98

Energy-momentumtensor

• Drawaboxaroundapointinspace-timecontainingabunchofparticlescarryingenergyandmomentum

• Iftheboxcontains4-momentum𝑑𝑝" andismovingwithvelocityDE

i

Dg,wedefine𝑻𝝁𝝂 = 𝒅𝒑𝝁

𝒅𝑽𝒅𝒙𝝂

𝒅𝒕

• Notethat𝑇"# isa“Lorentz-transformingquantity”becauseitisaproductoftwo4-vectorsandaLorentzscalar(thespace-timevolumeelement𝑑𝑉𝑑𝑡)

• Whatarethedifferentcomponentsof𝑇"#?

Energydensityandflow

• 𝑇44 istheenergydensityatapoint

• 𝑇4q = 𝑇q4(𝑖 = 1,2,3) isthefluxofenergyinthe𝒊-direction orthe𝒊-momentumdensity(×𝒄)

• 𝑇qu = 𝑇uq is the flux of 𝒊-momentum in the 𝒋-directionor the flux of 𝒋-momentum in the 𝒊-direction

• Hence the tensor is symmetric,𝑇"# = 𝑇#"

• Let’s get abetter sense of what 𝑇qu means …

Energydensityandflow

“Thefluxof𝑖-momentuminthe𝑗-direction”?Whatdoesthatmean??

• ConsidertwoadjacentcubesoffluidAandB.IngeneralAexertsaforce�⃗� onBthroughtheinterface𝑑𝑆 (and Bexerts anequal-and-opposite force onA)

• �⃗� is equal tothe rateatwhich momentum is pouring from Ainto B,such that the flux of momentum is �⃗�/𝑑𝑆

• So 𝑇qu is the force per unit area between adjacent elements

A B

Perfectfluids

• Someforces,suchasviscosity,actparalleltotheinterfacebetweenfluidelements

• Foraperfect fluid,we only consider forceswhich act perpendicular tothe interface,such that 𝑇qq = pressure 𝑃,and 𝑇qu = 0

• Foranon-relativistic perfect fluid,

• This applies tothe Universe!(see later!)

𝑇"# =𝜌𝑐U 00 𝑃

0 00 0

0 00 0

𝑃 00 𝑃

Energyconservation

• Wecanexpressenergy-momentumconservationusingtherelation

• Thisisfourequationsinoneagain– 1forenergyand3formomentum

• It’salocalrelationwhichappliesateverypointofspace-time

𝜕"𝑇"# = 0