Class8:Tensors

Inthisclasswewillexplorehowgeneralco-ordinatetransformationsmaybedescribedbyatensorcalculususingindexnotation,leadingto

ageneralizednotionofcurvature

Class8:Tensors

Attheendofthissessionyoushouldbeableto…

• … understandhowtheLorentztransformationsmaybereplacedbygeneralco-ordinatetransformations

• ...describewhysuchtransformationsarefundamentaltoformulatingthelawsofphysics

• … applyindexnotationtomanipulategeneraltensorobjects,suchasbyraisingorloweringanindex

• … describehowthenotionofparallel-transportinacurvedspaceleadstothegeneralizedRiemanncurvaturetensor

Thelawsofphysics

• AfundamentalideaofRelativityisallreferenceframesareequallysuitablefortheformulationofthelawsofphysics

• Areferenceframeisaspace-timeobservingsystem,suchastheEarth’sframe,orafreely-fallingframe,oraninertialframeinSR

Thelawsofphysics

• Physicsdoesnotdependonourchoiceofco-ordinateframe

• Anequationrepresentingaphysicallawinco-ordinateframe𝑥,suchas𝐴# = 𝐵#,musttransformtoadifferentframe𝑥′suchthat𝐴′# = 𝐵′#

• Weneedsomepowerfulmathematicstoensurethatthiswillhappen– thisisthemathematicsoftensorcalculus

https://comic.hmp.is.it/comic/tensor-calculus/

SpecialRelativityrecap

• InSpecialRelativityweintroducedtheideaofa4-vector,agroupoffourquantities𝐴# whosevaluesin2inertialframesarerelatedbytheLorentztransformations:

• Itwasconvenientforustodefinea“down”4-vector𝐴#:

• Thisisbecause𝐴#𝐴# isaninvariant

𝐴′# = 𝐿#(𝐴(

𝐴# = 𝜂#(𝐴(

𝐿#( =

𝛾 −𝑣𝛾/𝑐−𝑣𝛾/𝑐 𝛾

0 00 0

0 00 0

1 00 1

𝜂#( = 𝜂#( =−1 00 1

0 00 0

0 00 0

1 00 1

𝐴# = 𝜂#(𝐴(

Generaltransformations

• TheLorentztransformationbetweeninertialframesisaspecialcase– wemustdevelopmathematicstodescribeanarbitrarytransformationbetween2co-ordinateframes (e.g.theEarth’sframe,andafreely-fallingframe)

• Aco-ordinatetransformationprovidesrelationsforsome𝑥′co-ordinatesintermsof𝑥 co-ordinates,𝑥2 = 𝑓(𝑥)

https://math.stackexchange.com/questions/1228106/how-can-i-transform-coordinate-systems-based-on-quaternion-data

Generaltransformations

• Westartbytransformingsimpledifferentialsandgradients

usingthechainrule:𝑑𝑥′# = 789:

78;𝑑𝑥( and 7<

789:= 78;

789: 7<78;

• Usingthistemplate…

• A“generalup4-vector”𝐴# isanarraywhosevaluesinthe2

framesarerelatedby:𝑨′𝝁 = 𝝏𝒙9𝝁

𝝏𝒙𝝂𝑨𝝂 (theLorentz

transformationisaspecialcaseofthiswith𝑥′# = 𝐿#(𝑥()

• A“generaldown4-vector”𝐴# isanarraywhosevaluesinthe

2framesarerelatedby:𝑨′𝝁 =𝝏𝒙𝝂

𝝏𝒙9𝝁𝑨𝝂

Generaltransformations

• Thegeneraltransformationofan“up”indextoa“down”indexusesthespace-timemetric:

• 𝒈𝝁𝝂 istheinversematrixof𝒈𝝁𝝂,sinceapplyingbothoftheseoperationsinturnto𝐴# mustrestoretheoriginalquantity

• Inaninertialorfreely-fallingframe,𝑔#( = 𝜂#(,andwerecoverthepreviousrulesforraising/loweringanindex

𝐴# = 𝑔#(𝐴( 𝐴# = 𝑔#(𝐴(

Tensors

• Somephysicalquantitiesaregroupedintolargerstructures

• Moregenerally,atensor 𝐴#( transformsbetween2framesas:

• Weraiseandlowerindicesusingthemetric,forexample:

• Wecangeneralizetheserelationstohigherdimensions

𝐴′#( =𝜕𝑥2#

𝜕𝑥E 𝜕𝑥2(

𝜕𝑥F𝐴EF 𝐴′#( =

𝜕𝑥E

𝜕𝑥2# 𝜕𝑥F

𝜕𝑥2( 𝐴EF

𝐴F( = 𝑔F#𝐴#(

𝐴#F = 𝑔F(𝐴#(

𝐴EF = 𝑔E#𝑔F(𝐴#(

Tensors

Withthismathematicalapparatusinhandwecanderiveanumberofusefulrelationsoftensorcalculus:

• If𝐴# = 𝐵# then,inanyotherframe,𝐴′# = 𝐵′#

• 𝐴#𝐵( isatensor𝐶#(

• 𝐴#𝐵# isascalarinvariantinallframes

• 𝐶#(𝐴( isa4-vector𝐷#

• Wecanre-arrangesummedindices,e.g.𝐴#𝐵# = 𝐴#𝐵#

Wehavealreadymetsometensorsinthecourse,suchas𝑔#(and𝑇#(.Weareabouttomeetsomemore!

Generaldescriptionofcurvature

• Howcanwedescribethecurvatureofaregionofspace?

• Onthesurfaceofasphere,carryanarrowfromtheEquatortothePoleandbackonapath𝐴 → 𝑁 → 𝐵 → 𝐴 shownbelow

• Supposeweparallel-transport thearrow,meaningthatitscomponentsareunchangedinalocalCartesiansystem

https://commons.wikimedia.org/wiki/File:Parallel_transport.png

Vectorsparallel-transportedaroundaclosedpathonacurvedsurfacearerotated!

Generaldescriptionofcurvature

• Supposeyouareatapoint𝑥# inspace-time

• Travelindirection𝑖 untilyourco-ordinate𝑥M isincreasedbyasmallamount𝑑𝑥M,withoutchangingtheotherco-ordinates

• Nowtravelindirection𝑗 untilco-ordinate𝑥O isincreasedby𝑑𝑥O,withoutchangingtheotherco-ordinates

• Nowmovebackwardsby−𝑑𝑥M

• Finally,movebackwardsby−𝑑𝑥O

• Youare,ofcourse,backat𝑥#! 𝑑𝑥M

𝑑𝑥O

• Nowtravelthesamerouteagain,parallel-transportingavector𝐴P whichinitiallypointsindirection𝑘

• Vectorsparallel-transportedaroundaclosedpathinacurvedsurfacearerotated– letthechangeineachcomponentbe𝑑𝐴R

• ThisthoughtexperimentallowsustodefinetheRiemanntensor𝑹𝝀𝝁𝝂𝜿 ,whichprovidesageneralmeasureofcurvature:

• TheRiemanntensormaybeexpressedintermsoftheChristoffel symbols: 𝑅F#(

E = 𝜕#ΓF(E − 𝜕(ΓF#

E + Γ#YE ΓF(Y − Γ(YE ΓF#

Y

𝑑𝐴R = 𝑅PMOR 𝐴P𝑑𝑥M𝑑𝑥O

TheRiemanntensor

TheRiemanntensor

• Inan𝑁-dimensionalspace,wehave𝑁Z

possibleloops,and𝑁 finalcomponentsof𝑁 initialvectors,i.e.𝑁[ componentsateachpoint,= 256 for𝑁 = 4!!

• Itisnotthatbad,owingtosymmetries(e.g.,goingbackwardsroundtheloop).Actually,thenumberofindependentcomponentsis𝑵𝟐(𝑵𝟐 − 𝟏)/𝟏𝟐

• Sothecurvatureateachpointisdescribedby1numberwhen𝑁 = 2(ona2Dcurvedsurface),and20numberswhen𝑁 = 4