Einstein’sHappiestThought

DavidHodgsonCarolian AstronomySociety

8th Nov2017

QuizWeightlessnessisexperiencedbyaperson:

1. anywhereinspaceregardlessofmotion2. inspacebutoutsidetheearth’satmosphere3. infreefallinagravitationalfield4. ontheMoon5. onlyinanorbitingsatellite

...ledEinsteintoGeneralRelativity

...whichisEinstein’stheoryofgravity

..butwhyshouldwecare?What’swrongwithNewton’stheoryofgravity?

NewtoniangravityOKaslongas:

b)speedsmuchlessthanspeedoflighta)gravityisweakasinsolarsystem

Hencenogoodfor:neutronstars,blackholes,supernovae,gammaraybursts,gravitationalwaves,theoriesoftheuniverse

Conclusion:AstronomersneedGeneralRelativity

Firstthebadnews...

...andnowthegoodnews

OnecanunderstandthefundamentalsofGeneralRelativitythroughsimplephysicalargumentsandalittlemaths

Butfirstletustakeabrieftourthroughthehistoryofgravity

Aristotle350B.C.Heavyobjectsfallfasterthanlightones

Galileo’sDiscorsi 1638Heavyobjectsfallatthesamespeedaslightones

...and

• GalileostatedNewton’sFirstLawbeforeNewton:i.e.Abodywillcontinueinitsstateofrestoruniformmotioninastraightlineunlesssloweddownbyfriction.

...moreaccurately:withthesameacceleration

g=10(m/s)/stime dist0 01 5

2

3

20

45

speed010

20

30

Newton1670

What’swrongwiththispicture?

2dMmGF =

M

m

Newtonhimselfwrotein1692:“Thatonebodymayactuponanotheratadistancethroughavacuum,withoutthemediationofanythingelse,byandthroughwhichtheiractionandforcemaybeconveyedfromonetoanother,istomesogreatanabsurditythatIbelievenoman,whohasinphilosophicalmattersacompetentfacultyofthinking,caneverfallintoit.”

What’swrongwithNewton’stheoryofgravity?

FixedbyEinsteinin1915withhisGeneralTheoryofRelativity

• Actionatadistance• Forcetransmittedinstantaneously• i.e.ifSunsuddenlyceasedtoexistEarthwouldinstantlyshootoffatatangent• Timeisabsolute

FieldfromAcceleratedMass

SomethingelsewrongwithNewton’stheory

OK,Newtonwasn’tquiterightbuthowdidEinsteincorrecthis

theory?

Answer:HishappiestthoughtplussomefancygeometryduetoGaussandRiemann

CarlFriedrichGauss1777-1855Inventeddifferentialgeometrywhichdealswithcurvedspacese.g.thehyperboloid

Gaussaged8Addnumbersfrom1to100

51495248

964973982991

!!

5050490010050 =++=Sum

Sphere(2Dspace)withGeodesic(shortestdistancebetweentwopoints)

Whentwoinitiallyparallelgeodesicsmeet,itindicatesintrinsiccurvature

Locallyflat

Cylinderwithsimplegeodesics

Geodesicsstayparallelas“curved”surfaceisanintrinsicallyflatspace

Howtocharacteriseaspace

OrusePythagorastotestflatness

4 5

3

25169..543 222

=+=+

ei

Pythagorasin3D

a

c

b

d

d2= a2+b2+c2

Iftrianglesmallenoughthena2+b2=c2i.e.locallyflat

a

b

c

AnotherformofPythagoras

y

x

dx

dyds ds2 =dx2 +dy2 (METRIC)

dx =asmalldistanceinthex-direction

Characteristicofflatspace

Minkowski andSpace-Time

Space-TimeDiagramt(hours)

x(miles)

1.00

0.75

0.50

0.25

1 2 3 4

Space-time

x

y

tworldliney

x

SuttonParkRd

InSpecialRelativity

ds2 =dx2 +dy2 – c2 dt2

x

y

ct

cdtdx

dyworldline

Flatspace-time

PropertiesofCurvedSpaces(andspace-times)

1. Ageodesicistheshortest“distance”betweentwopointsandisthereforethenearestthingtoastraightline.

2. Ifinitiallyparallelgeodesicsintersect,thenthespaceiscurved.

3. Locallygeometryisflat.

FramesofReference

y

x

O

Time(s)

0

1

2

3

(x,y)(x’,y’)

y’ v

x’

O’

Viewedfromtrain

sxlightofspeed

xheightTime 6.118.022===

0.8light

seconds

Tickingclock

V=0.6

dx’=0,dt’=1.6 ds2 =dx’2 –dt’2 =02 -1.62 =-2.56

ViewedfromPlatform

ViewedfromPlatform

dx=1.2,dt=2 ds2 =dx2 –dt2 =1.22 -22 =-2.56

Einstein’sHappiestThought

IwassittinginachairinthepatentofficeinBernewhensuddenlyathoughtoccurredtome:“Ifapersonfallsfreelyhewillnotfeelhisownweight.“Iwasstartled.Thissimplethoughtmadeadeepimpressiononme.Itimpelledmetowardatheoryofgravitation.

A.Einstein1907

Einstein’sHappiestThought1907

Infreefallalleffectsofgravitydisappear.Canaccelerationcancelgravity?

Thyssen-KruppTowerRottweil

28July1945BettyLouOliversurvives75floorfall

FallingLiftandWeightlessness

Orbitingsatellitealsoinfreefall

FallenthisfartowardsEarth

Gravity

Fallingobjectinspace-time

Infallingliftgeodesicsbecomestraightlines

t

Distancefromground,y

Ingravity

geodesict

Distancefromfloor

Inlift

LimitedsizeofEinstein’sLift

LimitedsizeofEinstein’sLift

ExplainsTides

StarinellipticgalaxyRXJ1242-11tornapartbytidalforcesduetoblackhole

(artist’simpression)

NASA’sChandraX-rayandESA’sOpticalObservatories

Earth-MoonTides

Lifeinafreelyfallingframe

Foranygravitationalfieldtheeffectsofgravityarenullifiedinafreely-fallingframeoflimitedextent.Howeverthetidalforcesarenoteliminated.

Insuchaframethespace-timegeometryisflat.

Howtotellaspace-timeiscurved

x

y

Intersectionofinitiallyparallelgeodesicsmeanscurvature

y

x

t

Earth’scentre

Einsteindevelopshishappiestthought

Einstein’sEquivalencePrinciple

Itisimpossibletodistinguishbetweenastationaryframeofreferenceinagravitationalfieldgandaframeacceleratingwithaccelerationgbyanyphysicalexperiment.

Let’sapplyittoabeamoflight

FirstuseEinstein’shappiestthought

...thentheEquivalencePrinciple

Gravitydeflectsabeamoflight

GravitationalRedShift

E

R

h

R v

Accelg

InthistimeRacquiresanextravelocityv=gt =gh/c

EERcgh

cv lll )1()1( 2+=+=

Timetforphotontotravelheighth=h/c

HenceRisrecedingfromthesourcewithvelocitygh/cwhenitreceivesthephoton.Henceredshift.

ApplyPrincipleofEquivalence

MustalsobeagravitationalredshiftonEarth’ssurface h

...whichinturnimplies

ACLOCKPLACEDHIGHINAGRAVITATIONALFIELDRUNSFASTERTHANANIDENTICALCLOCKLOWERDOWN

Whathappensnearablackhole?

ERcgh tt )1( 2+=

Space-timedistortedbygravity

x

t

PoundandRebka,Harvard1959

mhsmg 5.22,/8.9 2 ==

152 105.2 -= xcgh

15105.2 -=D xll

Gravitymodifiesspace-timemetric

222

222 )21( dtcrc

GMdydxds --+=

x

ct

y

Newtonianorbit

Einsteinconfirmed

222

222 )21( dtcrc

GMdydxds --+=

Largemassmodifiesthesetermstogivecorrectiontonewtonian orbite.g.CorrectpredictionoftheprecessionofMercury’sorbit.