Functional Renormalization Group analysis of relativistic ...
16. Einstein and General Relativistic...
Transcript of 16. Einstein and General Relativistic...
16.EinsteinandGeneralRelativisticSpacetimesProblem:Specialrelativitydoesnotaccountforgravitationalforce.
Tworequirements(1) Newtheory("generalrelativity")mustreducetospecialrelativityin
sufficientlyflatregionsofspacetime:
• Replaceds2= ημνdxμdxνwithds2= gμνdxμdxν.
flat Minkowski metric non-flat metric
• Requiregμν toreducetoημν insmallregionsofspacetime.
arbitrarily curved surface
Any sufficiently small piece looks flat
• Toincludegravity...Geometricizeit!Makeitafeatureofspacetimegeometry.
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1. GeometrizationofGravity2. Conventionalityof
Geometry3. Mach'sPrinciple4. HoleArgument
16.EinsteinandGeneralRelativisticSpacetimes
Tworequirements
• Toincludegravity...
(2) Curvatureofspacetimemustberelatedtomatterdensity:
• TheEinsteinequations(1916):
Einstein tensor = encodes curvature of spacetime as a function of gμν
Stress-energy tensor =encodes matter density
Gμν(gμν)= κTμν
• Consequence:TheMinkowskimetricisthesolutionforzerocurvatureGμν= 0(i.e.,spatiotemporalflatness).
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Problem:Specialrelativitydoesnotaccountforgravitationalforce.
• Toincludegravity...Geometricizeit!Makeitafeatureofspacetimegeometry.
1. GeometrizationofGravity2. Conventionalityof
Geometry3. Mach'sPrinciple4. HoleArgument
Ageneralrelativisticspacetime =4-dimcollectionofpointssuchthatbetweenanytwopoints(infinitesimallyclose)points,thereisadefinitespacetimeintervalgivenbyds2= gμνdxμdxν,wheregμν isaLorentzianmetricthatsatisfiestheEinsteinequations.
"reducestotheMinkowskimetricatanypoint"
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ArbitraryGeneralRelativisticSpacetime
Variablelight-conestructureateachpoint:light-conescantwistandturnduetocurvature.
MinkowskiSpacetime
Invariantlight-conestructureateachpoint:light-conesallhavesamesizeandorientation.
• Idea:Thelight-conestructureconstrainsthemotionofphysicalobjects(travelingontimelikeworldlines).
• And:Inanarbitrarygeneralrelativisticspacetime,thematterdensitydeterminesthelight-conestructure.
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1.TheGeometrizationofGravityinGR
Twokeyobservations:(i) Geometry
Equationforastraightline:
orx(t)= v0t+ x0,wherev0,x0= constants
t
x
x(t)= v0t+ x0
x0•
•−x0/v0
𝑑!𝑥𝑑𝑡! = 0,
• Ininertialframes,Newton's2ndLawis
• Intheabsenceofexternalforces(F= 0)anobject'spositionx(t) asafunctionoftimeistheequationofastraightline!(Newton's1stLaw.)
𝐹 = 𝑚𝑎 = 𝑚𝑑!𝑥𝑑𝑡! .
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(ii) Physics• Considerwhentheexternalforceanobjectexperiencesisduetogravity:
TheNewtonian gravitationalforce onanobjectofmassmgduetoanotherobjectofmassM adistancer away.
gravitational mass inertial mass
measure of inertia of an object -- tendency of object to obey Newton's 1st Law
⇓
Φ=−GM/ristheNewtoniangravitationalpotentialfield (describestheparticulargravitationalfieldproducedbymassM).
• Newton's2ndLawbecomes:−mg∂Φ=mia
measure of degree to which an object experiences the Newtonian gravitational force
⇓
• Ismg thesameasmi?- Conceptuallyandmathematically,no!- Physically,yes! Allknownexperimentsindicatethatmg=mi .
𝐹 =𝐺𝑀𝑚"𝑟!
= −𝑚"𝜕Φ
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Consequenceofmg=mi :
UniversalityofGravitationalForceInanygivengravitationalfield(describedbysomeΦ),all objectsfallwiththesame accelerationa=−∂Φ.
• Thisisregardless oftheobject'sinternalproperties(it'smass,charge,etc.).
The gravitational force is universal: it affects all objects in the same way.
• Notuniversal! HowmuchanobjectacceleratesingivenE- andB-fieldsdependsonitschargeanditsinertialmassintheratioq/mi.
Different objects will experience different electromagnetically-induced accelerations.
Electromagnetic force experienced by an object with electric charge q moving at speed v in the presence of electric E and magnetic B fields.
Constrastwiththeelectromagneticforce:
�⃗�#$ = 𝑞(𝐸 + �⃗�×𝐵)
or
• Newton's2ndLawbecomes:
𝑞 𝐸 + �⃗�×𝐵 = 𝑚%�⃗� �⃗� =𝑞𝑚%
𝐸 + �⃗�×𝐵
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Sincegravityisuniversal,let'sincorporateitintothestructureofspacetime!- Let's"geometrize"it.
• Wecanviewtheseparticular "∂Φ"-straightlinesinthecurvedspaceasthepathsofobjectsthatareundergoinggravitationally-inducedacceleration.
• The"extra"term∂Φ canbeencodedintoa"non-flat"metric.
• Themotionofanobjectinagravitationalfieldisgivenby
A curved line in a flat space.𝑑!𝑥𝑑𝑡! = 𝑎 = −𝜕Φ
• Wecanrewriteitas:
A straight line in a curved space!𝑑!𝑥𝑑𝑡! + 𝜕Φ = 0
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Inflat GalileanandMinkowskispacetimes,thereisadistinctionbetween:
Earthatrest
straight non-accelerated trajectory
positivelychargedplateatrest
+
straighttrajectories;noforcespresent
ma= 0
𝑑!𝑥𝑑𝑡! = 0
spacestation
gravitationally-accelerated curved trajectory
electron
−
EM-accelerated curved trajectory
vs.
curvedtrajectories;forcespresent
ma= F
𝑑!𝑥𝑑𝑡! ≠ 0
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Incurved generalrelativisticspacetimes:
• No distinctionbetweenstraightsandgrav.-acceleratedtrajectories.• Still adistinctionbetweenstraights/grav.-acceleratedtrajectories,andallotherforce-inducedacceleratedtrajectories.
+
"old" straight still considered as straight
gravitationally-accelerated curved trajectory now considered as straight, too
straighttrajectories;noforcespresent
ma= 0
𝑑!𝑥𝑑𝑡! + 𝜕Φ = 0
−
EM-accelerated curved trajectory still considered curved
vs.
curvedtrajectories;forcespresent
ma= F
𝑑!𝑥𝑑𝑡! + 𝜕Φ ≠ 0
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ConsequencesofGeometrizingGravity1. Inertialreferenceframes(definedbythefamiliesofstraighttrajectoriesin
spacetime)nowincludeobjectsatrest,inconstantmotion,or gravitationallyaccelerating.
2. Gravitationally-induced accelerationisthusrelative(inexactlythesamewaythatpositionandvelocityarerelative):Whetherornotyouaregravitationallyacceleratingdependsonyourframeofreference.
3. Allother typesofaccelerationarestillabsolute:Whetherornotyouarenon-gravitationally-acceleratingisindependentofyourframeofreference(suchaccelerationsalwayscome"packaged"withattendantforces).
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(a) Underasubstantivalistinterpretation:Thegravitationalfieldisnolongeraphysicalfieldthatexistsin spacetime;ratheritisnowpartofthecurvatureofspacetimeitself.- We'vedemoted thestatusofthegravitationalfieldfromphysicstogeometry.
(b) Underarelationist interpretation:Themetricfieldisphysicallyrealandjustiswhatwaspreviouslycalledthegravitationalfield.- We'vepromoted thestatusofthemetricfieldfromgeometrytophysics.
• Bothinterpretationsagreethatthestructure ofspacetimeisnolongerflat,asinSpecialRelativityandNewtoniandynamics.
InterpretationofGeometrizingGravity
• Theydisagreeoverhowspacetimestructuremanifestsitself.- Asubstantivalistsaysit'sthestructureofarealspacetime.- Arelationalistsaysit'sthestructureofarealphysicalfield(themetricfield).
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2.TheConventionalityofGeometryinGR
• Canthegravforcebethoughtofasanundetectabledeformationforce?- Presentinthe"simple"flatgeometry,butabsentinthecomplicatedcurvedgeometry.
• Suppose:Thereisaunique splitbetweeninertialstructure andgravity inGR.- Supposethecontentsoftheparenthesisin(B)canalwaysbewrittenuniquelyastwodistinctterms).
• Then:Sinceallobservationsindicatemi=mg,therewouldbenoobservationaldifferencebetween(A)and(B).
• RealistResponse:Thecurvedgeometrydescriptionis,arguably,muchsimplier.
• Isitamatterofconventionwhetherornottogeometrizethegravitationalforce?
(A)Flatgeometry& grav.force (B)Curvedgeometry& nograv.force
𝑑!𝑥𝑑𝑡! 𝑚" = −𝑚#𝜕Φ
𝑑!𝑥𝑑𝑡! 𝑚" +𝑚#𝜕Φ = 0
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Claim:ClockB ticksslowerthanClockA.
Oneconsequenceofmi=mg:Thegravitationalredshiftingofclocks.Observationsofclocksinauniformlyacceleratingframe...
B
A
Why?Tocompareclocks,sendlightsignals:
(i) Correlatefrequency ofalightsignalwithClockB.
B
tick tick tick...
(iii) SinceA isacceleratingawayfromlightsignal,itwillreceivesignalatlower frequency(i.e.,shiftedtothered);henceAwillmeasureB astickingslower.
(ii) SendcorrelatedlightsignalfromB toA.
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...shouldbeindistinguishablefromobservationsofclocksina(homogeneous)gravitationalfield:
Oneconsequenceofmi=mg:Thegravitationalredshiftingofclocks.
B
A
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Prediction:Gravityslowsclocks(gravitational"red-shift").
ExperimentalEvidence: 1956Pound-RebkaexperimentintoweratJeffersonLabonHarvardcampus.
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Twowaystoexplainthegravitationalred-shiftingofclocks
Δt0= emissiontimeforlightsignalΔt1= absorptiontimeforlightsignal
Δt0Δt1
path of trailing edge of light signal
path of leading edge of light signal
B A
t
xFlatGeometry
• ExperimentsindicateΔt1> Δt0.
B A
Δt0
Δt1
t
xCurvedGeometry
• Ifspacetimeiscurved,thentheresultisexplainedbythefactthatthepathstakenbyleadingandtrailingedgesoflightsignalarenot"parallel".- Theexperimentalresultisexplainedwithoutreferencetoaforceactingonclocksinawaydifferentfromhowitactsonotherthings.
- Rather,wecansaythatgravity,asspacetimecurvature,affectsallobjectsinthesameway.
• Ifspacetimeisflat,weshouldhaveΔt1= Δt0.- Theexperimentalresultmustbeexplainedbyclaimingthatgravityaffectsclocksinawaydifferentfromitseffectsonotherobjects(soitdoesn'taffectallthingsinthesameway).
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BUT:Thereisn'tauniquesplitbetweeninertialstructureandgravityinGR.
• Underastandardcondition(thattheconnectionbesymmetric),thetermschematicallyrepresentedby(d2x/dt2 +∂Φ) cannotbesplitintoaninertialpartd2x/dt2 andagravitationalpart∂Φ.
• So:Underthisstandardcondition,geometryisnot conventionalinGR!
Supposewerelaxthisstandardcondition.- Wegetatheory("teleparallelgravity")thatlookslikeGRandinwhicha"split"between(whatlookslike)inertialstructureand(whatlookslike)gravitationcanbeachieved.
- But:TheverdictisstilloutonwhetherthisisanequivalentwayofformulatingGR,orwhetheritcountsasanentirelydifferenttheory!
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RecallMach:Waterisrotatingwithrespecttothe"fixedstars"inNewton'sBucket.
3.Mach'sPrincipleandGR
indistinguishablefrom
Wateratrest;fixedstarsrotating.Waterrotating;fixedstarsatrest.
sphere of fixed stars = matter density of universe
Mach'sPrinciple:Thematterdensity intheuniverseisthecauseofinertialforcesonobjectsundergoingnon-inertialmotion
• Details? How doesthematterdensityoftheuniversecauseinertialforces?- Machprovidesnoexplanation.- Einsteinthinksgeneralrelativitysuppliestheexplanation!
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InGR:Thestructureofspacetime......determines theinertialframesofreference(i.e.,thefamiliesofstraights)....isdeterminedby thematterdensity.
• Newton'ssubstantivalist(Einstein'sinterpretation):
Thestructureofspacetimeisthecauseofinertialforcesonacceleratingobjects.
• InGR:Thematterdensityintheuniversedeterminesthestructureofspacetime,whichthendeterminestheinertialframesofreference.
• Is"determiningtheinertialframesofreference"thesameas"beingthecauseofinertialforces"?
• Mach'srelationalist:
Thematterdensityintheuniverseisthecauseofinertialforcesonacceleratingobjects.
Does GR agree with Newton's substantivalist or Mach's relationalist?
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(1)DoestheGRaccountsupportsubstantivalismorrelationalism?
(a) Asubstantivalistmaysay:"Thestructureofspacetimeisgivenbypropertiesofrealspacetimepoints."- Takeallphysicalfieldsoutoftheuniverseandrealspacetimewouldbeleft.
(b) Arelationalistmaysay:"Thestructureofspacetimeisgivenbypropertiesofthemetricfield,whichisarealphysicalfield."- Takeallphysicalfieldsoutoftheuniverseandnothingwouldbeleft.
Dependsonhowyouinterpretthe"structureofspacetime":
ThreeQuestionsofInterpretation
Shouldthemetricfieldalsobeconsideredamatterfield?Substantivalist:No!Relationalist:Yes!
Gμν(gμν) = κTμν
metric field matter fields
• InGR,thereare"vacuum"solutions totheEinsteinequations.- Non-flatsolutionsinwhichthematterdensityiszero(Tμν =0)!- "Gravitationalwaves"withnosources.
(2)DoestheGRaccountsupportMach'sPrinciple?Dependsonhowyouinterpretwhatmatteris!
Gμν= κTμν= 0Doesn't necessarily mean zero curvature!
(a) Asubstanativalistmaysay:"InGR,therecanbeinertialforces(asexperiencedbygravitationalwaves)inauniversedevoidofmatter!- SoMach'sPrincipledoesnotholdingeneral."
(b) Arelationalistmaysay:"Invacuumsolutions,theinertialforcesarestilldeterminedbyamatterfield;namely,themetricfield!- Moreover,such'vacuum'solutionsdon'treallydescribeuniversesdevoidofmatter;whattheydescribeareuniversesinwhichtheonlymatterfieldisthemetricfield!
- SoMach'sPrincipledoesholdingeneral!"
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(b) Arelationalistmayrespond:"Thissupportsmyview:Gravitationalwavesarepropagationsinthemetricfield."
(3)Dovacuumsolutionssupportsubstantivalismorrelationalism?
(a) Asubstantivalistmaysay:"Thissupportsmyview:Gravitationalwavesarepropagationsofspacetimeitself.
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"One hundred years after Albert Einstein predicted the existence of gravitational waves, scientists have finally spotted these elusive ripples in space-time. In a highly anticipated announcement, physicists with the Advanced Laser Interferometer Gravitational-Wave Observatory (LIGO) revealed on 11 February that their twin detectors have heard the gravitational 'ringing' produced by the collision of two black holes about 400 megaparsects (1.3 billion light-years) from Earth."Casteivecchi & Witze (2016) 'Eintein's Gravitational Waves Found at Last', Nature News.
SymmetriesandEquationsofMotionSymmetries= transformationsthatleaveequationsofmotionunchanged.- Newton'sequations:Symmetries= Galileantransformations- Maxwell'sequations:Symmetries= Lorentztransformations- Einsteinequations:Symmetries= "diffeomorphisms"
4.LeibnizShiftsinGR:TheHoleArgument
• ManifoldSubstantivalism:The4-dimcollectionofpoints(amanifold)ofageneralrelativisticspacetimerepresentsrealsubstantivalspacetimepoints.
• Claim(HoleArgument):IfweadoptamanifoldsubstantivalistinterpretationofGR,thenwehavetoconcludethatGRisindeterministic!
d
diffeomorphism = transformationbetweenpointsonamanifold= arbitrary coordinatetransformation
=d(p)•p
•q
M
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Whatthismeans:
gμν
p • q•
M
• Let d∗gμν bewhatyougetwhenyouactwithd ongμν .
• Then:IfGμν(gμν) = κTμν,thenGμν(d∗gμν) = κTμν .- Ifgμν isasolutiontotheEinsteinequationswithmatterdistributionTμν ,thensoisd∗gμν .
• d∗gμν isobtainedfromgμν by"shifting"ittoadifferentsetofpoints.
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Whatthismeans:
d∗gμν
p • q•
M
• Let d∗gμν bewhatyougetwhenyouactwithd ongμν .
• d∗gμν isobtainedfromgμν by"shifting"ittoadifferentsetofpoints.
• Then:IfGμν(gμν)= κTμν,thenGμν(d∗gμν)= κTμν .- Ifgμν isasolutiontotheEinsteinequationswithmatterdistributionTμν ,thensoisd∗gμν .
• Recall:Leibnizproposedstaticandkinematicshifts.- TheappropriatetypeofshiftinGRisashift-by-a-diffeomorphism!
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Now:Constructa"hole"diffeomorphismhsuchthat:(1) h= identityoutsidearegionH (the"hole")ofM.(2) h≠ identityinsideH.
M
H
gμν
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Now:Constructa"hole"diffeomorphismhsuchthat:(1) h= identityoutsidearegionH (the"hole")ofM.(2) h≠ identityinsideH.
M
H
h*gμν
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Now:Constructa"hole"diffeomorphismhsuchthat:(1) h= identityoutsidearegionH (the"hole")ofM.(2) h≠ identityinsideH.
- gμν and h∗gμνdisagreeinside H.
- gμν and h∗gμν agreeoutside H.
- h shiftsthemetricgμνonlyinthehole.
• Manifoldsubstantivalistsmustclaimthatgμν andh∗gμν describedifferentstatesofaffairs.
• But:gμν andh∗gμν arephysicallyindistinguishable (botharesolutionsforthesamematterdistribution).
• So:ManifoldsubstantivalistsmustconcludethattheEinsteinequationsareindeterministic.
M
H
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A complete specification of the matter distribution outside the hole fails to uniquely determine the metric inside the hole.
SomeOptions:
2. Modifyyourspacetimesubstantivalism.- Claimthatspacetimepoints(orregions)arereal,butthisdoesn'tnecessarilymeangμνandh∗gμν describedistinctstatesofaffairs.
- Maybespacetimepointsobtaintheir"identities"instrangeways.- Maybetheyobtainthemonlyafterafieldhasbeen"spread"overthem,andnotbefore.)
3. Modifyyourspacetimerealism.- Claimthatspacetimestructurecanbethoughtofasrealwithouthavingtoadditionallyclaimthatspacetimepointsarereal(orthatmanifoldregionsarereal).
Non-Trivial options: they influence how you might attempt to reconcile GR with quantum theory!
1. AdoptarelationistinterpretationofGR.- Sincegμν andh∗gμν describethesamespacetimerelationsbetweenobjects,anddifferonlyonwhatpointstheyarespreadover,arelationistwillclaimtheyarenot distinct:theyrepresentthesamestateofaffairs.
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