16. Einstein and General Relativistic...

30
16. Einstein and General Relativistic Spacetimes Problem : Special relativity does not account for gravitational force. Two requirements (1) New theory ("general relativity") must reduce to special relativity in sufficiently flat regions of spacetime: • Replace ds 2 = η μν dx μ dx ν with ds 2 = g μν dx μ dx ν . flat Minkowski metric non-flat metric • Require g μν to reduce to η μν in small regions of spacetime. arbitrarily curved surface Any sufficiently small piece looks flat • To include gravity... Geometricize it! Make it a feature of spacetime geometry. 1 1. Geometrization of Gravity 2. Conventionality of Geometry 3. Mach's Principle 4. Hole Argument

Transcript of 16. Einstein and General Relativistic...

Page 1: 16. Einstein and General Relativistic Spacetimesfaculty.poly.edu/~jbain/spacetime/lectures/16.Einstein... · 2018-10-30 · 16. Einstein and General Relativistic Spacetimes Problem:

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.

1

1. GeometrizationofGravity2. Conventionalityof

Geometry3. Mach'sPrinciple4. HoleArgument

Page 2: 16. Einstein and General Relativistic Spacetimesfaculty.poly.edu/~jbain/spacetime/lectures/16.Einstein... · 2018-10-30 · 16. Einstein and General Relativistic Spacetimes Problem:

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).

2

Problem:Specialrelativitydoesnotaccountforgravitationalforce.

• Toincludegravity...Geometricizeit!Makeitafeatureofspacetimegeometry.

1. GeometrizationofGravity2. Conventionalityof

Geometry3. Mach'sPrinciple4. HoleArgument

Page 3: 16. Einstein and General Relativistic Spacetimesfaculty.poly.edu/~jbain/spacetime/lectures/16.Einstein... · 2018-10-30 · 16. Einstein and General Relativistic Spacetimes Problem:

Ageneralrelativisticspacetime =4-dimcollectionofpointssuchthatbetweenanytwopoints(infinitesimallyclose)points,thereisadefinitespacetimeintervalgivenbyds2= gμνdxμdxν,wheregμν isaLorentzianmetricthatsatisfiestheEinsteinequations.

"reducestotheMinkowskimetricatanypoint"

3

Page 4: 16. Einstein and General Relativistic Spacetimesfaculty.poly.edu/~jbain/spacetime/lectures/16.Einstein... · 2018-10-30 · 16. Einstein and General Relativistic Spacetimes Problem:

ArbitraryGeneralRelativisticSpacetime

Variablelight-conestructureateachpoint:light-conescantwistandturnduetocurvature.

MinkowskiSpacetime

Invariantlight-conestructureateachpoint:light-conesallhavesamesizeandorientation.

• Idea:Thelight-conestructureconstrainsthemotionofphysicalobjects(travelingontimelikeworldlines).

• And:Inanarbitrarygeneralrelativisticspacetime,thematterdensitydeterminesthelight-conestructure.

4

Page 5: 16. Einstein and General Relativistic Spacetimesfaculty.poly.edu/~jbain/spacetime/lectures/16.Einstein... · 2018-10-30 · 16. Einstein and General Relativistic Spacetimes Problem:

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! .

5

Page 6: 16. Einstein and General Relativistic Spacetimesfaculty.poly.edu/~jbain/spacetime/lectures/16.Einstein... · 2018-10-30 · 16. Einstein and General Relativistic Spacetimes Problem:

(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

6

Page 7: 16. Einstein and General Relativistic Spacetimesfaculty.poly.edu/~jbain/spacetime/lectures/16.Einstein... · 2018-10-30 · 16. Einstein and General Relativistic Spacetimes Problem:

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

7

Page 8: 16. Einstein and General Relativistic Spacetimesfaculty.poly.edu/~jbain/spacetime/lectures/16.Einstein... · 2018-10-30 · 16. Einstein and General Relativistic Spacetimes Problem:

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

8

Page 9: 16. Einstein and General Relativistic Spacetimesfaculty.poly.edu/~jbain/spacetime/lectures/16.Einstein... · 2018-10-30 · 16. Einstein and General Relativistic Spacetimes Problem:

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

9

Page 10: 16. Einstein and General Relativistic Spacetimesfaculty.poly.edu/~jbain/spacetime/lectures/16.Einstein... · 2018-10-30 · 16. Einstein and General Relativistic Spacetimes Problem:

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

10

Page 11: 16. Einstein and General Relativistic Spacetimesfaculty.poly.edu/~jbain/spacetime/lectures/16.Einstein... · 2018-10-30 · 16. Einstein and General Relativistic Spacetimes Problem:

ConsequencesofGeometrizingGravity1. Inertialreferenceframes(definedbythefamiliesofstraighttrajectoriesin

spacetime)nowincludeobjectsatrest,inconstantmotion,or gravitationallyaccelerating.

2. Gravitationally-induced accelerationisthusrelative(inexactlythesamewaythatpositionandvelocityarerelative):Whetherornotyouaregravitationallyacceleratingdependsonyourframeofreference.

3. Allother typesofaccelerationarestillabsolute:Whetherornotyouarenon-gravitationally-acceleratingisindependentofyourframeofreference(suchaccelerationsalwayscome"packaged"withattendantforces).

11

Page 12: 16. Einstein and General Relativistic Spacetimesfaculty.poly.edu/~jbain/spacetime/lectures/16.Einstein... · 2018-10-30 · 16. Einstein and General Relativistic Spacetimes Problem:

(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).

12

Page 13: 16. Einstein and General Relativistic Spacetimesfaculty.poly.edu/~jbain/spacetime/lectures/16.Einstein... · 2018-10-30 · 16. Einstein and General Relativistic Spacetimes Problem:

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

13

Page 14: 16. Einstein and General Relativistic Spacetimesfaculty.poly.edu/~jbain/spacetime/lectures/16.Einstein... · 2018-10-30 · 16. Einstein and General Relativistic Spacetimes Problem:

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.

14

Page 15: 16. Einstein and General Relativistic Spacetimesfaculty.poly.edu/~jbain/spacetime/lectures/16.Einstein... · 2018-10-30 · 16. Einstein and General Relativistic Spacetimes Problem:

...shouldbeindistinguishablefromobservationsofclocksina(homogeneous)gravitationalfield:

Oneconsequenceofmi=mg:Thegravitationalredshiftingofclocks.

B

A

15

Page 16: 16. Einstein and General Relativistic Spacetimesfaculty.poly.edu/~jbain/spacetime/lectures/16.Einstein... · 2018-10-30 · 16. Einstein and General Relativistic Spacetimes Problem:

Prediction:Gravityslowsclocks(gravitational"red-shift").

ExperimentalEvidence: 1956Pound-RebkaexperimentintoweratJeffersonLabonHarvardcampus.

16

Page 17: 16. Einstein and General Relativistic Spacetimesfaculty.poly.edu/~jbain/spacetime/lectures/16.Einstein... · 2018-10-30 · 16. Einstein and General Relativistic Spacetimes Problem:

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).

17

Page 18: 16. Einstein and General Relativistic Spacetimesfaculty.poly.edu/~jbain/spacetime/lectures/16.Einstein... · 2018-10-30 · 16. Einstein and General Relativistic Spacetimes Problem:

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!

18

Page 19: 16. Einstein and General Relativistic Spacetimesfaculty.poly.edu/~jbain/spacetime/lectures/16.Einstein... · 2018-10-30 · 16. Einstein and General Relativistic Spacetimes Problem:

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!

19

Page 20: 16. Einstein and General Relativistic Spacetimesfaculty.poly.edu/~jbain/spacetime/lectures/16.Einstein... · 2018-10-30 · 16. Einstein and General Relativistic Spacetimes Problem:

20

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?

Page 21: 16. Einstein and General Relativistic Spacetimesfaculty.poly.edu/~jbain/spacetime/lectures/16.Einstein... · 2018-10-30 · 16. Einstein and General Relativistic Spacetimes Problem:

21

(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

Page 22: 16. Einstein and General Relativistic Spacetimesfaculty.poly.edu/~jbain/spacetime/lectures/16.Einstein... · 2018-10-30 · 16. Einstein and General Relativistic Spacetimes Problem:

• 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!"

22

Page 23: 16. Einstein and General Relativistic Spacetimesfaculty.poly.edu/~jbain/spacetime/lectures/16.Einstein... · 2018-10-30 · 16. Einstein and General Relativistic Spacetimes Problem:

(b) Arelationalistmayrespond:"Thissupportsmyview:Gravitationalwavesarepropagationsinthemetricfield."

(3)Dovacuumsolutionssupportsubstantivalismorrelationalism?

(a) Asubstantivalistmaysay:"Thissupportsmyview:Gravitationalwavesarepropagationsofspacetimeitself.

23

"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.

Page 24: 16. Einstein and General Relativistic Spacetimesfaculty.poly.edu/~jbain/spacetime/lectures/16.Einstein... · 2018-10-30 · 16. Einstein and General Relativistic Spacetimes Problem:

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

24

Page 25: 16. Einstein and General Relativistic Spacetimesfaculty.poly.edu/~jbain/spacetime/lectures/16.Einstein... · 2018-10-30 · 16. Einstein and General Relativistic Spacetimes Problem:

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.

25

Page 26: 16. Einstein and General Relativistic Spacetimesfaculty.poly.edu/~jbain/spacetime/lectures/16.Einstein... · 2018-10-30 · 16. Einstein and General Relativistic Spacetimes Problem:

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!

26

Page 27: 16. Einstein and General Relativistic Spacetimesfaculty.poly.edu/~jbain/spacetime/lectures/16.Einstein... · 2018-10-30 · 16. Einstein and General Relativistic Spacetimes Problem:

Now:Constructa"hole"diffeomorphismhsuchthat:(1) h= identityoutsidearegionH (the"hole")ofM.(2) h≠ identityinsideH.

M

H

gμν

27

Page 28: 16. Einstein and General Relativistic Spacetimesfaculty.poly.edu/~jbain/spacetime/lectures/16.Einstein... · 2018-10-30 · 16. Einstein and General Relativistic Spacetimes Problem:

Now:Constructa"hole"diffeomorphismhsuchthat:(1) h= identityoutsidearegionH (the"hole")ofM.(2) h≠ identityinsideH.

M

H

h*gμν

28

Page 29: 16. Einstein and General Relativistic Spacetimesfaculty.poly.edu/~jbain/spacetime/lectures/16.Einstein... · 2018-10-30 · 16. Einstein and General Relativistic Spacetimes Problem:

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

29

A complete specification of the matter distribution outside the hole fails to uniquely determine the metric inside the hole.

Page 30: 16. Einstein and General Relativistic Spacetimesfaculty.poly.edu/~jbain/spacetime/lectures/16.Einstein... · 2018-10-30 · 16. Einstein and General Relativistic Spacetimes Problem:

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.

30