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1
[email protected],France
Pertuis,onDecember31,2019
toMr.T.Damour
IHES,routedesChartres
91440BuressurYvette
France
Recommendedwithacknowledgementofreceipt
Copiesto:
EtienneGhys,PermanentSecretaryoftheAcademyofSciences
DirectoroftheInstitutdesHautesEtudesScientifiques
Sir,
More than three months have passed since my letter of 26 September 2019, whichremainedunanswered.A letter that included all the elements of a reply to the articlethatyoupostedonJanuary7,2019onyourpageoftheIHESsite,whichconcluded"tothephysicalandmathematicalincoherenceoftheJanusmodel",basedonthedifficultyofdescribingthegeometryinsidethestars.
Ibroughtall theelementsthatputthismodelbackonits foundationsatthepriceofaminimal modification of the tensors of the second member, which ensures thesatisfaction of Bianchi's identities, without changing anything to the model'sachievements,namelythesatisfactionofabouttensetsofobservationaldata.ThisworkwaspublishedinMarch2019inthejournalProgressinPhysics.
http://www.jp-petit.org/papers/cosmo/2019-Progress-in-Physics-1.pdf
Iimmediatelyaskedyoutoputalinkonyourpagetowhatcanbeconsideredascientificrightofreply.
Noanswer
Thinking thatyouhadprobablynotreadthisarticle, Icomposedadetailed forty-pagepresentationofitscontents,withallthecalculationdetails,whichIforwardedtoyouonSeptember26,withthesamerequest.
Noresponse.
2
I am rephrasing this request for the last time. If youdonot reply, Iwill then take allnecessary steps to denounce this serious breach of scientific ethics and the resultingdamagetomyreputationasascientist.
Yourssincerely
Jean-PierrePetit
3
[email protected],France
Pertuis,onseptember26,2019
toMr.T.Damour
IHES,routedesChartres
91440BuressurYvette
France
CopytoG.D’Agostini,N.Debergh,S.Michea,NathalieDeruelle,YvesBlanchet
TotheDirectorofIHESandtothePermanentSecretaryoftheAcademyofSciencesJointfiles:
Article«ThephysicalandmathematicalconsistencyoftheJanusCosmologicalModel».ProgressinPhysics2019Vol.15issue1
Appendix1:Calculationdetails
Appendix2:theEnglishtranslationofyourarticle.
Sir,
OnJanuary4,2019,youplacedanarticle[1]onyourpageoftheIHESwebsiteentitled:
Onthe"Janusmodel"byJ.P.PETIT
Where you point out "the physical and mathematical inconsistency of our model". Irepliedinasimpleletter,drawingyourattentiontoanarticleofmine[2]thatappearedinthejournalProgressinPhysics(attachment),entitled:
PhysicalandmathematicalconsistencyoftheJanusCosmologicalModel
ProgressinPhysics2019Vol.15issue1
which, while agreeing on the relevance of your criticism brings the solution to theproblem,moduloaveryslightmodificationofthesystemofJanusfieldequationswhichin no way invalidates everything that had already been obtained and published asresultsandmanyagreementswiththeobservationalresults.
I hadaskedyou, in a simple letter, either to include the contentof this articleon thispage,orsimplytheaddresswhereitisaccessible,asalegitimaterightofscientificreply,
4
even if youmight formulate new criticisms on this paper, in order to maintain yournegativeopinionofourapproach.Thisispartofthenormalgameofscientificactivity.
But I believe that you have not read it, and in any case have not taken seriously theargumentsthatweredevelopedinit.Itisapity,becauseindoingso"youarethrowingthebabyoutwiththebathwater"atatimeofcrisisincosmologyandastrophysicswhentheexaminationofnewideaswouldbe,itseemstome,opportune.
Wehavereceivedseverallettersfromforeignresearcherswho,havingbeeninformedofthepresenceof your reviewonyour IHESpage, have translated this text intoEnglishand Russian, and were surprised not to see any links to a possible right of reply. Acolleague also informs me that your colleague Marc Lachièze-Rey tells anyone whowantstohearit"thatDamourhasshownthattheJanusmodeldoesnotmakesense".
I am therefore republishing my approach, this time by registered mail withacknowledgementofreceipt,onceagainattachingthecontentofmyarticle.ButasIamnotsureyouwillreadthisdocument,Iwillsummarizeit.
The firstmembersofyourownsystemofcoupled fieldequations [13]are identical tothoseinSabineHossenfelder's2008paper[3]andtoour2014systemofequations[4].The common denominator being to choose to include the Lagrangian densities
− g(+ ) R(+ ) and − g(− ) R(− ) (noted by you "right" and "left") in the action integral,whichimmediatelyproducesthisform
withtheLagrangian
With"Janus"notations,byoptingforanullityofthetwocosmologicalconstantsandbytaking χ = 1 thiscanbewritten:
(1) Rµν
(+ ) − 12
R(+ ) gµν(+ ) = Tµν
(+ ) + tµν(+ )
(2) Rµν
(− ) − 12
R(− ) gµν(− ) = Tµν
(− ) + tµν(− )
5
In the second members the sources of the fields determining the geometries of thesectors"+"and"-"or"Right"and"Left"accordingtoyournotations.
Yourterms tµν
(+ ) and tµν
(− ) reflecttheinteractionbetweenthesetwosectors
- tµν
(+ )
represents the contribution to the field,which determines the geometry"+"("right")duetothepresenceofmasses"-"("left").
- tµν
(− )
represents thecontribution to the field,whichdetermines the "-" ("left")geometry due to the presence of "+" ("right") masses.
The"Janus"writingconventionistranslatedas:
(3) Rµν
(+ ) − 12
R(+ ) gµν(+ ) = Tµν
(+ ) + tµν(+ )
(4) Rµν
(− ) − 12
R(− ) gµν(− ) = − Tµν
(− ) + tµν(− )⎡⎣ ⎤⎦
Theformofthefirsttwomembersthenrequiresthatthedifferencesofthesecondtwomembersbenullandvoid.
InordertodemonstratetheinconsistencyoftheJanussystemyouchoosetooptforthe:
-Stationarysituation
-Presenceofapositivemass,ofconstantdensity,locatedinsideasphere(thatistosay,schematically,a"star").
-Zeronegativematerialdensity("left").
Thesystemthenbecomes,withyournotations:
(5) Rµν
(+ ) − 12
R(+ ) gµν(+ ) = Tµν
(+ )
(6) Rµν
(− ) − 12
R(− ) gµν(− ) = − tµν
(− )
Itshouldbenotedatthisstagethatthereisnodefinitionofhowthetensor tµν
(− ) shouldbeconstructed. It is the effect of "induced geometry" created in the "left" sector by the"right"material. All that could be said is that this tensor should be a function of the"right"content,i.e.
(7) tµν(− ) ≡ψ ( ρ (+ ) , p(+ ) )
6
Theproposalofthe"Janus"modelamountedtogivingthistermtheform:
(8) tµν
(− ) = g(+ )
g(− ) Tµν(+ )
Toshowthat the inconsistencyappearseven inanalmostLorentiansituation, inyourarticle,page2,equation(5)youintroduceatensor
Tµν
(+ )
accordingto:
(9) Tµν
(+ ) = − g(+ )
g(− ) Tµν(+ )
Theconditionsofzerodivergenceofthetwoequationsarethenwritten(yourequations(7)and(8),page3ofyourarticle):
(10) ∇ν (+ )Tµν
(+ ) = 0
(11) ∇ν (− )Tµν
(+ ) = 0
Where theoperators ∇ν (+ ) and ∇
ν (+ ) areconstructed fromthe twodifferentmetrics gµν
(+ )
and gµν
(− ) .
What is the physical meaning of these zero divergence conditions? These areconservationequations.Itisthereforenotsurprisingthatequations(10)and(11)leadto Euler-type equations, which express the fact that, in the star, the force of gravitybalancestheforceofpressure.
Thecalculationleadsto:
(12) ∂i p(+ ) = + ρ (+ ) ∂i U
(13) ∂i p(+ ) = − ρ (+ ) ∂i U
Equationswhich,asyourightlypointout,contradicteachother.
Let's now go back to physics by deciding towrite the Janus equations in theirmixedform:
(14) R(+ )
µν − R(+ )δ µ
ν = T(+ )µν + g(− )
g(+ ) T(− )µν
(15) R(− )
µν − R(− )δ µ
ν = − g(+ )
g(− ) T(+ )µν + T(− )
µν
⎛
⎝⎜
⎞
⎠⎟
Likeyou,ItookEinstein'sconstantequaltounity.
Thetensorsthenwrite:
7
(16)
T(+ )µν =
ρ (+ ) 0 0 0
0 −p(+ )
c2 0 0
0 0 −p(+ )
c2 0
0 0 0 −p(+ )
c2
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟⎟
T(− )µν =
ρ (− ) 0 0 0
0 −p(− )
c2 0 0
0 0 −p(− )
c2 0
0 0 0 −p(− )
c2
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟⎟
InthiscasetheJanussystemisreducedto:
(17) R(+ )
µν − R(+ )δ µ
ν = T(+ )µν
(18) R(− )
µν − R(− )δ µ
ν = − g(+ )
g(− ) T(+ )µν = T(+ )
µν
Thecontradictionisthenexpressedwhencalculatingthedifferentialequationgivingthepressureasafunctionoftheradialvariable.ThiscorrespondstoTolmannOppenheimerVolkoff'sequation.Forequation(17)weobtain:
(19)
p(+ ) 'c2 = − m + 4π G p(+ ) r3 / c4
r r − 2m( ) ρ (+ ) + p(+ )
c2
⎛⎝⎜
⎞⎠⎟
With m = G M
c2 whereMisthemassofthestar.
WhenwepasstotheNewtonianapproximationthisequationbecomes
(20) p(+ ) ' = − ρ (+ ) mc2
r2 = − G M ρ (+ )
r2
We'rebacktoEuler'sequation.
Thesamething,appliedtoequation(18)provides:
(21)
p(+ ) 'c2 = + m − 4π G p(+ ) r3 / c4
r r + 2m( ) ρ (+ ) − p(+ )
c2
⎛⎝⎜
⎞⎠⎟
TheNewtonianapproximationthenprovides:
(22) p(+ ) ' = + ρ (+ ) mc2
r2 = + G M ρ (+ )
r2
It'sanequivalentwayofbringingoutthiscontradictionthatyouraise.
8
Butitisalsoawayofdiscoveringitsorigin,whichcomesfromthechoicemadetoexpressthetensor
T(+ )
µν
responsiblefortheinducedgeometryeffect.
However,thereisnoaprioriphysicalreasonforthistensortobewritten:
(23)
T(+ )µν = − g(+ )
g+ ) T(+ )µν = − g(+ )
g+ )
ρ (+ ) 0 0 0
0 −p(+ )
c2 0 0
0 0 −p(+ )
c2 0
0 0 0 −p(+ )
c2
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟⎟
WearegoingtoconsidermodifyingthesystemofJanuscoupledfieldequationsasfollowsandthisiswhatIdidinthearticleIpublishedinmarch2019inthepeer-reviewedjournalProgressinPhysicsandwhichyoudidnotconsider(IaskedyoutoputalinkinyourpageoftheIHESsite):
Remaining in the expression of the equations in their mixed form, let us considermodifyingthetensorsresponsiblefortheeffectsofinducedgeometry,whichamountstosuggestmovingfromthesystem(14)+(15)tothesystem:
(24) R(+ )
µν − R(+ )δ µ
ν = T(+ )µν + g(− )
g(+ )
⌢T(− )
µν
(25) R(− )
µν − R(− )δ µ
ν = − g(+ )
g(− )
⌢T(+ )
µν + T(− )
µν
⎛
⎝⎜
⎞
⎠⎟
Let's remember: no physical imperative imposes a particular choice of the form ofthesetensors
⌢T(− )
µν and
⌢T(+ )
µν .Ontheotherhand,theformofthefirstmembersimposes
themathematical imperativesofzerodivergence thatwehavepointedout,andwhichwecannotescape.
9
Let'sshowthatthechoice:
(26)
⌢T(+ )
µν =
ρ (+ ) 0 0 0
0 +p(+ )
c2 0 0
0 0 +p(+ )
c2 0
0 0 0 +p(+ )
c2
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟⎟
(27)
⌢T(− )
µν =
ρ (− ) 0 0 0
0 +p(− )
c2 0 0
0 0 +p(− )
c2 0
0 0 0 +p(− )
c2
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟⎟
allows us to satisfy this mathematical imperative conditions. Let us take again theconfigurationyouhaveconsideredinyourarticle, i.e. thesituationofastarofpositivemass,surroundedbyvacuum:
(28) R(+ )
µν − R(+ )δ µ
ν = T(+ )µν
(29) R(− )
µν − R(− )δ µ
ν = T (+ )µν = − g(+ )
g(− )
⌢T (+ )
µν
everything is in order (details of the calculations are provided in the appendix). Theseconddifferentialequationbecomes:
(30)
p(+ ) 'c2 = − m + 4π G p(+ ) r3 / c4
r r + 2m( ) ρ (+ ) + p(+ )
c2
⎛⎝⎜
⎞⎠⎟
which, in Newtonian, restores the Euler equation, reflecting the balance betweenpressureandgravityinthestar.:
Thephysicalandmathematicalinconsistencydisappears.
Both equations satisfy (asymptotically, in Newtonian approximation) Bianchi'sidentities.
10
Atthispoint,someonemightsay:
-That'sveryclever.TomakethisdifficultydisappearPetithastinkeredwiththetensorspresentinthesecondlimbssothattheinconsistencylinkedtotheemergenceofEuler'sequation, translating the balance between the forces of pressure and gravity into themasses,disappears.
But, as we have pointed out, the incoherence linked to the emergence of the Eulerequation,whichreflects thebalancebetween the forcesofpressureandgravity in themasses,willdisappear.
what determined the shape of the tensors tµν
(+ )
and tµν(− )
responsible for the inducedgeometryeffects?Here,usingyourformulation:
(31) Rµν
(+ ) − 12
R(+ ) gµν(+ ) = Tµν
(+ ) + tµν(+ )
(32) Rµν
(− ) − 12
R(− ) gµν(− ) = Tµν
(− ) + tµν(− )
Nothingapriori!
In the Newtonian approximation (linearization) the effect of pressure is neglected inrelationtothedensityterm ( p << ρc2 ) .Bysayingthatthissystemwillonlybevalidforlinearizedsolutions,itprovidesabouttenresultsinagreementwiththeobservations.
Inthislinearizationviewpointwewillhavetensorsintheform:
(33)
t(+ )µν ∼
ρ (+ ) 0 0 00 ... 0 00 0 ... 00 0 0 ...
⎛
⎝
⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟
t(− )µν ∼
ρ (− ) 0 0 00 ... 0 00 0 ... 00 0 0 ...
⎛
⎝
⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟
Thethreediagonaltermsbeingfinallyneglected.
Howthentocompletethesetensorsbyaddingthesemissingdiagonalterms?
Answer (from physicist): by making sure that the Euler equations (equilibrium, inregions where masses are present, between the force of gravity and the force ofpressure) are satisfied. This is equivalent to wishing that the equations satisfy(asymptotically)theBianchiconditions.
Thisleadstothechoice(26)+(27).
ThisistheanswerIgaveyouinthisarticlepublishedinProgressinPhysics,whichyouprobablyhaven'tread.
11
I saw that Nathalie Deruelle had been your adviser in the making of your article. Iproposedtoyouandtoherameetinginaroomwithablackboard,withoutwitnessesorrecording, which would allow me to expose this work and answer your questions.Neitherofyouhadthesimplecourtesyofsimplyansweringme.
The text,which still appears on your IHESpage, discreditsme as a scientist, not onlywithFrenchpeople,butwithintheentireinternationalscientificcommunity.Youcanofcoursechoosenottosubscribetomyrequests. Inthiscase,whatIcantellyouisthat,failingtoobtainalegitimatedebatewiththepeoplewhoaresupposedtobespecialistsin these matters, this whole affair will ultimately be brought to the attention of thegreatestnumber,inFrenchandinEnglish,viaoneormorevideos,withallthedetailsofthecalculationsprovidedinattachedpdffiles.
Anewsituationisemerging.ThroughtheseriesofaboutthirtyJanusvideos,usingmytalents as a teacher, I have exposed all the ins and outs of the approach we haveundertaken for so many years, underlining in passing the contradictions in whichcontemporarycosmologyandastrophysicsaresinkingdeeperanddeeper,byresortingtotheundefinedconceptsofdarkmatteranddarkenergy.
You are the only one to have reacted in a constructed and argued way through thearticlethatyouhavepositionedinyourpageoftheIHESandwearegratefultoyouforthat.
Everyoneknowsthatmodelsdon'tcomeintobeingallatonce, intheirmostelaborateform. Your comment therefore prompted a necessary reworking of the model, withpublication in a peer-reviewed journal (whichwas in progress at the time). This is apurely mathematical reworking, which, by the way, does not in any way change theresults already obtained and published and the many points of agreement with theobservations. From this point of view, we can only be grateful to you for havinghighlightedthisshortcomingandforhavingpromptedthisprogress.
-IthereforerequestthatyouaddthecontentsofthislettertothispageoftheIHES,asanexerciseofmyscientificrightofreply.Evenifyouputforwardanyargumentsthatcontradictmyarguments.
UnlessyouwouldprefertoputthislinkonyourpageoftheIHESsite/
-Iaskyoutoputthelinktomyprogressinphysicsarticle:
-IaskyoutoputalinktothetranslationofyourownarticleinEnglish,throughthelink:
OrtoreproducethistextinyourpageoftheIHESwebsite.
-Insofaraswehaverespondedtoyourlegitimateobjection,itwouldbeappropriateforustobeabletopresentthiswork,"revisited",inaseminarattheIHES,andIamrephrasingthisrequesttoyou.
12
SincerelyyoursJean-PierrePetit
References:
[1]T.Damour:Surlemodèle«Janus»deJ.P.Petithttp://www.ihes.fr/~damour/publications/JanusJanvier2019-1.pdf
[2]J.P.Petit:PhysicalandmathematicalconsistencyoftheJanusCosmologicalModel.ProgressinPhysics291Vol.15issue1.(http://www.ptep-online.com)
[3]S.HossenfelderAntigravitationPhysicalLettersBvol.636issue24may2006pp.119-125
[4]J.P.Petit,G.D’Agostini:NegativeMasshypothesisincosmologyandthenatureofdarkenergy.AstrophysicsAndSpaceSccience,.A29,145-182(2014)[5]G.DAgostiniandJ.P.Petit:ConstraintsonJanusCosmologicalmodelfromrecentobservationsofsupernovaetypeIa,AstrophysicsandSpaceScience,(2018),363:139.https://doi.org/10.1007/s10509-018-3365-3
[6]J.P.PETIT,P.MIDY&F.LANDSHEAT:Twinmatteragainstdarkmatter.Intern.Meet.onAtrophys.andCosm."Whereisthematter?",Marseille2001june25-29
[7]J.P.Petit,Astrophys.SpaceSci.Twinuniversecosmology226,273(1995).
[8] W.B.Bonnor : Negative mass and general relativity. General Relativity andGravitationVol.21,N°11,1989
[9]J.P.Petit&G.D’Agostini:LagrangianderivationofthetwocoupledfieldequationsintheJanusCosmologicalModel.AstrophysicsandSpaceScience2015,357:67
[10]G.DAgostiniandJ.P.Petit:ConstraintsonJanusCosmologicalmodelfromrecentobservationsofsupernovaetypeIa,AstrophysicsandSpaceScience,(2018),363:139.https://doi.org/10.1007/s10509-018-3365-3
[11]J.P.PETIT:Cosmologicalmodelwithvariablevelocityoflight.ModernPhysLettersA3,1988,pp.1527
[12]TheDipoleRepeller:YHoffman,D.Pomarède,R.B.Tully,H.Courtois.NatureAstronomy20171,0036
[13]T.DamourandIIKoganEffectiveLagrangiansanduniversalityclassesofnonlinearbigravityPhysRevD2002
13
APPENDIX1Puttingelementsofyourownarticleintoperspective
andthewaywe'ddealtwithit.
Quotationsfromexcerptsofyourtextareindented.
Inred,themodificationofyouranalysis,basedonthenew2019systemoffieldequations[2]whichcorrespondsto(28)-above.
Younote[1]notesthatduetothestructureofthefirstmembersoftheJanusfieldequationswehavetherelation:
∇ +
ν Eµν+ = 0 (2)
∇−
ν Eµν− = 0 (3)
AddingthattheseBianchiidentitiesimplyconservationlawsforthecorrespondingsources.Yourtext:
Sincethe(Janus)equationsconsistoftwoEinstein-typeequations,theseequationsimplytwoseparateconservationlawsfortheirtworight-handmembers.
Thisiswherethereasoningwillbetakenupagain.
YoustartfromtheJanussystemof2015[9].
w+ Eµν
+ = χ w+Tµν+ + w−Tµν
−( ) (1a)
w− Eµν
− = − χ w+Tµν+ + w−Tµν
−( ) (1b)
with: Eµν
± = Eµν (g± ) = Rµν± − 1
2R ± gµν
±
andyou'reposing: w± = −det g±
Youwrite:
Thetwotensorsources Tµν+and Tµν
−aresupposedtorepresent,
respectively,theimpulseenergyofordinarymatter(called"positivemass")andofanewmattercalled"negativemass".
14
Inthe2019paper[2]thefieldequationshavebeenmodifiedand,alongwithyournotations,theyshouldread::
w+ Eµν
+ = χ w+Tµν+ + w−
⌢Tµν−( ) (1a’)
w− Eµν
− = − χ w+⌢Tµν+ + w−Tµν
−( ) (1b’)Inthesecondmembersthesourcetermsof"inducedgeometry"(i.e.managinghowthegeometryofapopulationisinfluencedbytheenergy-matterdistributionofthesecond)arereplacedby
⌢Tµν− and
⌢Tµν+ .
Youthenmoveontothecasewherenegativemassisabsent:
Eµν+ = χ Tµν
+ (4a)
Eµν− = −
w+w−
Tµν+ (4b)
Whichistobereplacedbythesystem
Eµν+ = χ Tµν
+ (4a’)
Eµν− = −
w+w−
⌢Tµν+ (4b’)
Youthenwrite
Tµν+ = Tµν w+ = w w− = w
and:
Tµν = − w
wTµν (5)
ThismustbereplacedbythechoicemadeinJanus2019[2]:
Tµν = − w
w⌢Tµν (5’)
Youremindusthatwehavetohave:
∇νTµν = 0 (7)
∇νTµν = 0 (8)
True,butnowmodulothemodification(5')
15
Note: Please note your choice of signature: ( − + + + ) . I choose ( + − − − )But itdoesn'thaveanyconsequences.
Page5Youwrite:
"LetmefirstrecallthatthelinearizedsolutionofEinstein'sequationsin the usual Einstein equation (say the first system in (6)) can bewritten:
goo = − 1− 2U
c2
⎛⎝⎜
⎞⎠⎟
; gi j = + 1+ 2Uc2
⎛⎝⎜
⎞⎠⎟(19)
wherethequasi-NewtonianpotentialUsatisfiesthePoissonequation
ΔU= − 4πG
Too
c2 1+O( 1c2 )
⎛⎝⎜
⎞⎠⎟= − 4πG ρ 1+O( 1
c2 )⎛⎝⎜
⎞⎠⎟(20)
Becauseoftheformalsymmetrybetweenthetwosystemequations(6),alinearizedsolutionoftheEinstein-typeequationsformetrics g = g− iswrittenas:
goo = − 1− 2U
c2
⎛⎝⎜
⎞⎠⎟
; gi j = + 1+ 2Uc2
⎛⎝⎜
⎞⎠⎟(21)
where the quasi-Newtonian potential U satisfies the modifiedPoisson'sequation
ΔU= − 4πG
Too
c2 1+O( 1c2 )
⎛⎝⎜
⎞⎠⎟(22)
Accordingtoequation(5)thesourceofthismodifiedPoissonequation(denotedhere ρ ) is,at the lowestapproximationwhich sufficeshere(since the ratio w / w = 1+ O(1/ c2 ) , simply the opposite of the usualsource.
ρ≡
Too
c2 1+O 1/ c2( )( )= −Too
c2 1+O 1/ c2( )( ) = − ρ 1+O 1/ c2( )( ) (23)There I still agree, although in Janus 2019 [2], if
⌢Too = Too this second tensor becomes
Tµν = − w
w⌢Tµν .
Icontinue.
Asaresult,thequasi-Newtonianpotentialenteringthesecondmetricisalsotheoppositeoftheusualpotential:
U = − U 1+ O(1/ c2 )( ) (24)
16
Itisatthebeginningofpage6thatyouwrite(basedontheJanus2015equations[9]):
ThespatialpartofthesourcetensorforEinstein'ssecondequationis:
Ti j = − w
wTi j = − 1+ 4U
c2 + O(1/ c4 )⎛⎝⎜
⎞⎠⎟
Ti j (25)
Andthere,ifwebaseourselvesontheJanusequationsofMarch2019[2],whichare:
R(+ )
µν − 1
2R(+ )g(+ )
µν = χ T(+ )
µν + g(− )
g(+ )
⌢T(− )
µν
⎡
⎣⎢⎢
⎤
⎦⎥⎥
R(− )
µν − 1
2R(− )g(− )
µν = χ g(+ )
g(− )
⌢T(+ )
µν + T(− )
µν
⎡
⎣⎢⎢
⎤
⎦⎥⎥
with:
⌢T(+ )
µν =
ρ (+ ) 0 0 0
0 p(+ )
c2 0 0
0 0 p(+ )
c2 0
0 0 0 p(+ )
c2
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟⎟
⌢T(− )
µν =
ρ (− ) 0 0 0
0 p(− )
c2 0 0
0 0 p(− )
c2 0
0 0 0 p(− )
c2
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟⎟
Ce que je suis parfaitement en droit de choisir, alors le signe de la partie spatiale dutenseursourcedelagéométrieinduiteestinversé.
Thenyouwriteonpage6[1]:
Irecalltheexplicitformulaof ∇ . T (whereIcallthat w ≡ −det g )
∂ν Tµ
ν = − 1w∂ j w Tµ
ν( ) − 12∂µ gα β Tα β (26)
En appliquant cette formule au cas statique d’une étoile et pour unindicespatialunindicespatial µ = i valant ( 1 , 2 , 3 )
∂ν Ti
ν = − 1w∂ j w Ti
j( ) − 12∂i gα β Tα β (26)
Inthelasttermthecontributionofdominance α =β =0 , inthequasi-Newtonian case ( because T
oo = O(c2 ) while To1 = O(c1) and
Ti j = O(co ) .Wethenfind:
17
0=∇νTiν = ∂ j(Ti
j) − Too
c2 ∂i U + O(1/ c2 )
= ∂ j(Tij) − ρ ∂i U + O(1/ c2 )
(28)
ItisthisequationthatreflectsEuler'srelationshipofstaticequilibriuminausualfluid,let’srecall i = 1, 2 , 3( )
∂i p = ρ∂i U (32)
0=∇νTi
ν = = ∂ j(Tij) − ρ ∂i U + O(1/ c2 ) (30)
WiththeJanusequationsof2015wewillhavewell,asheindicatesatthetopofhispage7:
In this second Euler equation we can replace Tiν , ρ and U by their
values,i.e.inthelowestorderby- Tiν ,− ρ and −U .Thisgives
0=∇νTi
ν = = − ∂ j(Tij) − ρ ∂i U + O(1/ c2 ) (31)
A contradiction then appears between two contradictory Euler equations. But thiscontradiction disappears with the Janus 2019 equations [2] where the equivalentsentencewillbe:
Dans cette seconde équation d’Euler on peut remplacer Tiν , ρ et U
par leurs valeurs, c’est à dire à l’ordre le plus bas par +Tiν ,− ρ et
−U .Celadonne:
In this secondEuler equationwe can replace Tiν , ρ and U by their
values,i.e.inthelowestorderby +Tiν ,− ρ ,and −U .Thisgives:
0=∇νTi
ν = = + ∂ j(Tij) − ρ ∂i U + O(1/ c2 ) (31)
andthecontradictiondisappears.
Andthereweseethesufficientreasonpresidingoverthechoiceofthesourcetermsofthe"inducedgeometry"whichservesasaguidefortheJanus2019equations[2]:
InordertocancelanycontradictionintheEulerequations
18
Inaddition:
Whathasjustbeenestablishedforaregionoftheuniversewherenegativemasswouldbe practically absent, in a negligible quantity, can be extended to the opposite: to aportion of space where, in a situation considered stationary, it is on the contrarynegative mass that dominates and where positive mass can be neglected. This willcorrespondtothesystemofcoupledfieldequations:
(32)
R(+ )µν − 1
2R(+ )g(+ )
µν = χ g(− )
g(+ )
⌢T(− )
µν
(33)
R(− )µν − 1
2R(− )g(− )
µν = χ T(+ )
µν
Bianchi'srelationreferringtothesecondequationwillprovidetheequivalentofaEulerequationforthisnegativematter,reflectingthebalancebetweentheforceofgravityandtheforceofpressure.
But this same constraint, referring to the first equation of the system will have nophysical meaning and will only express the necessary mathematical compatibilitybetween the two solutions
( g(+ )
µν , g(− )
µν ) , which will be ensured if the induced
geometry effect (in the sector of positivemasses, due to the present negativemassescorrespondstotheexpressionofthetensorofthesecondmemberintheform:
(34)
⌢T(− )
µν =
ρ (− ) 0 0 0
0 p(− )
c2 0 0
0 0 p(− )
c2 0
0 0 0 p(− )
c2
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟⎟
Bianchi'srelation(commontobothequations)willcorrespond,withyournotations,to
(35) ∂i p = ρ ∂i U
wherethegravitationalpotential U isthencreatedbynegativemasses.
Bypushingtheconstructionofmetricsolutions,wewillobtaininparticular,fortheonedescribingthebehaviourofpositiveenergyparticles:
Innermetric gµν
int :
(36)
19
ds 2 = 32
1+rs
2
R̂2
⎛
⎝⎜⎞
⎠⎟
1/2
− 12
1+r2
R̂2
⎛
⎝⎜⎞
⎠⎟
1/2⎡
⎣⎢⎢
⎤
⎦⎥⎥
2
c2dt2 − dr 2
1+ r2
R̂2
− r2 dθ 2 + sin2θ dϕ 2( )
with:
R̂ 2 = 3c2
8π G ρ
ExteriorMetric gµν
ext :
(37)
ds2 = 1− 2G Mc2 r
⎛⎝⎜
⎞⎠⎟
c2dt2 − dr2
1− 2G Mc2 r
− r2(dθ 2 + sin2θ dϕ 2 )
with M < 0
Inlinearform:
(38) ds2 = 1+
2G Mc2 r
⎛
⎝⎜
⎞
⎠⎟ c2dt2 − 1−
2G Mc2 r
⎛
⎝⎜
⎞
⎠⎟ dr2 − r2(dθ 2 + sin2θ dϕ 2 )
Whichcorrespondstoaphenomenonofrepulsion.ThisexplainsthephenomenonoftheGreatRepeller,discoveredinJanuary2017[12] . Ithasbeenshownthatinadirectionroughly opposite to that of the Shapley attractor there existed an apparently emptyregionthatseemedtorepelallmatter.
20
Figure:TheeGreatRepeller
As suggested as early as 1995 years these negative mass conglomerates create anegative gravitational lens effect which has the effect of reducing the brightness ofdistant,backgroundsources.Effectwhich,accordingtous,explainsthelowmagnitudeofgalaxiesatz>7.
Thisbeingthecase,afineanalysisofthemagnitudesofthedistantsourceslocatedinthedirection of theGreatRepeller shouldprovide access to the diameter of this negativemassconglomerate,whichisinvisiblesinceitemitsnegativeenergyphotons.
Tosumup:
- So we have a system of two Janus coupled field equations, whose scope islimitedtolinearized,quasi-Newtoniansolutions.-Whichderivesfromanaction-ThatsatisfiesBianchi'sidentities-WhichtakescareofalltheclassicsituationsoftheRG-It'sanadvantageoussubstitutefordarkmatteranddarkenergy.-Fitsadozenorsoobservationaldata.
Inspiteoftheprogressrepresentedbythefirstevidenceoftheexistenceofgravitationalwaves,cosmologysuffersfromnotbeingabletohighlightthehypotheticaldarkmatternorbeingable toprovideanymodel for thisother component representedby thisnolesshypotheticaldarkenergy.
TheJanusmodelistheonlyonetoprovideawell-foundeddescriptionofthenatureoftheseinvisiblecomponentsofthecosmos,namelyantimatter(antihydrogenofnegativemass). The model explains in passing the non-observation of primordial antimatter,
21
givingsubstancetoAndréSakharov'sinitial1967idea.Itisconsistentwithagooddozenobservationaldatasets.
ItisshockingthatallthedoorsofFrenchseminarsinthefieldhavebeenclosedtousforthe last five years. In your registered letter of 7 January 2019, you confirmed yourrefusaltoseemepresentthisworktotheIHES.Iamrephrasingthisrequestonceagaininthehopethatmyletterwillhavemadeyouchangeyourmind.
I alsoaskyou to reproduce these clarificationson the Janusmodel inboth languages,FrenchandEnglish,accompanyingtheEnglish translationofyourownarticle,which Ihave attached. My foreign colleagues arewaiting to read the criticisms/responses inordertobeabletoformtheirownopiniononthismodel.
If there is no real debate on these issues a situation will continue to develop wherefinallynon-specialistsenduphavingaclearerglobalvisionthanspecialists,theattitudeofamanlikeLachièze-Reybeinganexampleofthisirrationalandabsurddeafness.
https://www.youtube.com/watch?v=Vl541wUXsSs&feature=youtu.be
Wehopethatthismailingwillhelptocleanupthissituation,whichisinurgentneed.
Jean-PierrePetit
References:
[1]T.Damour:Surlemodèle«Janus»deJ.P.Petithttp://www.ihes.fr/~damour/publications/JanusJanvier2019-1.pdf
[2]J.P.Petit:PhysicalandmathematicalconsistencyoftheJanusCosmologicalModel.ProgressinPhysics291Vol.15issue1.(http://www.ptep-online.com)
[3]S.HossenfelderAntigravitationPhysicalLettersBvol.636issue24may2006pp.119-125
[4]J.P.Petit,G.D’Agostini:NegativeMasshypothesisincosmologyandthenatureofdarkenergy.AstrophysicsAndSpaceSccience,.A29,145-182(2014)[5]G.DAgostiniandJ.P.Petit:ConstraintsonJanusCosmologicalmodelfromrecentobservationsofsupernovaetypeIa,AstrophysicsandSpaceScience,(2018),363:139.https://doi.org/10.1007/s10509-018-3365-3
[6]J.P.PETIT,P.MIDY&F.LANDSHEAT:Twinmatteragainstdarkmatter.Intern.Meet.onAtrophys.andCosm."Whereisthematter?",Marseille2001june25-29
[7]J.P.Petit,Astrophys.SpaceSci.Twinuniversecosmology226,273(1995).
[8] W.B.Bonnor : Negative mass and general relativity. General Relativity andGravitationVol.21,N°11,1989
[9]J.P.Petit&G.D’Agostini:LagrangianderivationofthetwocoupledfieldequationsintheJanusCosmologicalModel.AstrophysicsandSpaceScience2015,357:67
22
[10]G.DAgostiniandJ.P.Petit:ConstraintsonJanusCosmologicalmodelfromrecentobservationsofsupernovaetypeIa,AstrophysicsandSpaceScience,(2018),363:139.https://doi.org/10.1007/s10509-018-3365-3
[11]J.P.PETIT:Cosmologicalmodelwithvariablevelocityoflight.ModernPhysLettersA3,1988,pp.1527
[12]TheDipoleRepeller:YHoffman,D.Pomarède,R.B.Tully,H.Courtois.NatureAstronomy20171,0036
23
Appendix2Thiscontainsall thecalculations (ohso tedious,as isalways thecase in differential geometry) that support the reasoningpresentedinthebodyofthearticle.
Asageneralruleweareinthecaseofasphericallysymmetricalgeometry.
Inthiscasethetwometricsarewrittenasfollows:
(1) ds(+ )2 = eν (+ )
dx° 2 − eλ (+ )
dr 2 − r2 dθ 2 + sin2θ dϕ 2( ) (2)
Inthefollowing,tolightenthewriting,wewillask:
gµν
(+ ) ≡ gµν gµν(− ) ≡ gµν
Rµν
(+ ) ≡ Rµν Rµν(− ) ≡ Rµν
R(+ ) =R R(− ) =R
Eµν
(+ ) ≡ Eµν Eµν(− ) ≡ Eµν
ρ(+ ) = ρ ρ (− ) = ρ
gµν
(+ ) = gµν gµν(− ) = gµν
ν(+ ) = ν ; λ (+ ) = λ ν (− ) = ν ; λ (− ) = λ
We will perform the calculations starting from an expression of the field equationspresentedinmixedform:
(3) Eµ
ν = Rµν − 1
2Rgµ
ν = χ Tµν + g(− )
g(+ )
⌢T(− )
µν
⎡
⎣⎢⎢
⎤
⎦⎥⎥
(4) Eµ
ν = Rµν − 1
2R g µ
ν = − χ g(− )
g(+ )
⌢Tµ
ν + T(− )µν
⎡
⎣⎢⎢
⎤
⎦⎥⎥
Wewill thenopt for theconfigurationenvisagedbyDamour,consideringapartof thespacewherenegativemassisabsent,i.e.theequations:
24
(5) Eµ
ν = R µν − 1
2R g µ
ν = χ T µν
(6) Eµ
ν = Rµν − 1
2R gµ
ν = − χ gg⌢T(+ )
µν
- The first equation can then be identified with Einstein's equation withoutcosmologicalconstants.
-Thesecondequationtranslatesan"inducedgeometryeffect"(ongeodesicsofthenegative mass species, due to the presence of positive mass inside a sphere ofradius,density ρ
(+ ) = ρ
WewilltrytostickwiththenotationsusedbyT.Damour[1]inhispaper.Hewritesoursystem(5)+(6)accordingtohisequation(4),page1:
Eµν+ = χ Tµν
+
Eµν− = − χ w +
w −Tµν
+
thenheasks(hisequation(4))
Tµν = − χ w +
w −Tµν−
Thisleadshimtowritethesystemofequations(hisequations(6)):
Eµν = + χ Tµν
Eµν = + χ Tµν
And thenweget to the sourceof his critiqueof the two-equation system. Indeed, thestructureofthefirstmembersimposesthat:
(7) ∇ν Eµν = 0
(8) ∇ν Eµν = 0
Thereforewemusthavetheconservationlaws(hisequations(7)and(8)onpage3ofhispaper):
(9) ∇ν Tµν = 0
(10) ∇ν Tµν = 0
25
WewillresumethethreadofitscalculationattheendofthisAppendix1.Still,bygivingthetensortheformcorrespondingtotheunmodifiedJanusequations,equations(9)and(10) led to contradictory Euler equations (equations (32) and (33) on page 7 of hispaper).
Howtogetoutofthisimpasse?
Bynoticingthatwearetotallyfreeinthechoiceoftensorsreflectingtheeffectsinduced(byamaterialonthematerialofoppositesign).Aswewillshowbytakingagainallitscalculation by the menu, a light modification of the tensor
Tµν brings the solution,
withoutmodifyingbyoneiotaalltheaspectsrelatedtothesolutionsemergingfromthetwo coupledequations ("interior"metrics, that is to say inside the star and "exterior"metrics,outsidethestar).
When we start to calculate the exact solution of this system, if we do not take thisprecaution,wewouldalsoseethiskindofcontradiction, insidethestar, intheformoftheemergenceof twoequationsof theTolmannOppenheimerVolkoff type,whicharealso contradictory. In what will follow, which translates the construction of the twometrics as a whole, modulo this precaution, this problem will not appear. But toconvince the reader, we will take up this whole scheme according to the approachfollowedbyDamour[1].
BelowisthecalculationofthecomponentsoftheRiccitensorandthefirstlimb,forthepositivespecies.
Wehave:
(11)
gµν =
e−ν 0 0 00 −e −λ 0 00 0 − r−2 00 0 0 − r−2 sin−2θ
⎛
⎝
⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟
2
2 2
e 0 0 00 e 0 0
g0 0 r 00 0 0 r sin
−
⎛ ⎞⎜ ⎟⎜ ⎟= ⎜ ⎟−⎜ ⎟⎜ ⎟−⎝ ⎠
ν
λ
µν
θ
gµν = δ µ
ν
Withmetricsinthisformthenon-zerocomponentsoftheRiccitensorare:
(12)
2
oo" ' ' ' 'R e2 4 4 r
ν λ ν ν λ ν ν− ⎡ ⎤= − + − −⎢ ⎥
⎣ ⎦
200
" ' ' ' 'R e2 4 4 r
− ⎛ ⎞= − − + +⎜ ⎟
⎝ ⎠λ ν ν λ ν ν
2
11" ' ' ' 'R2 4 4 rν ν λ ν λ= − + −
211
" ' ' ' 'R e2 4 4 r
− ⎛ ⎞= − − + −⎜ ⎟
⎝ ⎠λ ν ν λ ν λ
22' r ' rR e 1 12 2
− ⎡ ⎤= + − −⎢ ⎥⎣ ⎦λ ν λ
22 2 2
1 ' ' 1R er 2r 2r r
− ⎛ ⎞= − + − +⎜ ⎟⎝ ⎠λ ν λ
26
233 22R R sin= θ 3 2
3 2R R=
Ricciscalar:
(13)
2
2 2
" ' ' ' ' ' 2 2 ' 2 ' 2R R e 22 4 4 r r r 2r 2r r
− ⎡ ⎤⎛ ⎞= = − + − − + − − + +⎢ ⎥⎜ ⎟
⎝ ⎠⎣ ⎦µ λµ
ν ν λ ν ν λ ν λ
WhichgivestheEinstein’stennsor:
(14)
00 2 2
1 ' 1E er r r
− ⎛ ⎞= − −⎜ ⎟⎝ ⎠λ λ
(15)
11 2 2
1 ' 1E er r r
− ⎛ ⎞= + −⎜ ⎟⎝ ⎠λ ν
(16)
222
" ' ' ' ' 'E e2 4 4 2r
− ⎡ ⎤−= − + +⎢ ⎥⎣ ⎦
λ ν ν λ ν ν λ
Let'swritetheequationscorrespondingtothefirstofthetwofieldequations,inDamour'snotations[1],inamixedwriting
(17)
Eµν = χ Tµ
ν
(18) e−λ 1
r2 − λ 'r
⎛⎝⎜
⎞⎠⎟− 1
r2 = χ T00
(19) e −λ 1
r2 +ν 'r
⎛⎝⎜
⎞⎠⎟− 1
r2 = χ T11
(20) e−λ ν "
2− ν 'λ '
4+ ν ' 2
4+ ν '− λ '
2r⎡
⎣⎢
⎤
⎦⎥ = χ T2
2
Etaussi:
(21) χ T0
0 − χ T11 = − ν '+ λ '
re−λ
Wewillnowconsidertheoutermetric,wherethesecondmembersoftheequationsarezero.Themethodisdescribedinreference[2],inchapter14,anditcorrespondsto:
eν = e−λ = 1 − 2m
r
27
(22)
ds2 = 1− 2mr
⎛⎝⎜
⎞⎠⎟
dx° 2 − dr2
1− 2mr
− r2 (dθ 2 + sin2θ dϕ 2 )
Avec
(23) 2
GMmc
=
Mbeingthe(positive)massofthestar.
Let'smoveontotheclassicalconstructionoftheinnermetric[2].Wehave:
(23)
2
2
2
0 0 0p0 0 0c
T p0 0 0c
p0 0 0c
−
−
−
⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟=⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠
νµ
ρ
Theequationsarewritten:
(24) e−λ 1
r2 − λ 'r
⎛⎝⎜
⎞⎠⎟− 1
r2 = χ ρ
(25) 2 2 2
1 ' 1 per r r c
− ⎛ ⎞+ − = −⎜ ⎟⎝ ⎠λ ν χ
(26) 2
2
" ' ' ' ' ' pe2 4 4 2r c
− ⎡ ⎤−− + + = −⎢ ⎥⎣ ⎦
λ ν ν λ ν ν λ χ
(27) 2
' ' per c
−+ ⎛ ⎞− = +⎜ ⎟⎝ ⎠λν λ χ ρ
Whence:
(28)2
2 2
1 ' 1 " ' ' ' ' 'e er r r 2 4 4 2r
− − ⎡ ⎤−⎛ ⎞+ − = − + +⎜ ⎟ ⎢ ⎥⎝ ⎠ ⎣ ⎦λ λν ν ν λ ν ν λ
(29)2
2 2
e 1 ' ' ' ' ' "r r 4 4 2r 2
+= − + + −λ ν ν λ ν λ ν
Tosolve,wewrite:
28
(30) e−λ ≡ 1− 2m(r)
rsoit
2m(r) = r 1− e−λ( )
Whence:
(31) 2m' = 1− e−λ( ) + rλ 'e−λ
(32) − 2m'
r2 = −1+ e−λ − rλ 'e−λ
r2 = − 1r2 + e−λ 1
r2 −λ 'r
⎛⎝⎜
⎞⎠⎟
(33) m ' = − r2χ ρ
2= 4π r2 G
c2 ρ
i.e.:
(34) m(r) = m'(r)dr
0
r
∫ = 43π r3ρ G
c2
(35) ν ' = r
r r − 2m( ) −χ pc2 r2 +1
⎛⎝⎜
⎞⎠⎟−
r − 2m( )r r − 2m( )
(36) ν ' = 2 m + 4π G pr3 / c4
r r − 2m( )
Wewilleliminatebyderivingequation(25)
(37) 2 3 2 3 2
p ' 2 1 ' 2 '' ''e ec r r r r r r
− − −⎛ ⎞ ⎛ ⎞− = − + + + −⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠λ λν ν νχ λ
2 3 2 3 2
p ' 2 ' ' ' 2 '' 'ec r r r r r r
− ⎛ ⎞− = − + + − +⎜ ⎟⎝ ⎠λ λ λ ν ν νχ
2 3 2
p ' 2 e ' ' ' 1 '' '2c r r 2r 2 r 2 2r
− ⎛ ⎞− = − + + − +⎜ ⎟⎝ ⎠
λ λ λ ν ν νχ
2 2
2 3 2
p ' 2 e 1 ' ' ' ' ' '' ' ' '2c r r r 4 4 2r 2 4 4
− ⎛ ⎞+− = − − + + − + +⎜ ⎟⎝ ⎠
λ ν λ ν λ ν ν ν λ νχ
Combiningtorequation(29)weget:
29
(38)
2
2 3 2
p ' 2 e e e ' ' '2 2c r r r r 4 4
− − ⎛ ⎞− = − − +⎜ ⎟
⎝ ⎠
λ λ λ ν λ νχ
(39)
( )2
p ' 'e ' 'c 2r
−− = − +λ νχ ν λ
Using(27)weget:
(40) ( )2 2
p ' e ' p '' 'c r 2 c 2
− ⎛ ⎞− = − + = +⎜ ⎟⎝ ⎠
λ ν νχ ν λ χ ρ
with
(41) 2 2
p ' ' pc 2 c
⎛ ⎞= − +⎜ ⎟⎝ ⎠ν ρ
Attheendwegetthe«TOV»1equation(Tolmann-Oppenheimer-Volkoff):
(42)
p'c2 = − m + 4π G pr3 / c4
r r − 2m( ) ρ + pc2
⎛⎝⎜
⎞⎠⎟
WhenweapplyNewtonianapproxximatio ( p<<ρc2 2m << r ) weget
(43)
Insphericalsymmetry, thegravitational fieldwhichprevailsatadistance r < rs (insidethestarofsupposedconstantdensity) isequal to the fieldwhichwouldbecreatedbythemasscontained inasphereofradiusrs,concentrated inthecentre.Thusequation(43)canbeidentifiedwiththeconservationequation(32)onpage7ofDamour'spaper:
∂i p = + ∂i U
Although it is terribly tedious it is essential to resume, line after line, all thesecalculations (here, classical) in order to extend them to the calculation of the innermetricdescribingthenegativespecies.Whenthiswillbedone, furtheron,wewillseethat without this precaution taken concerning the tensor we would end up with thesamecontraction.
1Whichcorrespondstoequation(14.22)inreference[2].
p' = − ρmc2
r2 = − G M ρr2
30
Continuingthecalculation,wewillnowexplainthecompletecalculationoftheinteriormetric
( gµν
(+ ) identifiée à gµν ) .
Takingthenotationofthereference[2]weask:
(44) R̂ = 3c2
8πGρ
Ashasbeenestablishedabove(34)that:
(45) m(r) = 4π G ρ r3
3c2
Thiswillimmediatelygiveusoneofthetermsofthemetric:
(46) e−λ = 1− 2m(r)
r= 1− 8πGρ r2
3c2 ≡ 1− r2
R̂2
Andsoourinnermetriciswritten:
(47)
ds 2 = eν dx° 2 − dr 2
1− r2
R̂2
− r2 dθ 2 + sin2θ dϕ 2( )
Thefunction ν (r) hasyettobedetermined.Thedensityisassumedtobeconstant.Wehave..:
(48) 2
2p ''c p
= −+
νρ
è ν ' = −
2 ρc2 + p( ) 'ρc2 + p
= − 2Log(ρc2 + p)'
(49) −ν
2= Log(ρc2 + p)+ cte
è 2
2 2 2
8 G p pDec c c
− ⎛ ⎞ ⎛ ⎞= + = − +⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
ν π ρ χ ρ
Weuse(25)tosolve
(50) − ν '+ λ '
re−λ = χ ρ + p
c2
⎛⎝⎜
⎞⎠⎟= − De
−ν2
è
r De−ν
2 = ν 'e−λ + λ 'e−λ = ν 'e−λ − e−λ( ) '
(51) r De
−ν2 = ν ' 1 − r2
R̂2
⎛⎝⎜
⎞⎠⎟− 1 − r2
R̂2
⎛⎝⎜
⎞⎠⎟
' = ν ' 1 − r2
R̂2
⎛⎝⎜
⎞⎠⎟+ 2r
R̂2
Writong 2e (r)≡ν
γ è 2'' e2
=ννγ
(52) r D = ν 'e
ν2 1 − r2
R̂2
⎛⎝⎜
⎞⎠⎟+ 2r
R̂2 eν2 = 2γ ' 1 − r2
R̂2
⎛⎝⎜
⎞⎠⎟+ 2r
R̂2 γ
31
Aparticularsolutionoftheequationis2
pR̂ D2
=γ
Wemustfindageneralsolutiontothehomogeneousequation:
(53) u ' 1 − r2
R̂2
⎛⎝⎜
⎞⎠⎟+ r
R̂2 u = 0
è u = B 1 − r2
R̂2
⎛⎝⎜
⎞⎠⎟
1/2
whence:
(54) γ ≡ e
ν2 = R̂2D
2− B 1 − r2
R̂2
⎛⎝⎜
⎞⎠⎟
1/2
(55)
g00 = eν = A − B 1− r2
R̂2
⎛⎝⎜
⎞⎠⎟
1/2⎡
⎣⎢⎢
⎤
⎦⎥⎥
2
where:
(56)2
22
R̂ D A 2 8 G 2A D 2 A Aˆ2 3 c 3R= ⇒ = = = −ρ π ρχ
Letusnowexpressthatthepressureonthesurfaceofthesphereiszero:
(57)11/22 2
22 2
ˆp 2 R D rDe A B 1 ˆc 3 2 R
−− ⎡ ⎤⎛ ⎞⎛ ⎞= − + = − − −⎢ ⎥⎜ ⎟⎜ ⎟⎝ ⎠ ⎢ ⎥⎝ ⎠⎣ ⎦
ν ρχ ρ χ
(58) 1/22 2
2
p 2 Ac 3 rA B 1
R̂
+ =⎛ ⎞
− −⎜ ⎟⎝ ⎠
ρρ
QWhen r = rs wegetp=0
(59)
1= 23
A
A − B 1−rs
2
R̂2
⎛
⎝⎜⎞
⎠⎟
1/2
è A = 3B 1−
rs2
R̂2
⎛
⎝⎜⎞
⎠⎟
1/2
Il reste à déterminer B, ce que nous allons faire en imposant que les métriquesintérieuresetextérieuresseraccordentsurlasurfacedelasphère.Cequisetraduitpar:
(60)
g00int rs( ) = eν rs( ) = A − B 1−
rs2
R̂2
⎛
⎝⎜⎞
⎠⎟
1/2⎡
⎣⎢⎢
⎤
⎦⎥⎥
2
= g00ext rs( ) = 1− 2G M
rs c2
⎛
⎝⎜⎞
⎠⎟
32
(61)
B2 3 1−rs
2
R̂2
⎛
⎝⎜⎞
⎠⎟
1/2
− 1−rs
2
R̂2
⎛
⎝⎜⎞
⎠⎟
1/2⎡
⎣⎢⎢
⎤
⎦⎥⎥
2
= 1− 2G Mr0 c2
⎛
⎝⎜⎞
⎠⎟
(62) 4B2 1−
rs2
R̂2
⎛
⎝⎜⎞
⎠⎟= 1− 2G M
rs c2
⎛
⎝⎜⎞
⎠⎟
(63) 4B2 1−
8π Gρ rs2
3c2
⎛
⎝⎜⎞
⎠⎟= 1− 8π G
3c2 ρ rs2⎛
⎝⎜⎞⎠⎟⇒ B = 1
2
(64) A = 3
21−
rs2
R̂2
⎛
⎝⎜⎞
⎠⎟
1/2
(65)
g00int r( ) = 3
21−
rs2
R̂2
⎛
⎝⎜⎞
⎠⎟
1/2
− 12
1−r2
R̂2
⎛
⎝⎜⎞
⎠⎟
1/2⎡
⎣⎢⎢
⎤
⎦⎥⎥
2
Following,theinnermetric2:
(66)
ds 2 = 32
1−rs
2
R̂2
⎛
⎝⎜⎞
⎠⎟
1/2
− 12
1−r2
R̂2
⎛
⎝⎜⎞
⎠⎟
1/2⎡
⎣⎢⎢
⎤
⎦⎥⎥
2
dx° 2 − dr 2
1− r2
R̂2
− r2 dθ 2 + sin2θ dϕ 2( )
Wearenowgoingtodeploythesamecalculationscheme,butthistimeadaptingittothemetricdescribingthenegativemassspecies,whichisthenthesolutionoftheequation:
(67) Eµ
ν ≡ Rµν − 1
2gµν R = − χ −g
−gTµ
ν ≡ − χ ww⌢Tµ
ν
Thedeterminantsratiocanbewritten:
(68)
− g− g
=−det (gµν )
−det (gµν )=
eνe λ r4 sin2θ
eν e λ r4 sin2θ= e
ν2 e
λ2 e
− ν2 e
− λ2 ≡ kD
kDisclosetountitybecausewedealwithNewtonianapproximation.
Thistimewecalculatetheimpactofthepresenceofthepositivemassesonthegeometry
gµν ofthenegativesector.Werecallthatweareperfectlyfreetochoosethis
tensor ⌢Tµ
ν ,asthischoicecanresultfromaLagrangianderivation.Andwehaveseen,choiceXVIII,thatweoptfor:
(69)
2Equation(14.47)delaréférence[2]
33
⌢Tµ
ν =
ρ 0 0 0
0 pc2 0 0
0 0 pc2 0
0 0 0 pc2
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟⎟
hypothesis which does not weight on the whole model since in the Newtonianapproximationthepressuretermsarealwaysnegligible.ThisthereforelimitsthescopeofthemodeltothisfieldoftheNewtonianapproximation.Butthisonecoversallknownobservations.
WewillshowthatthisoptionnolongerleadstotheinconsistencyreportedbyDamourinhispaper.
Onceagain,wedeclinetheconstructionofthefirstmemberfromametricwhichthistimeis:
(70) ds 2 = eν dx° 2 − eλ dr 2 − r2 dθ 2 + sin2θ dϕ 2( )
Thefirstmembersoftheequationsarethesame,simplyreplace (ν , λ ) with (ν , λ ) .Wethenget
(71) e−λ 1
r2 − λ 'r
⎛⎝⎜
⎞⎠⎟− 1
r2 = − χ ρ
(72) e −λ 1
r2 + ν 'r
⎛⎝⎜
⎞⎠⎟− 1
r2 = − χ pc2
(73) e−λ ν "
2− ν 'λ '
4+ ν ' 2
4+ ν '− λ '
2r⎡
⎣⎢
⎤
⎦⎥ = −χ p
c2
(74) − ν '+ λ '
re−λ = − χ ρ − p
c2
⎛⎝⎜
⎞⎠⎟
(75) 2
2 2
e 1 ' ' ' ' ' "r r 4 4 2r 2
+= − + + −λ ν ν λ ν λ ν
Tosoovewewrite
(76) e−λ ≡ 1− 2m
rsoit
2m = r 1− e−λ( )
(77) 2m' = 1− e−λ( ) + rλ 'e−λ
è
− 2m'
r2 = − 1r2 + e−λ 1
r2 −λ 'r
⎛⎝⎜
⎞⎠⎟
34
Using(71): m ' = − 4π r2 G
c2 ρ è
m(r) = m'(r)dr
0
r
∫ = − 43π r3ρ G
c2 = −m
Inconclusion,atthisstage:
(78) m (r) = − m(r)
Weget
(79) ν ' = 2 −m + 4π G pr3 / c4
r r + 2m( )
Toeliminate ν '' wedérive(72)
(80) −χ p'
c2 = 2r3 − λ 'e −λ 1
r2 +ν 'r
⎛⎝⎜
⎞⎠⎟+ e −λ −2
r3 + ν ''r− ν '
r2
⎛⎝⎜
⎞⎠⎟
−χ p'
c2 = 2r3 − e −λ λ '
r2 + λ 'ν 'r
+ 2r3 −
ν ''r+ ν '
r2
⎛⎝⎜
⎞⎠⎟
−χ p'
c2 = 2r3 − 2 e −λ
rλ '2r
+ λ 'ν '2
+ 1r2 −
ν ''2+ ν '
2r⎛⎝⎜
⎞⎠⎟
−χ p'
c2 = 2r3 − 2 e −λ
r1r2 −
ν ' 2
4+ λ 'ν '
4+ λ '+ν '
2r− ν ''
2+ ν ' 2
4+ λ 'ν '
4⎛⎝⎜
⎞⎠⎟
Combiningto(75)weget
(81) −χ p'
c2 = 2r3 − 2 e −λ
reλ
r2 + ν ' 2
4+ λ 'ν '
4⎛
⎝⎜⎞
⎠⎟= −2 e −λ
rν ' 2
4+ λ 'ν '
4⎛⎝⎜
⎞⎠⎟
(82) −χ p'
c2 = − e −λν '2r
ν '+ λ '( )
Using(74)
(83): −χ p'
c2 = −ν '+ λ '( )
re −λ ν '
2= −χ ρ − p
c2
⎛⎝⎜
⎞⎠⎟ν '2
Finally:
(84):
p'c2 = − m − 4πGpr3 / c4
r r + 2m( ) ρ − pc2
⎛⎝⎜
⎞⎠⎟
35
Tobeomparedwithwhatemergedfromtheanalysisforpositivemasses,i.e.equation(43):
p'c2 = − m + 4πGpr3 / c4
r r − 2m( ) ρ + pc2
⎛⎝⎜
⎞⎠⎟
Thesedifferentialequationsarenotidentical,unlesstheNewtonianapproximationisused,thentheyleadtothesameresult
(85): p' = − m ρ c2
r2
Whichisequivalentto(32): p' = − m ρ
r2fromDamour’paper[1],page7.
Thephysicalandmathematicalinconsistencyofthemodeldisappears.Onecouldobjectthat this limits the solutions to those that fit this Newtonian approximation. But incosmology,whatmoredoyouaskfor?
Better a model that provides calculation results limited to the conditions of theNewtonian approximation (i.e. to all the data available observationally) than anextremely ambitious model (Damour and Kogan 2001) that promises us non-linearsolutions but which, in the end, does not offer a possible confrontation with theobservations.
Wearegoingtofinalizethecalculationoftheinnermetricofthenegativespecies,aswedidearlier.Wewillnotomitanyintermediaryofcalculationtobesurethatanerror(ithappenedquickly)willnotslipintotheprocess.
(86) ν ' = 2p'
ρc2 − p( ) Toexpressinnermetric:
(87) e−λ = 1− 2m
r= 1+ r2
R̂2
Giventhatbyassumption ρ isconstant.
(88)
ν '= −2p'− ρc2 + p( ) = − 2
ρc2 − p( ) 'ρc2 − p( ) = − 2 Log( ρc2 − p )'
(89) −ν
2= Log(ρc2 − p)'+ cte
Write:
36
(90) !De
− ν2 = − χ ρ − p
c2
⎛⎝⎜
⎞⎠⎟
Tosooveweuse(74)
(91) De
−ν2 =− χ ( ρ − p
c2 ) = −ν '+ λ 'r
e−λ
(92) −r De
−ν2 =ν 'e−λ − e−λ( ) '
(93) −r De
−ν2 =ν ' 1+ r2
R̂2
⎛⎝⎜
⎞⎠⎟− 1+ r2
R̂2
⎛⎝⎜
⎞⎠⎟
' = ν ' 1+ r2
R̂2
⎛⎝⎜
⎞⎠⎟− 2r
R̂2
Let:
(94) eν2 ≡ γ (r) è
γ ' = ν '
2e
ν2
whence:
(95) −r D = 2 ν '
2eν2 1+ r2
R̂2
⎛⎝⎜
⎞⎠⎟− 2r
R̂2 eν2 = 2γ ' 1+ r2
R̂2
⎛⎝⎜
⎞⎠⎟− 2r
R̂2 γ
Aparticularsolutiontothisdifferentialequationis:
(96) γ p =
R̂2 D2
Wehavetofindthegeneralsolutiontothehomogeneousequation:
(97)2
2 2
r ru ' 1 u 0ˆ ˆR R⎛ ⎞
+ − =⎜ ⎟⎝ ⎠
thatis:
(98) u = B 1+ r2
R̂2
⎛⎝⎜
⎞⎠⎟
1/2
Whencethegeneralsolutionis:
(99) γ ≡ e
ν2 = R̂2D
2+ B 1+ r2
R̂2
⎛⎝⎜
⎞⎠⎟
1/2
Let’scalculatethecomponentseofthemetric gµν :
37
(100)
g00 = e ν= A + B 1+ r2
R̂2
⎛⎝⎜
⎞⎠⎟
1/2⎡
⎣⎢⎢
⎤
⎦⎥⎥
2
Weget:
(101)
R̂ 2D2
≡ A ⇒ D = 2 AR̂2 = 2 8π G ρ
3c2 A = − χ 2ρ3
A
Weknowthat:
(102)
De−ν
2 = −χ ( ρ − pc2 ) = −χ 2ρ
3A e
−ν2 = −χ 2ρ
3A
A + B 1+ r2
R̂2
⎛⎝⎜
⎞⎠⎟
1/2
(103)
( ρ − pc2 ) = 2ρ
3A
A + B 1+ r2
R̂2
⎛⎝⎜
⎞⎠⎟
1/2
Thepressureonthesurfaceofthesphereisexpressedaszero.
(104) A = − 3 B 1+
rs2
R̂2
⎛
⎝⎜⎞
⎠⎟
1/2
TodetermineBwewillmakesurethatthereisacontinuousconnectionbetweentheinnermetricandtheoutermetric,wherer=rs
Weknowwehave:
(105) g11
int = − e λ = − 1+ r2
R̂2
⎛⎝⎜
⎞⎠⎟
−1
(106)
g00int r0( ) = e ν rs( ) = A + B 1+
rs2
R̂2
⎛
⎝⎜⎞
⎠⎟
1/2⎡
⎣⎢⎢
⎤
⎦⎥⎥
2
= g00ext rs( ) = 1+
rs2
R̂2
⎛
⎝⎜⎞
⎠⎟
(107)
− 3 B 1+r0
2
R̂2
⎛
⎝⎜⎞
⎠⎟
1/2
+ B 1+rs
2
R̂2
⎛
⎝⎜⎞
⎠⎟
1/2⎡
⎣⎢⎢
⎤
⎦⎥⎥
2
= 1+r0
2
R̂2
⎛
⎝⎜⎞
⎠⎟= 4 B2 1+
rs2
R̂2
⎛
⎝⎜⎞
⎠⎟
(108) 1B̂2
=
38
(109) A = − 3
21+
rs2
R̂2
⎛
⎝⎜⎞
⎠⎟
1/2
(110)
g00int r( ) = eν = − 3
21+
rs2
R̂2
⎛
⎝⎜⎞
⎠⎟
1/2
+ 12
1+r2
R̂2
⎛
⎝⎜⎞
⎠⎟
1/2⎡
⎣⎢⎢
⎤
⎦⎥⎥
2
Let’swritethefinalexpressionoftheinnermetriv gµν
(111)
ds 2 = 32
1+rs
2
R̂2
⎛
⎝⎜⎞
⎠⎟
1/2
− 12
1+r2
R̂2
⎛
⎝⎜⎞
⎠⎟
1/2⎡
⎣⎢⎢
⎤
⎦⎥⎥
2
dx° 2 − dr 2
1+ r2
R̂2
− r2 dθ 2 + sin2θ dϕ 2( )
whichfitstheexteriormetric
(112)
d!s 2 = 1+ 2G M
c2 r⎛⎝⎜
⎞⎠⎟
c2dt2 − dr2
1+ 2G Mc2 r
− r2(dθ 2 + sin2θ dϕ 2 )
Linearizedforms:
(113) ds 2 = 1+ 3
2rs
2
R̂2 − 12
r2
R̂2
⎛
⎝⎜⎞
⎠⎟dx° 2 − 1− r2
R̂2
⎛⎝⎜
⎞⎠⎟
dr 2 − r2 dθ 2 + sin2θ dϕ 2( )
(114) d!s 2 = 1+ 2G M
c2 r⎛⎝⎜
⎞⎠⎟
c2dt2 − 1− 2G Mc2 r
⎛⎝⎜
⎞⎠⎟
dr2 − r2(dθ 2 + sin2θ dϕ 2 )
References:
[1]T.Damour:Surlemodèle«Janus»deJ.P.Petithttp://www.ihes.fr/~damour/publications/JanusJanvier2019-1.pdf
[2]Adler,SchifferetBazin:IntroductiontoGeneralRelativity.http://www.jp-petit.org/books/asb.pdf
39
Annexe3:
ThibaudDamour,IHES2019Januarythefourth
Aboutthe«JanusCosmologicalModelofJ.P.Petit
(translatedbyJ.P.Petit)
Beforeallletusgiveourconclusion:
The«JanusCosmologicalModel»isphysically(andmathematically)unconsistent
TheJanusequationsarethefollowing:
(1a) G µν
(+ ) = χ Tµν(+ ) + g(− )
g(+ ) Tµν(− )
⎡
⎣⎢⎢
⎤
⎦⎥⎥
(1b) G µν
(− ) = − χ − g(+ )
g(− ) Tµν(+ ) +Tµν
(− )⎡
⎣⎢⎢
⎤
⎦⎥⎥
With G µν
(+ ) = Rµν(+ ) − 1
2R(+ ) gµν
(+ ) G µν(− ) = Rµν
(− ) − 12
R(− ) gµν(− )
Theclassicaldefinitionof Tµν
(+ ) whichensuresitstensorialconservationwithrespectto
gµν
(+ ) is:
−g(+ ) T µν
(+ ) ≡ −2δSmatter(+ )
δ g(+ )
Where Smatter(+ ) referstotheactionoftheordinarymatter.Thereisnoneedtogivethe
definitionof Tµν
(− ) ,whichwasnotprecisedintheworksofPetitandd’Agostini.
The«JanusModel»doesnotfittheBianchiidentities.Ineffectthesystem(1a)+(1b)goeswith:
(2a) ∇(+ )
ν G µν(+ ) = 0
(2b)
∇(− )
ν G µν(− ) = 0
40
Tµν = − w
wTµν Considerthecase Tµν
(− ) = 0sothattheJanussystembecomes:
(3a) G µν
(+ ) = χ Tµν(+ )
(3b)
G µν
(− ) = − χ Tµν(+ )
Letuswrite:
gµν
(+ ) = gµν gµν(− ) = gµν
−g(+ ) = w −g(− ) = w
G µν
(+ ) = G µν G µν(− ) = G µν
Tµν
(+ ) = Tµν Tµν = − ww
Tµν
ThetheJanussystembecomes:
(4a) G µν = χ Tµν
(4b)
G µν = χ Tµν
with(4c):
Tµν = − w
wTµν
The authors have introduced the factor
ww is order to cure a difficulty to some
unconsistency linked toa simplifiedmodelbut aswillbe shown further thisdoesnotprevent the severe unconsistency in the case of the hydrostatic equilibriumwhenweconsiderthecasofaself-gravitatingstar,intheNewtonianlimit c→∞
Thecentralpointisbasedontheconstainsts
(5a) ∇νTµν = 0
(5b)
∇νTµν = 0
where∇ istheconnectionlinkedto gµν .
Toillustratesuchpointletusconsiderthesimplecasewherethe«positive»mattercomesbothfromabackgroundsource
Tµν
o (forexampleastar,orthesuninoursolar
41
system),consideredasaspherefilledbyauniformdistributionof«dust»,i.e
Tµν
1 = ρ1 uµ uν ,then:
(6a) Tµν = Tµν
o + ρ1 uµ uν
(6b) Tµν = Tµν
o + ρ1 uµ uν
where
(7) uµ =
uµ
Nwith N2 ≡ − gµν uµ uν
(8) ρ1 = − N2 w
wρ1
(9) Tµν
o = − ww
Tµνo
Herethecovariant4-velocityfield uµ is,definedwithrespecttothemetric gµν ,sothat
gµν uµ uν = − 1 .Consideredwithrespecttothesecondmetric
gµν theco-vectorialfield
definesinauniquewaytheequivalent4-velocityfield g − unitary uµ (with
gµν uµ uν = − 1 )asdefinedabove.
Nowconsiderthetwoconservationlaws(5a)and(5b).
Letusfirstconcentrateonthemovementofthetestdustmatter.Thelaws(5a)and(5b)thefollowingconstrainst:
(10) ∇µ uµ = 0
(11) ∇µ (ρ1u
µ ) = 0
(12) ∇µ uµ = 0
(12) ∇µ (ρ1u
µ ) = 0
Thephysicalmeaningoftheequation(10)isthefollowing.Itshowsthatthelinesoftheuniverse of the matter (defined by u
µ = gµν uν ) are geodesics of gµν ≡ gµν(+ ) , while the
thirdequation (12) says that the samepositivematter is also ruled (by theequations
"−" )toobeyanotherequationsofthemovement ∇µ uµ = 0 whichshowsthattheline
of the universe defined by uµ = gµν uν must be geodesics derived from the gµν ≡ gµν
(+ )
metric.Butthe4-velocityfield uµ isnot independentof u
µ .Consideredasacovariant
42
field it isbasically thesame througha renormalization factor uµ = uµ / N , equation, so
that uµ = gµν uν / N = gµσ gσνuν / N. As the two metrics
gµν ≡ gµν
(+ ) and gµν ≡ gµν
(+ ) are aprioridifferentIdon’tseehowitcouldbepossible(consideringacomplexgeneraltimedependentsolution,definedbyarbitraryCauchydatafor
gµν and gµν )tohavethesame
matter following different motion equations. If we consider for example some initialvelocitydata foraa testdust, suchvelocitywouldbesupposed to folllowat thesametime two distinct rules of evolution, which is mathematically absurd for a classicaltheory!
Anotherphysico-mathematicalcontradictionmayarise fromequations(4a)and(4b)applying such system to the structure of a self-gravitating star, in Newtonian limit.Considerabackgroundsourcecorrespondingtoaperfectfluid:
(13) Tµν = T µν
o(+ ) = ( ρc2 + p ) uµ uν + pgµν
IwilllimittheanalysistothealmostNewtonianconditions.Iwillshowthatthistheoryisselfcontradictoryanddoesnotleadtoanyphysicalsolution.
IrecallthatthelinearizedsolutionoftheEinsteinequationsmaybewritten:
(14) goo = − (1− 2
Uc2 ) ; gi j = + (1+ 2
Uc2 )δ i j
whereUisthenewtonianpotentialfromPoissonequation:
(15) ΔU = − 4π G
Too
c2 1+ 0( 1
c2)⎛
⎝⎜⎞⎠⎟= − 4π G ρ 1+ 0( 1
c2)⎛
⎝⎜⎞⎠⎟
Duetotheformalsymmetryofthesystem(4a)+(4b)wegetthecorrespondinglinearizedsolution:
(16) goo = − (1− 2
Uc2 ) ; gi j = + (1+ 2
c2 )δ i j
wherethequasiNewtonianpotential U obeys:
(17) ΔU = − 4π G
Too
c2 1+ 01
c2( )⎛⎝⎜
⎞⎠⎟= − 4π G ρ 1+ 0
1
c2( )⎛⎝⎜
⎞⎠⎟
from(9)with w / w = 1+ 0( 1
c2) ρ issimply- ρ .Sothat:
(18) U = − U 1 + 0 1
c2( )⎛⎝⎜
⎞⎠⎟
NowIshifttoanotherthingthatshowstheunconsistencyofthe«JanusModel».Afterequation(4c)
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(19) Ti j = − w
wTi j = − 1+ 4 U
c2 + 0( 1c4 )
⎛⎝⎜
⎞⎠⎟
Ti j
Itisnowveryimportanttotakeinchargetheconsequencesoftheequations(5a)and(5b)whichactonthesameenergy-impulsiontensor.
Irecall:
(20) ∇ν Tµ
ν = 1w
∂ν (w Tµν ) − 1
2∂µ gα β Tα β
Ifireferstospace:
(21) ∇ν Ti
ν = 1w
∂ν (w Tiν ) − 1
2∂i gα β Tα β
IntheNewtonianapproximation,inthelasttermthecontributionfrom α = β = 0 isdominantbecause T
oo = 0(c2 ) while Toi = 0(c1) and T
i j = 0(co ) .Then
(22) 0 = ∇ν Ti
ν = ∂ j(Tij) − Too
c2 ∂i U + 01
c2
⎛⎝⎜
⎞⎠⎟ = ∂ j(Ti
j) − ρ∂i U + 01
c2
⎛⎝⎜
⎞⎠⎟
IrecallthatintheNewtonianapproximationtheorderofmagnitudeof Ti j isunity,i.e.is
when c→∞ .
Forexample,foraperfectmovingfluidwehave Ti j = ρ vi v j + pδ i j + 0(1/ c2 ). Thenthe
aboveequation(whenfullfilledby
1w∂o (w T i
o) = ∂t (ρ vi ) + 0(1/ c2 ) )isnothing(when
c→∞ )buttheclassicalhydrodynamicalEulerequation.Ihaveconsideredastaticcase,withtheequilibriumofaself-gravitatingstar.
Now,considerthesecondconservationlaw(5b).Weshallhave:
(23) ∇ν Ti
ν = 1w∂ j(w Ti
j) − 12∂i gα β Tα β
Thus,finally:
(24) 0 = ∇ν Ti
ν = ∂ j(Tij) − ρ ∂i U + 0(1/ c2 )
InthissecondEulerequation: T ij →− T i
j ρ → − ρ U →− U then
(25) 0 = ∇ν Ti
ν = − ∂ j(Tij) − ρ∂i U + 0(1/ c2 )
whichcontradictstheclassicalEulerequation(22).
Ifthestarisfilledbyaperfectfluidthisstaticequilibriumimpliesboth
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(26) ∂i p = + ρ∂i U and ∂i p = − ρ∂i U
CONCLUSION:Thesystemofcoupledequationsofthe«JanusModel»aremathematicallyandphysicallycontradictory.
