The time independent spherically symmetric solution of the ... · revisited with coordinates which...

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The time independent spherically symmetric solution of the Einstein equation revisited

Jean-Pierre Petit, Gilles d’Agostini and Sebastien Michea +

Manaty Research group

_________________________________________________________________________________________ Key words: Non contractible hypersurface. Throat sphere, space bridge. Sphericallysymmetricsolution.____________________________________________________________________________________________________

Abstract: Spherical symmetry does not immediately mean central symmetry. The timeindependent, spherically symmetric solution of the homogeneous Einstein equation isrevisitedwithcoordinateswhichkeepthesignature invariantandprevent timeandradialcoordinateinterchange.Theassociatedhypersurfaceisnotcontractibleandcorrespondstoa space bridge linking two Minkowski spacetimes though a throat sphere. As thedeterminant of the metric vanishes on that sphere one gets an orbifold structure. WhencrossingthatspheretheparticlesexperienceaPT,massandenergyinversions.____________________________________________________________________________________________________

Introductionandmainideaofthisarticle.In 1916 Karl Schwarzschild publishes [1] a solution of the vacuum Einstein equations(withoutsecondterm)correspondtotimetranslationinvarianceandsphericalsymmetry.Itis only in 1999 that an English translation of this article will be available [2] thanks toS.AntocietA.Loinger.Schwarzschilddecidestoexpressthis solutionbyusingafirstsetofrealvariables(a) t , x , y , z{ }∈!4

Thesolutionisthengivenintheform:(b) ds2 = Fdt2 − G (dx2 +dy2 +dz2 ) − H( xdx + ydy + zdz )2 Hethenintroducesanintermediaryvariable:(c) r = x2 + y2 + z2 Which,given(a)isessentiallypositive.He performs a new coordinates change that allows him to simply express the sphericalsymmetryhypothesis:(d) x = r sinθ cosϕ y = r sinθ sinϕ z = r cosθ

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Hethenobainstheform:(e) ds2 = Fdt2 − (G + H r2 ) dr2 − G r2 (dθ 2 +sin2θ dϕ 2 ) Inorderforthemetrictobelorentzianatinfinityitisnecessarythat:(f) r →∞ implies F→1, G →1 , H → 0 Heintroducesnextanewchangeofcoordinates:

(g) x1 =

r3

3, x2 = − cosθ , x3 =ϕ , x4 = t

Notethat(a)+(c)implythat x1 ≥ 0 Inthosenewcoordinatesthemetricbecomes:

(h) ds2 = f4 dx4

2 − f1 dx12 − f2

dx22

1− x22 − f3 dx3

2 ( 1− x22 )

where f1 , f2 = f3 , f4 are three functions of x1 ( hence of r ) that must satisfy theconditions:

-Forlarge x1 : f1 =

1r4 = (3x1)− 4/3 , f2 = f3 = r2 = (3x1) 2/3 , f4 = 1

-Thedeterminant f1 f2 f3 f4 = 1

-Themetricmustbeasolutionofthefieldequation.-Exceptfor x1 = 0 theffunctionsmustbecontinuous.

Hiscomputationleadshimto:

(i) f1 =

( 3x1 +α3 )− 4/3

1− α ( 3x1 +α3 )− 1/3 f2= f3 = ( 3x1 +α

3 ) 2/3 f4 = 1− 1− α ( 3x1 +α3 )− 1/3

α beinganintegrationconstant.Using(g)wecanrewritehissolutionas:

(j) ds2 = 1− α

( r3 +α 3 )1/3

⎡

⎣⎢

⎤

⎦⎥ dt2 − r4

( r3 +α 3 ) ( r3 +α 3 )1/3 − α⎡⎣ ⎤⎦dr2 − ( r3 +α 3 )2/3( dθ 2 + sin2θ dϕ 2 )

Thatwecanrewrite:

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(k)

ds2 = gt t dt2 − gr r dr2 − gθθ dθ 2 − gϕϕ dϕ 2

Thecoefficients

gt t , gr r et gθθ arefunctionsoftheonlyvariabler,thequantity gϕϕ beinga

functionofbothrandϕ .Thevariablervariesfrom0toinfinitywhilebeingstrictlypositivebydefinition(c).Letuslimitourselftothecase α > 0 andletconsiderapathcorrespondingtodt=dr=0.Ithasaperimeter:(l) p = 2π ( r3 +α 3 )1/3 Thisperimeterhasaminimalvalue p = 2πα .

Thehypersurfacesolutionisthereforenoncontractible.Thervariablecannotbeconsideredasa«radialdistance».Thishypersurfacedoesnothavea«centre»,theobjectcorrespondingtor=0hencetoàx=y=z=0isnotadotbutasphereofradiusα .WewillnowreplicateScharzschild'scalculationsbyreplacingthex,y,zandtvariablesbyGreekcharacters( ρ ishoweverkeptforalaterpurpose),sothatthereadernotbetemptedtoassimilatethemtodistances(x,yandz)ortime(t)andlosesightofwhattheyreallyare:realnumbers.

RevisitingSchwarzchildcomputation.LetusconsiderthezerosecondmemberEinsteinequation

Rµν = 0 intimeindependentand

sphericallysymmetricalconditions.Let ξ1 ,ξ2 ,ξ3 standforrectangularcoordinates,and ξo

as the time marker, with ( ξ0 , ξ1 , ξ2 , ξ3 )∈ R4 which stands real values for allcoordinates.Inadditionweassumethattherearenocrossedtermsinthelineelement,sothatthislastcanbewritten:(1)ds

2 = Fdξo2 −G (dξ12 + dξ22+ dξ32 )−H (ξ1dξ1 + ξ2dξ2 + ξ3dξ3 )2

whereF,G,Harefunctionsof ξ12 + ξ2

2 + ξ32 .

Atinfinitywemusthave(2) F and G →1 H → 0 Introducethefollowingcoordinatechange:

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(3) ζ = ξ12 + ξ2

2 + ξ32

(4) θ = arcos

ξ3

ζ⎛⎝⎜

⎞⎠⎟

(5)

ϕ = arccosξ1

ξ12 + ξ2

2

⎛

⎝⎜⎜

⎞

⎠⎟⎟

whichgoeswith: ζ ∈ R + θ ∈ R ϕ ∈ R and:

ξ1 =ζ sinθ cosϕ ,ξ2 =ζ sinθ sinϕ ,ξ3 =ζ cosθ

thatwewillcall«pseudosphericalcoordinates».Itgives:(6) ds2 = Fdξo

2 − ( G + Hζ 2 ) dζ 2 − Gζ 2 ( dθ 2 + sin2θ dϕ 2 ) Introducethefollowingadditionalcoordinatechange::

(7)η1 =

ζ 3

3 , η2 = − cosθ , η3 =ϕ

Thenwehavethevolumeelement ζ2sinθ dζ dθdϕ = dξ 1 dξ 2 dξ 3 .Thenewvariablesare

thenpseudopolarcoordinatewith thedeterminant1.Theyhave theevidentadvantageofpolarcoordinatesforthetreatmentoftheproblem.

Inthenewpseudopolarcoordinates:

(8) ds2 = Fdξo

2 − ( Gζ 4 + H

ζ 2 )dη12 − Gζ 2 dη 2

2

1− η 22 + dη3

2 1− η 22( )⎡

⎣⎢⎢

⎤

⎦⎥⎥

forwhichwewrite:

(9)ds 2= f4dξo

2 − f1dη 12 − f2

dη 22

1−η 22 − f3 dη 3

2 (1−η 22)

Thenf1,f2=f3,f4arefunctionsofη1 whichhavetofullfillthefollowingconditions:

1–Forξ1 =∞ : f1 = 1

ζ 4 = (3η1)− 4

3 , f2 = f3 =ζ2= (3η1)

23 , f4 =1

2–Theequationofthedeterminant: f1 . f2 . f3 . f4 = 1

3–Thefieldequations.

4–Continuityofthef,exceptforη1 =0

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In order to formulate the field equations one must first form the components of thegravitational fieldcorrespondingtothe lineelement(9).Thishappensinthesimplestwaywhen one builds the differential equation of the geodesic line by direct execution of thevariation,andreadsoutthecomponentsofthese.Thedifferentialequationsofthegeodesiclineforthelineelement(9)immediatlyresultfromthevariationintheform:

(10)

0= f1

d 2η 1d s 2

+ 12∂ f4∂η1

dη4d s

⎛

⎝⎜⎞

⎠⎟

2

+ 12∂ f1∂η1

dη1d s

⎛

⎝⎜⎞

⎠⎟

2

− 12∂ f2∂η1

11−η22

dη2d s

⎛

⎝⎜⎞

⎠⎟

2

+ (1−η22)dη3d s

⎛

⎝⎜⎞

⎠⎟

2⎡

⎣

⎢⎢

⎤

⎦

⎥⎥

(11)

0= f2

1−η22d 2η2

d s 2+∂ f2∂η1

11−η12

dη1d s

dη2d s

+f2η2

(1−η12)2dη2d s

⎛

⎝⎜⎞

⎠⎟

2

+ f2η2dη3d s

⎛

⎝⎜⎞

⎠⎟

2

(12)

0= f2(1−η22)

d 2η3

d s 2+∂ f2∂η1

(1−η22)dη1d s

dη3d s

−2 f2η2dη2d s

dη3d s

(13)

0= f4

d 2η4

d s 2+∂ f4∂η1

dη1d s

dη4d s

Thecomparisonwith

(14)

d 2ηα

ds 2= − 12 Γµν

α

µ ,ν∑

dηµ

d sdην

d s

givesthecomponentsofthegravitationalfield:

(15a)Γ1 1

1 = − 12∂ f1∂η1

(15b)Γ2 2

1 = + 121f1

∂ f2∂η1

11−η22

(15c)Γ3 3

1 = + 121f1

∂ f2∂η1

(1−η22)

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(15d)Γ2 1

2 = − 121f2

∂ f2∂η1

(15e)Γ2 2

2 = −η2

1−η22

(15f)Γ3 32 = −η2 (1−η22)

(15g)Γ3 1

1 = − 121f2

∂ f2∂η1

(15h)Γ3 2

2 =η2

1−η22

(15i)Γ4 1

4 = − 121f4

∂ f4∂η1

The other ones are zero. Due to rotational symmetry it is sufficient to write the fieldequations only for the equator (η2 =0) , therefore, since they will be differentiated onlyonce, in the previous expressions it is possible to set everywhere since the begining

1−η22 =0 .Thenthecalculationofthefieldequationgives:

(16a)∂∂η1

1f1

∂ f1∂η1

⎛

⎝⎜⎞

⎠⎟= 12

1f1

∂ f1∂η1

⎛

⎝⎜⎞

⎠⎟

2

+ 1f2

∂ f2∂η1

⎛

⎝⎜⎞

⎠⎟

2

+ 1f4

∂ f4∂η1

⎛

⎝⎜⎞

⎠⎟

2

(16b)∂∂η1

1f1

∂ f2∂η1

⎛

⎝⎜⎞

⎠⎟=2+ 1

f1 f2

∂ f2∂η1

⎛

⎝⎜⎞

⎠⎟

2

(16c)∂∂η1

1f1

∂ f4∂η1

⎛

⎝⎜⎞

⎠⎟= 1f1 f4

∂ f4∂η1

⎛

⎝⎜⎞

⎠⎟

2

Besides these three equations the functions f1 , f2 , f3 must fullfill the equation of thedeterminant:

(17)f1 f2

2 f4 =1 i.e. 1f1

∂ f1∂η1

+ 2f2

∂ f2∂η1

+ 1f4

∂ f4∂η1

=0

Fornowweneglect(16b)anddeterminethethreefunctions f1 , f2 , f4 from(16a),(16c)and(13).Theequation(16c)canbetransposedintotheform:

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(18)∂∂η1

1f4

∂ f4∂η1

⎛

⎝⎜⎞

⎠⎟= 1f1 f4

∂ f1∂η1

∂ f4∂η1

Thiscanbeintegratedandgives

(19)1f4

∂ f4∂η1

= α f1 (α beinganintegrationconstant)

Theadditionof(12a)and(12c’)gives:

(20)∂∂η1

1f1

∂ f1∂η1

+ 1f4

∂ f4∂η1

⎛

⎝⎜⎞

⎠⎟= 1

f2

∂ f2∂η1

⎛

⎝⎜⎞

⎠⎟

2

+ 121f1

∂ f1∂η1

+ 1f4

∂ f4∂η1

⎛

⎝⎜⎞

⎠⎟

2

Bytaking(17)intoaccountweget:

(21)−2 ∂

∂η1

1f2

∂ f2∂η1

⎛

⎝⎜⎞

⎠⎟= 3 1

f2

∂ f2∂η1

⎛

⎝⎜⎞

⎠⎟

2

Byintegrating:

(22)

11f2

∂ f2∂η1

= 32η1 +

σ2 (σ integrationconstant)

or:

(23)1f2

∂ f2∂η1

= 13η1 +σ

Afterasecondintegration:

(24)f2 = λ (3η1 +σ )23 (λ integrationconstant)

Theconditionatinfinityrequiresλ =1.Then

(19)f2 = (3η1 +σ )23

Henceitresultsfurtherfrom(22)and(17)

(20)

∂ f4∂η1

= α f1 f4 =αf22 =

α

(3η1 +σ )43

Byintegrating,whiletakingaccounttheconditionatinfinity

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(25) f4 = 1−α (3η1 +σ )−13

Hence,from(17)

(26)

f1 =

(3η1 +σ )− 4

3

1−α (3η1 +σ )−13

As it canbe easily verified the equation (16b) is automatically fullfilledby the expressionthatwefoundforf1andf2

Thereforealltheconditionsaresatisfiedexcepttheconditionofcontinuity.

f1willbediscontinuouswhen1=α (3η1 +α3 )

−13 , 3η1 =α 3 −σ

Inorderthisdisconinuitycoincideswiththeorigin,itmustbe:

(27)σ =α 3

Therefore the condition of continuity relates in thisway the two integration constants σ andα .Wehave:

(28) 3η1 +α = ζ 3 +α

(29) f4 = 1− α

(ζ 3+α 3 )1/3

(30) f2 = (ζ 3+α 3 )2/3

(31)

dη 22

1− η22 = dθ 2

Finally:

(32)

ds2 = 1− α(ζ 3+α 3 )1/3

⎛⎝⎜

⎞⎠⎟

dξo2 − ζ 4

α 3 +ζ 3( ) α 3 +ζ 3( )1/3−α⎡

⎣⎢⎤⎦⎥

dζ 2 − α 3 +ζ 3( )2/3dθ 2 + sin2θ dφ 2( )

Whenζ → ∞ thelineelementtendstoLorentzform.

gt t tendstozerowhenζ tendstozero.

When ζ → 0 gζ ζ !

00.Anexpansionintoaseriesshowsthat

gζ ζ !

3ζα

→ 0

Consideralooplocatedintheplane θ = 0 ,with dξ0 = 0 .Itisnocontractile:forζ =0the

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perimeterofthelooptendsto 2πα .

Notice that ζ = ξ12 + ξ2

2 + ξ32 is definitely not a «radius». The point ζ = 0 does not

correspond to some «center of symmetry». The spherically symmetric requirement doesnotidentifyautomaticallytoacentralsymmetry,assuggestedbyDavidHilbert[5]1ζ isjustoneofthe«spacemarkers»,nothingelse.It’sanumber,notalength.Theonlylengthtobeconsideredisthequantitys.

Expressingthemetricinabettercoordinatesystem.

Letusconsiderthecoordinatesystemintroducedin[3].

Introducethenewspacemarker ρ throughthefollowingcoordinatechange:

(33) ζ = α 1+ Logcos hρ( )

(34) ρ = ± argch ( e1+ ζ

α

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟

3

3 −1

)

ζ = 0 → ρ = 0

With ξo = c t thelineelementbecomes:

(35)

ds2 = Logcos hρ

1+ Logcos hρc2dt2 − 1+ Logcos hρ

Logcos hρα 2 tan h 2ρ dρ2 − α 2 (1+ Logcos hρ )2 ( dθ 2 + sin2θ dϕ 2 )

When ρ → ±∞ wegetthefollowingLorentzform:

(36) ds2 = c2dt2 − α 2 dρ2 − α 2 ρ2 ( dθ 2 + sin2θ dϕ 2 )

Wecan figure this geometricalobjet as a spacebridge linking twoMinkowski spacetimes,though a throat sphere whose perimeter is 2πα . We cannot think about its «radius»becausethatspherehasnocenter.

When ρ→ 0

1Wequote,page67ofthegermanedition:«DieGravitation

gµν istzentrischsymmetrischinBezugaufdenKooridinatenanfangspunkt.«

Englishtranslation:«Thegravitation gµν iscentrallysymmetricwithrespecttotheoriginof

coordinates.»

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(37) gt t =

Logcos hρ1+ Logcos hρ

c2 → 0

(38) gρ ρ = − 1+ Logcos hρ

Logcos hρα 2 tan h 2ρ → 0

0

(39) gθθ → α 2 gϕϕ → α 2 sin2θ

(40) gϕϕ =α 2 sin2θ

Wemayeasilyovercometheindetermination(33)throughanexpansionintoaseries,whichshowsthatwhen ρ → 0

(41) gρ ρ = − 1+ Logcos hρ

Logcos hρα 2 tan h 2ρ → − 2

Thedeterminantis:

(42) det gµν = − c2α 6 tan h2ρ sin2θ

Itvanisheson the throatsphere.Asaconsequence,on this lastwecannotdefinegaussiancoordinates,sothattheobjectisnolongeramanifoldbutanorbifold.Onthethroatspherethearrowoftimeandthespaceorientationcannotnedefined.Thiscanbeinterpretedasageometricstructurewherespaceandtimearereversedthroughthethroatsphere:whenaparticle crosses the throat sphere it experiences a PT-symmetry. According to Souriau’stheorem[4]thisT-inversiongoeswithamassinversion.

Inafuturepaperthephysicalinterpretationofsuchsolutionwillbeinvestigated.

References:

[1] K. Schwarzschild : Über das Gravitionalsfeld einer Kugel Aus incompressibler Flüssigkeit nach der Einsteinschen Theorie. Sitzung der phys. Math. Klasse v.23 märz 1916 [2] K. Schwarzschild : On the gravitational field of a sphere of incompressible fluid according to Einstein theory. Translation by S.Antoci and A.Loinger. arXiv :physics/9905030v1 [physics.hist-ph] 12 may 1999.

[3]J.P.Petit&G.D’Agostini:CancellationofthesingularityoftheSchwarzschildsolutionwithnaturalmassinversionprocess.Mod.Phys.Lett.Avol.30n°92015

[4]J.M.Souriau:Structuredessystèmesdynamiques.DunodEd.France,1970andStructureof Dynamical Systems. Boston, Birkhaüser Ed. 1997. For time inversion see page 190equation(14.67).

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[5] D.Hilbert : “DieGrundlagender Physij. (ZweiteMitteilung)” inNachchrichten vonderKöniglichenGesellschaftzuGöttingen.Math.-Phys.Klasse.1917,p.53-76.Presentedinthesessionof23december1916