A metric of Yukawa potential as an exact solution to the fie

It is shown that, by defining a suitable energy momentum tensor, the field equations of general relativity admit a line element of Yukawa potential as an exact solution. It is also shown that matter that produces strong force may be negative, in which case

AmetricofYukawapotentialasanexactsolution

tothe eldequationsofgeneralrelativity

arXiv:hep-th/9506154v3 25 Jun 1995

VuBHo

DepartmentofPhysicsMonashUniversityClaytonVictoria3168

Australia

Abstract

Itisshownthat,byde ningasuitableenergymomentumtensor,the eldequa-tionsofgeneralrelativityadmitalineelementofYukawapotentialasanexactsolution.Itisalsoshownthatmatterthatproducesstrongforcemaybenegative,inwhichcasetherewouldbenoSchwarzschild-likesingularity.

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It is shown that, by defining a suitable energy momentum tensor, the field equations of general relativity admit a line element of Yukawa potential as an exact solution. It is also shown that matter that produces strong force may be negative, in which case

1.Approximatesolution

IthasbeenshownthatthelineelementofYukawapotentialoftheform

e

ν

=1 α

e βr

gµνR=κTµν,2

assumingacentrallysymmetricspacetimemetric

ds2=eµdt2 eνdr2 r2(dθ2+sin2θdφ2).

(2)(3)

Ingeneral,thequantityR=1/βspeci esarangesothatthelineelementofYukawapotentialcanbeapproximatedasasolutiontothe eldequationsofgeneralrelativityfortheregionr R.ItisseenthatforthemaximalpossiblerangeofR→∞,themetricofYukawaformreducestothefamiliarSchwarzschildmetric,whichisusedtodescribethegravitational eld.Inthecaseofshortrangeofnuclearphysics,thequantityRcanbeassignedavalueintermsofthefundamentalconstantsh¯andc,andtherestmassofYukawaforcecarrier.Theresulthasshownthatwithintheshortrangeofstrongforce,the eldequationsofgeneralrelativityadmitalineelementthattakestheformofYukawapotentialforstronginteraction.Thisleadstotheconclusionthatifthereisnootherformofmatter,besidesthemassandthecharge,thatcharacterisesstronginteraction,thenitwouldbepossibletoconsiderstronginteractionalsoamanifestationofgeneralrelativityatshortrange.

Assumethemotionofaparticleinastrongforce eldisgovernedbythegeodesicequations

d2xµdxσ

ds

r

1

cdt 1 α

2

22

e βr

dr

dτ

r

+r2sin2θ

dφ

r

2

dt

dτ

r2

dθ

dτ

dd

r

=0,=0,

(7)(8)

dτ dt

It is shown that, by defining a suitable energy momentum tensor, the field equations of general relativity admit a line element of Yukawa potential as an exact solution. It is also shown that matter that produces strong force may be negative, in which case

Bychoosingsphericalpolarcoordinatesandconsideringthemotionintheplaneθ=π/2,theequations(8)and(9)arereducedto

dφ

r,2

dt

r αe βr

,

(10)

wherelandkareconstantsofintegration.Withtheserelations,theequationfortheorbit

canbeobtainedfromtheequation(6)as

ldφ

2

+

l2

r

e

βr

+

αl2

drr2

r2

=c(k (1+αβ))+

22

αc2

r3

.(12)

Bylettingu=1/randdi erentiatingtheresultingequationwithrespecttothevariableφ,itisfound

3αl2d2u

+2l2

e βrκ

κ

2κ

2κ

e βr

e βr

e βr

It is shown that, by defining a suitable energy momentum tensor, the field equations of general relativity admit a line element of Yukawa potential as an exact solution. It is also shown that matter that produces strong force may be negative, in which case

Withthisenergymomentumtensor,the eldequationsofgeneralrelativityreducetothesystemofequations[3]

r2

+

µ

e ν

ν

+1

, e ν

r µ

r

r

+

1

r

,

ν

µ

2

r

r

µ r

+e

µ

2

νc

r

+

√

g

gλσ2

(15)(16)

2

t

ν t

=αβ

2e

βr

It is shown that, by defining a suitable energy momentum tensor, the field equations of general relativity admit a line element of Yukawa potential as an exact solution. It is also shown that matter that produces strong force may be negative, in which case

[2]Lawden,D.F.AnIntroductiontoTensorCalculus,RelativityandCosmology(John

Wiley&Sons,1982).[3]Landau,L.D.&Lifshitz,E.M.TheClassicalTheoryofFields(PergamonPress,

1975).

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