Solution of a Spherically Symmetric Static Problem of General Relativity for an Elastic Solid Sphere

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1 Applied Physics eseach; Vol. 9, No. 6; 7 ISSN E-ISSN Published by Canadian Cente of Science and Education Solution of a Spheically Symmetic Static Poblem of Geneal elativity fo an Elastic Solid Sphee Valey V. Vasiliev & Leonid V. Fedoov Institute of Poblems in Mechanics, ussian Academy of Sciences, Venadskoo, Moscow 956, ussia Coespondence: Valey V. Vasiliev, Institute of Poblems in Mechanics, ussian Academy of Sciences, Venadskoo, Moscow 956, ussia. Tel: vvvas@dol.u eceived: Septembe, 7 Accepted: Septembe, 7 Online Published: Octobe 9, 7 doi:.559/ap.v9n6p8 UL: Abstact The pape is concened with the spheically symmetic static poblem of the Geneal elativity Theoy (GT. The classical inteio solution of this poblem found in 96 by K. Schwazschild fo a fluid sphee is enealized fo a linea elastic isotopic solid sphee. The GT equations ae supplemented with the equation fo the stesses which is simila to the compatibility equation of the theoy of elasticity and is deived usin the pinciple of minimum complementay eney fo an elastic solid. Numeical analysis of the obtained solution is undetaken. Keywods: eneal elativity theoy, theoy of elasticity, spheically symmetic static poblem. Intoduction. Theoy of Elasticity Solution To intoduce the poposed appoach to GT poblem fo elastic solid, conside the poblem of the classical theoy of elasticity fo a sphee whose avitational field is descibed by the Newton theoy. Fo a solid sphee with constant density μ and adius, the Newton avitational potential φ is the solution of the Poisson equation ( φ = 4π Gμ ( Hee, is the adial coodinate (, ( = d( / d and G is the classical avitational constant. Fo the extenal ( space, μ = and Equation ( has the followin well known solution: Gm φ e = ( Whee index e coesponds to the extenal space and 4 m = πμ ( is the mass of a homoeneous solid sphee whose intenal space is Euclidean. Fo the intenal ( space the solution of Equation ( which satisfies the eulaity condition at the sphee cente is φi = πμg + C (4 Hee, index i coesponds to the intenal space. The inteation constant C is detemined fom the bounday condition on the sphee suface accodin to which φe( = φi(. Usin Equation (, we can pesent Equation (4 in the followin final fom: Gm φi = (5 The avitational body foces which act inside the sphee ae 4 f = μφ i = k, k G Then, the theoy of elasticity equilibium equation fo the sphee element can be pesented as = π μ (6 8

2 ap.ccsenet.o Applied Physics eseach Vol. 9, No. 6; 7 + = (7 ( k Hee, and ae the adial and the cicumfeential stesses which ae accompanied by the coespondin elastic stains followin fom Hooke s law, i.e. ε = ( ν, ε = [ ( ν ν ] (8 E E in which E and ν ae the elastic modulus and the Poisson s atio of the sphee mateial. The stains ae expessed in tems of the adial displacement u as ε = u, ε = u/ (9 Substitutin u fom the second of these equations in the fist one, we aive at the followin compatibility equation fo the stains: ε + ε ε = Usin Equations (7, we can wite this equation in tems of stesses, i.e. ( ν ν + ( ν( = ( Thus, we have two equations, Equations (7 and (, fo two unknown stesses. To educe these equations to one equation with espect to the adial stess, expess usin Equation (7, i.e. = + ( k ( and substitute it in Equation ( to et ν + 4 k = ( ν The solution of this equation must satisfy the followin bounday conditions: ( = = ( =, ( = = ( Usin Equation (, we can tansfom the fist of these conditions to ( = =. The final solution fo the stesses is ( ν k ( ν k = ( + ν, = (4 ( ν ( ν ν Havin in mind to obtain the solution of the poblem unde study within the famewok of GT, we should take into account the GT equations ae fomulated in the iemannian space in which the displacement u, as well as the stain-displacement equations, Equations (9, do not exist. Thus, we cannot deive the compatibility equation, Equation ( usin the taditional appoach. Howeve, theoy of elasticity povides anothe way to obtain this equation not attactin Equations (9. As known, the compatibility equation fomulated in stesses follows fom the pinciple of minimum of the complementay eney unde the condition that the stesses satisfy the equilibium equations. The elastic eney of the solid sphee is w d U = 4π (5 whee w = ( ε + ε is the elastic potential. Substitutin the stains fom Hooke s law, Equations (8, in Equation (5, expessin in tems of with the aid of Equation ( and thus satisfyin the equilibium equation, we can educe the complementay eney to the followin functional: π U = F(,, d E (6 whee 9

3 ap.ccsenet.o Applied Physics eseach Vol. 9, No. 6; 7 ν 4 F = ( ν( + k + ( k + k (7 The Eule equation which povides the minimum value of the complementay eney is F d F = d (8 Substitutin F fom Equation (7, we aive at the compatibility equation, Equation (. In conclusion, tansfom the obtained esults intoducin the followin dimensionless paametes:, = =, = (9 μc Hee, c is the velocity of liht and mg = ( c is the so-called avitational adius. Usin Equations ( and (6 fo m and k and applyin Equations (9, we can pesent the compatibility equation, Equation ( in the followin fom: d ( d ν + 4 = ( d d ( ν The nomalized stesses become ( ν ( ν + ν = (, = ( ( ν ( ν ν. Spheically Symmetic Static Poblem of Geneal elativity fo an Elastic Sphee Fo a spheically symmetic static poblem, the line element is taditionally taken in the followin fom coespondin to the classical Schwazchild solution: ds = d + ( d + sin dφ c dt ( Hee,, φ and t ae space spheical and time coodinates, ij ae the metic coefficients that depend on the adial coodinate only. Fo the spheically symmetic static poblem and the line element in Equation (, the consevation equation which is analoous to the equilibium equation, Equation (7 of the theoy of elasticity, is (Syne, 96 ( ( c + μ = (4 Hee, the stesses and the density ae expessed in tems of the metic coefficients with aid of Einstein s equations which can be pesented as (Syne, 96 χ = + (5 χ = + (6 χμc = (7 whee χ πg c 4 = 8 / (8

4 ap.ccsenet.o Applied Physics eseach Vol. 9, No. 6; 7 is the GT avitational constant. As known, substitution of Equations (5-(7 in Equation (4 identically satisfies this equation. Pefom some tansfomations. Fist, conside Equation (7. The solutions of this equation fo the extenal (, μ = and intenal (, μ = const spaces ae specified by the well known exteio and inteio Schwazchild solutions which have the fom (Syne, 96 e = /, i = / (9 Hee, indices e and i coespond to extenal and intenal spaces, wheeas is the avitational adius specified by Equation (. Second, intoduce function f ( as = f, = f + f ( Finally, substitutin the second of Equations (9 and Equations ( in Equations (5 and (6, we aive at χ = f ( f χ f = f + Followin the appoach descibed in Section, expess f in tems of usin Equation ( ( f = / χ and substitute this esult in Equation (, i.e. = 4 + χ 6 4( / χ Accodin to the basic idea of GT, the stesses specified by the Einstein equations, Equations ( and (, identically satisfy the equilibium equation, Equation (4. Tansfomin this equation with the aid of Equations (, ( and (8 fo m,, χ and substitutin fom Equation (, we can eadily pove that the equilibium equation is satisfied. Thus, to detemine, we can apply the pinciple of minimum of the complementay eney discussed in Section. In the iemannian space, Equation (5 fo the complementay eney is enealized as U = 4π w d Usin Equation (, we can educe it to the functional in Equation (6. Omittin the explicit expession fo the function F which is athe cumbesome, pesent the Eule equation, Equation (8, which takes the followin final fom: d d ( ν ( + ( ν( (4 4( ( ν + ν d d + 7ν + (+ ν 9 ( ν ( ν = (4 To simplify this equation, we use dimensionless paametes in Equations (9 and Equations(, ( and (8 fo m, and χ. Nelectin the tems with in compaison with unity and omittin nonlinea tems, we aive at equation ( of the theoy of elasticity.. Numeical Analysis Fo the numeical analysis, we take ν = and educe Equation (4 to (

5 ap.ccsenet.o Applied Physics eseach Vol. 9, No. 6; 7 d d ( + ( (4 + (8 7 + (+ d d = (5 9 ( This equation is numeically inteated unde the bounday conditions that follow fom Equations (, i.e. ( = = and ( = =. The cicumfeential stess is found fom Equation ( which can be tansfomed to d = ( + 4 d 4( (6 It is inteestin to compae the esults that follow fom Equations (5 and (6 with the theoy of elasticity solution specified by Equations ( and with the inteio Schwazchild solution obtained fo a sphee of pefect incompessible fluid. This solution has the followin fom (Syne, 96: p p = = μc and demonstates a specific behavio. Takin = in Equation (7, detemine the pessue at the sphee cente, i.e. (7 p = (8 The denominato of this expession is zeo fo the sphee with adius = 8/9=.8888, and the pessue becomes infinitely hih at the sphee cente. This esult is sometimes used to suppot the existence of the objects efeed to as the Black Holes (Thone, 994. The esults of the numeical analysis ae pesented in Fiues,. Fiue demonstates the distibutions of the nomalized adial (solid lines and cicumfeential (dashed lines stesses ove the adial coodinate fo =.,.4,.6,.8,.99. (a (b Fiue. Distibutions of the nomalized adial (solid lines and cicumfeential (dashed lines stesses ove the adial coodinate fo (a =.,.4,.6 and (b.8,.9,.99 Fiue shows the nomalized stesses at the sphee cente as functions of the nomalized avitational adius.

6 ap.ccsenet.o Applied Physics eseach Vol. 9, No. 6; 7 - the obtained solution - the theoy of elasticity solution - pessue in the fluid sphee Fiue. Dependences of the nomalized stesses at the sphee cente on the adial coodinate As can be seen, in contast to the pessue specified by Equation (8 which becomes infinitely hih at = 8/9, the stesses ae finite fo the sphee with the adius = 9/8. Fo =, the numeical solution does not convee. Howeve, it does not look like the stesses ae sinula in this case. It seems that the toleance of the applied numeical pocedue is not as hih as should be. The last esult =.5886 is obtained fo =.99. Dashed line in Fiue coesponds to the theoy of elasticity solution specified by Equations (. 4. Conclusion The solution of the Schwazchild spheically symmetic static poblem fo a fluid sphee is enealized fo a linea elastic sphee. The equation fo the stesses missin in GT is deived usin the minimum complementay eney pinciple of the theoy of elasticity. In contast to the sinula Schwazchild solution fo the pessue in the fluid, the stesses do not demonstate sinula behavio fo the elastic sphee whose adius is equal to the avitational adius. efeences Syne, J. L. (96. elativity: the Geneal Theoy. Amstedam, Noth Holland. Thon, K. S. (994. Black Holes and Time Waps Einstein s Outaes Leacy. New Yok, London, W.W. Noton and Company. Copyihts Copyiht fo this aticle is etained by the autho(s, with fist publication ihts anted to the jounal. This is an open-access aticle distibuted unde the tems and conditions of the Ceative Commons Attibution license (