The Tolman-Bondi Model in the Ruban-Chernin Coordinates. 1. Equations and Solutions.

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1 The Tolman-Bondi Model in the Ruban-Chernin Coordinates.. Equations and Solutions. arxiv:gr-qc/9638v 5 Dec 996 Alexander Gromov March, 8 Abstract It is demonstrated that the system of coordinates {M, t} defined by Ruban and Chernin is unique, two undetermined functions of the Tolman-Bondi (TB) model, f(m) and F(M), are dependent. Two initial conditions of the model are calculated through an initial profiles of density and energy, both are given as function of coordinate M. It is shown that in a general case of an arbitrary co-moving system of coordinates {r, t} two undetermined functions f(r) and F(r) define the transformation rule r M, which is given by the definition of invariant mass. It is shown that the Bonnor s flat solution of the TB model is reduced to an explicit dependence on the co-moving coordinate M in the Ruban-Chernin system of coordinates; non-flat solutions are dependent on M through initial conditions. The number of initial conditions and transformation rules are studied. PACS number(s): 95.3.Sf, r, k key words: cosmology:theory gravitation relativity The Introduction The observations [], [] show that in the large scale the Universe is not homogeneous. At the same time it is also supposed that the secreted centre is missing. These two properties separately are presented in the Friedmann-Robertson-Walker (FRW) and Tolman-Bondi (TB) models: the FRW model is homogeneous and does not include the secreted centre; the TB is nonhomogeneous and includes one. As the simplest nonhomogeneous model the TB model is used for interpretation of the observation data. The TB model is used to calculate the redshift [3], [4], [5], [6] to describe the local void [7], Submitted to New Astronomy St. Petersburg State Technical University, Faculty of Technical Cybernetics, Dept. of Computer Science, 9, Polytechnicheskaya str. St.-Petersburg, 955, Russia. gromov@natus.stud.pu.ru

2 [8], [9] to interpret the fractal structure of the matter distribution in the Universe [5], [6], []. A place of the TB model among the models consistent with the modern observation data is shown in []. The TB model [3], [] describes the spherical symmetry dust motion with zero pressure in a co-moving system of coordinates. The solution of the Tolman s equations obtained by Bonnor [3], [4] contains two undetermined functions f(r) and F(r) of the co-moving coordinate r which are defined by an initial conditions of the model. But the co-moving coordinate is defined up to a continuous transformation r = Φ(r), so the functions f(r) and F(r) are defined nonuniquely. This makes difficulty in an interpretation of the observation data. The article is based on the univalent definition of the co-moving coordinate given by Ruban and Chernin [5], a number of independent undetermined functions is decreased by one unit. All other examples of the co-moving coordinates may be obtained by the continuous transformation Φ. The article is dedicated to the formulation of the TB model in the Ruban-Chernin coordinates and to representation of the transformation rule Φ(r) : r M through the functions f(r) and F(r). It is shown that the Bonnor s flat solution of the TB model is reduced to an explicit dependence on the co-moving coordinate M in the Ruban-Chernin system of coordinates. The TB Model The TB model is represented by an interval, an equation of motion for the metrical function ω(r, t) with initial conditions for it and an equation for density []. In the spherical symmetry and co-moving system of coordinates the interval of the TB model has the form ds (r, t) = e ω(r,t) ω (r, t) 4 f (r) dr e ω(r,t) ( dθ + sin θdφ ) + c dt, () c is the velocity fo light, t is time, ω(r, t) is the metrical function, f(r) is the undetermined function, = c. The metric () is defined up to a continuous transformation Φ of the co-moving coordinate r [6]. The system of Einstein s equation is reduced to the equation of motion and the equation of density. The equation of motion is: e ( ω(r, ω(r,t) t) + 3 ) 4 ω (r, t) Λ + [ f (r) ] = () with initial conditions ω(r, t) t= = ω (r), ω(r, t) t= = ω (r), (3) Λ is a cosmological constant and ω (r) and ω (r) are given functions. Using the Bonnor [3] notation R(r, t) = e ω(r,t)/, (4) R(r, t) is an analog of the Euler coordinate of the particle with co-moving coordinate r, we rewrite the initial conditions in the form First integral of the equation () is: R(r, t) t= = R (r) = e ω(r)/, (5) Ṙ(r, t) = Ṙ(r) = t= ω (r)e ω(r)/. (6) ( ) R(r, t) = c f (r) + c F(r) 4 R(r, t) + Λ 6 c R (r, t), (7) F(r) is the second undetermined function. The function F(r) is defined from (5), (6) and (7). If Λ = then the equation (7) may be interpreted as an analog of the energy conservation law [3].

3 The integral of the equation (7) has the form ± t + F(r) = R(M,t) R (M) d R f (r) + F(r) R + Λ R, (8) 3 F(r) is the third undetermined function, the sign + corresponds to an expanding solution and the sign corresponds to the falling one. The definition of density is given in the TB model by the formula 8πG df(r) ρ(r, t) = c dr R (r, t). (9) R(r, t) r 3 The Definition of the Functions f, F and F in the Ruban-Chernin Coordinates First interpretation of the functions f(r) and F(r) has been presented in [3]. Following Bondi let us compare the equation (7) and the equation of the total energy in the Newtonian theory: ( R(r, t) ) = E(m) + Gm R(m, t), () m is the mass of the sphere the particle is located, E(m) is the full specific energy. In case of Λ =, the equations have the unique structure concerning R, so E (r) = c f (r) may be interpreted as an analog of a full specific energy and c Gm(r) F(r) = 4R(r, t) R(r, t) () () may be interpreted as an analog of a specific potential energy, m(r) is an effective mass which will be defined. In accordance with (9) the invariant density has the form ρ(r, t) g(r, t) = c F (r) 6 π G f(r) (3) and do not depend on time. The invariant mass is 4 π r M(r) = 4 π r ρ(r, t) g dr = ρ(r, t) R (r, t) R(r, t) r dr = c f(r) r 4 G F (r) dr. (4) f(r) Ruban and Chernin [5] have proposed to use the co-moving system of coordinates defined by the transformation r r = M(r) (5) 3

4 and shown that the use of M as a new independent coordinate allows to decrease a number of undetermined functions by one unit and to give the simple interpretation for M and R(M, t) [5], [7]: in the Newtonian limit M becomes the mass and R(M, t) becomes the radial (Euler) coordinate of the particle. From (4) it follows that F(M) = 4 G c f(m)dm. (6) This means that in the Ruban-Chernin coordinates it is not possible to choose the functions F(M) and f(m) independently. Now let us study the case of two system of coordinates: a system of an arbitrary coordinates {r, t} and the Ruban-Chernin system of coordinates {M, t}. Suppose the functions f(r) and F(r) are specified. The transformation rule from {r, t} to {M, t} is given by the formula (4), M(r) is the continuous transformation up to which the co-moving coordinate is defined. So, we have three functions f(r), F(r) and M(r), any two of which define the third function. From (3) and (6) it follows that the Ruban-Chernin system of coordinates is singled out by the fact that the invariant density is constant: ρ(r, t) g(r, t) = From () it follows that the effective mass m(m) is: m(m) = c 6 π G. (7) f( M)d M. (8) The initial condition E (M) defines the function f(m) by the formula (), so the formula (6) becomes the definition of the function F(M). We can now rewrite the equation (7) in the form ( ) R(M, t) = E (M)+ G R(M, t) + c E (M)dM + Λ 6 c R (M, t), (9) E (M) c. Tolman [] also uses the third undetermined function F to solve the equation (7). The function F(M) in the Ruban-Chernin coordinates has the form F(M) = R(M,t) R (M) d R. () E (M) + G m(m) + R c Λ 3 R 4 The Initial Conditions for the TB Model in the Ruban-Chernin Coordinates The equation of motion () for metrical function ω(m, t) requires two initial conditions: ω(m, ) and ω(m, ). These functions are obtained in this section. The idea of a possible dependence F(r) on f(r) has been proposed by Dr. Sergei Kopeikin [8]. 4

5 Let us suppose that an initial profile of the density is given as the function of the co-moving coordinate M. From (4) it follows that R(M, ) = R (M) = 3 4 π Substituting (4) into () we obtain the first initial condition ω(m, ) = 3 ln 3 4 π f( M) ρ( M, ) d M /3. () f( M) ρ( M, ) d M. () The second initial condition is obtained by the substitution (4), (5) and (6) into (9): ( ) ω(m, t) = c ω (M, ) = t= = 8 E (M) R (M, ) + 8 Gm(M) R 3 (M, ) c Λ. (3) For the initial profile of the velocity we obtain: R(M, ) = ± E (M) + Gm(M) R(M, ) + c Λ 3 R (M, ). (4) The equations () - (4) represent the initial conditions of the TB model through the initial profiles of density and energy. The last equation defines the initial profile of velocity. The full specific energy E (R) is limited by the meaning E min (R) when R(M,) = : E (R) E min (R) = Gm(M) R(M, ) c Λ 6 R (M, ). (5) 5 The Initial Conditions for the FRW Model in the Ruban-Chernin Coordinates The FRW model is the special case of the TB model which is specified by the condition ρ(m, t) M So, only the first initial condition is changed and reads: R FRW (M, ) = 3 4 π ρ FRW () =. (6) ω FRW (M, ) = 3 ln 3 4 π ρ FRW () f( M)d M /3, (7) f( M)d M. (8) 5

6 6 The Bonnor s Solution in the Ruban-Chernin Coordinates The solution of the equation of the TB model with Λ = has been obtained by Bonnor [3] and [4]. In this section we rewrite it in the Ruban-Chernin system of coordinates. From the equation (9) it follows that ( ) R(M, t) R(M, t ) E (M)R(M, t) = Gm(M). (9) This equation is studied together with the initial conditions (). The flat Bonnor s solution is obtained by the substitution of E (M) = and brings to the equation ( ) R(M, t) R(M, t ) = GM (3) which has the solution R 3/ (M, t) = R 3/ GM (M) ± 3 t, (3) satisfying the initial condition (). Now we will rewrite the Bonnor s solution for E (M) in the Ruban-Chernin system of coordinates. We will also use Ruban-Chernin notations. We obtain for E (M) < : R(M, t) = R m (M) { cos[η(m, t)]}, ± t + F(M) = Rm(M) E(M) {η(m, t) sin[η(m, t)]} E (M) <, (3) R m (M) = Gm(M) E (M), (33) R(M, t) η(m, t) = arcsin R m (M). (34) In the Ruban-Chernin system of coordinates the function F(M) has the form: F(M) = In case of E (M) > we obtain R m(m) E (M) {η (M) sin[η (M)]}, (35) η (M) = η(m, ) (36) R(M, t) = R m (M) {ch[η(m, t)] }, ± t + F(M) = Rm(M) E(M) {sh[η(m, t)] η(m, t)} E (M) >, (37) R m (M) = Gm(M) E (M), (38) R(M, t) η(m, t) = arcsh R m (M). (39) 6

7 Table : Examples of co-moving coordinates. author Φ : r new coordinate Ruban and Chernin,969 r M(r) Bonnor, 97 r α R (r) Bonnor, 97 r α ( + k r 3) /6 Bonnor, 974 r α F(r) In the Ruban-Chernin system of coordinates the function F(M) has the form: F(M) = R m(m) E (M) {sh[η (M)] η (M)}, (4) η (M) = η(m, ). (4) 7 The Examples of Functions f(r), F(r), Φ(r) and Initial Conditions r: The LTB model includes the following set of the functions of an arbitrary co-moving coordinate and the transformation f(r), F(r), R (r) Φ : r r. The function R (M) plays a role of the function F(r) in Tolman s notation []. Two functions f(r) and F(r) are defined by the initial conditions of the model. Φ is the transformation up to which the co-moving system of coordinate is defined. First we used the transformation Φ to define the Ruban-Chernin system of coordinates: r Φ r = M(r). In the system of coordinates {M, t} the definition of the invariant mass is reduced to the correlation between f(m) and F(M), so the number of independent functions is decreased by one unit. These independent functions are: f(m), R (M). A position of an arbitrary system of coordinates {r, t} with respect to the system {M, t} is defined by the functions f(r) and F(r). The definition of the invariant mass becomes the transformation rule r Φ M. Up to this moment we have used one transformation Φ, which has been produced by the definition of the invariant mass. In the Table we show several examples of co-moving coordinates given by Bonnor. Table shows the transformation rules from some system of coordinates {r, t} defined by two functions f(r) and F(r), to the Ruban-Chernin system of coordinates {M, t} produced by the functions f(r) and F(r). 7

8 Table : Examples of transformation rules. author f(r) F(r) M = Φ(r) Bonnor, 97 α R 3 (r) +k R 3 (r) 4 π α R 3 (r) +k R 3 (r) Bonnor, 974 α r 3 4 π α r 3 Bonnor, r α r 3 6 π α ( r + r Arsh(r) ) Bonnor and Chamorro 99 sin (r) k sin 3 (r) π k r Ribeiro 993 α r p 4 π α r p Ribeiro 993 cos(r) α r p 4 π α p r Ribeiro 993 cosh(r) α r p 4 π α p r r p d r cos( r) r p d r cosh( r) At the end of the section we will analyze two examples of initial conditions given by Bonnor [4] and Ribeiro [6]. They consider the flat Bonnor s solution in the form R(r, t) = [9 F(r)]/3 [t + β(r)] /3. (4) Independently from the fact how the function F depends on r, it follows from (4) that in case of f(r) = Comparing (4), (43) and (3) we find out that F(M) = 4 G M. (43) c R(M, t) = 9/3 ( ) /3 4 GM [t + β(m)] /3, (44) c β(m) = 3 R 3 (M) GM. (45) Using the initial coditions () we obtain β(m) = 6 π GM d M. (46) ρ ( M) Bonnor [4] studies one particular case of initial conditions It follows from (46) that β = β() =. (47) β() = 6 π Gρ () (48) 8

9 so, (47) means At the same time it follows from (44) and (47) that ρ () =. (49) R(M, ) =. (5) This means that the initail condition (47) produces the TB model with delta like distribution of dust: ρ (M) = δ(m). (5) The second example which we study is given by Ribeiro [6]. Ribeiro puts F(r) = α r p, β(r) = β + η r q, (5) α, β, η, p and q are constants. From (4) it follows that so, M = 4 π α r p (53) ( ) q/p M β(m) = β + η. (54) 4 πα From (54) and (46) the equation for initial density distribution it follows: 6 π G [ We obtain for ρ (): ρ (M) = ( ) ] q/p M β + η + π G η q 4 πα (4 π α) q/p p We also note that it follows from (55) that in case of q/p >. 8 Results [ ( ) ] q/p M β + η 4 π α M q/p. (55) ρ () = 6 π Gβ. (56) ρ (M) M = (57) M= The Ruban-Chernin system of co-moving coordinates {M, t} is used to describe the TB model. It was demonstrated that the system of coordinates is unique two functions, f(m) and F(M), are dependent. The dependence is fixed by the equation (6). It is shown that two undetermined functions of the TB model, f(r) and F(r), r is an arbitrary co-moving coordinate, define the transformation rule r M. The transformation rule is given by the definition of invariant mass (4). The third undetermined function becomes the initial condition () of the equation (9). This initial condition is calculated through the initial profile of density which is given as a function of coordinate M. The system of coordinates is also singled out by the fact that the invariant density is constant. The initial conditions for the TB model and for FRW model are obtained. The equations of the TB model with Λ = and their solutions, obtained by Bonnor, are rewritten in the Ruban-Chernin system of coordinates. We note here that the Newtonian analog of the Tolman- Bondi model has been stadied by Ruban and Chernin [5] has the same solution. It is shown that the Bonnor s solution is reduced to the explicit dependence on the co-moving coordinate M. The univalent definition of the Ruban-Chernin co-moving system of coordinates allows to simplify the interpretation of the observations using the model. 9

10 9 Acknowledgements This article has been written under the influence of the results obtained by Prof. William Bonnor, Prof. Arthur Chernin, Prof. Johne Moffat. I m grateful for encouragement and discussion to Prof. Arthur Chernin, Prof. Victor Brumberg, Prof. John Moffat, Dr.Yurij Baryshev, Dr. Andrzej Krasinski, Dr. Sergei Kopeikin, Dr.Sergei Krasnikov, Dr.Roman Zapatrin and Marina Vasil eva. Dr. Baryshev has initiated my interest to the modern Cosmology. Dr. Krasinski sent me his book Physics in an Inhomogeneous Universe. This paper was financially supported by COSMION Ltd., Moscow. References [] Davis M. 996, Is the Universe Homogeneous on Large Scales? Lecture presented at Critical dialogues in Cosmology, Princeton, June 4-8, USA. [] Pietronero L., Montuori M. and Sylos Labini F. 996, On the fractal structure of the visible universe, Lecture presented at Critical dialogues in Cosmology, Princeton, June 4-8, USA. [3] Bondi H. 947, MNRAS, 7, 4. [4] Moffat J.W. and Tatarski D.S. 99, Phys.Rev.D 45,. [5] Ribeiro M.B. 99, ApJ, 388,. [6] Ribeiro M.B. 993, ApJ 45. [7] Moffat J.W., Tatarski D.S. 995, ApJ, 453,. [8] Bonnor W.B.,Chamorro A. 99, ApJ, 36,. [9] Chamorro A. 99, ApJ, 383,. [] Ribeiro M.B. 99, ApJ 395,9. [] Baryshev Yu.V., Sylos Labini F., Montuori M. and Pietronero L. 994, Vistas in Astronomy, 38, 4. [] Tolman R.C. 934, Proc.Nat.Acad.Sci (Wash),. [3] Bonnor W.B. 97, MNRAS, 59, 6. [4] Bonnor W.B. 974, MNRAS, 67, 55. [5] Ruban V.A., Chernin A.D., 969, The isotropisation of the nonhomogeneous cosmological models. Proceeding of the 6th Winter School on the Cosmophysics, p.5, Appatiti. [6] Landau L.D., Lifshits E.M. 973, The Field Theory, Moscow, Nauka.

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