A Newtonian equivalent fo the cosmological constant Mugu B. Răuţ We deduce fom Newtonian mechanics the cosmological constant, following some olde ideas. An equivalent to this constant in classical mechanics it was found. Theefoe in ou development, the cosmological constant appeas in a natual way into Fiedmann cosmological model. But the theoetical context in which it appeas tells almost nothing about the natue of expanding univese foce. Intoduction The Newtonian deivation of the Fiedmann equations by Milne [1] and McCea [] came afte Fiedmann [] and Lamaite [4,5] demonstation. They, [, 4, 5] had used geneal elativity, fo an unbounded homogeneous univese, to show the validity of the Fiedmann equations. Milne and McCea, using a Newtonian fomalism, had obtained the same esults like elativistic cosmology theoy. Nevetheless this Newtonian cosmology can not explain all the obsevational data since it does not contain a theoy of light popagation. Even so, this appoach is quite legitimate, since the stuctual similaity of geneal elativity and Newtonian celestial mechanics wee pointed out by Catan [6,7]. Following his ideas [8] showed a coect deivation of the Fiedmann equations fom Newtonian theoy. The development of Milne [1] leads to some poblems:1) the foce on any paticle of an infinite homogeneous distibution is undetemined [9]; ) the finiteness of foce equies eithe the mass distibution to be finite, and thus homogeneous, o inhomogeneous and infinite, fo cosmological distances; ) the ecession speed of the mass distibution paticles is close to the speed of light, which pesume a elativistic teatment of the expanding sphee. These poblems wee patially solved in [10], consideing that the expanding sphee descibe a small egion compaed with the size of the obsevable univese. Since all egions of an unifom and isotopic univese expend the same way, the study of a small egion may give infomation about the whole univese, in which case the Newtonian teatment is coect [11]. Howeve, this solution solved some issues and leads to othe poblems egading the cente of a local isotopic and homogeneous expansion that is a point impossible to establish. Cate [1] and McCea [1] indicated that each point can be chosen as cente with local isotopy and homogeneity suounding, but thee is a zone at the bounday of the expanding sphee whee these popeties ae boken. But the same poblem occus in geneal elativity too. By taking into account all above consideations we can conclude that the Newtonian cosmological models deived fom Fiedmann s equations ae pefectly valid fo an homogeneous univese and fo small scale of length. We pesently develop a follow-up of Milne ideas by intoducing in a natual manne the cosmological constant. Empiically intoduced by Einstein in 1917 [14], the cosmological constant, afte the discovey of expanding univese [15], was epelled by its autho. Howeve, in Fiedmann model, appeas fom geneal elativity equations of field deduced by Lemaite in 197, [,4,16]. Milne, [1,], demonstated his Newtonian deivation, in which the cosmological constant appeas by postulation.
The main aim of this pape is to show that can be intoduced diectly, without postulation. Theoetical teatment 1 If two sphees attact each othe with a foce popotional with, as demonstated by Newton s theoem, then we can eplace the two sphees with points that have the mass of the associated sphees. In this case one can study [17], the geneal fom of the gavitational potential and the limits of this appoximation. Consideing a spheical suface with adius, density, situated at distance fom a cente O and an abitay exteio point P, one obseves that the gavitational potential in P is equivalent to one geneated by the mass m ( ) placed in O. We can wite this as a function of the gavitational potential due to a cental mass at distance, () : m ( ) ( ) ( ) ( ) d (1) whee ( ) is a constant which can be added to the potential without alteing its associated law foce. Equation (1) has two classes of solutions. The fist one, [18], is function of the Yukawa type potentials: A1e Ae ( ) A () with the equivalent mass: sh( ) m ( ) 4 () and A 1, A, A being abitay eal constants, is a constant R o C and ( ) A When A 0 and 0 we obtain the mass of sphee in the Newtonian paticula case: m ( ) 4 The second class of solutions of equation (1) contains the algebaic potentials [19]: B1 ( ) B B (4) with the same equivalent mass: m ( ) 4, B, B B abitay eal constants and 1, ( ) B B
1 The Newtonian potential is obtained by taking into account anothe popety of invese squae adius foce. Fo the potential the inteio of a spheical suface must be an equipotent egion. In geneal () will have this popety if: ( ) ( ) d, fo < (5) Equation (5) has the unique solution: B ( ) C (6) whee the constant C can be zeo without it s associated foce law being alteed. The potential (6) has been used by Milne to deive the fist Fiedmann equation fom the consevation of the total enegy: d / dt 8G kc c ( ) (7) whee the cosmological constant has been intoduced by postulation. Nevetheless if we obseve that equation (4) includes the tem B which is the Newtonian equivalent of a cosmological constant, one may econside Milne s deivation. Fist thing we must do is to pesume valid the potential (4). In othe wods to conside as valid the hypothesis that the inteio egion of the supposed spheical suface will not to be an equipotent egion. Consequently we have: B1 ( ) B (8) which is the potential (4) with B 0, an opeation which simplifies (4) without the associated foce law being alteed. Then we have: B1 GM, whee G is the Newton s gavitational constant and M is the entie mass within the sphee, 4 M, and is the mass density. The constant B is pesumed positive, it coespond to a epulsive foce. The gavitational enegy of a paticle of mass m on motion within the potential (8) is: GMm E g B m The kinetic enegy of the same paticle can be witten as: 1 E k m( d / dt) The consevation of the total enegy leads, afte an elementay calculus, to equation: d / dt 8G kc ( ) B (9) with k a dimensionless constant given by:
E k mc The total foce on a paticle with mass m is: 4G F m( B ) (10) If F 0, leading to d / dt 0 and const., we obtain: 4G B (11) Using the same pocedue like in the case of equation (7) we find the same esult c fo the quantity.this indicates that thee is a physical equivalence between the two quantities, obseved diectly fom similaities between equations (7) and (9). The only significant diffeence between them is the fact that one is deduced natually fom equation (8), the othe is postulated. In conclusion if we neglect equation (7) and set: c B we have been deduced the cosmological constant fom Newtonian theoy. Discussions The deivation of cosmological constant fom Newton s theoy doesn t solve the poblem of its physical natue. We can t conceive a Newtonian cosmology pue and simple based on this theoy. To establish a concodance with obsevational data we must exceed the Newtonian theoy and make the assumptions which lead us to s physical natue. Thus the constant B will esult fom othe theoies, as until now. Some ealy genealizations of geneal elativity theoy found that the cosmological constant aose natually fom the mathematics, [0, 1]. But these ideas don t have a natual suppot and wee soon abandoned. Inevitable we must conside B as an intinsic enegy density of the vacuum in which case the fom (11) it will be conseved. A positive cosmological constant is geneated by a negative vacuum pessue. The sign minus between the constant B and the cosmological constant legitimate this idea. A positive epelling foce coesponds only to a negative pessue. Hence the cosmological constant poblem is occuing, which is known like the most difficult situation of fine-tuning in physics. Thee is no coheent pocedue to deive the cosmological constant fom paticle physics, and also the moden field theoies ae pessimistic egading this matte. Even so the constant seams to be the best solution fo any complete cosmological model, like a cyclic one. Following [] we obseve that (6) is, fo l 0, the depending pat on solution of the Laplace s equation, []: l l1 R l( ) Al Bl, [ 0, ) l (1) but the potential (4) it is not. As illustated in ef. [], the potential (4) could have a distibution with absolute minima o maxima. It means that a Newtonian univese without a cosmological constant is instable and geneates paadoxes. Unde these cicumstances we have only an
expanding univese. By including the cosmological constant makes sense to a cyclic cosmological model. Fo this eason we need, othewise we don t have altenative to an etenal expanding univese. A coheent theoy could be build only if we accept the modified potential (4) instead the common potential (6). On the othe hand a Newtonian celestial mechanics govened by (4) it doesn t make sense at small distances. But the opposite situation of having two Newtonian mechanics, one fo small distances, (eq. 6) and anothe fo cosmological distances, (eq.4), intoduce too much ambiguity into the theoy. If we imagine the situation of two existing foce laws, only one opeable, this is completely unscientific. Thee is only one Newtonian mechanics, govened by only one potential, no matte what distances ae claimed. Compaing the potentials expessed by equation (4) and (6), one may obseve that the fome has a simple epesentation but thee is a econciliatoy way, egading to all aspects, fo potential (4): to make it looks like (6). In othe wods, one could fomulate a new potential with a modified gavitational constant. The cumulated effects of the attacting and epelling potentials within elation (4) can be witten as an effect of a single sensibly smalle gavitational constant. Futhe wok will conside this poblem. Conclusions In this pape we obtain the complete Newtonian deivation of the fist Fiedmann s equation. The cosmological constant was intoduced in a natual way, as it esulted fom (4), by calculus not by postulation. It is discussed then the impotance of in context of a cyclic cosmological model, in which is indispensable. As anothe consequence it appeas to be the modified gavitational constant which holds fo geat values of the univese mass, at any scale we want. Refeences [1] E. Milne- A Newtonian expanding univese, Quately J. of Math. 5 (194), 64 [] W.H. McCea and E. Milne- Newtonian univeses and the cuvatue of space, Quately J. of Math. 5 (194), 7 [] A. Fiedmann- Ube die Kummung des Raumes, Z. Phys. 10 (19), 77-86 [4] G. Lemaite-An. Soc. Sci. Buxelles A 47, 49, 197 [5] G. Lemaite-Mon. Not. Roy. Aston. Soc. 91, 48, 191 [6] E. Catan-Ann. Sci. de l Ecole Nomale Supeieue 40 (19), 5 [7] E. Catan-Ann. Sci. de l Ecole Nomale Supeieue 41 (194), 1 [8] C. Ruede and N. Staumann- On Newton-Catan Cosmology, [qqc/9604054] [9] D. Layze- On the significance of Newtonian cosmology, Aston. J. 59 (1954), 68
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