International Journal of Astronomy and Astrophysics, 2011, 1, 183-199 doi:10.4236/ijaa.2011.14024 Published Online December 2011 (http://www.SciRP.org/journal/ijaa)
An Astrophysical Peek into Einstein’s Static Universe: No Dark Energy
Abhas Mitra Theoretical Astrophysics Section, Bhabha Atomic Research Centre, Mumbai, India Homi Bhabha National Institute, Mumbai, India E-mail: [email protected] Received June 26, 2011; revised August 28, 2011; accepted September 13, 2011
Abstract
It is shown that in order that the fluid pressure and acceleration are uniform and finite in Einstein’s Static Universe (ESU), , the cosmological constant, is zero. being a fundamental constant, should be the same everywhere including the Friedman model. Independent proofs show that it must be so. Accordingly, the supposed acceleration of the universe and the attendant concept of a “Dark Energy” (DE) could be an illusion; an artifact of explaining cosmological observations in terms of an oversimplified model which is fundamentally inappropriate. Indeed observations show that the actual universe is lumpy and inhomogeneous at the largest scales. Further in order that there is no preferred centre, such an inhomogeneity might be ex- pressed in terms of infinite hierarchial fractals. Also, the recent finding that the Friedman model intrinsically corresponds to zero pressure (and hence zero temperature) in accordance with the fact that an ideal Hubble flow implies no collision, no randomness (Mitra, Astrophys. Sp. Sc., 333,351, 2011) too shows that the Friedman model cannot represent the real universe having pressure, temperature and radiation. Dark Energy might also be an artifact of the neglect of dust absorption of distant Type 1a supernovae coupled with likely evolution of supernovae luminosities or imprecise calibration of cosmic distance ladders or other systemetic errors (White, Rep. Prog. Phys., 70, 883, 2007). In reality, observations may not rule out an inhomogeneous static universe (Ellis, Gen. Rel. Grav. 9, 87, 1978), if the fundamental “constant”s are indeed constant.
Keywords: General Relativity, Cosmological Constant, Cosmology, Dark Energy, Fractal Cosmology, Static Universe, Big Bang Theory
1. Introduction hand side (LHS) of the same equation contains much
more tangible quantity, Tik , the energy momentum ten- The “cosmological principle” demands that not only does sor of the matter generating the space time structure gik the center of the universe lie everywhere, but also any or Gik . Thus any constraint imposed on gik from fundamental observer must see the universe as isotropic symmetry or other considerations must take its toll on the and homogeneous. This means that, the space must be admissible form of Tik as well. This means that con- “maximally symmetric” [1]. By this consideration, one straints imposed on Gik must be reflected on the Tik of can directly obtain not only the metric for Einstein’s sta- the fluid generating the Gik . This latter physics aspect tic universe (ESU) but the non-static Friedmann-Rober- however may remain hidden in a purely symmetry driven tson-Walker (FRW) metric as well [1,2] without using Eins- mathematical derivation of the cosmological metric tein’s equation Gik=8π T ik at all ( Gc==1). Let us in- which does not invoke Einstein equation at all. And in vert it in the form this paper, we would like to explore the tacit constraints imposed on T by the requirements of the cosmological 1 ik Tik = Gik (1) principle, “there is no preferred center”, or “center lies 8π everywhere”. All the results obtained during this process
The Einstein tensor Gik , on the right hand side (RHS) will be also ratified by physical considerations as well. of this equation, comprises only geometric quantities, the In effect we shall be dealing with spherically symmet- structure of space time gik . On the other hand, the left ric static solutions of Einstein equation with the inclusion
Copyright © 2011 SciRes. IJAA 184 A. MITRA of a cosmological constant , [3,4]. In recent times order that the timelike worldlines of test particles always several authors have shown keen interest in this impor- remain timelike and no trapped surface is formed, FRW tant problem [5-8]. However such studies mostly focus model too has to be vacuous with =0. Then the real attention on isolated objects or stars where there (1) must universe must be something entirely different from the be a pressure gradient, (2) in general, a density gradient, isotropic, homogeneous and continuous FRW/ESU mod-
(3) a natural boundary R= Rb where fluid pressure els. Indeed galaxies and structures are found to be dis- must vanish, pRb =0 and (4) where a discontinuity tributed in discrete, lumpy and inhomogeneous manner in the fluid density might also occur. Further in such even at the largest scales. Nonetheless such matter dis- generalized studies, has not been considered as a tribution can still satisfy the “Copernican Principle” of fixed fundamental constant contrary to what was in Ein- no unique centre if it would form infinite hierarchal stein’s mind when it was first introduced. fractal pattern. At the beginning, however, we focus at-
In contrast, in the present study Einstein is a basic tention on the ESU. constant and there is no question of modifying its value according to the needs of a general fluid solution. And of 2. Formulation course, there must not be any natural “boundary” at all for the universe so that Copernican principle remains We start with the assumption of spherical symmetry and valid. It may be recalled that the ancient Ionian philoso- consider a general form a static metric [2] pher “Aristarchos of Samos” had proposed the heliocen- d=dseteR22 d22Rd2 (2) tric theory much before Copernicus and hence we may 222 2 also call this as “Aristarchian Principle”. It will be found where d=d sin d and R is the area coor- that both the metric for a star and ESU have a “coordi- dinate. In particular, in a static universe, the luminosity nate singularity at R = , where R is the luminosity distance turns out to be exactly R [2] distance and is an appropriate integration constant. d=L R (3) This singularity gets propagated in the hydrostatic bal- We also assume the cosmic fluid to be perfect with ance equation as well. And in order to ensure that iso- tropic pressure gradient p does not blow up at this Tpik = uupi k gik (4) singularity, relativistic stars are constructed so that where, is the fluid density, p is the isotropic pres- R > . Here a prime denotes differentiation by R. Ac- i b sure, and u is fluid 4-velocity. As is clear from Equa- cordingly one can almost forget this singularity while tion (1), at the beginning, we do not consider any cos- studying static stars. But for the ESU one has to live with 0 mological constant ; and the 0 component of the this singularity. And in this paper, we would study the Einstein equation reads behavior of physical quantities like pressure and accel- eration despite the presence of this singularity. And the 0 1 1 8πTe0 =8π = 22 (5) conclusion is that ESU has to be vacuous with =0 in RRR order to avoid such singularities anywhere. This can be integrated to yield At the very beginning, let us remind the difference R between a fundamantal constant and a model parameter. e =1 (6) Suppose we are studying a room temperature homoge- R neous static gas and we have obtained some expression where R for pressure p , temperature T and particle number RR=8π 2dR (7) density n . But since pn= kT, in principle, we may 0 obtain a numerical value of the Boltzmann constant k . Here the condition (0) = 0 has been used to ensure Once we assume that k is a fundamental constant and that e is regular at R =0. Thus whether, it is the in- not just a model related parameter, the value of k must terior solution of a star or the static universe, general be independent of time t or position r or any other form of the metric is variable. Thus, if we would be studying a gas in a situa- dR2 tion where the gas may be inhomogeneous or may have d=dset22 R 2 d2 (8) 1 R bulk motion, we must be able to use the same numerical value of k obtained in an ideal static homogeneous Note, as of now, we need not necessarily interpret case. Similarly, here we would study the case of the ()R as related to the observed mass though what we supposed fundamental constant though it will be in the have done looks like finding the interior of a relativistic context of ESU. Therefore, we would expect =0 for star by working out the Schwarzschild interior solution. the dynamic FRW model too. It would be found that in The important difference, however, is that while for a
Copyright © 2011 SciRes. IJAA A. MITRA 185 star there is a unique center and a boundary where the to rewrite Equation (14) as density may be discontinuous, for the universe, center is 2222 2 2 d=dsetS dsin d (16) everywhere and there is no boundary, no exterior solu- tion, and no density discontinuity. Further even for a There is clearly a singularity in one of the metric coef- constant density star, there must be a pressure gradient ficients in Equations (12) and (14) at RS= or because there is a unique center. On the other hand, for R =1 or r =1. On the other hand, metric (16) has a an isotropic and homogeneous continuous universe, not similar singularity at =0. But one cannot make any a only p and , but all physically meaningful quanti- priori comment on the nature of such singularities with- ties, all geometrical scalars must the same everywhere. out studying the behavior of relevant scalar quantities. Such requirements may demand severe restriction on the Accordingly, we would try to study the behavior of sca- admissible equation of state (EOS) of the cosmic fluid. lars at appropriate regions instead of having any a priori In contrast for a star, at best only can be uniform in debate/discussion on these singularities. which case there would be a discontinuity at its boundary Let us suppose that somehow the density of the star
R= Rb . would be reduced to zero everywhere. In such a case, the Now we introduce the condition that the fluid must be space time must become flat with a metric homogeneous: = uniform and thus both for the uni- d=ddstRR22 22 d2 (17) verse as well as a constant density star, one has From the viewpoint of metric(8) it would immediately 8π = R3 (9) be apparent that one should then have e =1 and 3 R =0. But from the viewpoint of metric (14), it would and the metric assumes the form appear that in such a case one would have S = ,
2 r =0 and RS=r 0. Why would we have r =0 in 22 dR 22 d=sedt2 R d (10) the latter case? This is so because the general form of a 18 π 3 R space time with constant spatial curvature is 2 If we introduce a parameter 222 dr 22 d=dsetS 2 r d (18) 3 1 Kr S = (11) 8π where K =1,0,1 corresponds to closed, flat and open space time respectively. And once we presume Equation (10) would acquire the form K =1 and adopt metric (14), we presume >0 and dR2 then it becomes difficult to have a smooth transition to a d=dset22 R 2 d 2 (12) 1 RS22 0 case. Hence, if one would try to arrive at a flat space K =0 by imagining that it is due to a closed It may be noted that Tolman [4] too obtained the static space of infinite radius, one would land up with the con- cosmological metric in a similar way by using the scalar dition r =0. In fact, if in Equation (4), had we taken,
R as the radial variable. In the static case, R is also a 0 comoving coordinate for the interior solution because a TK0 = (19) given R = fixed shell encloses a fixed number of bar- instead of , we would indeed have obtained Equation yons/particles. (18) directly in lieu of Equation (14). Then we would Further, if we introduce a new coordinate have found that the condition for flatness is K =0. But 0 rRS= (13) when we do take T0 = and implicitly assume >0, we presume K =1 and exclude the possibility that one We can rewrite the above metric as might have K =0 too. In such a case, the condition for 2 flatness would appear as r =0 as mentioned above. 222 dr 22 d=setSd r d (14) 1 r 2 Note, the fact that the comoving coordinate r =0 sig- nifies that there is no baryon/particle at all; i.e., it is all Note, the time dependent Robertson-Walker metric vacuum. uses this r as the (comoving) radial coordinate. Hence, the spatial section of both the universe and the Cosmological Constant interior of a constant density star is that of a 3-sphere, a It is well known that in order to obtain a static universe space of constant curvature, a fact noted by Weyl [3]. which would be closed and finite radius, Einstein modi- This becomes clearer if we express fied Equation (1) into
rRS==sin (15) Ggik ik=8π T ik (20)
Copyright © 2011 SciRes. IJAA 186 A. MITRA apparently implying that either is a fundamental metric space time occurrence of (,)rt R > corre- constant like 8πGc4 or a basic scalar, like the Ricci sponds to a formation of a “trapped surface” and the scalar R appearing within Gik . It is also well known condition (,)rt R =1 marks the formation of an “ap- that, from a purely mathematical view point, one can parent horizon”. Though for a non-static system, it is incorporate the effect of , by replacing [7] conjectured to be possible to have trapped surfaces or horizons, for a static system they are not allowed. This is e = (21) so because once a trapped surface would be there, stellar 8π matter would be inexorably pulled towards the central and point of symmetry and thus matter would soon end up in a point singularity rather than as an extended static object. ppp= (22) e 8π While this is definitely not allowed for a static star, this problem would be much more severe for cosmology be- It should be borne in mind that such a mathematical cause “center of symmetry lies everywhere”. clubbing however does not really make a new form Following the case of a Schwarzschild black hole of matter. This is so because while real matter repre- space time, generally, it is believed that this R = sented by generates global negative self-gravita- singularity is a mere coordinate one even in the presence tional energy, pure vacuum represented by is not of matter. But as we would see below, the coordinate associated with any negative self-gravitational energy. So, as far as mathematics is concerned, instead of independent scalar acceleration of the fluid would blow Equation (7), one now obtains up unless severe constraints are imposed on the fluid EOS. ()R e =1 e (23) It may be recalled that Einstein was very much con- R cerned about this R = singularity and constructed a where static model of a fluid where test particles are moving in
R randomly oriented circular orbits under their own gravi- =8π RR2d (24) ee0 tational field. While the radial stress of the fluid is zero, tangential stresses are finite. This configuaration is One also finds known as “Einstein Cluster” and Einstein showed that, dR2 d=dset22 R 2 d2 (25) the speed of the orbiting particles would be equal to the 1e R speed of light c at R = [11]. Thus, he pleaded that there cannot be any R = or R < situation. Later And for a constant density case, one has it turned out that Einstein’s cluster indeed corresponded 2 22 dR 22 to some well defined interior Schwarzschild solution d=dset2 R d (26) 18 π 3 e R studied by Florides [12,13]. And now Bohmer & Lobo [14] have shown that Einstein’s intuition was correct, and and the R = singularity is indeed a curvature singu- 3 larity. This however does not at all mean that all R = S = (27) singularities are curvature singularities. In fact, in the pre- 8πe sent static case of isotropic pressure, it would be found Equations (12)-(16) remain unaltered in the presence that they should be regions with zero curvature singularity of . Thus now, if the space time has to be flat, in ad- to avoid an acceleration or pressure gradient singularities. dition to e =1, one must have For any static spherically symmetric fluid, one can easily find the acceleration [8] e 0 (28)
ik i This shows that if the original definition of “vacuum” au= k u (29) is =0, as if, in the presence of a , it gets modified In spherical symmetry, only one component of accele- to e =0. ration survives 3. Acceleration Scalar and Singularity e aR = (30) 2 The fact that for a spherically symmetric static system, one should have R <1 as has already been investi- How to evaluate this aR ? Again, for the sake of easy gated [8-10]. The basic reason for this is not difficult to understanding, we first do not consider any and R see. In general, static or non-static, for spherically sym- write down the R component of Einstein Equation (1),
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11 tion such as Equation (39) is needed to really ensure that 8πTpeR =8π = (31) R 22 R = . For a fluid with >0, R RR a is indeed finite at e e the safest way to avoid the R = singularity will sim- In view of Equation (30), let us rewrite this equation e ply be to ensure that R > e . The fluid can do so by as choosing an appropriate density profile and by ensuring e 8Rπp3 that its outer boundary = 3 (32) R R Rb> e (42) or For a constant density star or the ESU, we have 8πRp3 R Rpee 3 a = 2 (33) a = (43) 2R 2 12 e R Since there is only one component of ai , the scalar acceleration becomes where =4π 3 . In terms of r , we obtain
/2 rpee 3 iR e aS= (44) aaaag==||= (34) 2 iRR2 1 r Using Equation (32), we obtain It is clear that the sufficient condition for avoiding the R = singularity in this case is 8πRp3 e a = (35) RR>; (46) e 8πR2 8πRp3 b a = ee (36) 2 Thus, if one would imagine a region with =0, i.e., 21RRe e =8 π , one should restrict the radius of this vacuum It is now clear that if a static fluid would tread upon as the R = singularity, its acceleration could be infinite e 3 and it cannot be at rest. And this is the physical reason R <; (47) b that for a static fluid one must have R > e everywhere. And in case, the manifold would cover R = e , one And in case, one would have =4 π , then the must satisfy above restriction would become 3 ee 8πRp=0; R =e (37) 1 R < (48) b in an attempt to keep a regular. Using R = e in the foregoing equation, we have and which cannot be satisfied. Thus for a constant den- 2 sity star with Rb =1 , one must have >4 π . eee18 π p =0 (38) And in case this condition would be violated, one must i.e., one must have either critically analyze the additional constraint on the EOS =0 (39) which would prevent pe from blowing up at R = e . e It may be seen that the =4 π EOS corresponds to or, 3=0p (49) 1 ee pR== (40) ee 2 if p =0. 8πe or both of the above two conditions. If we assume that 4. TOV Equation the minimum value of p is zero and it cannot be nega- tive, we will have pe =8 π . Then Equation (40) Recall that local energy momentum conservation equa- would yield k tion Tik; 0 immediately leads to [2] 1 2 p RS== (41) = (50) p R But one is still not sure whether the additional condi- Further if we again consider the R component of
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Einstein Equation and combine it with Equation (50), we rp3 p pS = eee e (58) will obtain the TOV equation for hydrostatic balance for 1 r 2 any self-gravitating static fluid: And in term of a , we have pR8π 3 p p = (51) apee 2 p = (59) 21RR 2 12 e R Since the effect of gets included if one replaces and p and by their “effective” values[7]
3 apee eeepR8π p e p = (60) 2 p = 2 (52) 1 r 21RRe
Further since pp= e, in a complete symmetric man- 5. Constant Density Star ner, we rewrite the above equation as
3 For the interior solution of a constant density relativistic eeepR8π p e star one, one finds [7] pe = 2 (53) 21RR 2 e pc e = (61) This equation strongly suggests that if the definition of p a “dust” in the absence of is p =0, as if, in the where p is the central pressure. This relation is inter- presence of , dust EOS should be pe =0. Similarly, c if the original EOS of “vacuum” is =0, in the pres- esting because it does not involve any exterior bounday condition. Also note that since ence of , the vacuum EOS is e =0, a hint we have already found. pp = e e (62) Note, even now, there is no need to interpret in terms of any exterior boundary condition, and thus TOV we can rewrite Equation (61) as equation is valid in any spherically symmetric static GR c 2 p problem including ESU. By using Equation (36), it is e = ee (63) p interesting to rewrite the TOV Equation in terms of the ee acceleration scalar Since for a star, there must be a pressure gradient, c ap ppR , and one really cannot reduce eR to a p = ee (54) constant value independent of R . But if, from the e 1 R e mathematical viewpoint, one would still demand that it Clearly, there is a singularity in the denominator of should be possible to set up a time orthogonal Gaussian coordinate system where dt 2= d 2, one would un- TOV Equation at R = e . If we write knowingly kill the pressure gradient and set pc = p in x =1e R (55) Equation (61). In the absence of , for a static con- It is seen that while the singularity in acceleration figuration, =0 if p =0, and the star would vanish ax 1 , for the pressure gradient it is much stronger under the assumption of e =1! 2 When is present, the expression for pressure for a px . And if the fluid would cover the R = e sin- gularity, regularity of a may not be sufficient to ensure constant density star is [7] regularity of p . In fact Equation (54) suggests that one 4π 1cosC pR()= (64) might require the additional constraint 3cos
ee pR=0; =S (56) where to tame the p singularity at RS= . c c 34p π 3pe e Now let us use the condition = uniform which is C ==cc (65) e pp equally valid for the interior solution of a constant den- ee sity star as well as the ESU: Now it transpires that even when p =0 to honor p 3p e =1, there would still be a finite density =4 π . pR = eee e (57) But one attains this finite density at a huge cost because 12 R2 e it now turns out that the boundary of the star merges with
In terms of the normalized coordinate r , one finds the coordinate singularity RSb = where the metric
Copyright © 2011 SciRes. IJAA A. MITRA 189 behaves badly. For a real star, this singularity can always And at the singularity, RS= , by l’ Hospital rule, we be avoided by allowing e >1 which in turn means by obtain having >4 π . Thus a “world time” cannot be in- f troduced for a general relativistic star having a finite pe = lim (76) RS g density. From Equations (75) and (76) we obtain the required Verification from TOV Equation condition: Although the case of constant density stars in the pres- 3=ee p 4e (77) ence of cosmological constants has been studied in great i.e., detail, to our knowledge, nobody actually verified the validity of the solutions in case one would tread on the ee(=)=3(=)RS pRS (78) coordinate singularity at RS= . Since the denominator which looks like the EOS of incoherent radiation! By of Equation (53) becomes zero at RS= , in order that combining Equations (67) and (78), it becomes clear that p does not blow up there, one must have either in order that p does not blow up at the “coordinate
ee pR=0; =S (66) singularity” at RS= , the constant density star must or have 3=0;=pRS (67) ee ee(RS = )= pRS ( = )=0 (79) or both Thus instead of p =0 we would obtain pe =0 at RS= . This means that if indeed >0, the solution eeppR=3=0;= e e S (68) must avoid RS= singularity. More importantly, since If the last constraint would be satisfied, one would e = constant , we find that, if the solution would indeed immediately obtain ==0p . However, it is possible ee extend upto R= S, we must have e =0. Thus all that Equation (68) is not needed and only one of the less finite density stars must avoid the e R =1 or r =1 rigourous conditions (66) or (67) is satisfied. In particu- singularity. lar, to avoid occurrence of negative pressure, let us as- sume that only Equation (67) is satisfied at RS= . 6. Static Universe Since p attains a 00 form at RS= , let us study the nature of p at this singularity by using L’ Hospital In cosmological case, there must be a universal time rule. For this, let us first write which would be the proper time of all fundamental ob- f ()R servers, i.e., dt2= d 2. This demands that one must be p = (69) e g()R able to set e =1 so that, the Equation (14) would be- where come the ESU metric 2 222dr 22 f ()=RR eee p 3 p e (70) d=dstS 2 r d (80) 1 r and Note that the value of S for ESU is still given by gR=2 R2 1 (71) e Equation (27) and thus one is justified in deducing ESU Since =0, we find that metric indeed by setting e =1. Since the Einstein equa-
tion tells that the metric is essentially determined by Tik , fpp =3 eee e (72) such an important change of setting e =1 on the LHS of this equation must be endorsed by the RHS, i.e., by Rpeeeee33 p p the admissible forms of fluid EOS. What are those con-
Also, since, we have already considered ee 3=0p , ditions? To explore them, we note from Equations (50) we obtain a reduced expression for and (51) that the most general form of static, spherically symmetric Einstein Equation yields f =3Rpee p e (73) 8πRp3 ee On the other hand, = 2 (81) RR1e g =4e R (74) So that In constant density case, this condition becomes p Rp 3 fg=3 p ee (75) ee e =2 2 (82) 4e 12 e R
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And in term of r , this condition becomes we rewrite Equation (60) as rp 3 Sa p ee ee =2 S 2 (83) p = (94) 1 r 1 r 2 In order that e =1, one must have =0, and from so that the foregoing equation, it is immediately clear that then we must have atleast f ()=rSapee (95)
ee 3=0p (84) and
2 In any case, we must have pe =0 everywhere in- g()=r 1 r (96) cluding RS= or r =1. Then if we follow the L’ Hos- pital treatment of the previous section, we would find, Then we have we must have f = Sap (97)
e =0 (85) and everywhere because = constant . Although, we have r e g = (98) already obtained this important result, it would be inter- 1 r 2 esting to obtain this result from somewhat different routes. Again applying L’Hospital rule for the limit of p at First consider Equation (59) and write RS= , we find 2 f ()=Ra p (86) Sa1 r ee =1 (99) r and Inserting the expression for a from Equation (44), 2 g R=12e R (87) we obtain the interesting relation 2 so that Sp ee 3= 1 (100)
f = pae (88) or and 3 S = (101) 12 2 4πee 3p g =2ee 1 2 R (89) Now applying L’Hospital rule at RS= , we find Comparison of Equations (27) and (101) would again convey one of the hidden messages for the ESU as well aR12 2 as for a constant density star which is attemting to sup- pp= e (90) 2 R press its pressure gradient, namely 3=pee. The sum e and substance of the entire exercise is that for the ESU,
Since the limxx = 1 irrespective of whether x 0 in oder that pe is indeed zero everywhere, one must or x or x = finite , we can cancel p from both have sides of the foregoing equation. Then, using Equation =3p =0 (102) (43) in the above Equation , we find ee But unlike the constant density stellar case now we 3p ee=1 (91) cannot escape confronting this singularity by demanding 2e that >4 π . And hence we must accept the fact that
This again implies for the ESU fluid ee==0p everywhere. This means that e=3pe (92) 8π =0; = 8π (103) In conjunction with (68), this will lead to ee==p 0. It may be also of some interest to extend this study by atleast for the ESU. The weak energy condition demands directly considering the radial variable as r rather than that 0 and thus we find that =0! R . Note that 7. Einstein’s Solution 1 ppee =d dr (93) S If one would directly use Equation (80) into Einstein Now if we denote differentiation by r with a prime, Equation (20), one would be led to[4]
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4π 3=p (104) and accordingly and 1040 cm 2 before 1920 (113) 1 4π p = (105) As late as 1970, we hardly had any idea that the gal- S 2 axy clusters and superclusters are actually distributed as “filaments” and “walls” around huge voids whose di- Einstein chose, p =0 EOS which led to mensions could be as large as 280 Mpc [15]. The re- 1 cent Sloan Digital Sky Survey has revealed structures of 1=;=SS2 (106) dimension 500 Mpc [16]. Thus, we cannot rule out the possibility that eventually, it might be found that the Note that Equations (104) and (105) may be rewritten entire patch of presently observed universe also lies on as the wall of a larger void. If the mean density of observed
4πee 3=0p (107) universe would be revised by future observations, would we again revise the value of the fundamental constant and ? 1 4π p = (108) At the cost of sounding repetitive, let us again raise the eeS 2 old question: if would indeed be due to some quan- tum mechanical effect, why the value of obtained But, if we recall, Equation (92), actually, ee p =0 so that S = . Further Equations (100)-(101) showed that under the assumption of patch= true falls short of the theoretical value by an approximate factor of 10120 ? 3 And if the various field modes would cancel one another 4πee 3=p 2 (109) S to generate a small , why do they not cancel exactly And a comparison of Equations (108) and (109) again to result =0? In fact there are some theoretical esti- shows that, S = and e =0. As mentioned before, mates by which =0 [17]. for weak energy condition, this would mean ==0 . Note that it is indeed possible to have a situation where 8. Summary So Far
rest e 1 , where e is the proper internal energy den- sity. In such a case, one might approximately write Both a spherically symmetric static star and the ESU p 0 . But this does not mean that pressure is strictly result from the same spherically symmetrical form of R zero. A strict p =0 EOS is possible only when =0. Einstein equations. If R component of the Einstein In such a case, Equation (104) would again yield =0. equation would be studied, one would obtain, the expres- However, if we would ignore such physical and sion for acceleration scalar and condition for hydrostatic mathematical regularity or self-consistency considera- balance in both the cases. In fact, Tolman [4] did obtain tions, Einstein’s solution (106) would apparently suggest one of the constraints to be followed by the ESU fluid by =4πGc 2 >0. demanding that the pressure gradient p must vanish
Further, if one would assume that (1) ESU correctly for the ESU: e 3pe=0. We found that the singularity describes the real universe and (2) the true mean density appearing in the RR, component of the metric tensor of the real universe in its totality (about which we may in all spherically symmetric static configurations do never have an absolute knowledge) is equal to the mean propagate in the expressions for acceleration scalar a density of the patch of the observed universe, i.e., and pressure gradient p . While, for the former, the 1 31 3 nature of singularity is a x, for the latter it is much patch==10 true g cm (110) 2 stronger p x. In fact, the requirement e 3pe=0 one would obtain[2] is the basic condition for ensuring that the a does not blow up at x =0 or r =1. This basic condition how- 58 2 10 cm (111) ever need not be the sufficient condition for ensuring that Now ponder over the fact that until the advent of big a is finite at x =0. Further, since p tends to blow optical telescopes in 1920 or so, for hundreds of years, up much faster at x =0, one must require an additional most of the astronomers thought that total universe was condition to ensure the regularity of p . We found that nothing but our Milkyway galaxy. And if would have the latter requirement is e=3pe. This means that if the been possible to measure the mean density of the Mil- constant density static star would have RSb = , one kyway, one might have concluded that must have e =0. But one can always avoid this singu- larity by ensuring that > 4π . However, previous 13 3 patch==10 true g cm (112) detail studies of perfect fluid spheres were primarily in-
Copyright © 2011 SciRes. IJAA 192 A. MITRA tended for isolated systems and not for the universe, and one can define a “mass function” in terms of which ac- to our knowledge, nobody tried to study the remedy for celeration scalar either the x1 or x2 singularities at x =0. M 4πRp3 Incidentally, it has been found that even for a sup- a = (119) RM2 12 R posed constant density star with Rb < , the “constant density” is nothing but =0 [18]. we can now identify ()R in Equation (36) as twice the quasilocal mass-energy of the fluid 9. Buchdahl Inequality ()=2()RMR (120) It is well known that any spherically symmetric static In the presence of a , the quasilocal mass will be configuration, homogeneous or inhomogeneous, sup- M ee=2 . ported by isotropic pressure, satisfies a constraint much In general relativity, the gravitational mass of a sta- stronger than the Rb > or < Rb constraint. In the tionary system is[3] absence of a , this is known as Buchdahl inequality M =Tt00 gx d 3 (121) [19]: 00 k 8 where ti is energy momentum pseudo tensor and g < (114) is the determinant of the metric tensor. One can work out Rb 9 0 t0 from the metric gik of the universe. Probably start- If one would have =89 Rb , central pressure ing from Rosen [20], many authors have worked out the would blow up (assuming >0). For a uniform density mass energy of a closed universe and all of them have star this borderline would correspond to concluded that for a closed universe M =0 [21-25]. r =8 9 (115) This too would suggest that, in the absence of , =0, and, in general e =0. For the ESU, it is possi- But suppose there is a solution which appears to vio- 0 ble to confirm this result irrespective of the value of t0 . late this constraint, i.e., r 89. Then how to ensure In all coordinate systems, for a static fluid, one obtains that central pressure is still finite? The only solution out [2-4] of this dilemma would be to set =0 or =0. From 3 this more stringent condition, it should be clear why Ein- M e =34 pgπ dx (122) stein’s original static universe must have =0. =3 pgx d3 Does the fate of the ESU improve after the incorpora- ee tion of a positive ? From Equation (3.35) of ref.[7], it In view of Equation (84) we thus directly obtain turns out that now the modified Buchdhal constraint is M e =0 (123)
e 2 <4 (116) Further, from Equation (24), we obtain e =0 R 94π Thus with a positive , this constraint gets even 9.2. Poisson Equation tighter. With =4 π , one obtains It is known that the RHS of Poission equation indicates 2 e < (117) the source of gravity. For a spherically symmetric static R 3 fluid the Poission’s equation is [26,27]
In view of Equations (9), (11) and (33), this means 2 ggp=4π 34π that for a constant density star, one must have 00 00 (124) =4π gp 3 r <23 (118) 00 ee This shows that the source density of gravity is to ensure that pressure does not blow up at the center r =0! For the ESU, this would mean blowing up of g =g00 ee 3p (125) pressure everywhere because “center would lie every- where”. And the only solution to get rid of this problem And when g00 =1, one will obtain is is accept the fact that =0 for ESU. e g =3 ee p (126) 9.1. Mass Function In either case, the RHS of Equation (124) is zero when
ee 3=0p , which is the case for both a constant den- Since for a fluid in an asympotictially flat space time, sity star (if it would be extended up to RS= ) or the
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ESU. In all such cases, one must necessarily have and g00 ==e constant . And since source of gravity is zero, aKS r = (133) the space time must be flat. And a space time is flat when 2 ee 3p 1 Kr e =0, as obtained by us. Alternatively, if one would first focus attention on the LHS of Equation (124), it respectively. In such a case, it would have been seen seems that if one would set e = constant , the source of immediately that the requirement that the RHS of Equa- gravity will vanish, again in which case, one should have tions (132) and (133) are independent of r , one must have is K =0. And since we have already set e =1, it e =0. means, the space time must be flat to ensure that the 10. A Simple Reason RHS of Equations (127) and (128) or (132) and (133) are indeed independent of r . Hence one must have
Suppose one chooses to ignore most of the previous K e =0 in such a case. So will be the case for ESU. This means that in order that all physically meaningful proofs that e =0 for ESU in order to avoid any physically singular behavior. Even then, one can arrive quantities are position independent, the mean density of at the same conclusion by simply demanding that all the ESU must be zero. physical quantities must be position independent in ac- cordance with the inherent assumption in the following 11. Non-Static FRW Universe: Big Bang manner: Model For a constant density case, we may rearrange the TOV equation(58) as In physics, we have various fundamental constants like pSr G , c , h , me etc. But in no case there is any evidence = (127) that the values of such constants directly depend of am- ()(ppr3)1 2 eee e bient factors like mean density of universe. In fact, all We may also rewrite the acceleration Equation (44) as truly fundamental factors allow themselves to be deter- aS r mined with high precision by judicious combination of = (128) theory and experiments. The only exception here is the 3p 2 ee1 r supposed ! This anomaly seems to be resolved with The RHS of Equations (127) and (128) obviously de- the result that there is no ! On the other hand, cos- pends on r if r would indeed be a free parameter. mologists have found that distant Type 1a supernovae For a constant density star, the LHS of the same equa- appear to be fainter than expected if one would assume 1) tions too must depend on r . And they do depend o n r them to be standard candles and there is no evolutionary because ppree=() even though e = constant . But effect in the intrinsic luminosities of the distant superno- suppose we would like to freeze the r dependence of vae, 2) Assume that cosmological luminosity distance p to make e =1. Then the value of r on the RHS of measurements to be perfect; i.e., assume that standard Equations (127) and (128) too must be frozen. And since Big Bang Model (BBM) is correct, 3) Assume that there RSr=0 during the freezing process, the only solu- is no dust absorption of lights from distant supernovae. tion here would be to adopt And by ignoring such questions, in the paradigm of the SrR=; =0; 0 (129) BBM, such extra faintness was interpreted as the proof of existence of or “Dark Energy” (In fact, the Phys- We have already discussed that occurrence of Equa- ics Nobel for 2011 was due to such an interpretation): tion (129) actually signifies that the spatial section is flat: 4π 4π K =0. This would have been more transparent had we SS=(3)=3eep p (134) taken 334π TTT123=== Kp (130) The extra faintness is explained by assuming that the 123 galaxies are further away than expected (in a decelerat- alongwith Equation (19), i.e., had we written ing universe); i.e., by considering S >0 and which is possible for a fine tuned positive TKik = puupgi k ik (131) instead of Equation (4). Further had we also taken the 3p (135) 4π cosmological constant as K instead of , Equations (127) and (128) would have appeared as But if indeed =0, then BBM cannot explain a supposed S >0. Nonetheless, it has been claimed that if pKSr “Cold Dark Matter”, another crucial ingredient of BBM, = 2 (132) eeppe31 e Kr will have pressure, BBM would be able to explain
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S >0 even if =0 [28]. However, this suggestions is 2(,)GM r t <1 (139) patently false because 3p includes contribution Rc2 from both visible matter and DM, and any increase in the For the FRW case, one has RrSt=() and value of matter pressure p , can only give S <0. 3 Therefore, the faintness of distant supernovae need not 4πR M (,)=rte () t (140) imply that they are further away; and the faintness could 3 be due to unknown systemetic effects [29]. so that one should have The fact that the BBM cannot explain the physical 8πG 22 universe is apparent from the fact, for the BBM [30] rtSte () ()<1 (141) 3c2 2 p 2 e And in order that S does not blow up either in g00 =exp (136) e pee future or past, one needs to have 2 And in oder that the FRW metric indeed has g00 =1, e ()t S ()= t constant (142) one must have Again for this, first, it is necessary that =0 and 2 p e =0 (137) e = : p ee ()t S2 ()= t constant (143) Since p < , it is necessary here that ee But the FRW model generally obeys t2 = constant pp=8 π =0. Then, it is immediately seen from e so that one should have Equation (134) that, BBM model must (actually) have St() t (144) S 4πe =0 (138) And it is only for the Milne model (with k =1 ), one S 3 obtains St() t. But the Milne model is empty with Thus contrary to the popular belief, the BBM actually =e =0! Since we have already found pe =0, we cannot explain any accelerating universe! Further eventually obtain the vacuum EOS for the FRW metric: pp=8 π =0 would imply p =8 π . And if we e ee p= p=0. Does this result lead to any contra- would rule out such a fine tuning of matter pressure, we diction with Equation (137)? The answer is “no” because should have p =8 π =0 for the BBM irrespective of a 00 form could be anything including 0. One may the result obtained in the context of ESU. And with also point out that the result e =0 for the FRW model p =0, there can be neither any matter pressure nor any is not rigorous because we did not offer any proof for the microwave background radiation[30]. usual relation t 2 = constant . Such an objection would be valid, and thus now we focus on the de-Sitter case 2 12. Why the Big Bang Model Could Be with e =8 πc . Now Equation (141) will reduce to Vacuous 1 rS22()<1 t (145) 3 We have already pointed out that the total energy associ- In order that above constraint is satisfied, one must ated with the FRW metric is zero; and this is not because have =0. Thus, in a rigourous manner, we should of any negative self-gravitational interaction. The latter have =0 in order to satisfy the condition of “no can reduce total energy only when ggrt=(,)<1, 00 00 trapped surface” in the de-Sitter model. but for the FRW case, ggt00=() 00 =1 [26,27]. On the 0 So for the time being, let us ignore the possibility that other hand field energy density t0 0 . Thus vanishing the FRW metric subtly represents a vacuum spacetime of total energy indicates e =0. We probed this impor- and which is the reason that there is no density or pres- tant question independently and found that FRW metric sure gradient and there is an universal Newtonian time should intrinsically correspond to e =0 in order that despite the supposed presence of self-gravity. And even total energy is conserved [31]. In fact there can be a sim- if one would ignore the proof that =0, there are al- plified way to confirm this: ready many suggestion that and “Dark Energy” Long back Kriele showed that there cannot be any could be illusions created by an inhomogeneous lumpy trapped surface for a spherically symmetric homogene- universe which is significantly different from the sim- ous perfect fluid [32]. This seems to be a special case of plistic BBM [34-38]. However if there would be inho- the no trapped surface theorem obtained by demanding mogeneity in a monotonous and continuous manner, that timelike worldlines associated with material parti- there would be a preferred centre of the universe in vio- cles must always remain time like [33], i.e., lation of the “Copernican Principle” of no unique centre.
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On the other hand, the discrete fractal models could still where is the Newtonian potential and 21 c2 . satisfy the Copernican Principle. Then by further stting p =0, one can obtain the ap- proximate Newtonian Poission equation: 13. Fractal Model 2 4πG (149) In a fractal distribution, the number of galaxies increases But by starting from GR equation, one would never with R as [39-41] obtain an exact Poission’s Equation 2 =4πG if D 0 . Thus, GR can yield an exact Newtonian result NR(< ) R (146) only when the gravitational potential =0! This sug- and the number density of galaxies varies as gests that though, mathematically, one can conceive of D3 global clock synchronization and set g =1, physically, nRR< (147) 00 gravitation tends to manisfest itself with non-synchroni- Thus, in principle, for a fractal distribution over suffi- zation of clocks! In other words, a model assuming ciently large scale, one may have n 0 , 0 . Most global synchronization of clocks could be vacuous with of the fractal patterns however tend to have a preferred zero matter density! Thus one expects all dust models to center and thus not suitable for “Copernican Principle”. correspond to =0. And indeed, the famous Oppen- But there could be some fractal structures like Levy Dust heimer -Snyder dust collapse has explicitly been found to which may not have any preferred center. correspond to =0 [54]. The Schwarzschild “Black Indeed many cosmologists believe that within the Hole” exact solution too is illusory because the integra- patch of the observed universe, galaxies are distributed in tion constant involved there is actually zero implying fractal like pattern atleast on scales 10 Mpc [42-45], true black holes have zero gravitational mass; and the and one may recall here a succint comment by Wu, La- so-called “Black Hole Candidates” must be something hav & Rees [45] else[55]. Similarly, it has been found that a uniform den- “The universe is inhomogeneous—and essentially sity sphere cannot undergo any adiabatic collapse at all fractal—on the scale of galaxies and clusters of galaxies, and hence scores of exact solutions indicating collapse but most cosmologists believe that on larger scales it must actually correspond to =0 [56]. In general, becomes isotropic and homogeneous.” there cannot be any adiabatic gravitational evolution at This review written in 1999 however concluded that all, and which shows that all exact solutions indicating galaxy distribution is homogeneous on larger scales. But evolution must tacitly correspond to =0 [26]. Even now it is seen that galaxies are distributed in a roughly the famous interior solution for a static uniform density fractal pattern with dimension D 2.2 even on 100 spherical star is illusory in the sense that it actually cor- Mpc scale [46,48-52]. Note, many of the objections responds to =0 [18]. This may be suggesting that against the fractal model have already been addressed to self-gravity manifests itself not only through pressure [53]. gradient and non-syncronization of clocks, but also through density inhomogeneity. 14. Conclusions Thus it is indeed possible that FRW metric too subtly corresponds to a vacuum solution just like its static Since cosmological constant =0, so-called “Dark counterpart. In general, though, “exact GR solutions” Energy” too is expected to be absent. Simultaneously it may often be physically meaningful for the static cases, has been found that the Big Bang Model intrinsically they need not be so for the complex non-static cases; the corresponds to zero pressure and zero temperature [30]. complexities of a physical system, such as unknown and Note this result is in perfect agreement with the fact that evolving equation of state of matter, unknown radiation an ideal Hubble flow implies smooth radial motion with transport properties, unknown evolution of shear and no collision amongst the test particles. Thus even if one dissipation, may rarely allow physically meaningful would assume =0, in view of Equation (138), the non-static exact solutions. For instance, it has been found BBM cannot explain any cosmic acceleration. Note, that celebrated non-static Kerr solution is an illusion be- Equation (138) can be obtained by purely Newtonian cause the integration constants, namely the rotation pa- gravity! But when can a GR result be exactly synony- rameter a and mass m are actually zero: am==0 mous with a purely Newtonian result? To appreciate this, [57,58]. have a close look at GR Poission equation (124). Even if this suggestion that FRW metric too is tacitly a Assume gravitation is extremely weak: vacuum solution, would be ignored at this moment, there is now firm conclusion that galaxies are distributed in a g 12 (148) 00 c2 fractal pattern atleast on scales ~100 Mpc in violation of
Copyright © 2011 SciRes. IJAA 196 A. MITRA the assumptions of CDM model [46-52]. In fact the photon-electron plasma interaction cross-sections. Thus, latest Sloan Digital Sky Survey results confirm that the the excess dimming at high redshifts could simply be a observed universe is indeed lumpy on the largest scale consequence of the Compton scattering that accompanies [59], and Thomas et al. too have rightly questioned the the redshifting mechanism in the hot, sparse electron physical reality of “Dark Energy” [59]. The recent ob- plasmas that fill intergalactic space. servation that there is no hint of non-baryonic “Cold In Brynjolfsson’s cosmology, the universe is not ex- Dark Matter” in two dwarf galaxies Fornax and Scul- panding, and there is no time dilation affecting super- ptor raises serious questions about the validity of the nova light curves, and it claims to account for the red- CDM model [60]. shift/SNe Ia luminosity data. Note that it is possible that the distant Type 1a super- Though the BBM has had many successes to its credit novae may have different luminosities compared to local and it is the most developed cosmology, it fails to satis- ones because of different metallicities or other evolu- factory explain many cosmological facts [69]. And irre- tionary effects. And even if they would be standard can- spective of the presence or absence of an appropriate dles without appreciable cosmic evolution, there is some static model for the universe, there are innumerable chance that their faintness could be due to opacities of problems with the BBM. In particular, distant inter-galactic medium. For example, it has been BBM may be failing the angular size test; whereas a pointed out that Lyman Alpha clouds might introduce static model fits data better [70]. some non-transparency for optical emission from distant The age of extremely red and massive galaxies at universe [61]. It has been also argued that the atmos- very high redshift may be contradicting the CDM phere of planets could be additional sources of opacities cosmology [71]. And most importantly, for very distant supernovae lights [62]. Also, there could The Gamma Ray Bursts having zs as large as 9 be a fundamental non-accuracy in estimating the precise do not show signs of supposed space-expansion re- luminosity distances of distant supernovae because of lated time dilation [72,73]. missing gaps and extrapolations in the cosmic distance In fact a static model of universe (which must be en- measurement ladders. Further, consider the fact that we tirely different from a vacuous ESU) may fit at least are able to “see” galaxies and measure their redshifts some of the cosmological observations better [74,75] because they are radiating. But in the standard cosmolo- without any exotic assumptions. gies, galaxies are considered as non-radiating neutral Also, in an infinite and eternal universe, part of the “dust” particle. Consequently, one should leave here an starlight may get thermalized to generate the observed open window for some hitherto unknown physics or sur- microwave background radiation, as was first suggested prise for understanding this new phenomenon. In general, by Sir Fred Hoyle. However, there could also be alterna- there could be unknown systemetic effect behind the tive explanations for the same. For instance, the micro- interpretation of extra faintness of distant Type 1a su- wave background radition might be due to superposition pernovae [29]. of redshifted quiescent surface glow of the Eternally Col- And if the universe would indeed be unbounded and lapsing Objects, the so-called “Black Holes” [76]. infinite hierarchial fractal, then the basic assumption Note, when acoustic wave packets traverse a disper- behind the formulation of Hubble’s law, i.e., strict ho- sive medium for very long duration, they develop “red- mogeneity, gets invalidated. In such a case, the universe shift” [77]. And if the wave packet will have a Gaussian need not be expanding and the observed redshits could shape, it maintains this shape despite central energy loss; be of non-Doppler origin. Recall long back Ellis [63] and further, the attendant redshift obeys the Hubble’s argued that observations really cannot rule out a static Law [77]. Recall, all emitted electromagnetic lines are but inhomogeneous universe. Photons propagating over actually narrow “wave packets”, and their propagation in cosmic scales might undergo energy losses by yet un- the cosmos is likely to be dissipative. At a classical level, known feeble quantum electrodynamical interaction with one can think that the cosmos with matter and plasma the quantum vacuum [64]. Photons might also lose en- behaves as a “dissipative medium”; and on a Quantum ergy by interacting with plasma permeating the whole Electrodynamic level, one may take the QED vacuum as cosmos [65]. It is interesting to note that the plasma red- a “dissipative medium”. Can the electromagnetic wave shift proposed by Ari Brynjolfsson can simulate a value packets too undergo red-shift similar to the acoustic of “Hubble Parameter” 75 Km/s/Mpc by considering waves? Or can the electromagnetic wave packets spread a mean cosmic free electron density of 0.0002 cm–3 in the momentum space like the “de-Broglie matter [64-68]. The only “new physics” involved with his red- waves” and somehow simulate the Hubble’s law? shift mechanism is to include an electron-electron colli- In any case, from a purely observational view point, sion term in an otherwise standard quantum derivation of one can say that [78] “it is impossible to conclude either
Copyright © 2011 SciRes. IJAA A. MITRA 197 way whether the Universe is expanding or static. The ID: 044006. evidence is equivocal; open to more than one interpreta- [7] C. G. Boehmer, “General Relativistic Static Fluid Solu- tion. It would seem that cosmology is far from a preci- tions with Cosmological Constant,” Diploma Thesis, Pots- sion science, and there is still a lot more work that needs dam University, Potsdam, 2002, gr-qc/0308057. to be done to resolve the apparent evidences.” [8] C. G. Bohmer and G. Fodor, “Perfect Fluid Spheres with Cosmological Constant,” Physical Review D, Vol. 77, No. 15. Endnote 6, 2008, Article ID: 064008 [9] A. D. Rendall and B. G. Schmidt, “Existence and Proper- A very old version of this paper was submitted to Phys. ties of Spherically Symmetric Static Fluid Bodies with a Rev. D. in 2008. And though the assigned referee did not Given Equation of State,” Classical & Quantum Gravity, Vol. 8, No. 5, 1991, p. 985. point out any technical error, he would not accept the doi:10.1088/0264-9381/8/5/022 result =0 claiming it would be against observation! [10] T. W. Baumgarte and A. D. Rendall, “Regularity of Sphe- Simulataneously one moderator of arXiv.org also quietly rically Symmetric Static Solutions of the Einstein Equa- shifted it from astro-ph to phys-gen section; and in frus- tions,” Classical & Quantum Gravity, Vol. 10, No. 2, 1993, tration, I did not look at this manuscript for 3 years. Only p. 327. doi:10.1088/0264-9381/10/2/014 this yesr, another version was submitted to J. Cosmology [11] A. Einstein, “On a Stationary System with Spherical Sym- & Astroparticle Physics. The referee here accepted that metry Consisting of Many Gravitating Masses,” Annals the proofs claiming =0 are correct. 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