MNRAS 442, 382387 (2014) doi:10.1093/mnras/stu859

Why the Rh = ct cosmology is a vacuum solution in disguise and why allbig bang models should be so

Abhas MitraTheoretical Astrophysics Section, Bhabha Atomic Research Centre, Mumbai 400085, India

Accepted 2014 April 29. Received 2014 April 28; in original form 2014 January 30

ABSTRACTRecently, Melia and his coworkers have proposed the so-called Rh = ct cosmology where thescale factor of the universe is a(t) t and the spatial part is flat. Here, we look at this proposalfrom a fundamental angle. First, we note that Melia cosmology looks strikingly similar to theold Milne cosmology where a(t) t and the spatial part is negatively curved. It is known thatthough Milne cosmology is a valid Friedmann solution, it actually corresponds to = 0 andcan be described by a globally static Minkowski metric. Secondly, we note that for the Meliamodel, Ricci & Kretschmann scalars assume their perfect static form hinting that it too maytacitly correspond to vacuum. To compare Melia universe with the Milne universe, we expressMelia metric too in curvature/Schwarzschild coordinates. Finally, by using the fact for suchcoordinate transformations dx 4 = Jdx4, where J is the appropriate Jacobian, we explicitlyshow that Melia metric is static, which for k = 0 case implies vacuum. This shows that evenapparently meaningful general relativistic solutions could be illusory as far as physical realityis concerned. And since Melia model is the unique solution for the big bang model, eventually,all big bang models could be mathematical illusions.

Key words: gravitation dark energy inflation large scale structure of Universe.

1 IN T RO D U C T I O N

The dominant cosmological model for past several decades has beenthe so-called big bang Model (BBM) by which universe suddenlycame into being from a spacetime singularity. The spacetimegeometry of isotropic and homogeneous BBM is described by theFriedmann Robertson Walker (FRW) metric:

ds2 = dt2 a2(t)[

dr2

1 kr2 + r2d"2

], (1)

where a(t) is the scale factor, t is the universal cosmic time, r isa comoving radial coordinate, d"2 = d2 + sin 2 d2 and thenormalized spatial curvature parameter k can assume values of 0,+1, or 1. Thus, for the FRW metric, one has

g00 = 1; grr = a2(t)

1 kr2 . (2)

There are many apparent observational supports for the hypoth-esis that big bang was hot, corresponding to an extremely hightemperature and, hence, an equation of state (EOS) p=+(1/3)c2.Even this extreme radiation-dominated universe is expected to un-

E-mail: amitra@barc.gov.in

dergo decelerated expansion due to its self-gravity. However, itis believed that following the grand unified scale phase at tem-perature of 1015 GeV, universe went through a phase transitionwhich resulted in a vacuum like EOS of p = c2 and a briefhyperaccelerating inflationary phase of duration of 1032 s. Fol-lowing this anomalous phase, universe is supposed to have beenreverted to the radiation-dominated deceleration state until a cos-mological redshift of z 3600. Following this, matter-dominateduniverse sets in and which of course is expected to decelerate due toself-gravity:

The dynamical evolution governing the cosmic expansion is givenby the Friedmann equation (G = c = 1) (Longair 2008):

a

a= 4

3( + 3p), (3)

where an overdot denotes differentiation by t, is the proper densityand p is the isotropic pressure. The observation that the distantType Ia supernovae appear to be dimmer compared to the expectedbrightness may be interpreted as an evidence that in the recentepochs, the universe has been undergoing an accelerated expansionrather than an decelerated expansion (Riess et al. 1998; Perlmutteret al. 1999). In order to explain such a recent acceleration, thelong neglected cosmological constant & was reinstated. With its

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Vacuous Rh = ct cosmology 383inclusion, one would now have an effective matter density andpressure

e = (t) + &8 ; pe = p(t)&

8(4)

so that the dynamical Friedmann equation gets modified toa

a= 4

3(e + 3pe) = 43 ( + 3p &/4). (5)

In such a case, it is possible to explain the occurrence of a > 0 interms of & > 0, or an unknown dark energy (DE) endowed witha negative pressure. Hence, the most popular version of cosmologyis known as LCDM cosmology where L stands for & or DE andCDM stands for cold dark matter. However, in the past few years,Melia & his coworkers (MC) have proposed that universe neitherunderwent any deceleration nor any acceleration ever. On the otherhand, they have claimed that cosmic scale factor has a unique form:a(t) = Ft, where F is a constant (Bikwa, Melia & Shevchuk 2012;Melia & Schevchuk 2012; Melia 2012a).

For arriving at such a drastic conclusion, MC assumed the uni-verse to be spatially flat (k= 0) and which seems to be in agreementwith the observations. Then they defined the notion of a cosmo-logical horizon having an expression

Rh = 2GMc2

, (6)

where R = ar, the areal radius parameter and M(R) is the gravi-tational mass enclosed by a sphere of radius R. Next, MC notedthat at the present cosmic epoch t = t0, the numerical value ofRh 13.3 billion light-year for an assumed value of Hubble param-eter H = 70 km s1 Mpc1. And since t0 13.3 billion years, MCnoted that Rh(t0) t0. Finally, they claim that there must have anequality Rh(t) = ct at any cosmic epoch. Such a claimed equalityin turn requires a unique form of a(t) = Ft. By differentiation, itis seen that this constant is nothing but the supposed fixed speedof expansion: F = a. While arriving at such an Rh = ct relation-ship MC repeatedly invoked Weyls postulate and Corollary ofBirkhoffs Theorem. In this context, the following comments maybe in order:

Weyls postulate only demands that galaxies are moving alongrespective radial geodesics which do not intersect one another im-plying galaxies are in free fall without any mutual collision (Misner,Thorne & Wheeler 1973). And it is difficult to see how Weyls pos-tulate can dictate a unique form of a(t).

As to Birkhoffs theorem, it only tells that the vacuum spacetime around an adiabatically oscillating isolated self-gravitatingsphere is static and given by the vacuum Schwarzschild metric(Birkhoff 1923). In addition, an exterior observer is free to imaginethe entire central mass to be concentrated as a point at the origin.But neither is any section of the universe isolated, nor is there anyvacuum outside it. Thus, Birkhoffs theorem cannot be invokedfor the universe. MC also point out the fact that, in Newtoniangravity, for a spherically symmetric mass distribution, force on agiven shall is determined by the inner mass distribution and thismay be true in general relativity too. But for an unbounded non-isolated mass distribution, Newtonian gravitational force at anygiven point is ill defined and which was one of the major problemsfor Newtonian cosmology. So results pertaining to isolated objectscannot be applied even for Newtonian cosmology let alone for arelativistic cosmology.

Thus, even though one might arrive at an Rh = ct relationship fol-lowing the assumptions and arguments of MC, there is no denyingthat there are loose ends to various arguments, and hence one need

not accept the Rh = ct proposal at its face value. Accordingly, therealready have been considerable criticisms for this proposal (Bil-icki & Seikel 2012; Lewis & Oirschot 2012, 2013; Lewis 2013).Essentially, the lines of criticisms are of three kinds apart from inap-propriate application of Birkhoffs Theorem and vague referenceto Weyls postulate:

(1) The basic concept that Hubble sphere acts as the event horizonof a Schwarzschild black hole which limits our cosmic views isflawed; despite the apparent horizon like appearance of Rh in theFRW metric in Schwarzschild coordinates, physically there is nosuch barrier (Lewis 2013; Lewis & Oirschot 2013).

(2) The EOS implied by Rh = ct cosmology p = (1/3)c2 isunphysical (Lewis & Oirschot 2013). On the other hand, most ofthe cosmologists believe that there are strong circumstantial evi-dence from primordial nucleosynthesis and existence of microwavebackground radiation that except for the brief moment of inflation,the universe was radiation dominated with an EOS p = +(1/3)c2during its initial periods lasting up to z 3600.

It has been also suggested that existence of matter is extremelyproblematic in Rh = ct cosmology, and presence of matter is likelyto destroy the strict linear expansion property a(t) t (Lewis &Oirschot 2013).

(3) Eventually, it has been claimed that Rh = ct cosmology cannotsatisfactorily explain cosmic observations (Bilicki & Seikel 2012):

Our general conclusion is that the discussed model is stronglydisfavoured by observations, especially at low redshifts (z < 0.5).In particular, it predicts specific constant values for the decelerationparameter and for redshift derivatives of the Hubble parameter,which is ruled out by the data. Nonetheless, it has been found thatthe Melia solution is indeed the unique permissible solution of BBM(Mitra 2014a,b) and which negates the BICEP2 claim of confirmingcosmic inflation.

Yet, here we shall show that Rh = ct cosmology actually cor-responds to the globally static Minkowski vacuum the same wayMilne cosmology having a(t)= ct represents a Minkowski vacuum.

2 H I N T S T H AT Rh = ct UNI VERSEIS A STATIC O NE

First, we recall the Milne universe (Milne 1933) which technicallyis a valid FRW solution with k = 1 and a(t) = ct:

ds2 = dt2 c2t2[

dr2

1 + r2 + r2d"2

]. (7)

From a purely mathematical point of view, it represents galaxiesundergoing Hubble expansion with a fixed speed a = c without anyacceleration or deceleration. Also prima facie, the spatial part ofMilne universe is negatively curved. But it is known that Milne uni-verse can be represented by a globally static flat Minkowski spacetime by making use of Schwarzschild/curvature coordinates R, T(see later). So Milne metric is a very good example of how generalrelativistic exact solutions may lead to false pictures about com-plex physical reality. Note that the Rh = ct universe with a(t) = Ftand k = 0 looks even simpler to the Milne universe:ds2 = dt2 F 2t2(dr2 + d"2). (8)And since its spatial part is already flat, it could be more apt forbeing the Minkowski vacuum in the disguise of a curved FRWsolution.

In this context, first we note that for the vacuous Milne metric to,F = a = constant just like the Melia metric, and the only difference

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384 A. Mitra

is that while F = c for the Milne case the value of F is not knowna priori in the Melia case (Chodorowski 2005). And this hints thatMelia metric too may be vacuous.

To explore further, let us recall the expressions for the physicallyimportant Kretschmann and Ricci scalars associated with the FRWmetric (Muller & Grave 2011; Mitra 2014a).

The Kretschmann scalar is given by (G = c = 1):

K = 12 a2a2 + a4 + 2ka2 + k2

a4, (9)

while the Ricci scalar is

R = 6aa + a2 + k

a2. (10)

Note that for a static Einstein universe having k = +1, and a =a = 0, one would simply obtain

K = 12a4

(11)

and

R = 6a2. (12)

From equations (9) and (10), for the Milne metric with a = 0 andk = 1, one finds R = K = 0 despite a mathematical expansionexpressed through a(t) = ct. On the other hand, for a = 0, a = Fand k = 0 Melia model, one finds

K = 12F4

a4, (13)

while

R = 6F2

a2. (14)

Since for an assumed finite F, F can be absorbed in the rescaleddefinition of a, we see that for the Rh = ct case, Kretschmann andRicci scalars assume their static form. And this is a strong hint thatRh = ct spacetime too might represent a static spacetime like theMilne case.

2.1 Coordinate transformations to Minkowski spacetime?

Pure Hubble flow implies pure radial motion of test particles withoutany angular motion. Thus, effectively d"2 = 0 in all ideal FRWcases. So, tentatively let us use an effective metric

ds2 = dt2 a2dr2

1 kr2 (15)to extricate the secrets of Milne and Melia metrics. In the formercase, one obtains

ds2 = dt2 t2dr2

1 + r2 . (16)Now let us introduce the coordinates

T = t cosh

1 + r2; X = t sinh

1 + r2. (17)In terms of these new coordinates, one can rewrite equation (16) asds2 = dT 2 dX2, (18)which strongly suggests that Milne metric may be representing theMinkowski spacetime. As to the temporalradial Melia metric(Abramowicz et al. 2007), we haveds2 = dt2 F 2t2dr2. (19)

Again let us define

T = t coshFr; X = t sinhFr (20)in terms of which the Melia metric (19) assumes the 1DMinkowskian form

ds2 = dT 2 dX2. (21)While this strongly suggests that Melia metric too may be rep-

resenting the vacuum Minkowski metric just like the Milne case(Abramowicz et al. 2007), the foregoing example is actually onlytentative. This is so because the while the full Milne metric (7) canindeed assume the exact spherically symmetric Minkowski form(c = 1):ds2 = dT 2 dR2 R2d"2 (22)by means of the transformations

T = t

1 + r2; R = rt. (23)Melia metric (8) refuses to do so under the assumption t = finite orr = 0. Thus, one needs to probe more deeply to unveil the intrinsicnature of the Melia metric.

3 G E N E R A L F O R M O F F RW M E T R I CI N S C H WA R Z S C H I L D C O O R D I NAT E S

For exploring Melia metric in from a greater depth, we re-call the recently derived general form of the FRW metric inSchwarzschild/curvature coordinates (Mitra 2013a)

ds2 = 2 2

)2dT 2 dR

2

2 R2d"2, (24)

where K = k/a2 = tT = t/T , (25)

R(r, t) = ra(t), (26)

)2 = 1 kr2 = 1KR2, (27)

2 = 1 kr2 a2r2 = 1KR2 H 2R2; H = a/a. (28)Obviously, here

gRR = 1 2. (29)

Here, we let a subscript denote partial differentiation by the corre-sponding variable, and the condition that the metric (24) is diagonalrequired that (Mitra 2013a)

tR = tR

= ar 2. (30)

Then,

dt = tT dT + tRdR. (31)Since

dR = a dr + ra dt, (32)equations (25), (30), (31) and (32) lead to

dT = )2

2dt + aar

2dr. (33)

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Vacuous Rh = ct cosmology 385Also since

a dr = dR a rdt, (34)one finds from equations (25), (31) and (34)

rT = aar. (35)

Let us now recall the transformation rules for the metric tensorfrom an unprimed to a primed coordinate system:

gmn =xi

xm xk

xn gik. (36)

Then, as one transforms from the t, r coordinates to T, R coordi-nates by keeping and unchanged, one has

gRR = t2Rg00 + r2Rgrr (37)Following this, by using equations (2), (29), (30) and (37), one findsthat

rR = )2

a 2. (38)

By summing up these foregoing results, for the k = 0 and a = Ftcase, one has (see equation 27):)2 = 1; 2 = 1 F 2r2 (39)and

tT = , tR = Fr1 F 2r2, , rT =Fra

,

rR = 1a(1 F 2r2) (40)

so that, by using equations (25), (37) and (40), the Jacobian for thistransformation (t, r, , ) (T, R, , ) isJ = |rRtT rT tR| =

a. (41)

For these relevant coordinate transformations, now let us applythe fundamental transformation rule

dr dt d d = J dR dT d d. (42)By invoking equations (32), (33) and (41), the foregoing transfor-mation rule boils down to

dr dt = Fra(1 F 2r2) dt

2 + aFr1 F 2r2 dr

2 + 1 + F2r2

1 F 2r2 dr dt . (43)

And obviously, the above equation demands Fr = 0. Thus, thefundamental coordinate transformation properties impose a strongconstraint on the Rh = ct metric:Fr = 0. (44)Therefore, Rh = ct cosmology corresponds to F = a = 0. In otherwords, Melia cosmology too corresponds to an intrinsically staticmodel just like the Milne cosmology, and the apparent expansionwas just a coordinate effect.

Further, for a k= 0 case, the occurrence ofF = a = 0 too impliesa vacuum spacetime, e = 0, in view of the following Friedmannequation:

8e3

= k + a2

a2. (45)

Note that for the k=1 and a = 1 Milne case too, the foregoingequation suggests a vacuum solution with e = 0. Simultaneously,for a =, all FRW models degenerate into vacuum.

Hence, the real solution for both Milne and Melia metrics are thefollowing:

(i) While Milne metric is the asymptotic final state of a k = 1FRW universe with a = , t = and e = 0, the Melia metricrepresents the asymptotic final state of a k= 0 FRW universe whichhas attained a =, at t = so that e = 0.

In the Milne case, one can see more emphatically why it cor-responds to t = . Since Milne metric admits a globally staticMinkowski vacuum, obviously it cannot represent expansion of anyfluid. Thus from physical perspective, for the Milne metric, onemust have the expansion scalar (Misner et al. 1973) = aua, (46)the divergence of fluid 4-velocity, to be zero. However, if one woulduse the mathematical form of a(t) = ct, one would find

= 3 aa= 3

t. (47)

Thus, latently, one has t = for the Milne metric. For the Meliametric too, one obtains the same mathematical symbol for = 3/t .Again, for self-consistency, one must latently have t = here too.

4 W H AT A B O U T T H E C L A I M S O FO B S E RVAT I O NA L SU P E R I O R I T Y

To be fair with the Rh = ct proposal, it must be mentioned thatMC have tried to defend their model against various criticismsmentioned earlier (Bikwa et al. 2012; Melia & Schevchuk 2012;Melia 2012a,b,c, 2013a,b,c; Wei, Wu & Melia 2013). They have notonly tried to fend off various criticisms, but also claimed that Rh = ctproposal is superior to LCDM cosmology in explaining cosmicobservations, whether they are supernova data or gamma-ray burstdata or it is explanation of massive quasars at a very early stage ofthe universe. In fact, after the initial submission of this manuscriptMC have published three more papers claiming superiority of theircosmology over LCDM model (Melia 2014a,b; Wei et al. 2014).Thus, for appreciation of all readers, apart from mathematical proof,it would be in order to defend our result which tells that despite suchimpressive peer reviewed claims, Melia cosmology is bound to bevacuous.

Here, note that our aim is not to go into claims and counter-claimsabout better fitting of data to certain cosmological models; in fact,the present author is not competent to do so. But on the other hand,we note that cosmology is no laboratory science and is funda-mentally different from even many other branches of experimentalastronomy, for instance, solar physics.

Since cosmology is not an experimental science, but an obser-vational one, we must take this into account when we try to falsifyour theories in the sense advocated by Karl Popper (Kolb 2007).

If one would rely on claims to data fitting to various models, thenthere might be significant chance that universe may be static as well.This is so because in order to test the expansion of the universe andits geometry, Lopez-Correidora has recently carried out the Alcock& Paczynski cosmological test on six cosmological models (Lopez-Corredoira 2014). This test about the ratio of observed angular sizeto radial/redshift size. The main advantage of this test is that it doesnot depend on the evolution of the galaxies, but only on the geometryof the universe. And out of the six models tested, the data fit onlytwo models: (i) the concordance LCDM model and (ii) the model ofa static universe with tired-light redshift (Lopez-Corredoira 2014).

Note that since cosmic gamma ray bursts (GRBs) are the mostluminous sources in the universe they are considered as an important

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386 A. Mitra

cosmological probe. In fact, GRBs could be useful in testing variouscosmological models because we can detect them from as far asz 8. And obviously, in an expanding universe, the GRB lightcurves should show time dilation effect (Zhang et al. 2013). Butanother study of GRB light curves failed to detect any such timedilation effect (Kocevski & Petrosian 2013).

Interestingly, studies of GRBs also show that universe may havelumpy structure even at z 23 (Hovarah, Hakkila & Bagony2014).

Note that around 20 years back, most of the cosmologists thoughtthat there was no need for a cosmological constant and universewas best described by the simple Einsteinde-Sitter model wherea(t) t2/3. But now the Einsteinde-Sitter model has gone out offavour. In fact, as late as 2005, it has been claimed that the Einsteinde-Sitter model may explain all Type 1a supernova observationswithout postulating an accelerating universe (Vishwakarma 2005).

History of cosmology shows that there have been significantadjustments as well as paradigm shifts following big leaps in ourability to improve detection techniques and instrument sensitivities.There is no reason to believe that we have already exhausted allchances of qualitatively new realms of observational results. Alsoas pointed out by Kolb, Much has been written about the successesof the standard model of cosmology, but celebrations of its successesshould be tempered by the fact that it is based upon unknown physics(Kolb 2007).

The fact that we still do not have clear idea about the nature ofdark energy and dark matter which are supposed to comprise 9495 per cent of the universes energy budget is indeed very worryingand some authors have already called for the need for a ParadigmShift (Horvath 2009; Martinez & Trimble 2009). In fact, it has beenpointed out that the analysis of the Type 1a supernova data may befaulty in a universe which may be lumpy (Clarkson et al. 2012).

Thus, in view of the very nature of cosmology, both cosmologicalobservations as well claims of their matching to a given model mightbe misleading. And the claims of Rh = ct and LCDM cosmologiesthat they can explain the cosmological observations satisfactorilymust be viewed in this backdrop. Indeed DE is likely to be anillusion since cosmological constant is exactly zero (Mitra 2013b).

5 C O N C L U S I O N S

In spite of many apparently successful claims that Rh = ct cosmol-ogy can explain the universe in a very satisfactory manner, suchclaims must eventually be mirages. This is so because here weshowed that Rh = ct cosmology actually corresponds to an asymp-totic FRW model with k = 0, t = and = 0. This is almostsimilar to the case of the Milne cosmology which is another asymp-totic FRW model with k = 1, t = and = 0. In this context,it may be noted that there have been several claims that the vac-uum Milne model too can explain many cosmological observations(Benoit-Levy & Chadrin 2012).

Though from the viewpoint of general relativity, Milne modelcannot contain any real particle/galaxy ( = 0), one may interpret itas a special relativistic problem where test particles are expandingin a background Minkowski vacuum, in fact, this was the originalmotivation of the Milne cosmology (Milne 1933). And since thereare claims that Milne model can reasonably explain many observa-tions, in recent times, there have been a prolonged debate whethercosmological redshifts can be attributed to expansion of space orcould be of attributed to motion of galaxies in a non-expandingbackground space. Without making any elaboration on this debate,it may be just mentioned that purely Milne like kinematical origin

of cosmic redshift may not be satisfactory (Abramowicz et al. 2007,2009; Wiltshire et al. 2013) particularly when in a strict sense Milnemodel is pure vacuum with = 0.

It is known that most of the exact general relativistic solutionsare devoid of any physical significance (Stephani et al. 2003). Andhere we found that even some exact solutions which appear to bephysically meaningful are actually mathematical illusions or at bestasymptotic limiting solutions corresponding to vacuum. This is soeven when they appear to explain many cosmological observationswith reasonable or good level of satisfaction. Since Melia cosmol-ogy is the unique solution for BBM, it follows that all BBMs couldbe mathematical illusions.

N OT E A D D E D I N P RO O F

After this paper went into press, a new paper by E. J. Lerner,R. Falomo, R. Scarpa, IJMPD 2014, 23, 1450058, has claimedthat a proper Tolman test suggests that Universe is actually static.

ACK NOW L E DG E ME NT S

The author thanks both the anonymous referees for making a seriesof useful suggestions. In addition, the author thanks the anonymouseditor for making several suggestions which helped in an improvedpresentation of this manuscript.

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