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Abstract

Recently, Melia and his coworkers have proposed the so-called Rh = ct cosmology where the scale factor of the universe is a(t) ∝ t and the spatial part is flat. This model also implies a fixed equation of state (EOS) ρ = −(1/3)ρc2 of the universe. Here, we look at this proposal from a fundamental angle. First, we note that Melia cosmology looks strikingly similar to the old Milne cosmology where a(t) ∝ t and the spatial part is negatively curved. It is known that though Milne cosmology is a valid Friedmann solution, it actually corresponds to ρ = 0 and can be described by a globally static Minkowski metric. Secondly, we note that for the Melia model, Ricci & Kretschmann scalars assume their perfect static form hinting that it too may tacitly correspond to vacuum. This is also necessitated by the fact that even at the moment of ‘big bang’, Melia cosmology EOS is ρ = −(1/3)ρc2 rather than the radiation-dominated ρ = +(1/3)ρc2. To compare Melia universe with the Milne universe, we express Melia metric too in curvature/Schwarzschild coordinates. Finally, by using the fact for such coordinate transformations dx′4 = Jdx4, where J is the appropriate Jacobian, we explicitly show that Melia metric is static, which for k = 0 case implies vacuum. This shows that even apparently meaningful general relativistic solutions could be illusory as far as physical reality is concerned.

gravitation, dark energy, inflation, large scale structure of Universe

1 INTRODUCTION

The dominant cosmological model for past several decades has been the so-called big bang Model (BBM) by which universe suddenly came into being from a space–time singularity. The space–time geometry of isotropic and homogeneous BBM is described by the Friedmann Robertson Walker (FRW) metric:
(1)
where a(t) is the scale factor, t is the universal cosmic time, r is a comoving radial coordinate, dΩ2 = dθ2 + sin 2θ dϕ2 and the normalized spatial curvature parameter k can assume values of 0, +1, or −1. Thus, for the FRW metric, one has
(2)
 There are many apparent observational supports for the hypothesis that ‘big bang’ was hot, corresponding to an extremely high temperature and, hence, an equation of state (EOS) p = +(1/3)ρc2. Even this extreme radiation-dominated universe is expected to undergo decelerated expansion due to its self-gravity. However, it is believed that following the grand unified scale phase at temperature of ∼1015 GeV, universe went through a phase transition which resulted in a vacuum like EOS of p = −ρc2 and a brief hyperaccelerating inflationary phase of duration of ∼10−32 s. Following this anomalous phase, universe is supposed to have been reverted to the radiation-dominated deceleration state until a cosmological redshift of z ∼ 3600. Following this, matter-dominated universe sets in and which of course is expected to decelerate due to self-gravity:
The dynamical evolution governing the cosmic expansion is given by the Friedmann equation (G = c = 1) (Longair 2008):
(3)
where an overdot denotes differentiation by t, ρ is the proper density and p is the isotropic pressure. The observation that the distant Type Ia supernovae appear to be dimmer compared to the expected brightness may be interpreted as an evidence that in the recent epochs, the universe has been undergoing an accelerated expansion rather than an decelerated expansion (Riess et al. 1998; Perlmutter et al. 1999). In order to explain such a recent acceleration, the long neglected ‘cosmological constant’ Λ was reinstated. With its inclusion, one would now have an effective matter density and pressure
(4)
so that the dynamical Friedmann equation gets modified to
(5)
In such a case, it is possible to explain the occurrence of |${\ddot{a}} >0$| in terms of Λ > 0, or an unknown ‘dark energy’ (DE) endowed with a negative pressure. Hence, the most popular version of cosmology is known as ‘LCDM cosmology’ where ‘L stands for Λ or DE and CDM stands for ‘cold dark matter’. However, in the past few years, Melia & his coworkers (MC) have proposed that universe neither underwent any deceleration nor any acceleration ever. On the other hand, 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 universe to be spatially flat (k = 0) and which seems to be in agreement with the observations. Then they defined the notion of a ‘cosmological horizon’ having an expression
(6)
where R = ar, the areal radius parameter and M(R) is the gravitational mass enclosed by a sphere of radius R. Next, MC noted that at the present cosmic epoch t = t0, the numerical value of Rh ∼ 13.3 billion light-year for an assumed value of Hubble parameter H = 70 km s−1 Mpc−1. And since t0 ∼ 13.3 billion years, MC noted that Rh(t0) ∼ t0. Finally, they claim that there must have an equality Rh(t) = ct at any cosmic epoch. Such a claimed equality in turn requires a unique form of a(t) = Ft. By differentiation, it is seen that this constant is nothing but the supposed fixed speed of expansion: |$F={\dot{a}}$|⁠. While arriving at such an Rh = ct relationship MC repeatedly invoked ‘Weyl's postulate’ and ‘Corollary of Birkhoff's Theorem’. In this context, the following comments may be in order:

Weyl's postulate only demands that ‘galaxies’ are moving along respective radial geodesics which do not intersect one another implying galaxies are in free fall without any mutual collision (Misner, Thorne & Wheeler 1973). And it is difficult to see how Weyl's postulate can dictate a unique form of a(t).

As to Birkhoff's theorem, it only tells that the vacuum space–time around an adiabatically oscillating isolated self-gravitating sphere is static and given by the vacuum Schwarzschild metric (Birkhoff 1923). In addition, an exterior observer is free to imagine the entire central mass to be concentrated as a point at the origin. But neither is any section of the universe isolated, nor is there any vacuum outside it. Thus, Birkhoff's theorem cannot be invoked for the universe. MC also point out the fact that, in Newtonian gravity, for a spherically symmetric mass distribution, force on a given shall is determined by the inner mass distribution and this may be true in general relativity too. But for an unbounded non-isolated mass distribution, Newtonian gravitational force at any given point is ill defined and which was one of the major problems for Newtonian cosmology. So results pertaining to isolated objects cannot be applied even for Newtonian cosmology let alone for a relativistic cosmology.

Thus, even though one might arrive at an Rh = ct relationship following the assumptions and arguments of MC, there is no denying that there are loose ends to various arguments, and hence one need not accept the Rh = ct proposal at its face value. Accordingly, there already have been considerable criticisms for this proposal (Bilicki & Seikel 2012; Lewis & Oirschot 2012, 2013; Lewis 2013). Essentially, the lines of criticisms are of three kinds apart from inappropriate application of ‘Birkhoff's Theorem’ and vague reference to ‘Weyl's postulate’:

  1. The basic concept that Hubble sphere acts as the event horizon of a Schwarzschild black hole which limits our cosmic views is flawed; despite the apparent horizon like appearance of Rh in the FRW metric in Schwarzschild coordinates, physically there is no such barrier (Lewis 2013; Lewis & Oirschot 2013).

  2. The EOS implied by Rh = ct cosmology p = −(1/3)ρc2 is unphysical (Lewis & Oirschot 2013). On the other hand, most of the cosmologists believe that there are strong circumstantial evidence from primordial nucleosynthesis and existence of microwave background radiation that except for the brief moment of inflation, the universe was radiation dominated with an EOS p = +(1/3)ρc2 during its initial periods lasting up to z ∼ 3600.

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

  3. Eventually, it has been claimed that Rh = ct cosmology cannot satisfactorily explain cosmic observations (Bilicki & Seikel 2012):

     ‘Our general conclusion is that the discussed model is strongly disfavoured by observations, especially at low redshifts (z < 0.5). In particular, it predicts specific constant values for the deceleration parameter and for redshift derivatives of the Hubble parameter, which is ruled out by the data.’

    In the following, we shall resolve all such contradictions by showing that Rh = ct cosmology actually corresponds to the globally static Minkowski vacuum the same way Milne cosmology having a(t) = ct represents a Minkowski vacuum.

2 HINTS THAT Rh = ct UNIVERSE IS A STATIC ONE

First, we recall the Milne universe (Milne 1933) which technically is a valid FRW solution with k = −1 and a(t) = ct:
(7)
 From a purely mathematical point of view, it represents galaxies undergoing Hubble expansion with a fixed speed |${\dot{a}} =c$| without any acceleration or deceleration. Also prima facie, the spatial part of Milne universe is negatively curved. But it is known that Milne universe can be represented by a globally static flat Minkowski space–time by making use of Schwarzschild/curvature coordinates R, T (see later). So Milne metric is a very good example of how ‘general relativistic exact solutions’ may lead to false pictures about complex physical reality. Note that the Rh = ct universe with a(t) = Ft and k = 0 looks even simpler to the Milne universe:
(8)
And since its spatial part is already flat, it could be more apt for being the Minkowski vacuum in the disguise of a curved FRW solution.

In this context, first we note that for the vacuous Milne metric to, |$F={\dot{a}}=\rm{constant}$| just like the Melia metric, and the only difference is that while F = c for the Milne case the value of F is not known a priori in the Melia case (Chodorowski 2005). And this hints that Melia metric too may be vacuous.

To explore further, let us recall the expressions for the physically important Kretschmann and Ricci scalars associated with the FRW metric (Muller & Grave 2011; Mitra 2014).

The Kretschmann scalar is given by (G = c = 1):
(9)
while the Ricci scalar is
(10)
 Note that for a static Einstein universe having k = +1, and |${\dot{a}} = {\ddot{a}}=0$|⁠, one would simply obtain
(11)
and
(12)
 From equations (9) and (10), for the Milne metric with |${\ddot{a}}=0$| and k = −1, one finds |${\cal R} ={\cal K} =0$|despite a mathematical expansion expressed througha(t) = ct. On the other hand, for |${\ddot{a}}=0$|⁠, |${\dot{a}} =F$| and k = 0 Melia model, one finds
(13)
while
(14)
 Since for an assumed finite F, F can be absorbed in the rescaled definition of a, we see that for the Rh = ct case, Kretschmann and Ricci scalars assume their static form. And this is a strong hint that Rh = ct space–time too might represent a static space–time like the Milne case.

2.1 Coordinate transformations to Minkowski space–time?

Pure Hubble flow implies pure radial motion of test particles without any angular motion. Thus, effectively dΩ2 = 0 in all ideal FRW cases. So, tentatively let us use an effective metric
(15)
to extricate the secrets of Milne and Melia metrics. In the former case, one obtains
(16)
Now let us introduce the coordinates
(17)
In terms of these new coordinates, one can rewrite equation (16) as
(18)
which strongly suggests that Milne metric may be representing the Minkowski space–time. As to the temporal–radial Melia metric (Abramowicz et al. 2007), we have
(19)
 Again let us define
(20)
in terms of which the Melia metric (19) assumes the 1D Minkowskian form
(21)
 While this strongly suggests that Melia metric too may be representing the vacuum Minkowski metric just like the Milne case (Abramowicz et al. 2007), the foregoing example is actually only tentative. This is so because the while the full Milne metric (7) can indeed assume the exact spherically symmetric Minkowski form (c = 1):
(22)
by means of the transformations
(23)
Melia metric (8) refuses to do so under the assumption t = finite or r ≠ 0. Thus, one needs to probe more deeply to unveil the intrinsic nature of the Melia metric.

3 GENERAL FORM OF FRW METRIC IN SCHWARZSCHILD COORDINATES

For exploring Melia metric in from a greater depth, we recall the recently derived general form of the FRW metric in Schwarzschild/curvature coordinates (Mitra 2013)
(24)
where K = k/a2
(25)
(26)
(27)
(28)
Obviously, here
(29)
Here, we let a subscript denote partial differentiation by the corresponding variable, and the condition that the metric (24) is diagonal required that (Mitra 2013)
(30)
Then,
(31)
Since
(32)
equations (25), (30), (31) and (32) lead to
(33)
 Also since
(34)
one finds from equations (25), (31) and (34)
(35)
 Let us now recall the transformation rules for the metric tensor from an unprimed to a primed coordinate system:
(36)
 Then, as one transforms from the t, r coordinates to T, R coordinates by keeping θ and ϕ unchanged, one has
(37)
Following this, by using equations (2), (29), (30) and (37), one finds that
(38)
 By summing up these foregoing results, for the k = 0 and a = Ft case, one has (see equation 27):
(39)
and
(40)
so that, by using equations (25), (37) and (40), the Jacobian for this transformation (t, r, θ, ϕ) → (T, R, θ, ϕ) is
(41)
 For these relevant coordinate transformations, now let us apply the fundamental transformation rule
(42)
By invoking equations (32), (33) and (41), the foregoing transformation rule boils down to
(43)
And obviously, the above equation demands Fr = 0. Thus, the fundamental coordinate transformation properties impose a strong constraint on the Rh = ct metric:
(44)
Therefore, Rh = ct cosmology corresponds to |$F={\dot{a}}=0$|⁠. In other words, Melia cosmology too corresponds to an intrinsically static model just like the Milne cosmology, and the apparent expansion was just a coordinate effect.
Further, for a k = 0 case, the occurrence of |$F={\dot{a}} =0$| too implies a vacuum space–time, ρe = 0, in view of the following Friedmann equation:
(45)
 Note that for the k = −1 and |${\dot{a}} =1$| Milne case too, the foregoing equation suggests a vacuum solution with ρe = 0. Simultaneously, for a = ∞, all FRW models degenerate into vacuum.

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

  • While Milne metric is the asymptotic final state of a k = −1 FRW universe with a = ∞, t = ∞ and ρe = 0, the Melia metric represents the asymptotic final state of a k = 0 FRW universe which has attained a = ∞, at t = ∞ so that ρe = 0.

     In the Milne case, one can see more emphatically why it corresponds to t = ∞. Since Milne metric admits a globally static Minkowski vacuum, obviously it cannot represent expansion of any fluid. Thus from physical perspective, for the Milne metric, one must have the expansion scalar (Misner et al. 1973)
    (46)
    the divergence of fluid 4-velocity, to be zero. However, if one would use the mathematical form of a(t) = ct, one would find
    (47)
    Thus, latently, one has t = ∞ for the Milne metric. For the Melia metric too, one obtains the same mathematical symbol for |${\cal \theta } = 3/t$|⁠. Again, for self-consistency, one must latently have t = ∞ here too.

4 WHAT ABOUT THE CLAIMS OF OBSERVATIONAL SUPERIORITY

To be fair with the Rh = ct proposal, it must be mentioned that MC have tried to defend their model against various criticisms mentioned earlier (Bikwa et al. 2012; Melia & Schevchuk 2012; Melia 2012a,b,c, 2013a,b,c; Wei, Wu & Melia 2013). They have not only tried to fend off various criticisms, but also claimed that Rh = ct proposal is superior to LCDM cosmology in explaining cosmic observations, whether they are supernova data or gamma-ray burst data or it is explanation of massive quasars at a very early stage of the universe. In fact, after the initial submission of this manuscript MC have published three more papers claiming superiority of their cosmology 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 such impressive peer reviewed claims, Melia cosmology is bound to be vacuous.

Here, note that our aim is not to go into claims and counter-claims about 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 fundamentally different from even many other branches of experimental astronomy, for instance, solar physics.

‘Since cosmology is not an experimental science, but an observational one, we must take this into account when we try to falsify our theories in the sense advocated by Karl Popper (Kolb 2007).’

If one would rely on claims to data fitting to various models, then there might be significant chance that universe may be static as well. This is so because in order to test the expansion of the universe and its 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 size to radial/redshift size. The main advantage of this test is that it does not depend on the evolution of the galaxies, but only on the geometry of the universe. And out of the six models tested, the data fit only two models: (i) the concordance LCDM model and (ii) the model of a static universe with tired-light redshift (Lopez-Corredoira 2014).

Note that since cosmic gamma ray bursts (GRBs) are the most luminous sources in the universe they are considered as an important cosmological probe. In fact, GRBs could be useful in testing various cosmological models because we can detect them from as far as z ∼ 8. And obviously, in an expanding universe, the GRB light curves should show time dilation effect (Zhang et al. 2013). But another study of GRB light curves failed to detect any such time dilation effect (Kocevski & Petrosian 2013).

Interestingly, studies of GRBs also show that universe may have lumpy structure even at z ∼ 2–3 (Hovarah, Hakkila & Bagony 2014).

Note that around 20 years back, most of the cosmologists thought that there was no need for a ‘cosmological constant’ and universe was best described by the simple Einstein–de-Sitter model where a(t) ∝ t2/3. But now the Einstein–de-Sitter model has gone out of favour. In fact, as late as 2005, it has been claimed that the Einstein–de-Sitter model may explain all Type 1a supernova observations without postulating an accelerating universe (Vishwakarma 2005).

History of cosmology shows that there have been significant adjustments as well as paradigm shifts following big leaps in our ability to improve detection techniques and instrument sensitivities. There is no reason to believe that we have already exhausted all chances of qualitatively new realms of observational results. Also as pointed out by Kolb, ‘Much has been written about the successes of the standard model of cosmology, but celebrations of its successes should 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 of ‘dark energy’ and ‘dark matter’ which are supposed to comprise 94–95 per cent of the universe's energy budget is indeed very worrying and some authors have already called for the need for a ‘Paradigm Shift’ (Horvath 2009; Martinez & Trimble 2009). In fact, it has been pointed out that the analysis of the Type 1a supernova data may be faulty in a universe which may be lumpy (Clarkson et al. 2012).

Thus, in view of the very nature of cosmology, both cosmological observations as well claims of their matching to a given model might be misleading. And the claims of Rh = ct and LCDM cosmologies that they can explain the cosmological observations satisfactorily must be viewed in this backdrop.

5 CONCLUSIONS

In spite of many apparently successful claims that Rh = ct cosmology can explain the universe in a very satisfactory manner, such claims must eventually be mirages. This is so because here we showed that Rh = ct cosmology actually corresponds to an asymptotic FRW model with k = 0, t = ∞ and ρ = 0. This is almost similar to the case of the Milne cosmology which is another asymptotic FRW model with k = −1, t = ∞ and ρ = 0. In this context, it may be noted that there have been several claims that the vacuum Milne model too can explain many cosmological observations (Benoit-Levy & Chadrin 2012).

Though from the viewpoint of general relativity, Milne model cannot contain any real particle/galaxy (ρ = 0), one may interpret it as a special relativistic problem where test particles are expanding in a background Minkowski vacuum, in fact, this was the original motivation of the Milne cosmology (Milne 1933). And since there are claims that Milne model can reasonably explain many observations, in recent times, there have been a prolonged debate whether cosmological redshifts can be attributed to ‘expansion of space’ or could be of attributed to motion of galaxies in a non-expanding background 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 Milne model is pure vacuum with ρ = 0.

It is known that most of the exact general relativistic solutions are devoid of any physical significance (Stephani et al. 2003). And here we found that even some exact solutions which appear to be physically meaningful are actually mathematical illusions or at best asymptotic limiting solutions corresponding to vacuum. This is so even when they appear to explain many cosmological observations with reasonable or good level of satisfaction.

note added in proof

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

The author thanks both the anonymous referees for making a series of useful suggestions. In addition, the author thanks the anonymous editor for making several suggestions which helped in an improved presentation of this manuscript.

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