Does the Universe Really Expand, or Does the Size of Matter Shrink Instead?

Modern cosmology is built on the concept of the spatial expansion of the Universe. And the current astronomical observation is consistent with this paradigm. However, the expansion of the Universe is an expansion relative to matter. Therefore, in this work we try to switch the viewpoint from the spatial expansion of the Universe to the shrinkage of the size of matter during the evolution of the Universe, by employing Einstein’s general relativity and performing a conformal transformation of the metric. The effect of the size shrinkage of matter is then through the variation of the physical parameters/constants in a coordinated way. From this alternative viewpoint, there are advantages in realizing the evolution of the Universe, and also in better understanding Dirac’s large number hypothesis.


Introduction
Based on the astronomical observations, besides the Universe being composed of mostly dark matter and dark energy with only 4% ordinary matter [1], modern cosmology describes the evolution of the Universe as being expanding [2] after its big bang [3] and a cosmic inflation [4]. And the expansion rate of the Universe is increasing with time [5] [6].
Although this achievement in modern cosmology is so tremendous, there still exist many interesting questions to be answered. One of them is the concept of the Universe's spatial expansion. There could be two possible ways in realizing the spatial expansion of the Universe: One is that the Universe does expand spatially while the size of matter stays unchanged, and this is what we usually pic-How to cite this paper: Yo, H.-J. (2017) Does the Universe Really Expand, or Does ture; the other is that the Universe keeps its space held constant while the size of matter shrinks. The latter one has ever been considered before. Eddington discussed the possibility of "the shrinking atom" in his book [7]; Wetterich has ever considered in his work [8] a scalar-tensor model in which the size of atom, as well as the Universe, shrinks with its mass increasing. However, there is still a lack of serious and detailed illustration on this possibility from the standard theory-general relativity-without any new physics input. And the reason might originate from its foreseeable complication once the size of matter is linked to the evolution of the Universe.
In this work, we try to explore this possibility of and the mechanism for the size shrinkage of matter during the evolution of the Universe. We find that this paradigm is not only possible but can also be realized with the variation of the physical constants. This idea might remind people of the already existing scalar-tensor theories, especially the Brans-Dicke theory [9] in which the gravitational constant usually varies with a scalar field. Nevertheless, in this work we will mainly stick to the standard general relativity instead of the modified gravity theories. The result will show that, besides the gravitational constant, all the physical constants with the dimensionality of time should vary in a coordinated way such that the Universe still looks expanding in the shrinkage paradigm while all the physical laws remain intact.
The rest of this work is organized as follows: In the next section, we will give a brief description of general relativity applied to modern cosmology; We then conformally transform the metric g µν into g µν and demonstrate that these two metrics describe the same evolution of the Universe in Section 3; In Section 4, the variation of the physical "constants" in the conformal frame, thus in the shrinkage paradigm, are showed; The evolution of the Universe with the shrinkage paradigm and its relation with Dirac's large number hypothesis are described in Section 5. And the conclusion will be presented in the Section 6.

Standard Treatment
Let us start with a brief description of the general relativity applied to modern cosmology. In Einstein's general relativity, the connection with respect to the metric g µν is defined as Its action is and g L is the Einstein-Hilbert Lagrangian density, H where Φ is the matter field. The Euler-Lagrange equation where the 4-velocity ( ) Einstein's equation thus leads to And it gives Here we only consider the zero spatial curvature case, i.e., 0 k = , since the conclusion in this work will remain the same even in the 1 k = ± cases. universe case, 1 3 w = for the radiation-dominated universe case, and 1 w = − for the cosmological-constant-dominated universe case.

Conformal Treatment
Now let us switch our point of view by using a conformal transformation from g µν to g µν with g e g ψ µν µν = , then the relation between the Ricci tensors is ( ) where µ ∇ is the covariant derivative with respect to the conformal metric g µν , and so the Einstein-Hilbert Lagrangian density becomes where the symbol  means that the total derivative term is ignored after integrating by parts the second derivative term into one first derivative quadratic term and a total derivative term, and the matter Lagrangian density becomes Therefore, the total Lagrangian density becomes Here we encounter some difficulty. The second term in the RHS of Equation (21) has a "wrong sign" which makes its kinetic term become negative. Such a ghost term is usually disliked by theorists because it carries negative energy, although some theories allow its existence [10]. However, this issue in general relativity has long been identified in [11] [12] that ψ is related to the (global) timelike variable, with a negative hyperbolic signature. We can understand it as follows: Gravity is purely attractive and hence configurations are inherently physically unstable to expansion/contraction. And the sign of this term just reflects this unstable physical mode of gravitational dynamics. Since it is a characteristic of general relativity, we will not avoid this term by any modification during the comparison between the one with g µν and the other with the conformal metric g µν . Besides, it should do no harm within the familiar FLRW metric. After all, all we have done is simply a conformal transformation of the metric. We will come back to this point later.
The interval for the FLRW metric in Equation (10)

Varying Constants
After demonstrating the equivalence of using either g µν or the conformal one g µν for evolving the Universe, we can now turn back to the viewpoint  If we take G and c as the gravitational "constant" and the speed of light respectively in the shrinkage paradigm although they vary with respect to time via ( ) a t , then, by using them, we can equivalently explain the astronomical data without employing the concept of the spatial expansion of the Universe.
This can be understood from checking the conformal interval (33) in which the spatial part does not change with time. In the other words, instead of the Universe expanding with the size of matter unchanged, here we consider the Universe intact but the size of matter shrinking. And the shrinkage of matter is through the varying/decreasing of the physical constants, e.g., G and c. where ν is the related frequency. The rest mass 0 m is about matter and the Lorentz factor γ is dimensionless. Thus they will stay unchanged. Therefore it is easy to obtain the a-dependence of the Planck constant which indicates that the Planck constant will decrease as long as a increases with respect of time. We then have the a -dependence of the three fundamental physical constants most employed in general relativity and cosmology, where z is the so-called cosmological redshift 2 . One can easily deduce the a -dependence of the other physical quantities/constants by applying the known dependence, e.g., the ones in Equation (41), to the related physical laws.
There might be concerns on the details of the Universe's evolution along with the shrinkage paradigm, e.g., big bang nucleosynthesis, cosmic inflation, inhomogeneity and perturbation, cosmological constant, etc. As we have emphasized in the context, the shrinkage paradigm is only an alternative viewpoint on the Universe's evolution, compared with the expansion paradigm.
Both of the paradigms should give either the same or the equivalent physical result.

Discussion
The ( The following is one example that one could understand better and appreciate a controversial theory when it is reviewed with the shrinkage paradigm. In Dirac's large number hypothesis [14] [15] [16], he suggested that all large numbers obtained by combining the fundamental atomic constants and cosmic parameters must be related, and this implied a deeper meaning of Nature. The consequence of his hypothesis is that the gravitational constant should change with cosmic time, i.e., 1 .
However, Dirac's predictions have never been favored by modern cosmology.
Here we are able to address the constant variation issue with the viewpoint of the shrinkage paradigm. With this paradigm, it is natural for G to vary with time.
However, we have instead ( ) We check the large number from the ratio of the electromagnetic force to the gravitational force between a proton and an electron and find that the ratio remains constant during the evolution of the Universe,  and thus get rid of the concern of negative energy. However, the addition makes the theory deviate from general relativity to a scalar-tensor one in which ψ is promoted generically from a global timelike variable to an independent scalar field. It might be interesting to see how far this type of scalar-tensor theory of modified gravity could go in explaining the evolution of the Universe. However, this is beyond the scope of this work.

Conclusion
In this work, we switch the viewpoint from the spatial expansion of the Universe to the shrinkage of the size of matter during the evolution of the Universe. We find that this shrinkage paradigm can be realized by means of the variation of the physical constants. In spite of its slight complication, this paradigm is equally good with the viewpoint of the Universe's spatial expansion in describing physics. The shrinkage paradigm is advantageous to the expansion one in some special cases, especially, in reviewing the large number hypothesis.