Physico-mathematical foundations of relativistic cosmology

I briefly present the foundations of relativistic cosmology, which are, General Relativity Theory and the Cosmological Principle. I discuss some relativistic models, namely,"Einstein static universe"and"Friedmann universes". The classical bibliographic references for the relevant tensorial demonstrations are indicated whenever necessary, although the calculations themselves are not shown.


Introduction
The Cosmological Principle (CP) -the homogeneity and isotropy of the universe -and the General Relativity Theory (GRT) are the physico-mathematical foundations of relativistic cosmology. Summarizing all that will be presented in the following sections, one can simply state that the symmetries introduced by the CP reduce Einstein's full field equations of GRT to two straightforward differential equations in the scale -or expansion -factor of the universe [1, p. 260]. The most popular relativistic cosmological models are built from these two equations.
Einstein's field equations of GRT represent a mathematical description of a geometrical entity, the space-time, defined by three spatial coordinates and a temporal one. Such a four-dimensional entity is established by the existing matter and energy contents. On the left-hand side of the equations one has the geometrical description of space-time and on the right-hand side, the energy and momentum contents. Putting it in another way, GRT is Einstein's theory of gravitation. It can be understood, in a simplified way, by stating that "Spacetime grips mass, telling it how to move; and mass grips spacetime, telling it how to curve" [2, p. 275]. This is obviously an incomplete statement since not only matter curves space-time but also all kinds of energy [3, p. 229].
Einstein uses the tensorial formalism to express his field equations, therefore, GRT is a tensorial theory. Incidentally, the German mathematician Georg Friedrich Bernhard Riemann (1826-1866) was one of the main responsible for the development of tensor calculus, playing an extraordinary role in the formulation of GRT. But, what is a tensor? A tensor is a mathematical entity that has in each point of space n m components, where n is the number of dimensions of space and m is the order of the tensor. Hence, one can say that the scalar is a tensor of order 0 -therefore, it has 1 component -and that the vector is a tensor of order 1 -it has n components [3, p. 200]. Tensors used in GRT are tensors of order m=0,1 and 2 and "space" is the space-time of n=4 dimensions (three spatial coordinates and one temporal coordinate). Therefore, the second-order tensors of GRT have, in principle, 4 2 = 16 components. I say "in principle" because real physical problems impose symmetry constraints which reduce to 10 the really necessary components. The first-order tensors are the GRT vectors, called quadrivectors and have 4 components.
I shall make a simplified presentation of Einstein's field equations which consist of 10 nonlinear differential equations -a system of equations. This system is tremendously simplified when additional symmetry constraints imposed by the CP (see section 2 of [4]) are considered and reduce itself to 2 equationsonly 2 components of GRT's tensors are necessary to the complete formulation of the modern relativistic cosmology of the "Hot Big-Bang" theory.
Einstein's field equations of GRT can be qualitatively expressed [3, p. 229] as: space-time curvature = constant × matter-energy. (1) Space-time curvature is mathematically given by the Einstein tensor G µν : with indices µ and ν assuming the values of 0,1,2 and 3. The tensor R µν is called Ricci tensor, formed from the Riemann curvature tensor, which is a tensor of order 4, being the most general way of describing the curvature of any n-dimensional space. In the case of GRT, the space-time of 4 dimensions implies in the existence of 4 4 = 256 components. The Ricci tensor, of order 2, is the reduced form of the Riemann tensor to be used in Einstein's equations. The reduced form is obtained through the application of symmetry relations that eliminate the redundant terms in the Riemann tensor. The tensor g µν is the space-time metric tensor and play, in Einstein's equations, the role of the field [5, p. 179]. And that is why one says "Einstein's field equations". We do not speak, in GRT, of "action at a distance"; a test particle does not directly "feel" the sources of matter and energy, rather it feels the field, i.e., the metric -the geometry -that such sources generate in its neighborhood. The metric field transmits the perturbations in the geometry (gravitational waves) at the speed of light, a similar situation to what happens in electromagnetism [5, p. 179]. The metric field is the analogue of the gravitational field in Newtonian theory. Finally, the term R, in Eq. 2, is the scalar curvature, a scalar associated to the Ricci tensor and to the metric. In tensorial terms, the scalar curvature is equal to the trace of the Ricci tensor with respect to the metric tensor. The scalar curvature is also called the Ricci scalar [5, p. 219]. The part of matter and energy in Einstein's equations is given by the energymomentum tensor T µν . The full Einstein's field equations have, thus, the following compact form: where κ = 8πG/c 4 is the Einstein gravitational constant, G is the universal gravitational constant and c is the speed of light in vacuum. Finally, inserting Eq. 2 into 3, one has the explicit form of the 16 equations of Einstein: And in matrix format, still with the 16 components, one has: g 00 g 01 g 02 g 03 g 10 g 11 g 12 g 13 g 20 g 21 g 22 g 23 g 30 g 31 g 32 g 33 Before discussing Einstein's field equations, shown above, I describe, in the following section, the term that appears on the right-hand side of Eqs. 3, 4 and 5, namely, the energy-momentum tensor T µν . I discuss in section 3 the GRT equations, first, the vacuum equations, i.e., in the absence of sources of matter and energy, and then the full equations, which are the relevant equations for cosmology. Next, in section 4, I pass to the cosmological applications of the full equations. Here, I add a novelty, the inclusion of the so-called cosmological constant Λ in Eq. 4, which increases its generality. Einstein's static universe -where the cosmological constant explicitly appears -is discussed in section 4.1 and the Friedmann universes, in which Λ=0, are shown in section 4.2. In section 5, I present some final remarks.
It is necessary, at this point, to make an important warning. It is not possible a complete and satisfactory apprehension of GRT without the knowledge -even a rudimentary one -of the techniques of tensor calculus. But it is possible, nevertheless, to have a general idea of the theoretical picture, even without going deeply into the tensorial demonstrations. That's exactly what I intend, in this presentation of one of the most known applications of GRT, namely, modern cosmology. When they are necessary, I always give the appropriate references for the reader interested in the tensorial mathematical formalism.

Energy-momentum tensor
This is the tensor -the right-hand side of Eq. 4 -that describes the energetic activity in space. The energy-momentum tensor quantitatively gives the densities and the fluxes of energy and momentum generated by the sources present in space and which will determine the geometry of space-time -the left-hand side of Eq. 4.
The components of the energy-momentum tensor [6, p. 137] are the following: T 00 = density of matter and energy.
T 0ν = flux of energy (i.e., energy per unity of area, per unity of time) in the ν direction; ν = 0.
T µν = flux of the µ component of momentum in the ν direction (i.e., shear stress). Note that "flux of momentum" is the same as "force per area"; µ, ν = 0.
T µµ = flux of the µ component of momentum in the µ direction (i.e., force over the perpendicular area, that is, pressure, which differs from shear stress precisely for taking into account the component of the force perpendicular to the surface upon which it acts); µ = 0.
The tensor of energy-momentum is symmetric, that is, T µν = T νµ . Misner, Thorne and Wheeler [6, p. 141] show this using a physical argument. They consider the shear stresses upon a small cube of side L and mass-energy T 00 L 3 and go on to show that it would have infinite angular acceleration if the tensor was not symmetric. Being symmetric, the energy-momentum tensor has a maximum of 10 different components, instead of the 16 of any 4 × 4 tensor. As we shall see in what follows, the tensors related to the geometry of space-time, in the left-hand side of Eq. 4, are also symmetric.
We are ready now to discuss, in more details, Einstein's equations of GRT.

Einstein's field equations
The enormous success of Newton's gravitation in classical phenomena -weak gravitational fields and velocities much smaller than the speed of light -make it almost mandatory that any new theory of gravitation reduce itself, in those limits, to the Newtonian law of the inverse square. In other words, in the socalled "classical limit", in the absence of gravitational sources, the GRT must reduce to the Laplace equation for the Newtonian gravitational potential Φ, ∇ 2 Φ = 0, and to the Poisson equation, ∇ 2 Φ = 4πGρ, whenever there are sources, represented by the density of matter ρ. This was the course followed by Einstein, which I pass to discuss, first, with the field equations in the absence of sources and, then, with the full equations, that have as special case the first ones.
Besides the reduction to the Newtonian limits, Einstein used also, for the postulation of the field equations, the criteria of simplicity and of physical intuition. Einstein asks himself: What is the most simple form of the space-time metric, in the absence of sources, that produces, in the classical limit, the Laplace equation for the Newtonian gravitational potential? And if there are sources, how to get the most simple form of the metric tensor and, at the same time, the classical reduction to the Poisson equation? And yet more, in both cases of the general theory, he should obtain the conservation of energy and of momentum.
After the fulfillment of such criteria and of the reduction to the classical limits, the validity of the formulated equations must, of course, be verified by experiments. As we shall see, such a verification occurred in an extremely satisfactory way for the vacuum equations, but, apparently, has not yet occurred for the full equations.

Vacuum equations
GRT's vacuum equations are in grand manner -and to a certain point, paradoxically, for not being the full field equations -the great responsible for the extraordinary prestige enjoyed by GRT. That is what we shall see in what follows.
The vacuum equations are those valid for the metric field in vacuum, as for example, for the field around the Sun, where the density of matter ρ = 0. By investigating the symmetries of the Ricci tensor R µν in the classical limit of the metric tensor g µν [5, p. 222], Einstein postulates the following form for the vacuum field equations: Einstein put forward the vacuum equations of GRT in 1915, and in 1916 the German astronomer and physicist Karl Schwarzschild (1873-1916) derived the first and the most important exact solution of the vacuum field equations, known as the Schwarzschild metric [5, p. 228]. Such a solution has been applied with great success, for example, to planetary motion, giving the right explanation for the phenomenon of the precession of Mercury's orbit -which was not obtained with Newtonian gravitation -and predicting new phenomena, amongst them, the deflection of a light ray traveling next to a large matter concentration [5, p. 223]. These, and other tests, were realized with great experimental success and are responsible for the almost unanimous acceptance of GRT by the scientific community. Schwarzschild's metric is also responsible for the current and controversial discussion of phenomena such as gravitational radiation and black holes.
These are not, however, the field equations that will lead to the modern models of relativistic cosmology. The appropriate field equations need the presence of sources of matter and of radiation to be properly applied to the universe. Such equations, called full, had not yet definitive experimental confirmation (see discussion in [8]) and are presented below.

Full field equations
The first attempt for the field equations in the presence of sources would be obviously a modification of Eq. 6, i.e., R µν = constante × T µν , which does not work because the divergent of R µν is not null, implying that there is no conservation of energy and momentum. The second choice is simply substituting R µν by the Einstein tensor G µν (Eq. 2) which does have null divergent [5, p. 299]. The full field equations then take the form of Eq. 4 and, in matrix format, of Eq. 5. But there are additional simplifications. We have seen that the energy-momentum tensor is symmetric. And due to space-time symmetriessuch as, the least distance from A to B is the same as from B to A, and, the distance around a circle is the same in the clockwise and anticlockwise directions -the same is true for the Ricci and metric tensors [3, p. 240], and they have, just like it is the case for T µν , at most 10 different components in each event of space-time.
Einstein's full field equations are hence written as G µν ≡ R µν − 1/2g µν R = −κT µν , where κ is Einstein's gravitational constant, defined in section 1. In matrix format, we have: g 00 g 01 g 02 g 03 = g 11 g 12 g 13 = = g 22 g 23 It is interesting to note that the full field equation is reduced to the vacuum equation when T µν = 0. This happens in the following way. Einstein's full field equation can also be written in the form R µν = −κ(T µν − 1/2g µν T ) [5, p. 299]. It becomes apparent, therefore, that for T µν = 0 one has R µν = 0, that is, Eq. 6.

Cosmological models
The cosmological models are built up by means of Einstein's full field equations. One of the most tedious and laborious tasks in GRT is the calculation of the Ricci tensor, the scalar curvature and finally the Einstein tensor G µν , a calculation made for a given metric tensor g µν .
Rindler [5, p. 418] shows how these calculations must be done for a generic diagonal metric given by: where A, B, C and D are arbitrary functions of all space-time coordinates.
Let us now obtain the equations of relativistic cosmology. The CP, that is, the reduction of the real universe to a homogeneous and isotropic idealization, implies in a space-time with the Robertson-Walker metric [5, p. 367], which, in spatial spherical coordinates, is given by: S(t) is the scale factor of the universe and k is the spatial curvature constant. This equation is used, instead of Eq. 8, for the calculation of the Einstein tensor applied to cosmology, i.e., to the idealized universe of the CP.
In order to increase the generality of the field equations, one adds to the scalar curvature a constant, the so-called cosmological constant Λ, that will be essential for the discussion of Einstein's static universe, in the following section. Such a constant is called cosmological because it is only relevant in the context of cosmology, i.e., for the structure and evolution of the universe. The full field equation can be written then as The cosmological constant does not change in anything the formal validity of the field equations, and can be positive, negative or null. In the last case, of course, one recovers the usual formulation of the field equations (Eq. 4). According to Rindler [5, p. 303] "The Λ term seems to be here to stay; it belongs to the field equations much as an additive constant belongs to an indefinite integral." Like the scalar curvature R, Λ has dimensions of length −2 .
Just like the full field equation without Λ, the full field equation with Λ is reduced to the vacuum equation when T µν = 0. As before, Einstein's full field equation can also be written in the form R µν = −κ(T µν − 1/2g µν T ) + g µν Λ. For T µν = 0 one has [5, p. 303]: This equation, for the vacuum, that substitutes Eq. 6, without Λ, is only important for eventual cosmological studies. It is totally irrelevant, for example, in solar system studies. In such a case, Eq. 6, and its solution, Schwarzschild's metric, is perfectly satisfactory, even if there exists a cosmological constant.
In order to get the relativistic cosmological equations one has to make the fundamental assumption of the CP: all matter -including a possible "dark matter" -of the universe will be, so to speak, pulverized and redistributed in an uniform way throughout the universe. One has in such a way the physical requirements of the CP, i.e., the homogeneity and the isotropy of matter distribution. The energy-momentum tensor of these sources, namely, matter and radiation with the physical characteristics of uniformity, is reduced to the diagonal elements [5, p. 392] [6, p. 140]: where p is the isotropic pressure and ρ is the homogeneous density of the fluid. A fluid like this is called a perfect fluid. The negative sign that appears in p implies, in the field equation, that a positive pressure has an attractive gravitational effect [5, p. 156] [7, p. 172].
One must notice also that p represents the pressure of radiation and matter, and, in the same way, ρ must be split up into one part for radiation and one part for matter. Radiation pressure will only be significant in the initial stages of expanding models -a photon gas at high temperature with p = 1/3ρc 2 . Matter, which only appears in later stages, has negligible pressure. A perfect fluid of matter with zero pressure is often, technically, called dust. Such a dust stays at rest in the spatial substratum, because any random motion would result in the existence of pressure. Global motions of expansion or contraction are not excluded, though.
The main work, in order to obtain the cosmology equations, is to apply the left-hand side of the field equations given by Eq. 10 -the Einstein tensor -to the Robertson-Walker metric given by Eq. 9. It has been already said above that Rindler [5, p. 418] shows the detailed calculations needed to obtain each element of Einstein's tensor. Schwarzschild's metric, being the metric of a homogeneous and isotropic universe, with its innumerable symmetries, implies that the Einstein tensor, with the cosmological constant, G µν ≡ R µν − (1/2R − Λ)g µν , will only have the diagonal elements, precisely like the energy-momentum tensor: Rindler [5, p. 392] and Misner, Thorne and Wheeler [6, p. 728] give the results of the calculations for G µν : where S(t) is the scale factor of the universe (cf. Eq. 9).
With these values for G µν and T µν , the full Einstein equations, G µν = −8πG/c 4 T µν (Eq. 10), reduce themselves to only two non-linear differential equations for the scale factor S(t): 2S Eqs. 16 and 17 are the basic equations for the formulation of the majority of relativistic cosmological models, with or without the cosmological constant. Moreover, the simultaneous solution of this system of equations leads to the equation of mass and energy conservation of the cosmic fluid [5, p. 393].
Next, I show a cosmological model with Λ and a family of models without Λ.

Einstein's static universe
Soon after the final presentation of the GRT, in 1915, Albert Einstein (1879-1955) inaugurated the study of relativistic cosmology. He published, in 1917, an article with the suggestive title "Cosmological considerations on the General Theory of Relativity". He uses Eq. 16 with positive Λ in order to get a repulsive cosmic effect and, hence, exactly counterbalance the attractive effect of matter and radiation of the universe. Thus, he obtains a static universe, which was in accordance with the current ideas of the epoch. This model was of enormous importance in the history of the science of cosmology because it was a source of scientific inspiration for many investigators. The model had, however, an undesirable feature: it was unstable under small perturbations on the state of static equilibrium. Einstein's model is discussed in detail by Soares [9], including its instability. Eq. 2 of Soares [9, p. 1302-2] is the same Eq. 16 determined here.

Friedmann's universes
The Russian physicist, meteorologist and cosmologist Aleksandr Aleksandrovich Friedmann (1888-1925) was responsible for the next great contribution to relativistic cosmology. In 1922, he published an article, in a prestigious German scientific journal, with the title "On the curvature of space", where he solves Einstein's full field equations, with the hypotheses of homogeneity and isotropy of the universe -which would turn out to be later known as the "Cosmological Principle" -and obtains a model of positive spatial curvature (spherical space) with expanding and contracting phases. Subsequently, it was recognized that this was only one of the possibilities of dynamical universes -an oscillating closed universe -amongst others, namely, the open universes; these are one of negative curvature, hyperbolic space, and another of null curvature, flat or Euclidean space.
Friedmann's work was only recognized by the scientific community a long time after its publication. In his homage, the models resulting from Eq. 16, without Λ, were called Friedmann's models or universes. A detailed discussion, albeit at the elementary level, on Friedmann's models is presented in [10].

Final remarks
Einstein's static model was the first relativistic cosmological model and, also, the first to use the cosmological constant. Modern relativistic cosmological models also adopt the cosmological constant, and by its means are able to get consistency between the theoretical age of the universe and the limits imposed by stellar evolution. In other words, the cosmological constant is able to solve the so-called age of the universe dilemma (more details in [11]).
It is worthwhile pointing out that Friedmann (section 4.2) originally obtained only the closed model, by means of Einstein's full field equations and of the CP, which was a cosmological concept introduced by him. The three modern Friedmann models appeared with the generalization introduced by the Robertson-Walker metric (Eq. 9), which predicts still other global spatial topologies, specified by the spatial curvature constant k, besides the known open models, hyperbolic, with k=-1, and flat, with k=0 [5, p. 367].
As anticipated in [8], we saw, in section 3.1, that the great and decisive tests of GRT are done for one solution of Einstein's field equations for the vacuum, that is, in the absence of sources of energy and momentum. Such a solution has innumerable practical applications and is given by Schwarzschild's metric. The most known solutions of the full field equations are precisely the relativistic cosmological models, valid for a homogeneous and isotropic fluid, and they fail when confronted with observations. The relativistic models only survive when the existence of unobservable physical entities like dark matter and dark energy are postulated (cf. [8]).