Einstein's and de Sitter's model of the universe

AstroNuclPhysics Nuclear Physics - Astrophysics - Cosmology - Philosophy Gravity, black holes and physics

Chapter 5
5.1. Basic principles of cosmology
5.2. Einstein's and deSitter's universe. Cosmological constant.
5.3. Fridman's dynamic models of the universe
5.4. Standard cosmological model. Big Bang.
5.5. Microphysics and cosmology. Inflationary universe.
5.6. The future of the universe
5.7. Anthropic principle and existence of multiple universes 5.8. Cosmology and physics

5.2. Einstein's and deSitter's model of the universe. Cosmological constant.

Let's start with the simplest assumption about the static nature of the universe, which as we now know unrealistic, but has played an important heuristic role and still has a theoretical significance - Einstein and de Sitter cosmological models coming from it, are often used to compare and illustrate the properties of more complex and realistic models. In a homogeneous static universe, in which the conditions are the same everywhere at every point in time, it is natural to choose a coordinate system so that the space-time interval is spherically symmetrical with respect to any point. The space-time interval element will then have a general shape in spherical coordinates

ds 2   =   - A (r) c 2 dt 2 + B (r) dr 2 + r 2 (d J 2 + sin 2 J d j 2 ) , (5.5)

where A and B are functions only r ; for small r , this interval must take the form corresponding to the flat space-time of the special theory of relativity.

By directly calculating the components of the Ricci tensor Rik and substituting the energy-momentum tensor Tik of the form (5.3) corresponding to an ideal "liquid", the Einstein equations for the metric (5.5) can be converted to a system of ordinary differential equations (comma " ' " means derivative according to r)

A'/A.B.r - (1 - 1/B)/r2 = 8p p , B'/B2.r - (1 - 1/B)/r2 = 8p r ,
p' =
- A'.(r + p)/2A ;
(5.6a, b, c)

(the last equation can be most easily obtained from the law of conservation Tik; k = 0).

Since p' dp/dr = 0 (homogeneity), equation (5.6c) gives the condition A'.(R + p) = 0. If ommit the case of empty space r = p = 0 , Einstein's equations have a static homogeneous solution only if A'(r) = 0. However, according to the field equations (5.6a, b), this leads to the condition r + 3p = 0, which again means r = p = 0 for real matter . Thus, Einstein's equations in the common form (2.50) do not allow any homogeneous static solution other than the empty flat Minkowski spacetime STR; they are therefore incompatible with the concept of a homogeneous static universe filled with matter with a constant positive density r.

In order for equations (5.6) to have a static homogeneous solution for the realistic case r > 0, p> 0, it is necessary to introduce a suitable constant L into them. In Einstein's equations, this can be ensured by introducing an additional cosmological term L .g ik , as suggested by Einstein in 1917 :

Rik - 1/2 gik R - L.gik  =  8p Tik  ,         (5.7)

where L is a new (small enough) universal natural constant - the so-called cosmological constant, the value of which should result from a comparison of the relevant cosmological model with the results of astronomical observations.

For a static homogeneous metric (5.5), generalized Einstein's equations lead to a system of ordinary differential equations

A'/A.B.r - (1 - 1/B)/r2 + L = 8p p ,
2.r - (1 - 1/B)/r2 - L = 8p r ,
p' dp/dr = - A'.(r + p)/2A ;

Due to the homogeneity requirement, dp/dr = 0, so equation (5.8c) can only be satisfied if (r+p).A' = 0. Equations (5.8) are therefore solvable in three cases, which correspond to the following solutions :

A ' = 0
r + p = 0
A' = 0,
r + p = 0
Einstein's model;
de Sitter's model;
flat spacetime STR.  

Einstein's model of the universe
In the case of A' = 0, A(r) must be a constant, so that by appropriate choice of the unit of time (time coordinates) A=1 can be achieved; this ensures that for small r the interval ds2 is the same as in STR. From equation (5.8a) by substituting A'= 0 we get a solution for function B

B (r) = 1 / [ 1 - ( L  - 8 p p) ] r 2    = 1 / (1 - r 2 / a 2 ) ,  

where a new constant is introduced a using the relation

1 / a 2   =   L -   8 p p  .    (5.9)

The metric (5.5) therefore for Einstein's cosmological model of a homogeneous static universe has a form


By comparing with (5.4) we see that the spatial part dl2 = dr2/(1- r2/a2 ) + r2 (dJ2 + sin2J dj2) of this spatiotemporal interval can be interpreted as a metric of in a fictitious three-dimensional hypersphere * ) of constant radius a, nested four-dimensional Euclidean space (Fig.5.1). If we introduce coordinates in this auxiliary space

w1 = a. (1- a2/r2);  w2 = r.sin J cos j = x;  w3 = r.sin J sin j = y;  w4 = r.cos J = z  ,       

we get the sphere equation w1 2 + w2 2 + w3 2 + w4 2 = a 2, and the element of spatial distance has the form dl2 = (dw1)2 + (dw2)2 + (dw3)2 + (dw4)2. If we consider not only the spatial but also the temporal dimension, it is possible to display the overall space-time geometry of the Einstein universe as the geometry of a four-dimensional cylindrical surface nested in a fictitious (auxiliary) five-dimensional space - Fig.5.1b.
*) Again, it should be noted that the shape of the metric does not unequivocally determine the type of geometry, because it is possible to assume different global topological properties, as mentioned in 3.1. However, the choice of spherical geometry is the simplest and most natural here .

The total volume of space in Einstein's universe is (assuming a spherical topology) equal to


the "perimeter" of the universe (the length of the principal circle of a three-dimensional sphere) is

L =  0 2p a dj   =   2p a   .  

Thus, Einstein's universe is finite, spatially closed; only a finite number of stars and galaxies "fit" into it.

Fig.5.1. Einstein's cosmological model.
a ) The geometry of three-dimensional space in Einstein's model of the universe can be imagined as a three-dimensional hypersphere with a constant radius, embedded in a fictitious 4-dimensional Euclidean space.
b ) The total space-time geometry of the Einstein universe can be displayed as the geometry of a four-dimensional cylindrical surface nested in a fictitious five-dimensional space.
c ) The specific peculiarities of the spatial geometry and topology of a closed universe can be clearly illustrated on a spherical surface, eg on a globe - see the text.

The spatial closedness of the universe has interesting consequences, which can be easily imagined using a two-dimensional analogy on a spherical surface, for example on the surface of the globe (Fig.5.1c). If we stand on a pole (which from a geometric point of view we can place anywhere on a spherical surface) and describe circles with an ever larger radius, we find that the ratio of the length of the circle to the radius will be less than 2p and when crossing the "equator", the length of the circle decreases with increasing radius. Similarly, when an observer located anywhere in the closed universe demarcate spherical surfaces around himself in a thought experiment, their surface grows more slowly than the square of the radius, and after exceeding a certain distance, the size of the surface begins to decrease even as the distance (radius) increases. Another characteristic of the geometry of the enclosed space is the fact that the observer, still advancing straight in one direction, for a period of time returns to the starting point (from the opposite side). The same is true for rays of light: light sent from somewhere in a certain direction "orbits the universe" and returns to the starting point from the opposite direction. So if we look ahead in a closed universe, we can see our own backs in the distance after a certain time. Similar "ghosts" arise here when observing each luminous object *), so we could see some stars or galaxies twice in different places in the sky (however, the search for identical duplicate objects in opposite places in the sky was not successful).
*) By the way, this effect would lead to Olbers' photometric paradox in Einstein's cosmological model, similar to the earlier idea of an infinite static universe. This is because each ray from each star will constantly orbit the universe until it hits another star or scatters on interstellar matter. In a closed static universe, in which the average luminosity of the stars is the same for an infinitely long time, it will not be dark at night, the sky will be equally bright everywhere.

The relationships between the density, pressure, cosmological constant and radius of curvature of space in the Einstein cosmological model follow from equations (5.8) - (5.9) :

8p p = - 1/a2 + L ,  8p r = 3/a2 - L  ,
L = 4p (r + 3p) ,  1/a2 = 4p (r + p)  .

Assuming that the matter of the universe consists of incoherent dust causing no pressure, it will

L   = 1 / a 2   =  4 p r  , (5.13)

and the radius of curvature of space and its total volume is determined by the value of the cosmological constant :

a = 1 / L   ,  V = 2 p 2 / ( L 3 )  . (5.14)

The "total mass" of the universe is then equal to

M =   r . V  =   1 / 2 p a  =   p / (2 L )  ; (5.15)

however, the mass thus determined has only a formal significance from the point of view of non-gravitational physics as a measure of the amount of material particles filling the universe *). In the second extreme assumption that the universe is filled only with radiation for which p = r/3, we get

L  = 3 / 2a 2 ,  4 p r  = 3 / 4a 2 ,  4 p p = 1 / 4a 2   . (5.16)

The effect of the total gravitational field of the Einstein model on the test particle is given by the equation of geodesy (2.5a). Substituting static metrics (5.5) into the geodesic equation body which is at that moment stationary relative to the surrounding mass, we obtain d2xi/dt2 = 0, so the total gravitational field (space-time metric) in Einstein's universe cannot set a motionless body in motion.
*) In fact, the total gravitational mass, like the total electric charge of a closed universe, has no real significance - it must be equal to zero. The electric charge and mass (4-momentum) contained in a spatial region are given by the Gaussian integral fluxes (1.28) and (2.96) of the electric and gravitational field over a closed surface bounding this region. If we increase the spatial area in a closed space in which we determine the amount of matter and electric charge, the bounding box first increases, but then it begins to decrease until it retracts to a point - see Fig.5.1c. The surface of the closed area bounding the whole universe is therefore zero, so that the total 4-momentum (2.96) and the electric charge (1.28a) are therefore equal to zero. The laws of conservation of total energy, momentum and total electric charge of a closed universe are thus reduced to physically meaningless identities 0 = 0. From a physical point of view, the principal impossibility of determining the total mass or electric charge of a closed universe is clear: there is no external spacetime where the observer could stand and explore this universe "from the outside" - eg "weigh it on a dish of some gigantic scales" or to have a test body run around it.

De Sitter's cosmological model
Analogously to the previous Einstein case A'
= 0, equation (5.8) is solved for the case r + p = 0. Using the requirement that for small r the searched metric (3.45) changes to Minkowski's form, we get

1 / A  =  B  =  1 - r 2 ( L + 8 pr ) / 3   .      

So the metric of de Sitter's model of the universe is


where the constant a is defined by the relation

1 / a 2   =   ( L + 8 p r ) / 3   . (5.18)

For the motion of test particles and the propagation of light signals, which is generally given by the equation of geodesy (2.5a), for the deSitter metric after adjustments (thanks to spherical symmetry, it is enough to investigated the movement only in the plane J = p/2 without prejudice to generality) the equation is obtained

(H and L are integration constants); the speed of light in the de Sitter model is given by the relation dr/dt = (1 - r2/a2) for the case of purely radial propagation. From these equations it can be seen first of all, that at r = a the speed of motion of the particles and the coordinate speed of light become zero. By integrating from r=0 to r=a we find that from the point of view of the observer in the center r=0, every particle and light from the center r=0 to the place r=a arrives only in an infinitely long time. Thus, the observer in the deSitter model can never obtain any information about what happens at distances greater than a from it: in the de Sitter model there is a causal horizon of the universe at a distance r = a = (3/(L +8pr) (= (3/L) for r = 0).

It follows from the equations of motion that the originally stationary body will have a radial acceleration d2r/dt2 = r.(1 - r2/a2)/a, which increases with distance from the origin of local coordinates (which can be located at any point). If particles are homogeneously and isotropically distributed in the de Sitter universe, they will move away from each other at a speed proportional to their distance. The metric of the de Sitter universe is static (it does not depend on time in the given frame of reference), but in the interval (5.17) the coefficient at dt is no longer constant. Unlike Einstein's model, the total gravitational field (space-time metric) in de Sitter's universe causes the scattering of celestial bodies - as if each point were a repulsive center. The law of inertia does not apply here for large distances, the bodies will expand from each other with increasing speed. This variability in the own distances of the particles will cause a Doppler spectral shift of the light emitted by these particles; at not too large distances r, the Hubble's law dl/l H.r will apply approximately for this frequency shift , where the " Hubble constant" H = a-1 = [(L + 8pr)/3] (= (L/3) for r = 0).

Thus, since de Sitter's model captures the observed redshift of the spectrum of distant sources in space, it could at first glance be considered a realistic cosmological model. In reality, however, this model is not physically consistent. The basic condition on which de Sitter's universe is based is r + p = 0. The intrinsic density of matter r is (by its physical nature) always non-negative. Although the pressure p can in principle be negative, no form of matter creates such a negative pressure, the absolute magnitude of which would approach the density of matter r (in geometrodynamic units) - see also 2.6 *). The condition r + p = 0 can therefore be satisfied in practice only if r= 0 and p= 0. De Sitter's model thus corresponds to a completely empty universe , which does not contain any appreciable amount of matter or radiation. Existing stars and galaxies should be considered in this model as "test particles" that do not contribute to the overall cosmological gravitational field. And this is against the spirit of the general theory of relativity, which puts gravity and the geometry of space-time directly related to the distribution of matter.
*) However, current quantum unitary field theories allow for the possibility of large negative pressures leading to antigravity effects. According to this, the de Sitter expansion could actually have taken place in a very early universe (inflation expansion) - see 5.5 "Microphysics and cosmology. Inflationary universe.".

Government of cosmological constant
De Sitter's model represents a cosmological solution of Einstein's gravitational equations GTR for a universe in which there is virtually no matter and radiation. The only thing that determines the dynamics of the universe here - the behavior of the scale factor a - is the cosmological constant
L. From Fridman's equations (5.23) (in the following 5.3 "Fridman's dynamic models of the universe") under conditions r = 0, p = 0, L> 0, follows for time dependence of the scale factor a the exponential dependence

a (t)  ~  e H . t   =   e (L/3) . t   , (5.19)

where H is the instantaneous value of the "Hubble constant" (5.24) generated by the cosmological constant L.
  De Sitter's model is inadequate for the current universe in the era of matter, in the evolution of which luminous and dark matter play a significant role, as well as in the earlier era of radiation. However, it could be adequate in a very early period, in a time range aprox. 10-36 10-32 sec., when a short but powerful inflationary expansion is theoretically assumed (5.5 "Microphysics and cosmology. Inflationary universe."). And also in the distant future: if the currently observed accelerated expansion of the universe is caused by a cosmological constant generating dark energy, the universe will continue to expand, all matter and radiation will dilute to almost zero (5.6, section "Dark energy and accented expansion of the universe"). The vacuum energy generated by the cosmological constant will then dominate and the universe will expand exponentially (5.19) according to de Sitter's model.

Cosmological constant
Let us now notice the general nature of the cosmological term. When Einstein introduced the cosmological term, he placed it on the left side of the equation: Gik + L .gik = (8p G/c4) Tik , thus expressing that it is a geometric property of space itself (spacetime).
However, the physical significance of the cosmological member becomes clearer after its transfer to the right-hand side of Einstein's equations

Rik - 1/2 gik R  =  (8pG/c4) Tik + L.gik   ,  (5.7 ')

ie from its inclusion in the energy-momentum tensor Tik . If we consider the case of vacuum Tik = 0, it can be seen that L.gik represents a kind of immanent fundamentally irreversible curvature of empty space, which is applied even without any matter, radiation and gravitational waves (for the ability of gravitational waves to curve space-time and "imitate" matter 2.8 and B.3); in other words, the cosmological term expresses the gravitational effects of a vacuum. If were L 0, it means that the vacuum creates a gravitational field, as if it was (from the point of view of the usual approach L=0) filled with matter with effective density rcosm = c2 .L/8pG and effective pressure pcosm = -c4 L/8pG = - ecosm (ecosm is the effective energy density of this fictitious mass), which corresponds to the equation of state p = - r. c2 .
  The cosmological term can be considered as a manifestation of some "exotic" type of matter - vacuum energy. It penetrates the entire space and continuously fills it with a certain basic energy density, even without the presence of "normal" matter (in substance form). It does not dilute with the expansion of the universe, nor does it cluster as a substance form of matter, but maintains a constant density *), contributing to the general energy density, which gravitationally influences the dynamics of the evolution of the universe.
*) Truth be told, this is how a standard "geometrically induced" cosmological term behaves. In principle, a physically conceived cosmological term could change over time and also have a different value in different areas of the universe..?..
  According to current astronomical measurements, the value of this vacuum energy is very close to zero, less than about 10-9 J/m3, which corresponds to a mass density of about 10-26 kg/m3.
The physical nature and origin of the cosmological term?
From the point of view of the general theory of relativity, the introduction of the cosmological constant as another independent universal natural constant is purely phenomenological, although the cosmological term may be an organic part of field equations (3.5) - the introduction of the cosmological member L.gik is the only permissible modification of Einstein's equations in the sense that it does not violate the law of conservation of energy Tik; k = 0, because the covariant 4-divergence of the tensor Rik - (1/2) gik R + L.gik is identically equal to zero as for the tensor Gik Rik - (1/2) gik R.
  But what is the physical nature and origin of the cosmological term? Attempts have been made to relate L to the "vacuum physics" of quantum field theory : the cosmological term should be the result of polarization and quantum fluctuations of the vacuum. Immediate calculation (resp. dimensional estimation), encompassing all vibrational modes of energy with a wavelength greater than the Planck length (10-35 m), however gives an unimaginably high vacuum energy density, corresponding to a density rcosm ~ 1096 kg/m3..!.. In order for a vacuum to look like an empty space, far-reaching compensations must be applied between the vacuum fluctuations of the various fields, which cancel out the vast majority of the fluctuations.
  No satisfactory explanation of the cosmological constant based on microphysics yet exists; perhaps some hopes are promised by calibration unitary field theories, where spontaneous disruption of the symmetry of the Higgs scalar field could "generate" a cosmological constant [113] - see also 5.5.  

The history of the cosmological constant is quite varied, opinions on its significance have changed significantly during the development (from the beginning of the 1920s to the present). There have been periods when the cosmological member was completely rejected (eg after the creation of Fridman's model of the expanding universe and Hubble's discovery of the cosmological redshift), with periods of some "renaissance" when the cosmological member was to explain supposed or actual facts (such as the need to extend expansion time of universe in an overestimation of the value of the Hubble constant, or later an explanation of the accumulation of the redshift of quasars at the value of z = 1.95).

Current astronomical observations do not require L 0, but they do not strictly exclude this possibility *) . The study of extragalactic objects only increasingly limits the value of the cosmological constant (now |L| <~ 10-55 cm-2), so that the theory does not contradict the results of observations of the available part of the universe. It is obvious that laboratory determination of such a small value of L is completely hopeless. Even such a small cosmological constant, however, could significantly affect the structure and evolution of the universe as a whole. In the interest of objectivity, it is therefore necessary to remember the possibility of L 0 and to consider the cosmolological term when studying the global properties of the universe. In addition, it has recently been shouwn, that the cosmological term could play a significant role in the earliest stages of the universe's evolution, when the effects of quantum field theory and uniformity of fundamental interactions manifested themselves - the cosmological constant could be the "driving force" of inflationary expansion of the universe, as shown in 5.5 "Microphysics and Cosmology. Inflationary Universe.".
*) According to current astronomical observations of distant supernovae, there are indications that the expansion of the universe is currently accelerating, that in addition to dark (non-radiant) matter, there is also so-called dark energy in space, which shows "antigravity ". Thus, the evolution of the universe seems to take place under the influence of the cosmological constant L > 0 (see 5.6 "The Future of the Universe. The Arrow of Time.", the passage "Dark Energy and the Accented Expansion of the Universe").

5.1. Basic principles and principles of cosmology   5.3. Fridman's dynamic models of the universe

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