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13On the curious feature of the Friedman's solutions for the gravitation field equations of the Universe and its cosmological consequences
A.S. Kartashow (kart@meteo.nw.ru) Institute of Simulation Technologies, St. Petersburg
Introduction
It is common knowledge that the choice of coordinate system in the general theory of relativity is arbitrary. Observer's proper time t in homogeneous and isotropic gravitational field can be considered formally as universal time t within framework of the theory if to select such special coordinate system that the time component of the metric tensor will be equal to one al over ( g00 = 1) [1].
Nevertheless the question on adequacy of the universal time for a physical observer has debatable character. I. Jeans [2] and A.S. Eddington [3], for example, spoke out for the universal time as utilitarian way to agree comprehension of time in the modern gravity theory and Newtonian physics. De Sitter [4] has shown that coordinate system can be chosen also in such manner within the gravity theory that full invariance is providing relative to any transformations of all four coordinates including time one, and therefore proper time is differed from the universal time. K. Goedel [5] rejected the universal time on the whole as depending on the special type of connection between matter and motion in the Universe and therefore not satisfactory as philosophical concept.
Proper time for the physical observer can differ from the universal time as well as for the following reasons:
- the observer exists going with the time, and it means he observes the Universe relative to the transforming reference system possibly;
- the universal time is counted off from horizon of the observable Universe whereas the proper time mark off from the observer himself.
Latter reason is not as trivial as it seems at first glance because there is basic distinction between initial and boundary conditions. Placing the beginning of the Universe on its observable border we collide with a problem of initial conditions unavoidably. It is difficultly (if it is possibly in general) to define the actual cosmological boundary data, and for their completion usually it is using, for energy reasons only, seeming analogy with the boundary microcosm conditions, and high-energy physics are regarded as fundamental instrument for interpretation of the boundary data of the Universe. Such reasons are not indisputable and do not led yet to single-valued results on modeling of the Universe.
At the same time it is not excluded that the initial conditions of the big bang [6,7] can prove to be the observer's illusion caused by that he transfer his conditions for existence implicitly, including a background course of his own time, himself on properties of observable objects. An eloquent historical example for that was the Ptolemy System of the Universe.
In this paper is made an attempt to verify this suspicion on the basis of the classical Friedman's theory of the unsteady state Universe.
Scale-invariant transformation of the Friedman's equations
According to Friedman's model [8,9] the basic dynamic parameters of the Universe including: speed of light (c), radius of curvature (a), density of matter (p) and parameter of expansion (Hubble constant H = a '¡a ) - are connected among themselves by solutions of the gravitational field equations as follows [6, 7]:
kc2 8nG TTl Ac2
p -H +■
a2 3 ' 3
kc2
= (2q - 1) H2 + Ac2
a2
where G is gravitational constant, q - acceleration parameter, A - cosmological constant, k -index of a spatial curvature type.
The acceleration parameter is a model free parameter, as the Friedman's model does not contain sufficient conditions to define it unequivocally. However there is one curious feature of the solutions with acceleration parameter equal to one (q = 1, it is one of the closed models).
Let us examine the hypothetical coordinate system in which the time coordinate t is connected with time t by coordinate transformation dr = f (t )dt, leaving while without attention the problems of its existence. In this frame effective speed of light c is not a constant and depends on time as the time component of the invariant interval acquires the form:
c02 dr2 = c02 f (t )2 dt2 = c2 dt2.
If we have q = 1 and A = 0 then solutions of Friedman's equations are writing down as „ c 3H2 aa" ,
=a- p=n ' q=(0f=1
It is possible to be convinced that:
1) radius of space curvature as well as its first and second time derivatives are connected among themselves by equations:
a' = Ha, a' = -fcLi = -Ha', a
2) equation for parameter of expansion whence follows:
H' = -2H • H
3) effective speed of light together with its derivative can be written down as:
c = Ha = a', . c' = a" = -Hc,
4) derivative of density of matter can be appeared in the form:
p = -4Hp.
The resulted equations suggest that in the converting coordinate system the basic parameters of the Universe in this particular case of Friedman's model can be appeared as eigenfunctions of the operator:
S = H- d/dt.
Let £ is any of the dimensional cosmological parameters entering Friedman's equations. Then, with q = 1, A = 0 and dr = f (t )dt, we have:
S£ = n£,
Here integer n is eigenvalue of function £ in the spectrum of the S - operator, and we have n = -2 for the Hubble's constant, n = -4 for the density of matter, n = -1 for the speed of light and n = 1 for the radius of curvature.
It means that all the basic dynamic cosmological parameters, which are included in Friedman's equations, can be considered from the point of view the converting coordinate system as eigenfunctions of the S - operator and be written down as follows:
H = H0/1 + 2H0t, p = p0/(1 + 2H01)2, c = c0/^1 + 2H t r = r0A/1 + 2H0t.
where c0, Ho, ao, po are the contemporary meanings of the dynamic parameters of the Universe.
For this representation the solutions of the gravitational field equations prove to be scale-invariant relative to the time transformation and at any moment t they are reduced to the parity between observable parameters at present t = 0:
c 3H2
ho =-°, p0 = o
a0' 0 4nG
We can say therefore that action of the S -operator does not change the ratio of the observable parameters of the Universe (c, H, a, p), and their contemporary meanings prove to be fundamental physical constants. For the given dynamic conditions (q = 1) and representations (dr = f (t )dt) the Universe looks the same both for observer been in the deep past and for the observer locating on boundless distances from us - its dynamic properties are identical elsewhere and forever.
Evolution of the Universe for that special model case cannot be connected causally as to dynamic factors as to initial conditions, hence the nature of the S - operator can be relativistic only, as relativity is incorporated in bases of the general theory of relativity on which the classical theory of the unsteady state Universe is constructed. This conclusion gives grounds to assume that evolution of the Universe for the given special case of Friedman's model is represented by relativistic effect only for the contemporary observer if to admit really existence of the permanent time transformation his coordinate system.
It is necessary to note that in contrast to the well known relativistic effects being become apparent in space, such as time dilation during the movement of material object relative to observer, or bending of space by gravitational fields, the assumed relativistic effect is specific one - it is connected with permanent transformation of observer's coordinate system in time only, and such transformation could be named as "time displacement" of the observer.
Proper transformation for time and proper time of observer
Let us consider transformation written above for time dr = f (t )dt to define function f (t) for model q = 1 in terms of S - operator. Derivative of distance r on time coordinate t is written down as
dr = H 0 r0
dt + 2H0t '
and derivative of distance r on time r is determined by Hubble's law:
dr H — = H 0 r0 . dr
Consequently the function of the time transformation would look like
dz drldt f (t ) = " '
dt drjdr + 2 H 01
Therefore we can say that time is the eigenfunction of the S -operator with the eigenvalue n = 1 in exactly the same way as distance r, that is not surprising as time axis would not formally differ from spatial axes in the four-dimensional space-time. It yield St = nr, and with n = 1 we obtain:
dr H — = Hr, dt
t = r0A/l + 2H01 .
Substituting the second parity into the first: dr H0t0
dt ^l + 2H01'
and comparing result with the expression for the time transformation function f (t) we obtain that the dynamic time scale is determined by relation: r0 = 1/H0 . It means that lapse of time is defined on observer's clock by parity: dt
dr =
V1 + 2H 0
The form of the eigenfunctions for the operator of cosmological evolution S leads to the obvious enough conclusions about the existence of the time horizon for the observer:
r = -. 1
2H 0
It is necessary to note that the dynamic time scale twice exceeds the time horizon of observer. At approach to observer's time horizon, the lapse of time in his coordinate system aspires to infinity (i.e. time stops) when the time aspires to zero (i.e. to its natural reference point from the time horizon of the observer). Obviously, that is the way concerning the distances in space: the transformation of time transfers their reference point to horizon, and the scale of distances near to zero is indefinitely stretched. Figuratively speaking, space-time, observable from the coordinate system of the contemporary observer, is turned off on horizon and the infinite space-time scales of the Universe are packed into the narrow ranges of the observer's view field both in time and in space. In this, strictly speaking, the essence of any horizon consists and in this, possibly, concludes the assumed mechanism of the illusion of the big bang.
The existence of the time horizon means that at the point of space where the observer is located cannot exist objects older than T. This conclusion gives in hands to the observer the elementary way to estimate upper boundary of the Hubble's constant value. It consists that it is necessary to find the most ancient material, to determine its age tmax accurately using the appropriate methods of
the analysis of matter, and to calculate Hubble's constant having equated tmax and T. Most ancient
of the known today materials is meteoritic matter, age of which composes 4.55 billions years [9]. For Hubble's constant this way gives value 105 km/sec/Mpk.
It is represented that the time factor K = 1/^1 + 2H01 will be able to appear as the important factor
of the consecutively relativistic theory of the unsteady Universe. The time factor K in essence is Lorenz-factor with reference to movement through time, and its introduction means the propagation of the special theory of relativity for the displacement of the observer in time.
By virtue of action of the time Lorenz-factor the lapse of time in coordinate system of the observer is dissimilar trough time as well as the scale of distance dissimilar trough space. It means that
1
t
between time and space there would be stiffening joint. Asymmetric transformation of time would be compensated for the observer somehow by corresponding transformation of space, and this is the basic criterion for existence of any frames for the contemporary observer; otherwise it would be impossible for him to observe objects on the whole.
Transformation of observable space and Hubble's law
Observable space-time assumes the central position of observer; therefore the flat sections of the space-time mean planes passing through the observer himself. Initial assumption of the subsequent analysis is central symmetry of the space-time transformation. The central symmetry in this case means that if transformation of any flat section of the space-time takes place than accurately the same transformation must undergo the any others.
Acceptation of the hypothesis of the central symmetry of the observable space-time in the indicated sense leads, on the base of the special theory of relativity, to conclusion about the possibility of existence of the space transformation that is in some form like rotation. Actually, relative motion along any space coordinate x with the speed v0 is reflected in the space-time section tx as the rotation of coordinates in the form [1]:
x = x ' c0h(y) + c0r ' sh(y) c0r = x' sh(y) + cr' ch(y)
In this case there is a one-to-one correspondence between speed v0 and rotation angle y: v0/ c0 = th(W)
Detection of the regular isotropic movement of the universe matter in radial direction with speed v0 = H 0 r0, where H 0 is Hubble's constant and r0 is distance to a detected object, establishes the fact of regular rotation in any space-time section(tx, ty, tz). Transformations as of the space-time sections as of any space section(xy, xz, yz) would be corresponded one another according to the hypothesis, so that it is possible to assume the existence of the centrally symmetric transformation of space by which occurs rotation of any arbitrarily selected spatial plane (xy, xz, yz) . In this case the time component of metric tensor only is exposed to change, but spatial components at the same time do not change. Let us assume that this generalization of rotary motion in the macrocosm is possible being repulsed from a certain analogy with particle zero spin in the microcosm.
According to the theory of the unsteady Universe, the space-time invariant interval in the arbitrarily selected section of the observable space is written down as [8]
ds2 = c02dr2--dr° - r2dp2.
! kr0 1 2
a 0
Let us examine the equation for trajectory of the photons moving within the arbitrarily chosen rotating flat section of the observable space. The invariant interval would be equal to zero along the line of propagation of light signal (ds = 0) [1]. Without rotation the light rays are spread in radial direction (p = const, dp = 0) with the speed c0. The rotation of coordinates within the chosen plane with angular velocity H 0 = dp dr leads to deviation of photons, radiated by luminous source, from radial direction and to its radial propagation with the effective speed cf = c^1 -H02r02/c^ [1]. The same transformations occur in any other flat section of space, and it means that concentric light wave takes place with the phase velocity Cf variable in the radial direction. This wave can be described within two-dimensional representation applying for the any
section. Having excluded time r with the help of relation dr = dp/H0 from the equation ds = 0
applying for light propagation, and taking into account the above-indicated general Friedman's solutions of the gravitational field equations, it is possible to write down equation
dp = dz
V(1 -z2[- 1)+Ac02/H02 -z2] where z = kr0/ a 0 is the dimensionless radial coordinate.
The simplest form of the equation for trajectory of the photons is realized by performance of the equality:
q -1 = -AC"2
2H 02
Making integration with the initial conditions r0 = 0, p= 0, and taking into account that, from a physical point of view, the range for the dimensionless radial coordinate is z < 1 with p > 0, it is possible to obtain the equation:
k r0
z = ■
a 0
= th(pp).
This trajectory is a spiral emanating from the center r0 = 0 and reeling up the focal circle with the radius r0 = a0 limiting the observable Universe. It was mention above that the rotation angle y of the space-time coordinates (t, r0), according to the special theory of relativity, is connected with the speed of relative movement of matter in any radial direction by the parity v0/c0 = th(y), from which, in a view of the hypothesis of central symmetry of the space-time transformation (p = y) and also from the equation for trajectory of photons with expression for radius of curvature in the form a0 = c0/H0, directly follows Hubble's law v0 = H0r0.
It is impossible to obtain linear dependence of the speed v0 on the distance r0 (Hubble's law) with
the arbitrary values for the parameter of acceleration in this representation, so that with A = 0 (although this is a very debatable question at present [11]) we come to the closed model q = 1 once again (see p. 1 above). The uniqueness of cosmological model on parameter of acceleration gives essential advantages to the hypothesis of central symmetry of space-time in comparison with more general hypothesis of the expanded Universe.
Space-time dispersion and the time arrow
Having parameter of acceleration q = 1, in the presence of the time transformation within the model of the unsteady Universe, the requirement of invariance for Friedman's equations leads to appearance of the time factor K = 1/^1 + 2H01. In addition the assumption of the central symmetry
of the space-time transformation brings the rotation factor M = J 1 - H 02 r02 / c0 in the space-time
metrics and agrees with Hubble' law. For this reason the space-time wave, corresponding to the light wave, have to be created due to both these factors, and in its presence the specification of coordinates for the physical observer is possible only under specific conditions for his existence.
Generally speaking it is possible to imagine four types of coordinates in the transformed space-time according to the form for the time component of the invariant interval:
c0dr - time-universal local frame
c0dt - time-converted local frame cfdr - time-universal co-moving frame cfdt - time-converted co-moving frame
Contemporary observer is determined by his two fundamental abilities: contemporaneity and observability. The feasibility of these abilities can be expressed by two corresponding conditions:
Cf = c 1 - H 02 r2/ Co2 = , 1
a/1 + 2 H o t
1 - H2 r2 / c2 c . * c = c02 ^ 1 H 0 r°7 Co = 1 f 0 1 + 2H 0t
In accordance with the first expression, the time-converted co-moving frame must be equivalent to the time-universal local frame. This ensures displacement trough time of the time-converted proper frame of contemporary observer as a whole co-moving to the time-universal local frame, another speaking - contemporaneity of the observer.
In accordance with the second expression, the time-converted local frame must be equivalent to the time-universal co-moving frame being a formal basis for ability of observation in space. This condition of observability makes it possible for observer to present the time component of the invariant interval in the forms:
1 _ h 2 r 2 / c 2
c02dt = c fcdt2 = c02-^-dt2 = c02 (1 _ H02r02 /c02 )dr2 = cIdr2,
0 f 0 1 + 2H0t 0 0 0 0 f
and this possibility gives the way in principle for the observer with the proper time t to construct its frame, within transforming space and without disturbance of causality principle, on the basis of the space-time interval of Robertson-Walker, which was used above for the analysis of trajectory of a light ray.
The existence conditions for observer yield the dispersion relation for the space-time wave, transforming the time-converted co-moving frame, in the form
_H0r2/2c2, if t ^0 t = . H0r02/2c2(1 _ H2r2/ c2), if t > 0
The dispersion relation shows that the contemporaneity condition is feasible only in the positive range of time definition, consequently the contemporary observer can moves trough time in the future only. At the same time, on observability condition, he is capable to observe the sequence of the past events only as the universal co-moving frame can be constructed only in the negative range of time definition. It means that the time rate of the Universe is not arbitrary for the observer and it is assigned by the frequency dispersion of the space-time wave imposing two fundamental constraints on the observer:
1) his possibility to be moved trough time only into the future (the time arrow);
2) his possibility to observe only the past. Moreover, we obtain in the limit
r ^ a, t ^ ro, if t > 0 t ^-1/2H0, if t < 0
The expressions show that the transforming universe is limited by the horizon of the past events and potentially perpetual.
The dispersion relation for the space-time wave rigidly connects scale of the proper time of observer for observable object with its distance scale r0 and determines the reason for the universe
lapse of time, connecting it with the space transformation; this is a formal consequence of causality principle in the frame, which is permanently transformed by the space-time wave.
Taking into account Hubble's law it is follows from the dispersion relation with t < 0
2H01 = - H02r2/ c02 =- (vj c0)2,
it is possible to say therefore that recording red shift we measure the lapse of time at the different points of the Universe.
Here it is necessary to note in connection with this, that for the quite clear reasons the time factor implicitly was presented earlier in the spherical space-time of de Sitter [4].
Physical variables and the time factor
As it was shown above, the time factor K theoretically can be accompanied by the rotation factor M in the observable space, so that both factors completely compensate each other, and besides, observer does not find out them directly in the space-time metrics q00 = K2 M2 = 1. But if the time
factor exists for neutralizing the disturbance of the space metrics of observer, caused by action of the rotation factor, then the rotation factor works as the latent clockwork mechanism, and contemporary observer is compelled to be moved trough time in order to remain the contemporary capable to observe, and his replacement in time relative to the observable past would be accompanied by relativistic effects in time, as any other motion, which have influence on the observable dynamic variables of the Universe.
Moreover, taking into account that the operator S acts on the dynamic variables of the observable Universe, it is natural to assume that the time factor would influence on all observable dimensional physical quantities as well, i.e. all the physical dimensional variables would be the eigenfunctions of the S -operator having the discrete spectrum of the eigenvalues: s = 0, ± 1, ± 2,.... Formally this is evinced by appearance in the physical variables of the multipliers determined by various degrees of the time factor K; discreteness of the operator is directly connected with the dimensionality of physical variables, and the algebra of eigenvalues have to be corresponded to the algebra of dimensionality. Let us note in connection with this assumption that in that case the S -operator naturally brings, through physical variables, in all physical equations irreversibility in time.
The assumption about that any observable physical variables, including the locals, would be the eigenfunctions of the operator of cosmological evolution, as well as the universal variables of the dynamic equations of Friedman's model, has important significance for empirical validation of the model with q = 1. It is basis for the search of possible traces of cosmological evolution on the directly contactable objects, for example on the Earth [12].
At present it is hardly possible to find precisely the average density of matter of the Universe and its parameter of expansion by astronomical methods, and it is highly improbable that only this way would lead to the reliable result. There is sufficiently a lot of invisible matter in the Universe, and the doubt about whether everything matter is identified and took into consideration will always remain. At the same time, if the central-symmetric closed model of the Universe is adequate to reality, it is possible to expect that the S -operator can noticeably exhibit itself locally in the physical phenomena of geological time scale. The time scale of such phenomena would be order of billions of years.
In this respect geophysics, on the one hand, could render assistance for cosmology, astrophysics and astronomy, making it possible to select one of the many "scenarios" of the Universe available now and, on the other hand, it could strengthen its own theoretical bases. Preliminary analysis of geophysical, geological, paleomagnetic and paleontological data on evolution of the Earth, executed under the assumption of possible effect of the factor K on the geophysical characteristics, showed the surprising consistency of these different data and made it possible to estimate Hubble's constant with the value of 104 km/sec/Mpk, which practically coincides with the estimation made above on the basis of the age of meteorites. The results of this analysis will be given in more detail in the following paper.
Final remarks
On the idea presented in this paper the centrally symmetric non-stationary observable Universe can be considered dynamically as steady physical system - observable particle of macrocosm possessing of angular momentum. Centrifugal acceleration of matter being on the boundary of the Universe creates the force field compensating gravitational compression - the Universe as a whole is been in the dynamic equilibrium:
H2 a = (4/3)nGpa . Mechanical work
a 2
mo ^
A = f ■ v_dr,
W1 -(v/c)2 r
which must be carried out in order to transfer some particle with the rest mass m0 from behind the horizon of the Universe into the point of the observer's arrangement after overcoming centrifugal acceleration v2 / r, is exactly equal to energy of the rest mass of any transferred particle. Indeed, with the use of the relations: v/r = H and H = c/a, it is possible to obtain the expression:
A = - mc
-(r!a )2
= m0cL.
Consequently energy of the matter rest mass is the reason for transformation of space. The centrally symmetric rotatory transformation of space leads to that Universe's matter, locating in the centrally symmetric gravitational field, is moved on the outer boundary of the arbitrarily chosen flat section not along the field lines but along the equipotential surfaces as the inertial repulsive forces are present besides the gravitational attraction of the matter equal in magnitude to attracting force. It means that global movement of the matter probably occurs in such manner that total energy of the Universe is equal to zero, and moreover, according to the written above equation A = m0c02, if
there is no rotation, there is no matter. It is represented from this point of view that significance of the de Sitter's model [4] for the empty Universe with zero boundary conditions for metrics, which is invariant relative to transformation of all four dimensions including time, possibly is underestimated by modern science.
The mass of substance fulfills simultaneously two functions conformably to dynamics of the Universe - it creates the gravitational field (by means of bending space) and besides causes the isotropic transformation of space-time (finally it is something similar to the "bucket of Mach" revolving on a rope). Lapse of time for the contemporary observer is not arbitrary in the converting coordinate system but it is determined by the dispersion of the universal space-time wave caused by transformation, and the evolution of the Universe is represented as relativistic effect connected with replacement of the contemporary observer in time accompanying the wave. Figuratively speaking, and very conditionally, displacement in time of contemporary observer resembles sea surfing: being continuously rolled up along the slope of wave, a sportsman at the same time remains at its apex owing to opportune transformation of sea surface. The conditions for existences of the observer are implicitly perceived by the observer himself as the real properties of the observable objects, creating the illusion of big bang on the boundaries of the observable Universe. The space-time horizon of the centrally symmetric Universe for the observer is outlined by the time Lorenz factor, and there is no necessity at all to find out seeming initial conditions of the Universe.
Taking into account possibility in principle of permanent transformation of space-time, it appears that the non-stationary model of Friedman and the static model of de Sitter does not look alternative, and the hypothesis of the centrally symmetric space-time transformation, which consequence is affect of isotropic rotation of any flat spatial section, does not exclude the
a
0
hypothesis of expanded Universe but substantially supplements it on the contrary, creating the new possibilities for:
- selection of the dynamic model of the Universe at the theoretical level (q = 1),
- interpretation of observant data (astronomical and geological),
- development of the physical theory of time.
The time Lorenz factor introduces essential correctives into the physical properties of the observable objects near the horizon and moreover very sharply. First, expansion speed v = v 0 K
grows sharply in the limit, creating the effect of the boundary inflation of the linear Hubble's expansion (see [11, 13]). In the second place the linear sizes of the objects sharply decrease (inversely proportional to K), at the same time their energy characteristics are increased extremely sharp - in proportion to K3, that may be very useful for interpretation of observed data received from quasars.
Besides as the length of the light wave, coming from object at the distance r, is converted as Ar = Ar0 /K, there is an additional spectral shift of wavelength in the course of time and the corresponding apparent increase of expansion speed (acceleration [14, 15]) measured on the basis of the Doppler effect v0 = c0(Ar0/A0K -1); the essence of this effect does not change for the
relativistic Doppler effect. In connection with this it is necessary to make clear distinctions meanwhile what we observe and what we see in this case. This distinction has especially important significance for the interpretation of the astronomical observations of boundary objects.
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