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10On the curious feature of the Friedman's solutions for the gravitation field equations of the Universe, and its geophysical applications
A.S. Kartashow (kart@meteo.nw.ru) Institute of Simulation Technologies, St. Petersburg
In the previous paper [1] it was shown that, if we allow transformation of time in the model of the unsteady state Universe with the parameter of acceleration q = 1, the condition of the invariance of the solutions of Friedman's equations leads to the appearance of the time factor K = 1/^1 + 2H01, and it is possible to expect that this factor will noticeably display itself locally on the physical phenomena of geological scale of time.
The past is always examined relative to contemporary observer. As a result of that the contemporary observer, as shown above, can exist only being moved in time, and the past for him would change unavoidably. The same events, considered from points of view of observers existing in the different moments of time, would look differently, i.e. they have to be expressed differently in a quantitative sense.
Observer is any physical object as alive as not alive; contemporary physical processes affected it, and surrounding objects, had exerted influence during its existence. If the observer has memory, i.e. its nature is those, that physical processes are fixed by its structure anyhow for a long time, among relicts of the past there would be the traces allowing to estimate quantitatively influence of the time factor on various physical processes, and such traces are relativistic effects of its displacement in time.
For example, fossil corals can be examined as one of such observers because its growth rate in the course of their life depended on light exposure. Change of day and night is reflected on a body of such coral being fixed as intra-annual edges/rings of growth. Having counted up number of intra-annual edges at the corals concerning by various epoch, it is possible to define evolution of year duration [2-4].
The major observer of the past is the Earth having sufficiently many ways to fixate the contemporary physical processes; it may be for example distribution of continental and oceanic crust, geological structure of Earth interior, geography of the geomagnetic properties of rocks and many other things. One of the geophysical phenomena, on which the S -operator could render the perceptible effect, is delay of the angular velocity of the Earth rotation. The slowing-down effect of the Earth rotation is connected usually with nonconservation of the planet angular momentum exclusivly due to the lunar tidal friction, which was sufficiently accurately evaluated quantitatively by S. Newcomb. Let N is a serial number of year, than the year duration is determined by relation n = 365 + 6.14 * 10-8N, where n is the day number. However there are evidence of that the year duration actually changed faster.
Information about this (table. 1) gives the fossil corals [2-4 ]. Distinctions of the paleontological data on the fissil corals with Newcomb's formula are essential. It is indoubtable that the formula is accurate, therefor the distinctions can be connected only with the co-ordinated change both the angular velocitiy of the Earth and the orbital velocitiy of the Moon. Consequently the reason for the distinctions has far nonlocal nature. Let us suppose that it is bound up with cosmological evolution, i.e. with influence of the S -operator on celestial bodies.
Table 1.
Number of the intra-annual growth rings of fossil corals and year duration
№ Author paleochro- nology (million years) number of rings year duration (day)
observation approximation Newcomb's formula relativis-tic effect number of days
1 Wells J.W. (1937) 500 412 412 396 23 419
2 Scrutton C.T. (1965) 370-385 400 397-399 388-389 16-17 404-406
3 Scrutton C.T. (1965) 285-310 385390 388-391 382-384 13-14 395-398
4 Beauvais L., Chevalier J.P. (1980) 220-230 380 381-384 378-379 10-12 388-391
5 Wells J.W. (1937) 0 360 358 365 0 365
It is possible to expect that the duration of year in the course of cosmological evolution changes in inverse proportion to.^1 - 2H0N , and the data resulted in table 1 are described by function: n
n = . 0 + 6.14 X10-8 N Vl - 2H o N
The first term expresses in explicit form the effect of cosmological evolution (relativistic effect), the second is determined by nonconservation of the angular momentum of the Earth, and moreover the parameter H 0 would be equal to Hubble's constant.
Results of linear approximation of the data on calculation of the intra-annual growth rings of the fossil corals
0.2 0.15
Figure 1
Data processing of table 1 was carried out by linearization of function n (N), by means of the variable replacement, which is expressed for discrete points i as
( n
y< = 1---—
^ ^ n - 61.4 n ,
where Ni is time in billion years.
After fulfillment of the linear regression of the data, the dependence y = 0.225N - 0.0087 with the standard deviation of 0.010, which is resulted in figs 1, is obtained. Deviations have random nature. The coefficient of approximation 0.225 is equal to the doubled value of Hubble's constant expressed in dimensionality 1 yr. Processing data of table. 1 gives for the parameter of expansion the value H0 = 110 km/sec/Mpk, and reveals a systematic error of the data, in relation to the actual year duration, equal to 7th day (it is taken into account in the table).
Another track of cosmological evolution could be found in view of a possible change of the linear size of the Earth in the course of time. Different researchers regard a question about the evolution nature of the Earth during geological history ambiguously, and as a whole it remains open, although its history continues already during more than hundred years [5-9]. Disagreements in this question are connected mainly with the interpretation of geophysical, paleontological and other observational data, which have leaded to the appearance of different hypotheses about a possible change in the volume of the planet (invariability; pulsation, expansion).
Hypothesis of the expansion of the Earth basically allows resolving contradictions colliding both doctrines of the global tectonics - fixism and mobilism [6], but a weak point of this hypothesis is absence of single-valued answer to a question on the reasons of the expansion. In the context of this work there are grounds for assuming that, if the expansion of our planet and other global evolutionary effects actually occur, their reasons have not local nature, and they could be connected with the cosmological evolution. From this point of view the hypothesis of the expansion of the Earth can be used as working hypothesis for the purpose of the preliminary reply on the question as far as:
1 the geophysical phenomena treated by some researchers [15] for the benefit of expansion of the Earth will be coordinated quantitatively among themselves;
2 the Friedman's model of the Universe with parameter of acceleration equal to one will be coordinated to the hypothesis of expansion of the Earth;
3 geophysical characteristics of evolution will be coordinated to the astronomical data on expansion of the Universe.
There is a number of geophysical phenomena considered as the attributes of expansion of the Earth. It was in the beginning of the 70's when research on the paleomagnetism had showed that the position of magnetic pole in the past, found from the magnetization of the rocks in the different places of the planet, are differed in dependance of the contemporary position of the pole, and moreover the more age of species the more deviation of magnetic pole [5]. This phenomenon indicates a possible change in the radius of the planet in the course of time, as a result of which the earth's surface was extended, and the magnetized rocks gradually were moved away from the actual magnetic pole. If the vectorial angle of the deviation of the magnetized rocks from the magnetic pole a is known, a radius of the planet, at the moment when the rocks locating today on the magnetic pole and the previously magnetized rocks mentally converge at one point, is defined as: R = R0 cos(a/2) . Some paleomagnetic data and corresponding estimations of the radius are resulted in table 2.
Besides it is possible to include the estimations of the planet sizes in the past with the larger scale of time, for example as the estimations made by R.Dernly according to the results of investigating the Precambrian plicated belts, which had resulted in the book of W.Carey [5]. It follows from the Dernly's estimations that 650 million years ago radius of the Earth was 6000 km, and 2750 million years ago - 4400 km.
Furthermore if we assume that the contemporary continents really are the splinters of initially unbroken Earth's crust formed in the Archean era, which reported till our time information about the sizes of the Earth in the time when granites had formed, the ratio of total areas occupied by seas and oceans is the natural "memory unit" of cosmological evolution, and the contemporary specific area of continents equal to 0.29 is the estimation of the Earth's size approximately 3.8 billion years ago (age of the most ancient continental species).
Table 2.
Summary geophysical and paleomagnetic data on a change of specific radius of the Earth
Method of estimation Parameter Time Value of Specific rac ius
№ of the estimation (million years) the parameter Estimation Appro-ximatin Notes
1 Distance Vectorial 1 1.5 0.9999 0.9999 time mark
between the angle 3.5 3.5 0.9995 0.9993 relate to the
magnetized (deg) 14.5 6 0.9986 0.9984 middle of the
species and 84 12 0.9945 0.9894 corresponding
magnetic pole 176 16 0.9903 0.9822 era of the
[5] 226 20 0.9848 0.9771 geochronologi
265 25 0.9763 0.9732 cal table
2 Precambrian plicated belts [5] - 650 2750 - 0.9355 0.6875 0.9327 0.6708
3 Ratio of the Specific
overall sizes of continents and area of mainland 3800 0.29 0.5385 0.4900 Time mark is
world ocean crust referred to the
4 Size of Pacific Ocean Vectorial angle of age of the earliest
disclosure of the 3800 130 0.4226 0.4900 bedrock
ocean
(deg)
At the same point of view Pacific Ocean is most ancient oceanic formation on the Earth which beginning of formation can be attributed also to the age of the most ancient species on the continents equal to 3.8 billion years. If to take in account the effects of equatorial twisting, the form of Pacific Ocean appears round [5] with the polar angular size (expansion angle) about 130°, so that initial radius of the Earth evaluated according to the formula R = R0 cos(a/2) had to be 2700 km.
The summary geophysical and paleomagnetic data on a change of specific radius of the Earth according to the above-indicated signs are cited in table 2. It is possible to expect that these far incomplete estimated data will agree between themselves and it is reflected by overall dependence on the time of the form R = R0y]l - 2H0N , which enables to check up one more eigenfunction of the operator of cosmological evolution with the eigenvalue s = 1.
Standard regression analysis of the summary data with use of dimensionless variable y = 1 -(r/r0 )2 linearizing the dependence for radius of the Earth indicated above gives the linear approximation of the data y = 0.202x - 0.0065 with standard deviation 0.024. The factor 0.202
determines Hubble's constant, which is equal H 0 = 98 km/sec/Mpk. The data and corresponding
approximation illustrates figure 2.
The average value of Hubble's constant on two groups of data (eigenfunctions of the S -operator for time and distance) equal to 104 km/sec/Mpk, that practically coincides with the estimation of Hubble's constant made above on the basis of the maximum age of meteorites as the most ancient contemporary objects (105 km/sec/Mpk).
Results of the linear approximation on the summary geophysical and paleomagnetic data
Comparing the obtained results with the results of estimations made on studying of dynamics of astronomical systems (110 km/sec/Mpk) [10, 11], and also with the summary data on the parameter of Universe's expansion obtained on the basis of the optical methods of astronomy (95 km/sec/Mpk) [12], it is possible to be convinced that the geophysical and paleontological methods of evaluating Hubble's constant, based on the assumption of action on the physical variables of the S -operator, anyway do not contradict astronomical methods. Tendency toward the decrease of the value of Hubble's constant, outlined in astronomy today [13] in connection with studying of extremely removed sources, it can be explained by the optical effects possible for the transforming space and noticeable at the very great distances. But this is the theme of a separate study.
The geophysical data involved for the analysis are completely diverse and independent, and the fact that they not only agree rather well among themselves within the framework of the S -operator, but also they have no contradictions with data obtained in another field of knowledge -astronomy, it is represented by argument in favor of real existence of the operator of cosmological evolution S and adequacy of the model of the Universe with the parameter of acceleration equal to one.
The executed analysis of geophysical, paleontological and paleomagnetic data bears only preliminary estimated character; detailed researches of the interested experts in the direction considered in this work are necessary with more attraction of the various data which the Earth science has now.
Referenses
1. Kartashow A.S. Electronic Journal "Investigated in Russia", 8, 520-531, 2005, http://zhurnal.ape.relarn.ru/articles/2005/046.pdf
2. Wells J.W. // Nature, v. 197, No 4871, 948, 1963.
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5. S. Warren Carey. «Theories of the Earth and Universe», Stanford, California, Stanford University Press, 1988.
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8. O.G. Sorohtin, A.D. Ushakov. «Global evolution of the Earth», Moskow, MGTU, 1991 (in Russian).
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10. Lynden-Bell D. //Nature, v. 270, 396, 1977.
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