Cosmology and cosmogony with geophysical applications

Acta Geod. Geoph. Hung., Vol. 43(1), pp. 105–111 (2008)

DOI: 10.1556/AGeod.43.2008.1.8

COSMOLOGY AND COSMOGONY WITHGEOPHYSICAL APPLICATIONS

W Schroder and H-J Treder

Geophysical Commission, Hechelstrasse 8, D-28777 Bremen, Germany

[Manuscript received July 2, 2007, accepted August 2, 2007]

The different aspects in cosmology, cosmogony with geophysical application areillustrated and discussed by the authors in detail.

Keywords: cosmology; geodesy; Laplace; theoretical geophysics

I.

The basic problem of cosmogony is the age of the Universe and of its subsystems.The sense of these time intervals is very relative, it is based on the comparison ofrelaxation times of the systems and on the so-called world age. (The latter isaccording to order of magnitude the reciprocal of the Hubble constant H−1.)

The preconditions of the Kant- (and Kant-Laplace, respectively) cosmogony ofthe planetary system are very well valid, as the relaxation times of an “open clustersof stars” have a maximum value of 107 a. In contrast, the cosmogony of galaxiesoperates very poorly (or it has even no sense to formulate it, respectively) as therelaxation time for a galaxy of 1011 solar mass are of the order of magnitude 1016 a.

The so-called world age, the reciprocal of the Hubble constant H , is not morethan 1011 a (in reality, it is even somewhat less). Therefore there are no cosmicobjects which would be older than 1011 a.

At the beginning of this time interval, the cosmological singularity is in all worldscenarios: the big bang should have destroyed all previously existing structures, asfar as they are accessible in the metagalaxy. The metagalaxy is thus cosmologicallyvery young. Therefore one must not wonder about its peculiarities, as any regulari-

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106 W SCHRODER and H-J TREDER

ties could not clearly – present themselves. Such a situation is expected only 1017 a(see later II).

As mentioned, this is valid only for the complete structure of the cosmos, how-ever, it is not valid for its individual objects as the solar system, as the scale numbersare completely different.

The comparison of the relaxation times of large-scale cosmic systems (galaxies,clusters of galaxies, super-clusters) with the world age ∼H−1 excludes all meaningfulcosmologies. The metagalaxy is by far to young to possess any structural regularity.We are, however, sure that — how quickly or slowly something happens in thecosmos — certain conservation laws can be hurt. This is, regretfully no more validfor classical conservation laws of energy, impulse, spin and gravity center, as theselaws connected with the symmetries of the space-time world, which is just discussed.In contrast, the laws about the conservation of charge, of the lepton number, of thebaryon number and of the general relativistic cosmology, which can be substantiated“in all spaces”, are beyond doubt. Generalized cosmologies (inflating cosmoses)may make even these conservation laws questionable. (In contrast, there is noconservation law for the meson numbers.)

Well-based doubt about the conservation of the baryon numbers means theremark that it is not sure that protons are stable particles. Many authors supposethat the lifetime of the protons could be in the order of magnitude between 1031 aand 1034 a, i.e. being very long, but not infinitely long.

According to the former sentence, namely that there are absolutely no objectsolder than 1011 a, the statement that the protons have lifetimes between 1031 a and1034 a has for the moment no sense. As each g matter contains 6 × 1023 protons,sufficiently large mass (being a between few times 101 and 104) losses one protonin a year. That means that if somebody has doubt about the conservation of theproton number, he can this experimentally check, as the number of protons is inthe visible cosmos 1080.

The real problem of epistomology is what is the meaning of the “flight” into the“experimental recess”: it is possible to give limits, nobody can experimentally provethat a universal constant is really constant. (Consequently is in Planck’s sense: “itdepends on nothing but also on nothing at all”.) It is, however, always possibleto give an accuracy limit of the measurements and the necessary conditions withwhich the numerical value of its variability can be determined. No metrologicalobjection can be raised against the statement that the velocity of light c, Planck’sconstant h and especially the gravity constant f , further the baryon number N andso on can be variable in the case of a higher accuracy of the measurements andwith appropriate arrangement of the measurements. We arrive thus to the limit ofsensitive questioning. In a world being 1011 a old, an interval of 1034 a is practicallyinfinite.

There is another example for such flight experiments. Nobody can prove thatthe rest mass μ of the protons is exactly zero (de Broglie once supposed it to beapproximately 10−44 g).

Landau believed to be able to prove that due to the CPT theorem the rest massof the neutrino must be exactly zero. (An argument which is for us still plausible.)

Acta Geod. Geoph. Hung. 43, 2008

COSMOLOGY AND COSMOGONY 107

One can, empirically, however, never prove that neutrinos have no rest mass, it isonly possible to give an upper limit for this value, and these upper limits seem toget less and less with time, Boguljubov and Shirikov’ statement that all particlesmust have a rest mass μ > 0, is therefore with restriction questionable.

Everything written previously is invalid in the moment when somebody presentsa cosmic object the age of which is ≥ 1012 a, and in such a case is the relativis-tic cosmology similarly no more valid. Furthermore, if somebody would discoverprotons with a rest mass μ > 0, the Maxwell theory would be false and the non-calibration invariant Proca-equations are valid instead of the calibration invariantMaxwell equations

∂kF ik =μ2c2

h2Ai = −J i, Fik = ∂iAk − ∂kAi .

If somebody can prove that there are neutrinos with a finite rest mass μ, thenWeyl’s spinor theory for neutrinos false as it implies just that the rest mass μ ofthe neutrinos disappears.

The same is valid for the lifetime of the protons. If it can be proved that thelifetime of the protons is finite (be it as great as it is), then the conservation of thebaryon number is no more valid.

This, however, the same as the question: Is it possible to detect a dependence ofthe velocity of light c on the relative motion of source and observer, in which casethe special theory of relativity would be false, moreover, if it be proved could provethat the quantum of effect h depends on something, then the quantum mechanicsis disproved. One could Heisenberg’s uncertainty relation ΔpΔx ≥ h evade bymodifying h.

II.

The general theory of relativity is based in its cosmology on Weyl’s principle:the velocity of light is the carrier of information, which cannot be influenced. Thislaw tells us about the conform-invariance of all cosmic metrics and about the dis-appearance of the Weyl’s 4-dimensional conform curvature tensor

W hijk = Rh

ijk +12

(δhj Rik − δh

kRij + gikRhj − gijR

hk

)+

Γb

(υkgij − δh

j gik

)= 0 .

The present cosmology (Eddington, Friedmann, Lemaitre, Robertson etc.) stepsover Weyl’s principle and states a universal synchronism. Therefore the most generalcosmological line element is always according to the Robertson-Walker form

ds2 = c2dt2 − R2(t)dσ2

(in the case of an appropriate choice of the system of co-ordinates) where dσ2

means the metrics of a three-dimensional space of constant curvature. As ever, the“experiencibility of the cosmology” is defined (see later): One is confronted withthe problem of the particle horizon.



The expansion of the cosmos is an adiabatic and reversible process i.e. the solu-tions with time inversion are theoretically exactly so suitable as today the expansionknown from the Hubble-effect. The Robertson-Walker metrics have all the samemathematical sense: “The world is expanding for us”. Nevertheless, this expansionis reversible, i.e. a contraction of the cosmos would be physically similarly possible.

Everything cosmologically happening in the cosmos has to be, however, irre-versible; i.e. the sense (direction) of time is unambiguously defined. Such irreversibleprocesses should be based on interactions of cosmic particles (they may be of anyorder of magnitude), consequently, the problem of relaxation times emerges again.

Nevertheless, the interaction of particles is coupled to the finiteness of the ve-locity of light c. Interactions are namely between bodies only possible “which cansee each other”. (That is the problem of horizon.) It is always valid in standardcosmologies that the velocity of light is constant (Weyl’s principle), whereas thevelocity of expansion R decreases. Interactions between particles are possible if itis valid that

t0 ≥t0∫

t→0

dt/R(t) .

In the case of the Einstein-de-Sitter cosmos (three-dimensional plane space) one has

R ∼ t2/3 , R ∼ 23t−1/2

thus sometimes the light establishes the expansion of the cosmos, and together withit, the possibility of the interaction of the particles.

In contrast, cosmic electrodynamics tells us that (in the case of any small worldradius and correspondingly high temperature and density in cosmos) interactionsbetween particles take always place with relaxation times, which are generally largerby several orders of magnitude than the present particle horizons t0. — That is thefoundation for the statement that the cosmos is too young to possess a history asan entity.

Relativistic thermodynamics leads for an adiabatically expanding cosmos toTolman’s equations for mass (rest energy)

M =4π

3�R3 = const

for radiation energyμR4 = const

and for the kinetic energy of a non-relativistic one-atomic gas

iR5 = const

Everything can be attributed to von Laue’s law, namely that the impulse p isconnected with the world radius R by the equation

pR = const



for ultra-relativistic particles (e.g. photons) follows from this again Hubble’s relation

hc

λR = hvR = const

These processes are, however, all adiabatic and reversible: One experiences ahistory of the cosmos, as the initial conditions are as they had been. Hubble’srelation states corresponding to the Planck-Einstein photon theory that the blacktemperature T is the reciprocal to the world radius R:

TR = const

In the cosmological scenario, all particle numbers are conserved. Particles lossonly their impulse p and with it their energy corresponding to their kinematics.(That is just the definition of the adiabatic reversibility.)

III.

Heckmann and Zeldovich hint at the fact that the existing cosmos is isotropic(by taking the corresponding averages over “cosmic particles” and referred to thebackground radiation). They were right to declare this as a problem. If the cosmoswould be very old, such an isotropic state would be established according to thesecond law of thermodynamics.

There are two possible solutions, which were discussed by Heckmann, Zeldovichand by us. The first possibility is: the world is such, as it is, as it has been such,as it has been. Accordingly, the primeval structure of the cosmos has already beenhomogeneous an isotropic and thus it corresponded to Weyl’s principle. Never-theless, this idea seemed for Heckmann and Zeldovich too anthropomorphic. Thesecond possibility is that in the “primeval state of the cosmos” (i.e. in the frame ofthe “big bang”) something happened else as implied the relativistic cosmology ofFriedmann’s models.

There is no limit in the frame of Friedmann’s cosmology for the possibility toapply our up-to-date physics, these limits, respectively, impede to understand thehomogeneity and isotropy of the cosmos, as no argumentation is possible in a four-dimensional relativistic space against the particle horizon.

That is why an “inflating cosmos” was proposed as scenario. This inflatingcosmos has the property “to be energetically more dense”, than allowed by therelativistic physics. These scenarios contained accordingly in the early stadium ofthe universe degrees of freedom outside space-time, which are compactificied by thequantum conditions and by Planck’s orders of magnitude. These supplementarydegrees of freedom lead “to more material” than possible to accommodate in spaceand time. Interactions would be then much stronger than allowed by the four-dimensional general theory of relativity.

The early history of the cosmos can be continued outside of Planck’s times√hf/c5 and would lead to results, which are no more covered by Planck’s values.

According to these inflatory scenarios, all important properties of the cosmos cameinto being before its space-time existence. The early history of the universe is in



that case that of the demolition of these additional dimensions. According to ouropinion it is quite equivalent whether isotropy and homogeneity is set in advancealready among the initial conditions for the “world scenario”, or supplementary“parameters” are supposed which disappear during the scenario, therefore thesenon-space-time parameters are no more identifiable in the observable time-spaceson the basis of a horizon.

IV.

All irreversible processes in the cosmos presuppose that the sum

H = K + U

of the kinetic energy K and of the potential energy U (i.e. the Hamilton function H)is no more constant. The second main law of thermodynamics says that ΔH ≥ 0is always valid. — The first main law of thermodynamics is naturally always valid,but the second main law states that thermal energy cannot be transformed intokinetic energy (in the frame of Carnot’s principle).

Kant’s cosmogony of the solar system is based on the one hand on the laws ofconservation of impulse and of spin, on the other on the transformation of kineticenergy into interior thermal energy. In contrast if H = const, “nearly all” mechanicprocesses are cyclic ones. That means: each process where H = const is periodic,and the ergode-theory is valid with the exception of a “zero set” of initial values.

Laplace tried in his “Celestial Mechanics” to prove using analytic approximationmethods that all “disturbances” in the solar system are ultimately periodic. Theseries used by Laplace, however, converge only for certain time intervals Δt.

Poisson realized this situation and suspected a more general law: each mechan-ical system is cyclic, i.e. there is no “chaos”.

The decisive message is a theorem by Poincare.In a paper Poincare (1889) realized that all mechanical systems are cyclic accord-

ing to the principles of mechanics and to the topological consequences of Liouville’theorem about the conservation of the phase volume following from them. Anexception is an exotic minority of initial conditions.

Poincare’s theorem has nothing to do with the explicit possibility to compute theorbits. Each method of orbit computation breaks down after a certain time intervalΔt, as it can be traced further on neither analytically, or digitally. Nevertheless,these limits are barely those of the computing methods. Poincare’s theorem isindependent of all computing methods. It describes the topology of the phasespace.

The solar system is stable presupposing that e.g. the friction resistivity of amedium and from the unelastic collisions can be neglected (and that is valid forabout 4.6 billion years) and that rockets and similar reactors are neither considered.

There are exceptions (which give, however, mathematically a zero set) namelythat a celestial body is hurled out from the solar system, i.e. it gets a hyperbolicspeed (the so-called Schmidt’s case).



The destruction of the solar system takes place in its present stability exactlywhen the Sun explodes into a nova, and this will surely take place in a few billionyears. Else Poincare’s sentence is valid: “It will be proved that there are infinitelymany possibilities to select initial conditions so that the system returns infinitelyoften to its initial situation. There are also infinitely many solutions, which do notpossess this property. It will be, however, shown that these unstable solutions can beconsidered as “exceptions” and that it can be said, they have the probability zero.”— The initial conditions that result in stable systems of the size of the continuum.Those resulting in unstable solutions are in contrast countable.

Poincare’s periods may be much longer than the age of Sun, namely 1010 years.In contrast, the time of the emergence of the solar system is very short. Therelaxation times for open star systems of a few times 103 solar masses are of theorder 10 million years and the development of the solar system is determined onthe basis of the gravitational stability by the Helmholtz-Kelvin period of the sameorder of magnitude. During this time energy is dissipated, namely potential energyis transformed into kinetic, then the kinetic energy is transformed into thermalenergy due to inner friction or to inelastic impacts. In comparison of its time ofemergence, the solar system is very old. The time of emergence is only a fewtimes ten million years; the lifetime is in contrast some million years. Both aretheoretically reasonable, the first: the short time of emergence is based on the lawsof thermodynamics, the second on the practically “perpetual lifetime” deduced fromthe theorems of the Newtonian celestial mechanics.

References

Poincare H 1889: Sur le probleme des trois corps et les equations de dinamique. ActaMath., 13, 1–270. Reprinted in: Henri Poincare: Oeuvres, Vol. 7, Gauthier-Villars,Paris, 1952, 262–479.


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Cosmology and cosmogony with geophysical applications