Geophysics and cosmology — A historical review

Acta Geod. Geoph. Hung., Vol. 42(1), pp. 119–137 (2007)

DOI: 10.1556/AGeod.42.2007.1.7

GEOPHYSICS AND COSMOLOGY— A HISTORICAL REVIEW

W Schroder and H-J Treder

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

[Manuscript received June 28, 2005; accepted February 13, 2006]

Two types of connections exist between cosmology and geophysics: any funda-mental law of physics is incomplete without reference to initial and boundary con-ditions. That is true both for laboratory physics and for any cosmological physics afortiori. The new ideas in cosmology, relativity and unitary field theory, show strongconnections between the laws of elementary particle physics and space-time structureof the Universe. These connections give us new insight into the old ideas of Eddingtonand Ertel among others about cosmology and atom physics. The initial conditionswhich are resulting from the evolution and from the hot original state of the Universe,are also very important for the history of the Earth. On the other hand, geophysicalresearch into the history of the Earth gives us information about “self-experienced”cosmology and about the proper history of physical laws and constants.

Geophysical research is very important for these problems from the epistemogicalpoint of view, because geophysical experiments can be made about any physical inter-action — without the intervention of electromagnetic waves which have to overcomelong distances.

Keywords: cosmology; geophysics; history of Earth; relativity; unitary fieldtheory

1. Introduction

Einstein intended to develop a unitary field theory, or at least to find close con-nections between his geometric theory of gravitation and the physics of elementaryparticles. The experiments include a great amount of problems of the theory ofrecognition and it is interesting to note that geophysical research often give par-tial empirical criteria for such basic problems of physics. So basic recognition of

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120 W SCHRODER and H-J TREDER

cosmological research was initially the perfect Copernicanism, i.e. the statement for-mulated from cosmological principles, that there are no distinguishable directionsand areas in the Universe, and on average for a sufficiently large space, the Universeis completely isotropic and homogeneous corresponding both its geometry and itsphysical content. Deviations from homogeneity have thus only local significance andthey are without significance with respect to the global structure. The demandsof isotropy and homogeneity are otherwise of different importance. An isotropicuniverse is inevitably homogeneous. The opposite, however, is not valid.

2. Foundation of cosmology

The existence of a universal cosmic time is closely connected with isotropy.According to a principle formulated by Weyl, there exist universal systems at restin all perfect Copernican universes for the cosmic material. This means that areference system can be defined in which cosmic material is stationary and in whichall peculiar movements are neglected. (They are without exception small withrespect to the velocity of light, and therefore negligible.) The universe lines ofthis cosmic material at rest are simultaneously the universe lines of synchronouslyrunning clocks, i.e. periodic or quasi-periodic processes define a synchronous systemof clocks which indicate the same time in an arbitrary point of the world.

The space-time metrics of such a cosmos can be described by the use of theuniversal world time t

ds2 = c2dt2 − R2(t)dr2

1 +ε

4r2

= (dx4)2 − R2

(x4

c

)dσ2 . (1)

In Eq. (1) is

R2(t0)dσ2 = R2(t0)dr2

1 +εr2

4

(2)

the line element of the three-dimensional space of a certain world time t = t0. Thisthree-dimensional space is a homogeneous space with constant curvature. Accordingto the respective conditions

ε > 0 , (2a)

ε = 0 , (2b)

ε < 0 , (2c)

this three-dimensional space has spherical curvature, is plane and has hyperboliccurvature, i.e. it has a finite, an infinite and an imaginary radius of curvature. Theuniverse lines of the cosmic material are given as

dxi

ds= ui = δi

4 (i = 1, 2, 3, 4) (3)

thus its universe lines coincide with the co-ordinate lines of the world time t.

Acta Geod. Geoph. Hung. 42, 2007

GEOPHYSICS AND COSMOLOGY 121

This perfect Copernican space-time can be interpreted as a set of three-dimensionalspaces:

x4 = ct = const , (4)

which each represent the state of the cosmos at a certain world-time t.The function R(t) indicates in the line element of the three-dimensional space

that the distances between two points P1 and P2 are generally in these three-dimensional cosmoses not constant, but are functions of the world time:

S = P1P2 = R(t)

P2∫P1

dσ . (5)

Accordingly the volume of the three-dimensional cosmos is a function of the worldtime, too.

V ∼ R3(t) . (5a)

If the function R(t) increases monotonously with the world time, then all cosmicdistances also increase with time.

The validity of this statement is, however, restricted since the universe is actuallynot completely homogeneous, but has locally strong deviations from homogeneity.The expansion of all cosmic distances does not mean the expansion of the atomiclengths and of the size of macroscopic bodies, neither does it imply the increase ofdistances within the solar system and within our galaxy. Both atomic forces andgravitational interactions are in the solar system and in the galaxy to allow theexpansion of the atoms or the separation of the stars. In contrast, the distancesbetween galaxies and galaxy clusters does increase due to the increase of the cosmicdistance. The increase of the cosmic distances has accordingly a well defined physi-cal sense: cosmic distances increase measured e.g. in atomic radii or in astronomicalunits.

The above considerations are absolutely independent of which form gravitationaltheory the basis for the discussion. Einstein’s gravitational theory offers then thefollowing for cosmology. 1. It gives a differential equation for the computation ofthe function R(t) and 2. it gives information about the dynamics of cosmic massesand energies due to the expansion of space. The distribution of cosmic massesand energies must be homogeneous and isotropic in consequence of the previouslydiscussed homogeneity and isotropy. Masses and energies to be taken into accountare the following:

1. Those in average of stellar systems at rest (and their rest energy Mc2)

2. The thermal energy of cosmic masses I

3. The energy of the radiation fields US (electromagnetic fields, neutrino fields).These cosmic radiation fields must be homogeneous and isotropic too, i.e. theymust be such as can be treated as quantum gases (proton gases, neutrinogases).



The gravitational energy (including that of the gravitational radiation) must notbe taken into account in a relativistic gravitational theory as these phenomena areautomatically determined by the given geometry of space-time.

The important general characteristic of a metrics in the form (1) which is stillindependent of the actual form of the gravitational theory are the conformity of (1)with respect to the special relativistic Minkowski world and the non-existence ofa time-like Killing vector. The conformity means that the metrics can be broughtinto the form

ds2 = ϕ(τ, r)(dτ2 − dr2) (6)

by introducing a different time metrics — i.e. the time τ which does not correspondto world time t.

The line element (6) differs from Minkowski’s line element

ds2Min = dτ2 − dr2 (6a)

only by a conformity factorϕ(τ, r) . (6b)

This has the physical consequence that geometrical optics is the same in cosmicspace-time (Eq. 1) as in special relativistic theory and all known laws of geometricoptics are valid in the cosmos, however, the global light paths are different in thecosmos from those in Minkowski space, e.g. light follows closed paths in a sphericallyclosed space.

The non-existence of a time-like Killing vector is of special importance for themetrics (Eq. 1). This means especially that the tangential vector of the universe lineof the material, i.e. the vector of the world time δi

4 = ui does not fulfil a so-calledKilling equation. Namely

u(i;k) = u�∂�gik + ∂iu�gk� + ∂ku�gi� = −2RRδik i, k = 1, 2, 3 . (7)

There is only one conformal Killing vector of the time “type”.

R(t)ui = ξi , (8a)

which fulfils the equation

ξ(i;k) = 2Rgik =14ξ�;�gik . (8b)

Now, the existence of a Killing vector of the time-kind is necessary for theunrestricted validity of the law of conservation of energy. In mechanics, the energylaw is connected to time-independent (scleronomian) conditions. In the case oftime-dependent (rheonomian) conditions, the energy law in mechanics is no longervalid. Rheonomian conditions are interpreted in the mechanics of normal physics astransformations of the mechanical energy into other forms of energy. In cosmology,however, the space is time-dependent in the sense that the loss or production of



energy cannot be explained as the transformation of energy by adequate time-dependent geometry in the cosmos. Energy is rather absorbed or produced byspace-time.

In the general theory of relativity, the existence of the Killing vector (Eq. 8a) ofthe time-kind means still that the rest energy of masses stationary in the cosmos isconserved, while for all other energies (electromagnetic, radiative, thermal energy)the energy law is invalid (Heckmann and Schucking 1962, Treder 1968). Basicallythe time-dependent cosmos has a different dynamics to the stationary space-timepresupposed in normal physics.

If concrete laws of gravitation are substituted, then these general conclusionscan be made even more precise. The Einsteinian laws of gravitation

Rik − 12gikR + λgik = −χTik (χ = 8πf/c4) , (9)

— where λgik is the so-called cosmological term — are reduced for a homogeneousisotropic material

T ki =

⎛⎜⎜⎝

−p−p

−pu

⎞⎟⎟⎠ with p = p(t), u = u(t) (10)

into3R

R

c2= λ − 1

2χ(u + 3p) (10a)

and

u +3R

R(u + p) = 0 , (10b)

where Eq. (10b) determines the dynamics of cosmic material and Eq. (10a) definesthe dependence of spatial distances from cosmic time. (Equation (10a) is the so-called Friedmannian differential equation, which can be easily integrated if — aswas usual for a long time — the “cosmological constant” λ supposed simply to bezero).

The dynamical equation Eq. (10b) does not contain any statement about thecurvature radius of the world. The general dynamics is therefore independent ofthe character of the space, whether it is spherically or hyperbolically curved or isplanar. It is advantageous to split the energy density u and the pressure p

u = �c2 + uS + i, p = pS + π (11)

into rest energy density�c2 , (11a)

into radiation densityuS = 3pS (11b)



and into a term for the heat energy of mono-atomic gases

i =32π . (11c)

Thus the state equations are simultaneously given, too, and the dynamical equa-tion can be integrated supposing that no cosmologically significant exchange hasoccurred between the three different kinds of energy since a very early stage of thecosmos. The following equations are then obtained for the total energy (Tolman1934):

Mc2 ∼ �c2R3 = const, (12a)

RUS ∼ uSR4 = const, (12b)

R2T ∼ iR5 = const. (12c)

This means that the rest energy of the cosmos remains constant, as mentionedabove. In contrast, radiative energy and heat energy increase continuously with in-creasing R. Supposing that no quanta are produced or annihilated in cosmologicallysignificant quantities during this time, then it follows

RUS =∑

ν

NνhνR = const , (13a)

and from this, the universal red shift

ν ∼ R−1 , (13b)

as no interaction is possible between photons and the equation must be valid forphotons of arbitrary frequency ν.

In addition it follows from Eqs (12b) and (13b) that the frequency shift andthe decrease of the radiative energy are adiabatic processes which takes place withconstant entropy. If the radiation is black, i.e. the energy of the quantum is givenat a certain world time t = t0 by Planck’s radiation law, then it remains black foran arbitrary time t. — It follows for the radiative temperature TS and for the totaltemperature TG from the Planck-Wien law and from the gas law that

TS ∼ R−1 (14a)

TG ∼ R−2 . (14b)

In the present stage of the cosmos the rest energy in the stars and eventually ingas and dust clouds surpasses all other kinds of energy. Thermodynamic consider-ations show that this was not always the case, as the cosmological distances weresmaller earlier, than today. Together with the decrease of the cosmic distances boththe radiative and the heat energy increase and in a very dense stage of the cosmos,the (no longer distinguishable) heat and radiative energy will have surpassed therest energy by several orders of magnitude. If the cosmos is in its present stage apurely material cosmos, then in a densified stage it becomes an essentially radiation



cosmos. The stock of radiative energy does depend on its geometry, but its stockof heat energy does not.

The fact that cosmic distances have really been essentially smaller than theyare today, and that these distances are increase continuously, i.e. the cosmos isexpanding, follows from Friedmann’s differential equation. In the simple case ofa pure material cosmos (p = 0) of vanishing cosmological constant λ = 0 andof vanishing curvature ε = 0 the so-called Einstein-de Sitter model follows fromFriedmann’s differential equation:

R ∼ t2/3 . (15)

Similar connections follow with different functions R = R(t) with more generalpreconditions. All these functions imply that the universe is steadily expanding(R > 0) over long timescales (order of magnitude of some tens of thousand millionyears).

The time estimation is deduced from a combination of theoretical considerationswith observations. The expansion of the universe implies a radial movement of dis-tant galaxies relatively to the Earth with radial velocities increasing with increasingdistance from the Earth. Such a radial movement results in a fundamental shift ofall spectral lines towards the red according to the general formula

1 +Δλ

λ=

R(t2)R(t1)

(λ = wavelength) , (16)

from which follows approximately the Hubble-relation:

z =Δλ

λ=

R

RS =

−Δν

ν. (16a)

For a certain world time t = t0 R/R = 1/Θ is a constant. The value of thisconstant can be obtained from the observation of several not too distant galaxies.If the law of expansion is linear, i.e.

R = αt with α = const (17a)

would be valid, and together with it also

R = α, R/R = 1/t , (17b)

then the time distance at a time t = 0, when all cosmic distances have shrunk tozero, i.e. when R = 0, is given by

t = Θ . (18a)

In the case of a generally valid law of expansion one gets

t = AΘ (18b)

using a constant A depending on the law of expansion. If R(t) increases morequickly than linearly, then, in earlier epoch, the velocity of expansion R was higherthan today and therefore

A > 1 . (18c)



If the connection between R and t is weaker than linear (as it is in the Einstein-deSitter model), then the velocity of expansion was higher than it is today, and onegets

A < 1 . (18d)

The order of magnitude of the estimated value given earlier as 1010 years betweenthe time of minimum cosmic distances and the actual state of the world followsfrom these considerations and from the present value of the Hubble-constant basedon observations.

When computing back to densified states, it is to be taken into account that theprimeval state of the cosmos was an essentially radiative one, i.e. the Einstein-deSitter model was surely not valid some 1010 years ago. But even more general worldmodels with material and radiation (λ = 0) result in the consequence that all cosmicdistances were compressed to zero at a time of an order of magnitude given by Eq.(18), where the radiative and heat energy of the cosmos and also the density of therest energy were unlimited. A continuation of the theory beyond this densified stateis impossible. The world lines of cosmic particles (elementary particles and quanta)have realistic endpoints at the time t = R = 0. — We shall discuss these questionsin the following paragraph.

3. Initial and final conditions of the cosmos

Cosmological theory enables us to trace back the state of the cosmos througha time interval of several times ten thousand years on the basis of the gravitationequations and their connection with the equations of thermodynamics, of the elec-trodynamics and of the quantum physics. The present state of the cosmos, as faras it is accessible for observations, has to be deduced in the opposite sense fromappropriate assumptions about the initial state of the cosmos. The observationshave other problems, too, as discussed below. The arbitrariness of this suppositionis strongly limited by the presupposition of universal Copernicanism, i.e. by thehomogeneity and isotropy of the cosmos in all times.

As previously noted, the present ratio of different kinds of energy was not truein all times. In the primeval state thermal energy and radiative energy significantlysurpassed the rest energy. Moreover the temperature of the cosmos in the rimevalstate was so high that nuclear fusion processes occurred everywhere in cosmos,nowadays these occur only in stellar interiors.

A universal cosmic radiative energy was also discovered by radioastronomicalobservations some years ago in addition to the rest energy. This radiative energyhas a temperature of 3 ◦K and has been identified as a remnant of the primevalradiative energy of the cosmos. This idea is strongly supported by the fact thatthis radiation is black, i.e. it follows Planck’s law of radiation and appears to becompletely isotropic.

The idea is that the number of elementary particles and quanta did not essen-tially change since these primeval equilibriums. Therefore in later times they aregiven constants. The distribution of elementary particles in the cosmos would haveaccordingly a frozen-in equilibrium which corresponds to a cosmic temperature of



about T = 1013 ◦K which represent a world time t ≈ 10−5 s. This computationof the equilibrium corresponds to the fact that the unchanged number of quanta ofthe black primeval radiation surpasses the number of heavy elementary particles bya factor of the order of magnitude 109.

The nuclear reactions in primeval plasma led especially to the formation ofhelium and hydrogen. From this primeval material (103 to 106 years after theprimeval densification t = 0) started the development of the first stars and starsystems (stars of the elder, so-called extremal population II). The content of heliumincreased thereafter by “re-cooking” of the material in the stars and other heavierelements also came into being. The stars of the population I of which our Sun is anexample developed from this material.

It is important for all considerations and conclusions that the black radiation isa relic of the primeval structure of the cosmos and its present Planckian spectraldistribution was already conserved in a very early stage of the development of thecosmos. The recording of the quanta of the cosmic heat radiation leads us backdirectly in a very early stage of the cosmos. The realistic isotropy of this radiationproves that the cosmos was isotropic (and thus homogeneous) at this early stage.Models could be mathematically created in which this isotropy of the cosmos hasdeveloped during the process of cosmic expansion.

The isotropy of the black radiation also proves (this was mentioned e.g. byKorez (1966, 1968)) that the peculiar movement of the Earth (which is essentiallygiven by the movement of our Milky Way) relative to the cosmic rest system isvery small, as else the Doppler effect caused by this peculiar movement wouldsimulate an anisotropy of this black radiation. This insight is important, as it showsthat astronomical and geophysical time are synchronous with universal world timein a certain sense and therefore geological and astrophysical time determinationsapproximately express time intervals in world time t.

The image of the cosmos in the initial phase of the universe — from which ini-tial data for all cosmological, astrophysical and geophysical processes are deduced— is not without problems. An extremal densification of the cosmos results fromthis as a theoretically necessary conclusion only in the case when the cosmologicalconstant is supposed to be zero. This is true even in case when all other precondi-tions, especially the universal validity of the unmodified Einsteinian gravitationalequations are conserved. For a cosmological constant λ > 0 models are possible inwhich a densified state existed some ten thousands years in which, temperature andenergies did not have such extremely high levels as is presupposed in the model ofthe hot cosmos. Correspondingly it is a theoretically open question which of thetheoretically possible equilibrium temperatures was really occurred.

It is especially dubious whether temperatures were really so high since the pri-mary distribution of elements was determined by nuclear reactions in a primevalplasma. The model of the hot cosmos results in a primary helium content of morethan 30 percent, as Gamow and, later, Selmanow and Zeldovich et al. have shown.The primary helium content of the Sun which contains an already recooked material,would have to significantly higher than 30 percent.

However, experiments to determine the neutrino-radiation of the Sun show that



solar models are only acceptable in which the primary content of helium is less than30 percent (Bahkall et al. 1968). This conclusion is mainly of theoretical interest, aswide reaching cosmological conclusions are deduced from the state of space in theimmediate vicinity of the Earth. This conclusion is a prototype for the discussionin the following sections of the present paper.

It is at least of theoretical interest whether we have a closed or an open cosmos.The closed nature of the cosmos has consequences in quantum physics, as it pre-scribed a maximum wave length of the order of magnitude of the curvature of thecosmos, as Infeld remarked and would give a natural cut-off at the extremely lowfrequency part of the spectrum. The initial and boundary condition have, however,much greater importance which result from the fact that the cosmos is expanding.In a stationary cosmos a radiation equilibrium would have necessarily to developwhich would result in a bright night sky instead of the dark one and an averageirradiation to the unit surface corresponding to the surface density of the solar radi-ation. The expansion of the universe prevented the development of such a radiationequilibrium as it continuously rarifies the radiation. Moreover it diminishes thetotal energy of the radiation due to the Doppler effect.

The existence of “world horizons” is now introduced the particle horizons whichresults from the fact that the radiation of each star reaches the Earth, and “horizonsof events” which arise because expansion of the universe takes place so quickly thatthe velocities of distant sources relative to the Earth are greater than the velocity oflight, so that the light from these sources does not reach the Earth. Both horizonsrepresent limits of knowledge for cosmology.

Thus a horizon of events appears especially when law of expansion leads to acontinuously increasing velocity of expansion of the cosmos, i.e. if it is asymptoticallyvalid:

R ∼ tn with n > 1 (19a)

orR ∼ exp t . (19b)

The horizon of events is naturally bound to the observer and does not represent aboundary of the universe, but rather a boundary of the possible experience of anobserver. The horizon of events expands together with the expanding universe, sothat objects which were earlier in the horizon of events of the Earth remain in theirown horizon of events in spite of the radiation received by them becoming more redand its intensity weaker.

Such a horizon of events seems to coincide with a boundary condition of theelementary physical laws: Advanced, (i.e. those which depend on the future) so-lutions of the wave equations for the Planck-hypothesis of the natural radiation,and by the setting in advance of Sommerfelds condition of radiation respectively,are normally excluded from the physics. Both conditions mean that no incomingwaves are allowed, but only outgoing ones. These conditions are to be fulfilled bythemselves in the case of world models with horizons of events due to mathematicalconsistency, i.e. the hypothesis of natural radiation is, in models with horizons ofevents not an additional condition. but a necessary consequence of the dynamics of



the cosmos. Thus the structure of the cosmic world models directly influences thenormal physics (Gold 1968, Treder 1968b).

In addition to the great horizons related to the structure of the world modelsthere are also further horizons in the general relativistic theory of gravitation whichare connected to the Einsteinian geometrisation of the gravitation and which arespecial features of the Einsteinian theory. The normal space-time ideas fundamen-tally break down in the Einsteinian theory as mass and energy distributions do occurwhich have geometrical dimensions of the same or of smaller order of magnitudethan the length

a = (f/c2)M (f = Newtonian gravitational constant) (20)

corresponding to that deduced from their complete mass M , which is called theEinsteinian gravitational radius and is the Schwarzschild’s constant of the mass M .

The condition for the validity of normal space-time ideas and for a normal causalnexus can be expressed by an inequality:

r > 2a . (21)

This inequality may break down for very great masses and for very high densities.In such a case there are surfaces Σ in space-time which delineate a region which is inan anomalous causal connection with the surrounding world. It is even possible thatthe region enclosed by the surfaces Σ of the three-dimensional space has absolutelyno causal connection with the outside world. Such surfaces Σ then represent acertain kind of horizons which limits and even excludes the exchange of informationbetween the internal and external regions (Bergmann 1968b).

A kind of “causality anomaly” on the surface Σ appears when world lines of freelymoving elementary particles start or end on these (realistic or possible) surfaces,i.e. the particles assigned to these world lines spontaneously appear and disappearsimultaneously with the space filled by them. Thus there is a spontaneous creationor annihilation of particles and quanta which cannot be interpreted as the creationof particles in space-time, but as the emergence of space-time regions together withtheir material content.

There is a connection between the life-time and the invariant Δτ eigen-time Δtand the duration of the process in world time

Δτ =√

g44dt ≈√

1 − 2fm

c2rdt , (22)

due to Einsteinian gravitational dilatation of time which increases toward infinityin the vicinity of Schwarzschild’s limit. This process of emergence and annihilationgenerally needs an infinitely long time from the position of an observer — as judgedfrom the Earth. There are, however, types of computation of causality at which aninfinitely large Einsteinian dilatation arbitrarily expands the observer time.

It is remarkable that these computations of causality have to appear at the endof the history of a normal star, namely when the star is compressed over nuclear



density. There is an upper limit N0 of the number of heavy elementary particles(baryons) which can be contained in a compressed star. If the star contains a highernumber of baryons N > N0 at an early phase of its development then the surplusbaryons ΔN = N − N0 have to spontaneously disappear during the densification(gravitational collapse, Einstein and Oppenheimer 1939, Wheeler 1964).

The opposite case is also theoretically possible, too, when a cosmic object isexpanding and baryons are spontaneously produced the mass of the object increasesduring the expansion.

The primeval singularity of the cosmic models is basically no more else than aspecial form of this expansion at the moment of the singularity R = 0, i.e. all parti-cles and quanta of the universe come into being spontaneously and without cause atthe beginning of world time (t = 0) simultaneously with the space in which they are.Therefore it has been repeatedly proposed (Hoyle 1965, Jordan 1967) to dissolvethe cosmological primeval singularity into a large number of smaller processes ofthe spontaneous emergence of galactic nuclei or of stars either in form of explosivegalaxies or in form of supernovae and in the form of intergalactic material, respec-tively. (The Quasars are also related to the process of the spontaneous emergenceand subsequent expansion of new parts of cosmic space by several authors.)

These ideas lead far away from the usual conceptions of the physics of elementaryparticles as they involve the computation of both the macroscopic causality and ofthe conservation theorems of the number of elementary particles. They appear,however inevitably if the relativistic theory of gravitation (or Jordans’s or Hoyle’smodification) is accepted. The solution of this problem presupposes a well developedtheory of the connections between gravitational theory on the one hand, and onelementary particle physics on the other.

4. Cosmology and elementary particles

It is supposed that there are close connections between the global structure ofthe cosmos and the elementary laws of physics. Such connections may result fromthe fact, that the presupposed universal general validity of the elementary laws ne-cessitates a certain global structure of the cosmos, in which e.g. certain classes ofinitial and boundary conditions are only tolerated at the universal validity elemen-tary laws. (An example is the previously mentioned selection of world models withhorizon event by the hypothesis of the natural radiation.) The idea that the demandof the global validity, if the elementary laws of physics gives necessary and perhapseven sufficient conditions for the global structure of the cosmos, is the basis both forEinstein’s vision of a uniform geometrical physics (and Wheeler’s experiment of a re-alisation of this vision by the concept of a “geometrodynamics”) and of Eddington’sold attempts to synthesise relativistic and quantum theories in his “fundamentaltheory”. Eddington’s and Ertel’s experiments to deduce certain cosmological quan-tities and connections as thought necessities could find methodological justificationin that essential cosmological quantities can be logically developed.

A complementary line of thought has emerged recently, which postulates remarkthat the usual differential form of the fundamental equations of physics is insufficient



to describe all possible mutual interactions. For example, Heisenberg remarked thatthe differential elementary laws are supposedly only sufficient to describe the strongnear-interactions (i.e. nuclear forces). A complete theory of the far-acting weakerinteractions (electromagnetism, neutrino fields, gravitation) implies considerationof the cosmological initial and boundary conditions in addition to the differentialequations, where these initial and boundary conditions are contingent on the differ-ential equations. The complete physical laws are then a synthesis of the differentialequations and cosmic conditions.

A complementary concept would suppose a connection between physics of ele-mentary particles and cosmological quantities. Eddington, Ertel, Haas and Furthhinted at the existence of such a connection several decades ago.

Their considerations led to the so-called cosmic numbers. The connection be-tween the coupling constants of the strongest and the weakest known interactions,namely of nuclear forces and gravitation is:

g2/hc = α(fμ2/hc) (23)

(g = coupling constant of the nuclear forces, μ = mass of the baryons). The samenumber

α � 1040 (24)

appears in other empirical connections. The relationship between the characteris-tic length of the quantum theory of the gravitational field l (Planck’s elementarylength), of the physics of elementary particles (radius and Compton wavelength,respectively lc of a nucleon): lc = h/μc and the radius R of the principally visibleworld (as far as the horizons) is

α3/2l = αlc = R . (25)

Finally, the number N , the number of baryons, is according to cosmology in thevisible part of the cosmos:

N � α2 . (26)

Some of these relations can be deduced in other ways, as in discussed later. Thephysical significance of other connections is at present not yet clear.

Eddington, Ertel, Haak and Thurk (and more recently Weizsacker) supposedthat the relations 23–26 (and further analogous numerical equations) would neces-sarily result from experiments to merge quanta and gravitation theory and representrealistic constants. In contrast, Dirac and later Jordan suggested that the relation25 cannot have universal validity since the radius of the visible world continuouslyincreases due to the expansion of the cosmos. Accordingly the number α wouldnot be universal constant but a momentary numerical value of a function of worldtime t. Thus one has to assume dependence of some physical elementary laws andelementary quantities on world time according to Milne’s ideas. If the validity ofthe relations 23–26 in all times is not presupposed to be invariant with a numericalvalue depending on world time, then the Dirac-Jordan hypothesis follows with atime dependence of both the gravitational constant and the baryon number.



The value of the gravitational constant decreases correspondingly with worldtime:

f ∼ R−1 , (27)

while the baryon number increases:

N ∼ R2 . (28)

The development of these ideas results in Jordan’s (and later Dicke’s) gravita-tional theory with Eq. (11) defining the gravitational field, namely with components(Eq. 10) of the metric tensor gik introduced by Einstein as gravitational potentialand of the gravitational number χ = 8πf/c4. New baryons will emerge as presentedin the previous section for the general theory of relativity.

All these ideas supposedly contain a physical core and they could yield an ap-proximate qualitative description of the cosmological laws. The spontaneous emer-gence of baryons and the corresponding computation if the conservation law ofbaryon number is valid, represent a problem for the physics of elementary particlesas it is very difficult to understand how cosmological conditions influence stronginteractions with their extremal near effect.

The connections between the gravitational theory and elementary particle physicsin sense of Einstein’s and Heisenberg’s concepts of a unitary field theory are stillmuch discussed. These develop from Einstein’s remark that the general relativisticgravitational equation are only approximations in the frame of the general programof a unitary field theory. Einstein’s gravitational theory came into being at a timewhen knowledge about the elementary structure of matter was sparse. BasicallyEinstein’s theory starts from the phenomenology of the macroscopic material andthe only field explicitly introduced originally into the gravitational theory was theelectromagnetic field. Einstein formulated the basic principle of the gravitationaltheory requiring the geometrisation of the gravitational field (equivalence principle)for this material and for the electromagnetic fields as follows.

The influence of gravitation on all physical processes is obtained if the basicequations of physics (in “canonical” form) are written covariant against all trans-formations of the co-ordinates, and invariant against all Lorentz-rotations of thereference system.

This requirement makes it possible to understand the influence of gravitationon the phenomenological material and also on all tensorial fields (to which espe-cially belongs the electromagnetic field). It is impossible with this form of equiva-lence principle to understand the influence of the gravitation on the real elementaryparticles (baryons and leptons) as these particles are assigned to the spinor fieldsdiscovered by Dirac. If the equivalence principle should be extended to the spinorfields and with them to the elementary particles, then the equivalence principle isto be supplemented hy the following requirement (see Treder 1966b, 1967). Thefield equations of the spinor fields for elementary particles are invariant against co-ordinate transformations and covariant against Lorentz-rotations of the referencesystem. As all material can be built from spinor material. Einstein’s version followsfrom this form of the equivalence principle for the phenomenological material.



On the basis of our version of the equivalence principle the idea is developedthat the space-time structure is broader than supposed in the general relativistictheory. The structure of the space-time has not the usual Riemannian manyfold-ness, but a Riemannian manifoldness with a distinguishable (“core parallelised”)bundle of reference systems as investigated by Einstein in 1928–1931. The basicquantities of the gravitation field are correspondingly not the metric tensor gik, but16 components of the metric spin vectors from which gik can be obtained by themerging of Eq. (29):

gik =2∑

α,β=1

σiαβσαβk = −h1

i h1k − h2

i h2k − h3

i h3k + h4

i h4k . (29)

Instead of Einstein’s tensor equation of the metric field gik 4×4 vector equationshave to appear for the metric spin vector fields σβ

iα (or for the equivalent tetrad fieldhA

i , respectively). Thus the source of the gravity field cannot directly be T ki the

Einsteinian energy-impulse tensor, but is represented by 4 vectors. Such vectorscan be expressed by the merging of the energy-impulse tensor with the spin vectorand tetrad field, respectively, and they are e.g. (Treder 1967):

∼ χ∑

k

hAk (T k

i − 1/2δki T ) (A = 1, 1, 3, 4) . (30)

The basic difference with respect to the Newtonian and Einsteinian gravitationtheory is that the material is coupled “potential-like” with the gravitational field hA

i

and σαβi , respectively. The physical consequence is that the “effective gravitational

number” — which is a constant for Newton and Einstein, and a scalar field for Dickeand Jordan — is a quadratic 4×4 matrix and a function of the gravitational potentialfield itself. Correspondingly the effective field producing (active) gravitational massmeff is also a function of the gravitational potential. It follows that the effectivefield producing mass decreases with increasing local gravitational potential, andapproximately one gets:

meff ≈(

1 − fm

c2r

)m . (31)

The effective gravitational mass practically disappears when the dimensions of adense mass-rich body are such that the Schwarzschild radius becomes comparablewith the geometric radius:

meff → 0 r → a = fm/c2 . (31a)

Here, instead of gravitational collapse and anomalies of causality, which appear inthe general theory of relativity, the gravitational force is shielded by the dense mass

r < a .

If the same material were loosely packed, then its effective gravitational mass wouldbe heavier and would correspond to its Newtonian mass m.



The gravitational collapse of a star does not lead therefore to a causal anomalyand to disappearance of baryons from the cosmos, but rather the collapse is retardedby the commencement of this shielding effect. At the end of the collapse an effectiveactive gravitational mass meff results which is of the same order of magnitude asthe mass corresponding to the general theory of relativity. The baryon number,however, remains the same in contrast to the general theory of relativity and toJordan’s theory, but the surplus baryons do not contribute to the gravitational fieldof the densified star due to the shielding of their gravitational field.

The effective gravitational mass of a dense body continuously increases due toits expansion, without the entrance of new baryons into the cosmos. The maximum(Newtonian) mass is reached when the shielding effect disappears so that the grav-itational fields of single baryons are simply added. (Similar situations result fromthe cosmological primeval densification. This densification does not get singular andthere was a contraction phase of the cosmos before the expansion. The contractionbegan at the time t = −∞.)

The consequences of our theory of the gravitation and the hypothesis of thespontaneous emergence of mass are very similar. In both theories the productionand the annihilation on of mass can be dynamically detected, at the collapse andthe anti-collapse phases, respectively. This process is related to a disappearanceand an emergence of baryons in the general theory of relativity (and to an evengreater extent Jordan’s theory), while in our gravitational theory the gravitationaleffects of the baryons with a pre-defined constant number N does change.

The analogies and differences of the three theories, namely those of Einstein,Jordan and of ourselves is evident in the interpretation of the Mach-Einstein prin-ciple. Eddington commenting on Einstein’s and Mach’s proposed connection of theinertia of an arbitrary body with the global gravitational field of the cosmos noted,that it is valid for the visible universe at least to an order of magnitude:

mc2 = fmM/R , (32)

where m is the inertial mass of an arbitrary body and M is the total mass of thevisible cosmos, R the radius of the visible universe. Equation (32) tells us that thetotal energy of each body is equal in order of magnitude to its potential gravitationalenergy related to the cosmos. (This connection is closely related to Eddington’snumbers.) As, however, R is continuously increasing due to the expansion of thecosmos, the Eq. (32) only remains valid at constant gravitational number f , if themass of the visible world is also continuously increasing. In the Einsteinian theoryit would be required that

M ∼ R, N ∼ R . (33a)

If it is assumed with Dirac and Jordan, that the gravitational constant decreaseswith increasing world radius, then the Jordan-relation

M ∼ R2, N ∼ R2 (33b)

should be valid. Our theory gives, however, an increase of the effective gravita-tional number with increasing world radius corresponding to the thinning of the



cosmic material and of the absorption if the gravitation, decreasing together withthe density. One gets thus the connection

feff ∼ R (34)

and the relation 32 with constant N is valid at all times.

5. Geophysical contributions to cosmology

We found in the previous sections that theoretical cosmology can lead to verydifferent models of the cosmos even when considering all possible astrophysical ob-servations. These models have some common basic features. They have a commonbasis, as they start all from Einsteins geometric gravity theory, which is then furtherdeveloped and modified. Einstein’s theory, however, is compatible with many worldmodels taking into account available astrophysical observations. The questionsabout the physical significance of the cosmological constants and of the curvatureradius of the world remain open to discussion (see Schroder and Treder 2003).

There are several theoretical reasons of this situation. The first cause is thatastronomical observations foes not represent the state of the universe at a certainworld time t = const (as Milne expressed it, no “map of the world”), but thecosmic events observed by us order in time backwards, the farther away they occur,due to the finite propagation velocity of light (Milne’s “snapshot of the world”).According to these, theoretical assumptions are needed to separate distance effectsand development effects.

All kinds of astrophysical information about the cosmos are obtained by the re-ception of electromagnetic waves. We obtain accordingly only information which di-rectly or indirectly modulate electromagnetic waves. Information about the overallstructure of the cosmos is thus influenced by the laws of propagation of electromag-netic waves. These laws do not fix unambiguously the structure of space-time, asthe laws of the propagation of electromagnetic waves are invariant against conformaltransformations of the metric of space-time. Therefore all conformally transformedworld models with metrics ds2 are basically possible descriptions of the universetogether with a world model of metrics dσ2 = ϕ2ds2 if no additional theoreticalconclusions are added.

Moreover there is a possibility emphasised by Milne that some basic laws andfundamental constants relative to other laws and constants respectively change dur-ing the world time t. An example is the discussion about the constant or changingvalue of the gravitational number. Milne proposed among others that Planck’sconstant h is also changing. Houtermans and Jordan supposed that the thermalcoupling constant of the weak interactions depends on the age of the world, approx-imately with ∼ t−1/2. The dependence on world time t of the velocity of light in avacuum (in an inertial system) was also considered.

Since astrophysical data yield only information which is transmitted to us viaelectromagnetic waves, it is necessary to modify the laws of transmission and inter-action of the light in order to include different ideas about the temporal change oflaws and constants into cosmology. Such transformations of the models may hint at



a realistic difference of the physical content of the theory, they need, however, a dif-ferent mathematical language to express the same physical content. All invariantsof this transformation are now integers emphasised by Ertel and Eddington, i.e. allphysical and cosmological values are dimensionless integers and they have the samevalue in all isomorphic images of the world. This results from the non-continuousnature and from the steadiness of transformations in the same manner as Planckconcluded for the Lorentz invariance of the entropy from the number of moleculesin each state. It is a pity that these integers are hardly within reach of observations.

There is a second source of information about past states of the cosmos inaddition to astrophysical observations. Geophysics (and more generally the cosmicphysics of the environment of the Earth) yields us information on the state of anobject over more than five thousand million years. Further geophysics yields usinformation not only on the history of the electromagnetic interactions, but alsoof all interactions, including nuclear forces and gravitation. Nuclear physical andchemical methods in geology should be mentioned here as an expression of thehistory of both strong and weak interactions.

The Earth and its environment record several thousand million years of “self-experienced history” The interpretation of singular observations can be very com-plex, but, these observations are much more informative and more direct thanastrophysical information with its space-time retardation and with the necessarytransformation of the modulation of electromagnetic waves. Cosmological theo-ries and hypotheses should rather be controlled by geophysical and similar criteria(meteor physics) than by purely astrophysical observation.

6. Geophysics and cosmology

A decision about the time dependence of certain universal constants of naturewhich arise in the theories of Milne, Dirac, Jordan, Dicke and others may well beverifiable by geophysics. It is even possible that the supposed dependence of thegravitational number on space and time can be detected by gravimetric measure-ments.

Finally we refer to some earlier results of geophysical cosmology. Nernst con-cluded as early as in 1921 by combination of geophysical results with the then knownfacts on radioactive decay that the present age of the universe has to lie between109 and 1012 years and he remarked that the realistic age is clearly nearer to 109

than to 1012 years. Nernst’s conclusions were at that time rigorously disputed byastronomers. It turned out, however out that Nernst’s estimation is fully correct.

Some decades ago it was supposed that the age of the cosmos is less than twothousand million years in consequence of an erroneous determination of Hubble’sconstant 1/Θ. This estimation was in contradiction with the geophysical age de-terminations of the Earth. The logically correct consequence, namely that theastronomical age determination must be false was not reached at that time untilBaade showed that the determination of Hubbles constant contained a systematicerror. The present estimation of the age of the world of about ten thousand millionyears is compatible with geophysics.



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Geophysics and cosmology — A historical review