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If a cosmological term is included in the equations of general relativity, the linearized equations can be interpreted as a tensor-scalar theory of finite-range gravitation. The scalar field cannot be transformed away be a gauge transformation (general co-ordinate transformation) and so must be interpreted as a physically significant degree of freedom. The hypothesis that a massive spin-two meson (mass m 2 ) satisfied equations identical in form to the equations of general relativity leads to the prediction of a massive spin-zero meson (mass m 0 ), the ratio of masses being m 0 / m 2 = √3.

Clusters of galaxies are approximated by the Schwarzschild interior solution (with non-zero cosmological constant) embedded in a Robertson/Walker background. The conditions that the two metrics join up smoothly and that the cluster be stable imply either (a)k=−1 with Λ lying in the range −1.1×10−27≲Λ≲1.5×10−36 (s−2), or (b)k=0. Also, superclustering on a scale larger than 0.5 Mpc is unacceptable unless Λ

A cosmological substratum for energy propagation is defined in terms of a hypothesis by McCrea. It has been shown that the assumption of such a substratum for a uniformly expanding universe provides a cosmological interpretation of Special Relativity, and leads further to a theory of gravitation in terms of a universal acceleration field. Following a critical discussion of the bases of General Relativity, it is suggested that the proposed substratum model and its consequences are also compatible with the General Relativistic approach, providing that this is applied in a manner which recognises the centrally directed character of gravitational fields, and hence employs harmonic coordinates as proposed by Fock. It is shown that Fock's procedure leads to results which are consistent with the assumption of a uniformly expanding cosmological substratum. Finally, it is suggested that the cosmological substratum concept is also implied by the formulation of the Robertson-Walker metric.

The tetrad field equations of general relativity discussed in previous articles are applied to Robertson-Walker cosmological models. A generalized Friedmann equation is derived and some of its consequences are discussed.

An attempt is made to give a physi-cal basis for our modification of Einstein's gravita­ tional equations, which has been proposed in a previous paper to arrive at an isotropic model-universe that may bounce in the hadron era, tends rapidly to the Friedmann universe and remains regular with respect to the excitation of gravitational and rotational waves. The basic idea lies in reinterpreting the modified field equations as those for the mean-value field of a highly turbulent gravito-hydrodynamic field (obeying Einstein's equations) whose substratum is dominated by the hadronic matter, with large density fluctuation. In the case where the mean-value metric represents the regular model-universe, it is shown that, so far as the bounce epoch of the universe. is concerned, three characteristic mean-value quantities specifying metric, velocity and density fluctuations are compatible with the basic idea. Sev­ eral problems to be pursued further are also touched upon. § I. Introduction In a previous paper1> (which is referred to as [IJ hereafter), a phenomenol­ ogical modification -of Einstein's gravitational equations has been proposed, in order to arrive at a regular isotropic model-universe which asymptotically tends to the Friedmann universe and remains regular with respect to the excitation of gravitational and rotational waves. It has been shown that the substratum at its most compressed stage must be dominated by hot hadronic matter at an extremely

The Large Numbers hypothesis asserts that all the large dimensionless numbers occurring in Nature are connected with the present epoch, expressed in atomic units, and thus vary with time. It requires that the gravitational constant G shall vary, and also that there shall be continuous creation of matter. The consistent following out of the hypothesis leads to the possibility of only two cosmological models. One of them, which occurs if one assumes that the continuous creation is a multiplication of existing matter, is Einstein’s cylindrical closed Universe. The other, which occurs if one assumes the continuous creation takes place uniformly through the whole of space, involves an approximately flat Minkowski space with a point of origin where the Big Bang occurred.

In preparation for an investigation of whether field-theoretic effects helped to make the early universe become isotropic, we seek to determine the physical (divergence-free) energy-momentum tensor through which the geometry of spacetime is influenced by a quantized scalar field with conformal ("new improved") coupling to the metric. The cosmological models studied are the Kasner-like (type I) metrics (homogeneous, spatially flat, nonrotating, but anisotropic), and also the isotropic Robertson-Walker metrics. The methods employed have previously been expounded in the context of a minimally coupled scalar field and a Robertson-Walker metric. Three divergent leading terms are extracted from an adiabatic expansion of the formal expressions for the expectation values of the energy density and pressures. In the Kasner case a slight reshuffling of the leading terms in the energy density displays all divergences to be proportional to either the metric tensor or a second-order curvature tensor which vanishes when the spacetime is isotropic; hence a finite energy-momentum tensor remains after renormalization of the cosmological constant and one other coupling constant in a generalized Einstein equation. In the Robertson-Walker cases, because of conformal flatness, there is no divergence beyond the usual quartically divergent constant vacuum energy; when the mass is not zero, however, a finite renormalization of the gravitational constant is suggested. The correctness of the methods is tested by considering a coordinate system in which flat spacetime assumes the form of a Kasner universe: The adiabatic definition of particle number and vacuum, which is basic to our expansion and renormalization methods, is seen to be consistent with the usual flat-space concepts.

In the projective theory of relativity the 5-dimensional field equation $$_{\mu \nu } $$ and the resulting equation of motion Tμυ;μ = 0 are investigated. There Tμυ stands for the 5-dimensional tensor of macroscopic matter. The 4-dimensional field equations and equation of motion obtained by projection are a generalization of Einstein's theory of general relativity and Maxwell's electrodynamics, involving a scalar field φ.They contain a single constant φ0.The weak field approximation is investigated for the case of an ideal fluid and leads to Newton's mechanics, including Newton's gravitational law, and to Maxwell's electrodynamics. For the constant φ0 one obtains the approximate value φ0c4/γN with Newton's gravitational constant γN.For homogeneous and isotropic cosmological models consisting of matter only the general solution for the radius K of curvature is given. This solution is independent of the equation of state of matter For a pure dust universe the general solution for the scalar field φ is given. For a closed universe a power law φ ∿K−1 is valid which leads to Mach's principle. The calculation of the age of a closed universe yields over 7×109y,if one uses mean values of the present cosmological data.

An autonomous system of equations, describing uniform cosmological models, is formulated by using the perfect fluid approximation of Einstein’s equations. These equations contain an arbitrary function related to the matter content of the universe, which may include negative energy fields. This function, designated α , is assumed to depend on the density and expansion rate of the universe only. Geometrical methods of analysis are used to study the behaviour of all models described by this system. The analysis shows that there are only three possible modes of behaviour that can be exhibited by a uniform universe. Examples of the first two classes are well known in the ‘big-bang’ and ‘steady-state’ theories. However, it is shown that the familiar theories are not unique, but an infinite number of both such types of model exist for various α . It is also shown that all steady-state models in an expanding universe are stable. The third class of model, associated with periodic behaviour, is of two types. The first is demonstrated by a universe which oscillates between expansion and contraction but never achieves infinite density. The second consists of ever-expanding (or contracting) models in which the density and expansion rate oscillate between finite values. These latter models possess evolutionary characteristics on ‘short’ time-scales, while satisfying the ‘perfect cosmological principle’ in the large, and only arise in the presence of gross non-linearities introduced by the function α . Both the periodic and steady-state classes occur only in the case of negative energy fields.

The anomalies in relativistic theory resulting from the infinities present in Riemannian geometry as used by Einstein are discussed and a better solution proposed by introducing a discrete and indivisible time unit. In considering the approach of non-simultaneity, as recently published in this Journal (Thyssen-Bornemisza andGrayson [11]), all infinities are assumed eliminated in our universe. To fully develop the geodesic of a four-dimensional universe the perspective of discreteness seems a basic requirement.

We show the inconsistency of some reserves of which the unitarian Einstein theory, till now, was the object. Hence, by two quite different relativistic methods, we obtain almost perfectly coincident results for the age of the Universe. At last we prove that the beginning of the expansion is not a repeatible event.

It is shown that every type {22} vacuum solution of Einstein's equations admits a quadratic first integral of the null geodesic equations (conformal Killing tensor of valence 2), which is independent of the metric and of any Killing vectors arising from symmetries. In particular, the charged Kerr solution (with or without cosmological constant) is shown to admit a Killing tensor of valence 2. The Killing tensor, together with the metric and the two Killing vectors, provides a method of explicitly integrating the geodesics of the (charged) Kerr solution, thus shedding some light on a result due to Carter.

The exact solution is sought for the cosmological equations of Brans and Dicke's scalar-tensor theory when a power law exists between the gravitational constant and the radius of curvature of the universe. For the space of negative curvature no solution is possible. On the contrary for a closed space the gravitational constant and the radius of curvature increase linearly with respect to the age of the universe. The parameter ω of the scalar-tensor theory is necessarily negative and can be determined by the present values of the mass-density of the universe, the Hubble-constant and the gravitational constant. The solution has no analogy in Einstein's theory with vanishing cosmological constant, even when the deviations from Einstein's values of the solar relativistic effects are small.

The exact cosmological vacuum solutions of Brans and Dicke's scalas-tensor theory are derived when a power law is valid between the gravitational constant κ and the radius of curvatureR of the universe. There exist even in the case of the closed 3-dimensional space of positive curvature solutions with increasingR and κ with respect to the age of the universe, whereby the freely available parameter ω of the scalar-tensor theory can take all values greater than −3/2. Such solutions are contrary to Dirac's hypothesis as well as to Einstein-Mach's principle.

It is pointed out in the first place: (1) in Birkhoff's gravitational theory based on "flat" space-time, the "red shift" is accounted for by the energy change of the photon as it travels from the emitting body, whereas the photon plays no especial role in the Einstein theory; (2) the solution of the problem of two or more bodies is feasible in the new theory because of its simpler character. Four comments of H. Weyl concerning the Birkhoff theory are discussed, and it is concluded that these are to be taken with much reserve. In regard to the third of these comments it is pointed out that the "perfect fluid" used by Birkhoff as the ultimate carrier of mass and electric charge is to be characterized as the simplest fluid with disturbance velocity that of light ($c$). It is affirmed to be a glaring defect of earlier relativistic theories that the disturbance velocity in matter has been taken as arbitrary, although that of gravitation and of the electromagnetic field have been equal to $c$. The differential equations of the theory are then set up. An additional cosmological term in the gravitational potentials ${h}_{\mathrm{ij}}$ is suggested, namely, ${h}_{\mathrm{ij}}^{*}=(\frac{K}{8})({t}^{2}\ensuremath{-}{x}^{2}\ensuremath{-}{y}^{2}\ensuremath{-}{z}^{2}){g}_{\mathrm{ij}},$ where $x$, $y$, $z$, $t$ are Lorentz coordinates and $K$ is the (small) cosmological constant. The explicit formula for the rate of advance of periastron $P$ of two bodies (mass points) of masses $m$ and ${m}_{2}$ is given, as obtained from the solution of the two-body problem in the theory, and its possible application to double stars is referred to. The authors propose to give a detailed development of the theory and its applications in papers to be published shortly elsewhere.

Modern cosmological theories imply that the universe is filled with a shallow degenerate Fermi sea of neutrinos. In the steady state and oscillating models (and perhaps also the "big bang" theories) it can be shown rigorously that the proportion of filled neutrino levels (plus the proportion of filled antineutrino levels) is precisely one up to a finite Fermi energy ${E}_{F}$. The proof takes into account both absorption and the repressive effects of already filled levels on neutrino emission. Experiment shows that ${E}_{F}\ensuremath{\le}200$ eV for antineutrinos and ${E}_{F}\ensuremath{\le}1000$ eV for neutrinos. The degenerate neutrinos could be observed (if ${E}_{F}g10$ eV) by looking for apparent violations of energy conservation in ${\ensuremath{\beta}}^{\ensuremath{-}}$ decay. In the steady state and evolutionary cosmologies ${E}_{F}$ is much too low to ever be observed, but in the oscillating cosmologies ${E}_{F}\ensuremath{\simeq}5{R}_{c}$ MeV, where ${R}_{c}$ is the minimum radius of the universe in units of its present radius; thus experiment already shows that the universe will contract by a factor over ${10}^{3}$, if at all. Astronomical evidence plus Einstein's field equation (without cosmological constant) require in an oscillating cosmology that ${E}_{F}l2\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}3}$ eV (so ${R}_{c}l{10}^{\ensuremath{-}9}$) and suggest that higher energy neutrinos may represent the bulk of the energy of the universe. A model universe incorporating this idea is constructed.

Idealizations in cosmology with special theory of relativity - partitioned universe, microcosmic model, and similarity to Newtonian cosmology

The application of the general theory of relativity to the cosmological plane has led to a significant change in traditional notions on the space-time structure of the universe. This has been expressed in a new formulation and solution of cosmological problems such as that of a single world space and time, the problem of selection of a cosmological model for description of the universe, and the problem of the infinity of the universe.

Since a homogeneous isotropic universe can be embedded in a flat space of five dimensions, the question is considered under what conditions a more general universe can be embedded in a five-dimensional flat space. On the assumption that the deviation from homogeneity is small, it is found that real inhomogeneities can occur only in the case of a universe filled with radiation, or a universe containing at least two different substances with different equations of state, as for example radiation and matter. In the case of a radiation-filled universe, the inhomogeneities can be of arbitrary size and can conceivably be the precursors of galaxies.

The solution of the cosmological equations in isotropic and homogeneous closed-Universe models containing simultaneously matter and radiation, having in general the cosmological constantΛ≠0 has been obtained. The limiting cases (matter dominating and radiation dominating) have been studied also numerically. The general case is intermediate between the limiting cases. Some of the results obtained in those limiting cases are:a) The possibility of having a closed Universe already with the presently known matter density.b) The conclusion thatΛ must be positive if the age of the Universe exceeds a calculated minimum value.c) Reasonably accurate determination of the value ofΛ if the age of the Universe exceeds a minimum value.

It is shown that a cosmological model of light propagation associated with a uniformly expanding universe provides a physical significance to the hyperbolic velocity space by which this model can be described. Some implications of this result are discussed.

It is argued that Einstein's Theory of General Relativity as it stands incorporates Mach's Principle. The boundary conditions for Machian solutions are stated in a coordinate system in which the cosmological background is described by a conformally flat metric. The metric tensor g μν is then written as a product of the scalar density ϕ 2 and a tensor density γ μν with unit determinant. In the coordinate system that has been so chosen ϕ describes the cosmological structure, while γ μν refers to gravitational phenomena. This becomes clear when Einstein's fundamental equations are rewritten in terms of ϕ and γ μν . Then κϕ −1 is seen to play the role of the gravitational constant instead of κ in the weak field approximation. The quantity κϕ −1 can be expressed in terms of the radius and the total mass of the universe and the sign of the forces between inhomogeneities of the metric is determined by the requirements of Mach's principle. The forces which derive from ϕ are found to be repulsive for the cosmological background, leading to the expansion of the universe, while attractive gravitational forces arise from the deviations of γ μν from the Minkowski metric. Various statements associated with Mach's Principle are discussed in the light of this reformulation of Einstein's Theory.

Exact expressions are given for the properties of the general expanding Friedmann model universe with non-zero cosmological constant and containing non-interacting matter and radiation.

An attempt is made to develop a new theory of cosmology based on a scalar field, in view of several weak points proper to the general relativistic cosmology. In our treatment,. a kind of correspondence principle plays the important role, together with the consideration of the scope of applicability of general relativity. Our cosmological field equation thus proposed not only has an intimate linkage with Einstein's equations, but also satisfies Mach's relation for the universe as a whole in a covariant and intrinsic manner.

The classical dynamics of a test particle in a homogeneous closed universe is investigated. The canonical method is used to obtain dynamical quantities unambiguously. The picture of the motion is described in the « chronometric » three-space. The Newtonian inertial mass is decreasing with the expanding universe.

In recent years observational techniques at cosmological distances have been improved so that cosmology has become an empirical science, rather than a field for unchecked speculation. There remains the fact that its object, the whole universe, exists only once; hence we are unable to separate ‘general’ features from particular aspects of ‘our’ universe. This might not be a serious drawback if we were justified in the belief that presently accepted laws of nature remain valid on the cosmological scale. In the author’s opinion, however, there are grounds for doubting that belief. The three arguments presented are (1) the possibility that apparent constants of nature may during cosmological times turn out to vary (Dirac); (2) the effective break-down of the principle of relativity caused by the effects of the cosmological environment on local experiments; and (3) the fact that present theory leads to field singularities at the early stages of the expanding universe, which might be a signal that currently accepted theoretical concepts are inadequate for an understanding of highly condensed matter.

In this paper, we compare the magnitude-redshift data for brightest cluster galaxies with cosmological models of general relativity, including non-zero cosmological constant. Our study is parallel to that by Peach (1970). We use the same data but an independent method, and obtain somewhat different results. We also consider possible effects of evolution.