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The anomalous acceleration of Pioneer 10 and 11 spacecraft of (8.74 \pm 1.33) \times 10^{-8} cm. s^{-2} fits with a theoretical prediction of a minimal acceleration in nature of about 7.61 \times 10^{-8} cm. s^{-2}, Comment: 3 pages, accepted in Int. J. Theor. Phys

In cosmology one labels the time t since the Big Bang in terms of the redshift of light emitted at t, as we see it now. In this Note we derive a formula that relates t to z which is valid for all redshifts. One can go back in time as far as one wishes, but not to the Big Bang at which the redshift tends to infinity., Comment: 3 pages, 3 figures

The issue of the cosmological constant is discussed in details and a solution to the problem is suggested., Comment: 13 pages in LaTeX with 3 figures in eps files, paper presented at the Fifth Friedmann Seminar; the cls file necessary for successful PostScript generation is also attached

It is shown that the cosmological relativity theory predicts the value Λ = 1.934 × 10−35s−2 for the cosmological constant. This value of Λ is in excellent agreement with the measurements recently obtained by the High-Z Supernova Team and the Supernova Cosmology Project.

Using cosmological relativity theory, we derive the formula for the cosmological redshift written explicitly in terms of 1 − Ω, where Ω = ρ/ρc is the ratio of the average mass density to the critical “closure” density. Based on the present day data of observed redshifts, we conclude that Ω < 1, which means the universe is infinite and curved, and expands forever.

A new general-relativistic theory of cosmology, the dynamical variables of whichare those of Hubble's, namely distances and redshifts, is presented. The theorydescribes the universe as having a three-phase evolution with a deceleratingexpansion followed by a constant and an accelerating expansion, and it predictsthat the universe is now in the latter phase. The theory is actually a generalizationof Hubble's law taking gravity into account by means of Einstein's theory ofgeneral relativity. The equations obtained for the universe expansion are elegantand very simple. It is shown, assuming Ω0 = 0.24, that the time at which theuniverse goes over from a decelerating to an accelerating expansion, i.e., theconstant expansion phase, occurs at 0.03 τ from the big bang, where τ is theHubble time in vacuum. Also, at that time the cosmic radiation temperature was11 K. Recent observations of distant supernovae imply, in defiance of expectations,that the universe's growth is accelerating, contrary to what has always beenassumed, that the expansion is slowing down due to gravity. Our theory confirmsthese recent experimental results by showing that the universe now is definitelyin a stage of accelerating expansion.

For disk galaxies the fourth power of the circular velocity ν4 c of stars around thecore of the galaxy is proportional to the luminosity L, ν4 c ∝ L (Tully—Fisher law).Since L is proportional to the mass M of the galaxy, it follows that ν4 c ∝ M.Newtonian mechanics, however, yields ν2 c = GM/r for a circular motion. In orderto rectify this big difference, astronomers assume the existence of dark matter.We derive the equation of motion of a star moving in the central field of a galaxyand show that, for a circular motion, it yields a term of the form ν4 c ∝ GMc/τ,where G is Newton's gravitational constant, c is the speed of light, and τ is theHubble time. This puts in doubt the existence of halo dark matter for galaxies.

An amended formula for the decay of radioactivematerial is presented. It is a modification of thestandard exponential formula. The new formula appliesfor long cosmic times comparable to the Hubble time. It reduces to the standard formula for shorttimes. It is shown that the material decays faster thanexpected. The application of the new formula to directmeasurements of the age of the universe and its implications are briefly discussed.

In this paper we review cosmological relativity,a new special theory of relativity that was recentlydeveloped for cosmology, and discuss in detail some ofits aspects. We recall that in this theory it is assumed that gravitation is negligible.Under this assumption, the receding velocities ofgalaxies and the distances between them in the Hubbleexpansion are united into a four-dimensionalpseudo-Euclidean manifold, similarly to space and time inordinary special relativity. The Hubble law is assumedand is written in an invariant way that enables one toderive a four-dimensional transformation which issimilar to the Lorentz transformation. The parameter inthe new transformation is the ratio between the cosmictime to the Hubble time (in which the cosmic time ismeasured backward with respect to the present time). Accordingly, the new transformationrelates physical quantities at different cosmic times inthe limit of weak or negligible gravitation. Thetransformation is then applied to the problem of the expansion of the universe at the very earlystage when gravity was negligible and thus thetransformation is applicable. We calculate the ratio ofthe volumes of the universe at two different timesT1 and T2 after the big bang. Under theassumptions that T2 – T1≈ 10-32 sec and T2 ≪ 1 sec,we find that V2/V1 =10-16/√T1. For T1≈ 10-132 sec we obtainV2/V1 ≈ 1050. Thisresult conforms with the standard inflationary universe theory, but now it isobtained without assuming that the universe is propelledby antigravity. New applications of the theory arepresented. This includes a new law for the decay of radioactive materials that was recentlydeveloped by Carmeli and Malin. The new law is amodification of the standard exponential formula whencosmic times are considered instead of the ordinarylocal times. We also show that there is no need to assumethe existence of galaxy dark matter; the Tully-Fisherlaw is derived from our theory. A significant extensionof the theory to cosmology that was recently made by Krori, Pathak, Das, and Purkayastha isgiven. In this way cosmological relativity becomes ageneral theory of relativity in seven dimensions ofcurved space-time-velocity. The solutions of the field equations in seven dimensions obtained by Kroriet al. are given and compared to those of the standardFriedmann-Robertson-Walker result. A completely newpicture of the expanding universe is thus obtained and compared to the FRW one.

A new approach to quantize the gavitationalfield is presented. It is based on the observation thatthe quantum character of matter becomes more significantas one gets closer to the big bang. As the metric loses its meaning, it makes sense to considerSchrodinger's three generic types of manifolds —unconnected differentiable, affinely connected, andmetrically connected — as a temporal sequencefollowing the big bang. Hence one should quantize thegravitational field on general differentiable manifoldsor on affinely connected manifolds. The SL(2,C) gaugetheory of gravitation is employed to explore thispossibility. Within this framework, the quantization itselfmay well be canonical.

We describe the motion of a particle in acentral field in an expanding universe. Use is made ofa double expansion in 1/c and 1/τ, where c and τare the speed of light and the Hubble time. In thelowest approximation the rotational velocity is shownto satisfy v4 = 2/3 GMcH0, whereG is Newton's gravitational constant, M is the mass ofthe central body (galaxy), and H0 is theHubble constant. This formula satisfies observations of stars moving inspiral and elliptical galaxies, and is in accordancewith the familiar Tully–Fisher law.

In the limit of negligible gravity, a transformation that relates physical quantities at different cosmic times, similar to the Lorentz transformation which relates measurments at different velocities, is derived.

Under the assumption that Hubble's constant H0 is constant in cosmic time, there is an analogy between the equation of propagation of light and that of expansion of the universe. Using this analogy, and assuming that the laws of physics are the same at all cosmic times, a new special relativity, a cosmological relativity, is developed. As a result, a transformation is obtained that relates physical quantities at different cosmic times. In a one-dimensional motion, the new transformation is given by $$x' = \frac{{x - Tv}}{{(1 - T^2 /T_0^2 )^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} }},v' = \frac{{v - xT/T_0^2 }}{{(1 - T^2 /T_0^2 )^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} }}$$ where x and v are the coordinate and velocity, T is the cosmic time measured backward with respect to our present time T=0, tand T0 is Hubble's time. Some consequences of this transformation are given, and its applicability limitation is pointed out.

The Einstein universe is a simple model describing a static cosmological spacetime, having a constant radius and a constant curvature, and, as is well known, it does not describe our universe. We propose a model which is an extension of Einstein's. Our metric, havingR × S3 topology, describes a nonisotropic homogeneous closed (finite) universe of Bianchi type IX. This metric is similar to that of Taub, but is simpler. Unlike the Taub solution (which is a cosmological extension of the NUT solution), however, the universe described by our metric contains matter. Like the Taub metric, our metric has two positive constants (τ, T). The gravitational red shift calculated from our metric is given. Similarly to the Schwarzschild metric, which has a “singularity” atr = 2m, this metric has the same kind of “singularity” att = 2τ. The maximal extension of the coordinates in our metric is fairly analogous to that of the Schwarzschild metric.

A brief description of the ordinary field theory, from the variational and Noether's theorem point of view, is outlined. A discussion is then given of the field equations of Klein-Gordon, Schrodinger, Dirac, Weyl, and Maxwell in their ordinary form on the Minkowskian space-time manifold as well as on the topological space-time manifold R × S3 as they were formulated by Carmeli and Malin, including the latter's most general solutions. We then formulate the general variational principle in the R × S3 topological space, from which we derive the field equations in this space.

A five-dimensional cosmological theory of gravitation that unifies space, time and velocity is presented, 6 pages in LaTex, presented at Dark Matter 2002