Cosmic Time Transformations in Cosmological Relativity

Abstract

The relativity of cosmic time is developed within the framework of Cosmological Relativity in five dimensions of space, time and velocity. A general linearized metric element is defined to have the form $ds^2 = (1+ϕ) c^2 dt^2 - dr^2 + (1+ψ) τ^2 dv^2$, where the coordinates are time $t$, radial distance $r=\sqrt{x^2 + y^2 + z^2}$ for spatials $x$, $y$ and $z$, and velocity $v$, with $c$ the speed of light in vacuum and $τ$ the Hubble-Carmeli time constant. The metric is accurate to first order in $t/τ$ and $v/c$. The fields $ϕ$ and $ψ$ are general functions of the coordinates. By showing that $ϕ= ψ$, a metric of the form $ds^2 = c^2 dt^2 - dr^2 + τ^2 dv^2$ is obtained from the general metric, implying that the universe is flat. For cosmological redshift $z$, the luminosity distance relation $D_L (z,t) = r (1 + z) / \sqrt{1 - t^2 / τ^2}$ is used to fit combined distance moduli from Type Ia Supernovae up to $z < 1.5$ and Gamma-Ray Bursts up to $z < 7$, from which a value of $Ω_M = 0.800 \pm 0.080$ is obtained for the matter density parameter at the present epoch. Assuming a baryon density of $Ω_B = 0.038 \pm 0.004$, a rest mass energy of $( 9.79 \pm 0.47 ) \, {\rm GeV}$ is predicted for the anti-baryonic $\bar{Y}$ and the $Φ^{*}$ particles which decay from a hypothetical $\bar{X}_1$ particle. The cosmic aging function $g_1(z,t)= ( 1 + z) ( 1 - t^2 / τ^2 )$ makes good fits to light curve data from two reports of Type 1a supernovae and in fitting to simulated quasar like light curve power spectra separated by redshift $Δ{z} \approx 1$. We determine the multipole of the first acoustic peak of the Cosmic Microwave Background radiation anisotropy to be $l \approx 224 \pm 5$ and a sound horizon of $θ_{sh0} \approx (0.805 \pm 0.020 ) {}^{\circ}$ on today's sky.