The Derivative of Velocity Addition with Proper Time and What We Actually Measure in the Universe
Year: 2026
Type: article
Abstract: This paper derives the time derivative of Einstein's velocity-addition formula with explicit distinction between coordinate time $t$ and proper time $t'$. A previous version of this paper used a common coordinate time $t$ for both the local velocity and the system velocity. That version is superseded here: it is a simplified model, not the standard special-relativistic transformation. The corrected formula is $$a^* = \frac{a_v'}{\gamma_u^3\left(1+vu/c^2\right)^3} + a_u\,\frac{1-v^2/c^2}{\left(1+vu/c^2\right)^2},$$ where $a_v'=dv/dt'$ is the local acceleration measured with proper time, $a_u=du/dt$ is the acceleration of the system measured with coordinate time, and $a^*=dv^*/dt$ is the acceleration measured from Earth. At low velocities the correction is negligible; at high velocities ( $v,u\sim 0.5$-- $0.9\,c$) the measured acceleration is 40--99% smaller than the Galilean sum. This affects the interpretation of the Hubble constant, the estimated age of the Universe, and the dark-energy explanation of accelerated expansion. The paper does not change any physical constants, does not introduce tired light, it only differentiates Einstein's 1905 formula with proper time taken into account.
Source: Figshare
Author: Jozsef Kerenyi
Language: English
Subfield: Astronomy and Astrophysics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225803919
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