[Paper Review] In an expanding universe, what doesn't expand?

Richard H. Price, Joseph D. Romano|arXiv (Cornell University)|Aug 12, 2005

Relativity and Gravitational Theory9 citations

TL;DR

This paper investigates whether a classical hydrogen-like atom expands with the universe, using Newtonian and relativistic models. It finds that bound systems either fully expand with the cosmological background or remain unaffected—there is no partial expansion, a behavior explained by the competition between atomic binding forces and cosmological expansion forces.

ABSTRACT

The expansion of the universe is often viewed as a uniform stretching of space that would affect compact objects, atoms and stars, as well as the separation of galaxies. One usually hears that bound systems do not take part in the general expansion, but a much more subtle question is whether bound systems expand partially. In this paper, a very definitive answer is given for a very simple system: a classical "atom" bound by electrical attraction. With a mathemical description appropriate for undergraduate physics majors, we show that this bound system either completely follows the cosmological expansion, or -- after initial transients -- completely ignores it. This "all or nothing" behavior can be understood with techniques of junior-level mechanics. Lastly, the simple description is shown to be a justifiable approximation of the relativistically correct formulation of the problem.

Motivation & Objective

• To resolve the long-standing question of whether bound systems like atoms expand with the universe.
• To clarify the misconception that bound systems might partially expand due to cosmological stretching.
• To provide a clear, accessible model using classical mechanics that captures the essential physics of the problem.
• To validate the classical model against a relativistic formulation, showing its adequacy for non-relativistic atoms.
• To demonstrate that relativistic effects diminish as an unbound atom expands, reinforcing the stability of bound systems.

Proposed method

• Model a classical atom with a point nucleus and a light electron bound by Coulomb attraction in a cosmologically expanding space.
• Use two coordinate systems: physical coordinates (r, θ, φ) for proper distance and cosmological coordinates (R, θ, φ) tied to the expanding space.
• Introduce a heuristic 'stretching force' in Newtonian mechanics to model the effect of cosmological expansion on orbital radius r(t).
• Analyze the system using differential equations for r(t), with expansion governed by a(t), and study cases with exponential and power-law expansion.
• Compare the Newtonian results with a fully relativistic treatment using general relativity and Maxwell electrodynamics in curved spacetime.
• Use numerical simulations to track the evolution of the electron's velocity and relativistic factor (U⁰/c) in expanding, unbound atoms.

Experimental results

Research questions

• RQ1 Does a bound atomic system partially expand with the cosmological expansion, or does it remain unaffected?
• RQ2 What determines whether an atom fully expands or remains stable in an expanding universe?
• RQ3 How does the competition between Coulomb binding and cosmological expansion affect the orbital radius over time?
• RQ4 Can a non-relativistic classical model accurately describe the behavior of atoms in an expanding universe?
• RQ5 Do relativistic effects increase or decrease as an unbound atom expands with the universe?

Key findings

• Atoms either fully expand with the universe (constant R) or remain unaffected (bounded r), with no intermediate behavior—this is an 'all or nothing' effect.
• The transition between expansion and non-expansion depends on the initial ratio of atomic binding forces to cosmological expansion forces.
• For typical atoms, the atomic time scale T_atom ≈ 10⁻¹⁶ s is much shorter than the cosmological expansion time T_exp ≈ 4×10¹⁷ s, so atoms are stable against cosmological expansion.
• In the case of an unbound atom expanding with the universe, the relativistic factor U⁰/c decreases over time, indicating that relativistic effects diminish as the atom grows.
• The Newtonian model with a heuristic stretching force accurately captures the essential physics and agrees with the relativistic formulation for non-relativistic initial conditions.
• The system exhibits a dynamical instability: if expansion dominates initially, the atom grows without bound; if binding dominates, expansion is suppressed and the orbit stabilizes.