Einstein’s rationale for rapid motion preserving uniform stellar distribution

Ascertain how Einstein envisaged that very large stellar velocities could maintain the observed statistical uniform distribution of matter in the static de Sitter spacetime described by the line element ds² = −dr² − R² sin²(r/R)[dψ² + sin²ψ dθ²] + cos²(r/R) c² dt², despite large gravitational potential differences implied by non-constant g44. Reconstruct the intended mechanism or argument from Einstein’s 1918 correspondence and related discussions.

Background

In Section 4.7, the paper reviews Einstein’s physical criteria for admissible cosmologies, including a static line element and Machian considerations. Discussing the de Sitter line element with g44 = cos²(r/R), Einstein argued that if the world were as depicted, fixed stars would need tremendous speeds to sustain their statistical distribution given large potential differences; he then rejected the scenario on observational grounds (absence of great star speeds).

The author notes uncertainty about how rapid motion would sustain uniformity, indicating a gap in reconstructing Einstein’s precise reasoning or mechanism. Clarifying this would sharpen our understanding of Einstein’s objections to de Sitter’s static formulation and his broader criteria for cosmological models.