Generalisation of Hubble's Law

Abstract

RECENTLY, several new efforts to improve the upper bound on the possible rest mass (m₀) of the photon have been made^1–3. The theoretical basis is provided by Proca's equations¹

(□ + μ²)A_K = (4π/c)/J_K (1)

where charge is assumed conserved and μ = m₀c/ħ. The bounds vary from 2 × 10⁻⁴³ g to 3 × 10⁻⁵⁶ g. As Yourgrau and Woodward⁴ point out, however, a finite mass is implicit in a ‘tired light’ interpretation of the cosmological redshift. When equation (1) is combined with Hubble's law⁵

dν/dl = −Hν/c (2)

it yields μ = H/2c and hence m₀ < 10⁻⁶³ g. Here ν is the photon frequency, l the distance traversed and H is Hubble's constant. In view of this limit a direct demonstration of the existence of a finite photon rest mass does not seem practical at present.