An early competitor of the Big Bang theory postulates the “continuous creation” of charged matter at a (very small) constant rate R at every point in space. In such a theory, the continuity equation is replaced by

(a) For this to be true, it is necessary to alter the source terms in the Maxwell equations. Show that it is sufficient to modify Gauss’ law to 

and the Amp`ere-Maxwell law to

Here, λ is a constant and ϕ and Aare the usual scalar and vector potentials. Is this theory gauge invariant?
(b) Confirm that a spherically symmetric solution of the new equations exists with

where f (r, t) is a scalar function and ϕ₀ is a constant.
(c) Show that the only non-singular solution to the partial differential equation satisfied by f (r, t) is a constant.
(d) Show that the velocity of the charge created by this theory, v = j/ρ, is a linear function of r. This agrees with Hubble’s famous observations.

Transcribed Image Text:

V.j+ др at = R.