Machian derivation of the Friedmann equation

Despite all fundamental objections against Newtonian concepts in cosmology, the Friedmann equation derives from these in an astoundingly simple way through application of the shell theorem and conservation of Newtonian energy in an infinite universe. However, Friedmann universes in general possess a finite gravitational horizon, as a result of which the application of the shell theorem fails and the Newtonian derivation collapses. Hence, unlike the general relativistic derivation, the Newtonian derivation does not prove the Friedmann equation in general, but exclusively in the Newtonian case of an infinite horizon. We show that in the presence of a gravitational horizon the Friedmann equation can be derived from conservation of Machian energy, without invoking the shell theorem. Whereas in the Newtonian case total energy translates to curvature energy density, in the Machian case total energy takes on different identities, depending on the evolution of the horizon; we show that in the de Sitter universe Machian total energy density is constant, i.e. appears as cosmological constant.

Figure 1

Gravitational force $F$ inside an empty spherical cavity within the horizon ${\chi }_g$.