Mach's principle and isotropic singularities

In this contribution, I review the work of Dennis Sciama and his collaborators on Mach's Principle, saying both what Mach's Principle is, and more generally what we should expect a ‘Principle’ to be and to do. Then I review the notion of an isotropic singularity, and the evidence for a connection between isotropic singularities and Mach's Principle. I suggest that a reasonable formulation of the cosmological part of Mach's Principle is that the initial singularity of space-time is an isotropic singularity, and that Mach's Principle may become a ‘theorem’ of quantum gravity. WHAT IS MACH'S PRINCIPLE? Mach's Principle is the name usually given to a loose constellation of ideas according to which “the inertia of a body is due to the presence of all the other matter in the universe” (Milne 1952) and “the local inertial frame is determined by some average of the motion of the distant astronomical objects” (Bondi 1952). In Wheeler's aphorism “matter there governs inertia here” (Misner et al . 1973). The aim of Mach's Principle is to explain, without recourse to Absolute Space, the origin of inertia, inertial frames and the standard of non-rotation in Newtonian Mechanics, where the existence of these things is a basic assumption.