A Machian Version of Einstein's Variable Speed of Light Theory

TL;DR

This paper revisits Einstein's early idea of a variable speed of light to describe gravity, demonstrating its equivalence to standard general relativity and providing a Machian interpretation of gravity and the gravitational constant.

Contribution

It analytically formulates Einstein's 1911 variable speed of light theory using Green's functions and relates curvature to local light speed, offering a Machian perspective.

Findings

Variable speed of light approach is equivalent to Einstein's general relativity.

Derived a simple formula relating curvature to local light speed.

Provides a Machian interpretation of gravity and G.

Abstract

It is a little known fact that while he was developing his theory of general relativity, Einstein's initial idea was a variable speed of light theory. Indeed space-time curvature can be mimicked by a speed of light $c(r)$ that depends on the distribution of masses. Einstein's 1911 theory was considerably improved by Robert Dicke in 1957, but only recently has the equivalence of the variable speed of light approach to the conventional formalism been demonstrated (Broekaert, 2008). Using Green's functions, we show that Einstein's 1911 idea can be expressed in an analytic form, similar to the Poisson equation. Using heuristic arguments, we derive a simple formula that directly relates curvature $w$ to the local speed of light, $w= -c^2 \Delta \frac{1}{c^2}$. In contrast to the conventional formulation, this allows for a Machian interpretation of general relativity and the gravitational constant $G$. Gravity, though described by local equations, has its origin in all other masses in the universe.