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

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.

• The paper introduces a theoretical reformulation where gravity is expressed through a Machian Poisson-like equation linking variable speed of light to mass distribution.
• The authors derive an analytical expression, Δ(1/c²) = -2κρ, eliminating Newton's gravitational constant and challenging conventional metrics.
• The study revisits Einstein's 1911 hypothesis and Dicke's 1957 contributions, inviting further empirical tests in gravitational wave and cosmological research.

A Machian Exploration of Einstein’s Variable Speed of Light Theory

The paper "A Machian Version of Einstein’s Variable Speed of Light Theory" by Alexander Unzicker and Jan Preuss presents a revived examination of an alternative theory initially proposed by Einstein in 1911 concerning the variable speed of light (VSL). The authors elucidate how Einstein’s overlooked hypothesis can be mathematically formulated akin to the Poisson equation, providing a novel Machian perspective to Einstein's classical insights. This examination is extended to consider Robert Dicke's advancements in 1957 that align with Mach's principle.

Historical Context and Theoretical Framework

Einstein's original proposition suggested that the speed of light varies with the distribution of masses. This preliminary idea was influenced by his equivalence principle, postulating that gravitational fields affect the curvature of light rays—a relation that Einstein recognized necessitated a localized definition of the speed of light in gravitational contexts. However, this foundational variable was overshadowed by the geometric interpretation of general relativity validated by Eddington's eclipse observations in 1919. Consequently, Einstein's VSL model was largely sidelined until Robert Dicke revised it in 1957 by incorporating variations in length scales, thereby restoring the model’s alignment with classical gravitational tests.

Dicke's contribution introduced a variable index of refraction $c = c_0 (1 + \frac{2GM}{rc^2})$, indicating a marginal dependence on the Sun, with a further implication that this formulation might trace its origin to the cosmic mass distribution—echoing Mach's philosophical notions.

Analytical Reformulation and Implications

Building upon Dicke and Sciama's frameworks, the paper's authors provide an analytical formula for the VSL hypothesis, articulated through equations relating curvature to local light speed and following Mach’s principle. These calculations equally attempt to eliminate Newton’s gravitational constant $G$ from fundamental equations by reinterpreting Einstein's gravitational constant $\kappa$ in what can be described as Machian units. The transformations proposed in this analytical model portray gravity as fundamentally linked to a universe-wide context influenced by the distribution of massive bodies, thus paving the way for a Machian articulation of gravity without reliance on $G$.

Notably, the paper presents the expression $$\Delta \left( \frac{1}{c^2} \right) = -2\kappa \rho$$, signifying a departure from conventional formulations by emphasizing the interdependent nature of $c$ and the metric. This casts the metric itself as a function of the speed of light, rather than merely a constant in space-time transformations.

Speculation and Future Directions

The potential ramifications of the paper's assertions are profound, offering a reduction in fundamental constants—a fundamental criterion for theoretical advancement in physics. Speculating on future research, these considerations necessitate further scrutiny and potential experimental tests, including how the proposed Machian VSL theory aligns with phenomena like gravitational waves and their correspondence with general relativity's standard predictions.

Any future exploration could focus on empirical validation, exploring whether this theoretical reformulation can demonstrate its consistency and predictive advantage over established models. Additionally, the dynamic aspects of $c$ in the context of space-time events warrant further exploration, particularly in high-energy astrophysics and cosmology.

Conclusively, this exploration reflects an effort to incorporate Mach's ideas into a coherent framework for understanding gravity, challenging entrenched paradigms while inviting additional theoretical and empirical advancements in the understanding of gravitational phenomena.