Reformulation of general relativity in accordance with Mach’s principle

Feza Gürsey

It is argued that Einstein’s Theory of General Relativity as it stands incorporates Mach’s Principle. The boundary conditions for Machian solutions are stated in a coordinate system in which the cosmological background is described by a conformally flat metric. The metric tensor gμν is then written as a product of the scalar density ϕ2 and a tensor density γμν with unit determinant. In the coordinate system that has been so chosen ϕ describes the cosmological structure, while γμν refers to gravitational phenomena. This becomes clear when Einstein’s fundamental equations are rewritten in terms of ϕ and γμν. Then κϕ−1 is seen to play the role of the gravitational constant instead of κ in the weak field approximation. The quantity κϕ−1 can be expressed in terms of the radius and the total mass of the universe and the sign of the forces between inhomogeneities of the metric is determined by the requirements of Mach’s principle. The forces which derive from ϕ are found to be repulsive for the cosmological background, leading to the expansion of the universe, while attractive gravitational forces arise from the deviations of γμν from the Minkowski metric. Various statements associated with Mach’s Principle are discussed in the light of this reformulation of Einstein’s Theory.

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