Feynman Path Integral Formulation

9.6 Quantum Gravity and Mach’s Principle 321
9.6 Quantum Gravity and Mach’s Principle
The essence of Mach’s proposal can be summarized in the statement that faraway
galaxies provide a preferential reference frame with respect to which the motion of
an observer can be characterized as inertial or not (Mach, 1960). In such a framework
acceleration cannot be described in absolute terms, since in the absence of
such a distant mass distribution the very notion of acceleration would be meaningless.
Water in the Newtonian water bucket rotating by itself in an empty universe
could not possibly rise at the sides, since there would be no frame of reference for
such an isolated bucket. It would seem possible therefore that the origin of inertial
mass itself might be due to some new long range interaction between a test mass
and the faraway galaxy distribution (Sciama, 1953; Feynman, 1963).
Over the years, and starting with considerations by Einstein himself, extensive
discussion’s of Mach’s principle and its relation to General Relativity have appeared.
Some have argued convincingly that some aspects of Mach’s principle are indeed
incorporated in the framework of General Relativity, since for example the choice
of locally inertial frames is determined form the overall mass distribution, and in
particular the Einstein tensor is completely determined by the energy-momentum
tensor for matter. On the other hand the Weyl conformal tensor W μνλσ is not determined
by matter, and the field equations remain incomplete without the specifications
of suitable boundary conditions. Another disturbing aspect is the fact that in
a universe without matter, T μν = 0, there still exist flat space solutions g μν = η μν
describing the motion of a test particle in terms of Minkowski dynamics, whereas
based on Mach’s principle one would expect in this case a complete absence of inertia
(as discussed in some detail already in (Pauli, 1958), where he provides some
arguments in favor of the existence of a small cosmological constant λ).
A number of proposals have been put forward to address the issue of how to
incorporate some aspects of Mach’s proposal in General Relativity. One possibility
has been the introduction of boundary terms to account for non-trivial boundary conditions
at infinity. Another set of proposals (Sciama, 1953; 1971; Brans and Dicke,
1961; Feynman, 1963) postulate that the value of Newton’s constant G is in some
way related to a new cosmic field describing inertial “forces”, i.e. the natural tendency
of massive bodies to resist acceleration. In Sciama’s original theory, which
is non-relativistic and therefore in many ways incomplete, the key ingredient is the
fact that an acceleration of a test particle with respect to the universe should be
equivalently described by an acceleration of the universe as a whole, with the test
particle at rest.
In the Brans-Dicke relativistic scalar-tensor theory of gravity, G is assumed not
to be a constant of nature: instead it is described by a local average of the additional
Brans-Dicke scalar field φ, G −1 ≃〈φ〉. Of course even in this extended theory of
gravity on needs at some point to specify a consistent set of boundary conditions
at infinity. But unfortunately the theory, at least in its original formulation is not
favored by current solar system tests, which put stringent bounds on the value of
the Brans-Dicke parameter ω describing the coupling of the cosmic scalar field φ to
gravity.