**3. Carl Brans' "spectator matter"argument**

Brans did his doctoral work at Princeton in the late 1950s. His doctoral supervisor was the noted experimentalist, Robert Dicke. After passing his qualifying exam, Dicke tasked Brans with investigating the question of as the origin of inertia in general relativity, as Dennis Sciama and others had made "Mach's principle" a central question in general relativity several years earlier. When Brans read Einstein's remarks on Machian inertia in Einstein's 1921 comments quoted above, he noted a problem. If gravitational potential energy due to nearby matter contributes to the rest masses of test particles, the Equivalence Principle is violated. This is not a minor problem. The Equivalence Principle is the bedrock of general relativity. The solution to this problem adopted in Brans' graduate school days was the imposition of a "coordinate condition". (A "gauge" solution is/was not available as general relativity is not a gauge theory).

As Brans, responding to several published papers on Mach's principle, later wrote in 1977 [4]:

*Over the years, many and varied expressions of Mach's principle have been proposed, making it one of the most elusive concepts in physics. However, it seems clear that Einstein intended to show that locally measured inertial-mass values are gravitationally coupled to the mass distribution in the universe in his theory. For convenience I repeat the first order geodesic equations given by Einstein to support his argument:*

*[Brans inserted here Einstein's equations displayed above.]*

*… Einstein's claim is that "The inertial mass is proportional to l*ð Þ þ *σ , and therefore increases when ponderable masses approach the test body.*

Brans pointed out that having the masses of local objects, the unit mass test particle in this case, depend on their gravitational potential energies acquired by interaction with spectator matter must be wrong. Were it true, then the electric charge to mass ratios of elementary particles for example would depend on the presence of nearby matter. If this were true, gravity could be discriminated from accelerations without having to check for the presence of spectator matter by going to the window in a small lab and looking out to see if one were on Earth, or in a rocket accelerating at one "gee" in deep outer space– a violation of the Equivalence Principle. From this, Brans inferred that

… *global, i.e., nontidal, gravitational fields are completely invisible in such local standard measurements of inertial mass, contrary to Einstein's claim … Einstein ought to have normalized his local space-time measurements to inertial frames, in which the metric has been transformed approximately to the standard Minkowski values, and for which distant-matter contributions are not present. [Emphasis added.]*

This is the "coordinate condition" required by Brans' work: that the coordinates be compatible with the assumed approximate Minkowski metric applicable in small regions of spacetime. Since the absence of gravity is presupposed for Minkowski spacetime, this amounts to the assumption that the Newtonian potential due to exterior matter in such small regions of spacetime is effectively everywhere/ when equal to zero. That is, the locally measured value of the total Newtonian gravitational potential is universally zero. This certainly makes the localization of gravitational potential energy impossible in general relativity, a now widely accepted fact. And where there is effectively no gravity, there can be no gravitational induction of inertia. Accordingly, it would seem that Brans'spectator matter argument makes Machian gravitationally induced inertia incompatible with general relativity.
