Why was Mach’s principle abandoned & the anisotropy of inertia debate

Research in Mach’s principle, apart from early attempts after the work of Mach by Reissner, Schrödinger, and Einstein’s considerations in General Relativity, was largely abandoned during the period of stagnation in gravitational research from the 1920s to the 1950s. Its revival after the 1950s, mainly by Sciama and the Jordan-Brans-Dicke theory, slowly decayed after the failure of the Brans-Dicke theory, only brought up again by Treder in the 1970s and Julian Barbour’s 1993 Tübingen conference. We briefly covered Brans-Dicke theory in our previous post “Einstein’s true reason for the Cosmological Constant”, but there is another reason why Mach’s principle doesn’t have the same level of popularity today as it once had.

One of the objections often raised against Mach’s principle is the widespread false claim that it necessarily implies some sort of anisotropy of inertia: the idea that an object’s inertial mass (or its resistance to acceleration) depends on the direction in which it is being accelerated, which seems reasonable if mass distributions define inertial frames, following Mach. We covered Mach’s principle in our previous post “Mach’s principle explained”. Mathematically, this is expressed by the fact that inertial mass has a tensorial character. The most cited suggestion that Mach’s principle may result in anisotropy of inertia came from Cocconi and Salpeter in 1958 and 1960. Although they clearly state in their first abstract that ‘one of the possible consequences of Mach’s principle is that the asymmetric position of the Galaxy with respect to the Earth induces an asymmetry in inertia’, this was apparently picked up by many authors and misinterpreted as implying that Mach’s principle necessarily leads to anisotropy of inertia.

The preferred direction of the greatest nearby mass distribution of the center of the Milky way allows for testing the anisotropy of inertia by experiments on Earth. Cocconi and Salpeter named the Zeeman splitting in an atom, the Mössbauer effect, and nuclear magnetic resonance as possible ways to test any anisotropy of inertia. All these tests have yielded null results, or in other words, they proved the isotropy of inertia. In particular, the Hughes-Drever experiments were the most sensitive ones, with the last upper bound reaching as low as 10^-34.

Robert Dicke, co-author of the Jordan-Brans-Dicke theory, argued against Cocconi and Salpeter, claiming that in a relativistic model following Mach’s principle, a tensorial inertial mass anisotropy would be the same for all particles, and due to this universality, it would be unobservable locally. Consequently, the null result tests on the anisotropy of inertia could not be used to refute the validity of Mach’s principle. His argument, however, depends on the requirement that the coordinate system can always be chosen to be locally Minkowskian, as is usually assumed in standard General Relativity.

Hans-Jürgen Treder was the first to propose a model unifying inertia and gravity under Machian ideas without anisotropic inertial mass (Schrödinger noted in 1925 the problem of inertial mass anisotropy in his model, but avoided it by the low and underestimated mass of the Milky way considered at the time). We briefly covered his work in our post “History of Modified Inertia”. Treder avoided the anisotropy of inertial mass through his velocity-dependent gravitational potential based on the relative velocities between masses, which leaves inertial mass as a scalar. Therefore, the concern that Mach’s principle would inevitably result in anisotropic inertial masses is unjustified. Only a dependence on the rates of changes of the distances between masses, as in Reissner’s, Schrödinger’s, Assis’s and Barbour’s earlier models, would result in a tensorial and thus direction-dependent inertial mass (Sciama’s model has a scalar inertial mass). He further argued that General Relativity predicts anisotropic inertial mass in 2nd post-Newtonian order. In tensorial theories of gravitation, inertial mass is proportional to the spatial part of the metric tensor and thus, if the metric is not Minkowskian, it is anisotropic. In the usual interpretation of General Relativity it is argued that the coordinate system can always be chosen in such a way that it is locally Minkowskian and anisotropic effects appear as higher-order tidal effects, far beyond today’s constraints from the Hughes-Drever experiments. Treder, however, refuted this: The frame is not arbitrarily selectable but is given by the one in which the Hughes-Drever experiments are performed. This is the surface of the Earth, which is a stationary, accelerated frame, while the locally Minkowskian frame corresponds to a free-falling inertial frame. Thus, according to Treder and in opposition to Dicke, he claimed that the anisotropy of inertia should be observable, whether through the implementation of Mach’s principle or even within General Relativity alone.

Nowadays, Modified Newtonian Dynamics is one of the most promising ideas related to Mach’s principle, as we explained in our previous posts. But Mordehai Milgrom, the original author of MOND, has personally expressed a preference for a smooth quantum vacuum as the agent causing inertia because he thought that Mach’s principle necessarily implies the anisotropy of inertia in a not so smooth distribution of mass in the universe. Even though MOND can be written in its most pristine form as a modification to the inertial law based on the usual scalar inertial mass, the local mass distribution (for instance, a galaxy), and the global mass distribution (which defines MOND’s acceleration scale), which is very Machian, Milgrom only mentioned Mach’s principle in a couple of lines in his papers about MOND as modified inertia, suggesting a possible relationship of his work to that of Mach. We showed that, as long as inertial mass remains a scalar, it does not predict any directional dependence, and there is no reason to reject a Machian origin for MOND on the basis of anisotropy of inertia. Does MOND come from General Relativity not implementing Mach’s principle?

Written in colaboration with Dennis Braun