History of modified inertia

Ernst Mach, who challenged Newton’s ideas of absolute space, time and motion, never put into mathematical terms the implication of his critique. Although he was first to strongly claim that inertia must have an origin as an interaction with the rest of the universe, that local physical laws must be determined by the large-scale structure of the universe, and that inertial motion is not motion in the absence of causes but motion caused by background environment, other great physicists did develop his ideas further after him.

The first ones to relate Mach’s work to gravity were Benedict and Immanuel Friedländer, August Föppl, and Wenzel Hofmann. In 1896, the Friedländer brothers published “Absolute or relative motion” stating that “it seems to me that the correct form of the law of inertia will only then have been found when relative inertia as an effect of masses on each other and gravitation, which is also an effect of masses on each other, have been derived on the basis of a unified law”.

Einstein modified inertia through Special Relativity, which changes the classical single-particle kinetic action when accounting for relativistic effects by incorporating the velocity of the particle relative to the speed of light. This modification to the equation of motion slows the increase in velocity as a force is applied, up to the limit of the speed of light.

Erwin Schrödinger soon identified in his 1925 paper “The Possibility of Fulfillment of the Relativity Requirement in Classical Mechanics”, the same year in which he also found the famous Schrödinger equation, that General Relativity did not fully implement Mach’s principle, and proposed a relationship between the gravitational potential of distant masses and the speed of light: “This remarkable relationship states that the (negative) potential of all masses at the point of observation, calculated with the gravitational constant valid at that observation point, must be equal to half the square of the speed of light”. This conclusion was reached by imposing that kinetic energy, which arises directly from motion, had an origin in a potential-like interaction as the rest of forms of energy, which arise from position or configuration relative to forces. Classical kinetic energy is not based on interaction, even though motion often arises as a result of forces due to interactions. By making the classical definition of kinetic energy K=1/2 mv^2 relational through the gravitational Newtonian potential GM/R with the need to introduce a constant of speed squared as the speed of light squared, Schrödinger considered the kinetic energy of a moving mass an approximation K = (GM/Rc^2)(1/2 mv^2) with a missing term containing the mass of the rest of the masses in the universe M, and a distance between the moving mass and these others R, with GM/Rc^2=1

Schrödinger's & Reissner's interpretation of the gravitational constant — Schrödinger’s & Reissner’s interpretation of the gravitational constant

This relationship was afterwards credited to Hans Reissner by Schrödinger himself, so that Reissner was the first in his 1915 article “On the Possibility of Deriving Gravity as the Direct Consequence of the Relativity of Inertia” to formulate inertia as of gravitational origin. In other words, Schrödinger and Reissner were the firsts to independently derive the gravitational constant from the mass and size of the universe, which matches in order of magnitude with today’s measurements. The fact that the gravitational constant is not fundamental, but a derived parameter, aligns with our previous posts, in which we explain that the only two cases where the fundamental gravitational constant is used in Quantum Field Theory, as part of the Planck Energy, results in the vacuum catastrophe and in a hierarchy problem of the Higgs mass. It is also impossible to derive the units of mass through dimensional analysis from the dimensional constants of Quantum Field Theory, if one does not consider the gravitational constant or a mass parameter, such as Quantum Field Theory does with a Higgs parameter, as fundamental. If this relationship were to be true, then General Relativity would be incomplete.

Reissner’s and Schrödinger’s models were based on the velocity-dependent Weber potential depending on relative distances and on the rate of change of relative distances, which took the role of both kinetic and potential energy, explaining inertia as of gravitational origin.

Andre Koch Assis’s model from 1999 also follows Weber’s potential, in which a Weber-type force is an approximation of a simple retarded potential. These Weber potential formulations result in anisotropic tensorial inertial masses, which are ruled out by observations.

Weber’s potential

Schrödinger’s and Reissner’s relationship for the gravitational constant appeared in following articles, but was popularized by Dennis Sciama in his 1953 article “On the origin of inertia”, in which he derived a cosmological vector potential theory of modified inertia. Sciama’s approach explains inertia as a part of gravity by describing the latter using gravoelectromagnetic equations instead of just the usual Newtonian term, based on Mach’s principle. He postulated that “In the rest-frame of any body the total gravitational field at the body arising from all the other matter in the universe is zero”. Sciama shows how inertia can be derived from gravitoelectromagnetic equations, and the inertial law arises as a side effect of gravity, with inertial forces resulting from the gravitational induction of the gravitoelectric field from the whole universe. No gravitational constant had to be introduced a priori, but it naturally arises as the Reissner and Schrödinger’s relationship. It also predicted a non-infinite and expanding universe, since the potential of an infinite non-expanding eternal universe would also be infinite. Although Sciama did enormous progress into the development of Mach’s ideas, he stated that his theory was incomplete and non-relativistic, and that it probably required a tensor form with a metric instead of a vector form.

Sciama´s interpretation of the gravitational constant

Robert Dicke and Carl Brans developed in their 1961 article “Mach’s Principle and a Relativistic Theory of Gravitation” a scalar-tensor theory of gravity, known as Brans–Dicke theory. They considered a scalar field that determined the effective gravitational constant varying in space and time, which depended on the matter distribution of the universe. Although featuring a Machian behavior, it did not modify inertia, and it was constrained by observations, showing that its deviations from General Relativity would have to be extremely small.

Hans-Jürgen Treder developed in 1972 a formulation of inertia-free mechanics in his book “The relativity of inertia”, with inertia having an origin in gravitation, in which inertial mass depended on the gravitational potential of the universe as an analogy with gravitoelectromagnetism. In both Weber’s potential models and Treder’s theory, the crucial part that allows a description of inertia as a gravitational effect is the fact that the kinetic energy is a velocity-dependent part of the gravitational potential, which replaces the conventional Newtonian kinetic energy.

In contrast to the Weber potential models, Treder’s non-relativistic model was built on the Riemann’s electromagnetism theory potential depending on relative velocities, resulting in isotropic scalar inertial masses, for which acceleration is not direction dependent. Sciama’s idea is contained in Treder’s theory and can be derived from it. Treder’s conclusions are that Mach’s principle does not imply anisotropy of inertial masses, that the smallness of the strength of gravity is due to inertial mass induced by all other masses of the universe, and that total momentum of the universe is zero.

Riemann’s potential

Julian Barbour and Bertotti stated in 1977 that a Machian theory should be invariant under certain transformations that ensure the dependence of the theory on purely relative quantities. Only Weber’s potential models satisfy this invariance of a combination between arbitrary time-dependent translation and rotation. Treder’s Riemann potential model is only invariant under translations, but not under rotations.

Babour condition

They presented a non-relativistic model in their 1982 article “Mach’s principle and the structure of dynamical theories”, which respected these transformations, but didn’t incorporate the idea of inertia being of gravitational origin because the kinetic energy was not proportional to the gravitational potential, and were unable to explain the gravitational constant or naturally incorporate the weak equivalence principle. Indeed, Barbour and others abandoned these modified theories of inertia because they thought they would all result in anisotropic inertial mass.

Lynden-Bell and Katz explored in their 1995 article “Classical mechanics without absolute space” the notion of rotational invariance (independence of frame rotation) through a zero total angular momentum of the universe. This satisfied the Machian relativity of rotation, in which the inertial forces of a rotating bucket of water are indistinguishable from the ones observed in a rotating bucket of water and a static universe. But as Barbour, it did not explain inertia as having a gravitational origin or the gravitational constant, and it was not relativistic.

Treder’s theory with isotropic inertial mass can be made rotationally invariant, and thus invariant under the transformation proposed by Barbour, using the same method that Lynden-bell and Katz, but applied to the Riemann potential.

Today, the question arises: can a Machian theory of modified inertia based on relative quantities, invariant under translations and rotations, in which inertial forces have a gravitational and Machian origin, inertial mass is isotropic, and gravity and the weak equivalence principle arise naturally, be constructed to satisfy Lorentz invariance? Could this theory result in a Machian MOND modified inertia equations?

Written in collaboration with Dennis Braun: Gravinertia