What causes inertia? An intuitive framework

Newton’s first law of motion states that “a body remains at rest, or in motion at a constant speed in a straight line, except insofar as it is acted upon a force”. But with respect to what is that rectilinear and uniform motion defined? Newton would argue that it is defined with respect to his absolute, unobservable and unphysical space, which defines inertial frames. But Ernst Mach would argue that it is something physical that defines them: the background distribution of all masses of the universe. Following Mach’s principle, inertial motion is not the result of an absence of forces, but the interaction with the causally connected rest of the universe.

Newton’s third law states that for every force in nature there is an equal and opposite reaction. When a force is applied to a mass which is stationary or at constant velocity, an equal and opposite inertial force appears. According to Newton, this inertial force arises from motion in a non-inertial reference frame. But how can an absolute space without physical properties react with an observable and physical inertial force? The origin of the first force can be easily identified as of electromagnetic origin, but the inertial force can’t be of electromagnetic origin. Electromagnetic forces can cancel out between positive and negative charges, and that’s why planetary orbits and galaxy dynamics are not mainly governed by electromagnetic forces (there exists some kind of equilibrium), but they are governed by gravity, which cannot be shielded or counteracted. Curiously, inertial forces are proportional to inertial mass, which is equal to gravitational mass following the equivalence principle (an inertial force is locally indistinguishable from a gravitational force). If inertial forces are not fictitious or pseudo-forces, but forces arising from an interaction, that interaction must be gravitational. The way in which the rest of the universe is connected to an inertial mass must be through gravity.

The modified inertia models of Sciama or Treder, which implement the idea of inertia being of gravitational origin, are based on the same field equations as gravitoelectromagnetism and Maxwell’s equations. In Maxwell’s electromagnetism, the electric field of a moving charge with constant speed is perfectly synchronized with the moving charge. This comoving field, which satisfies the postulate of relativity by which there is no physical difference between a frame at rest and a frame in uniform motion, can be mechanistically explained through wave mechanics by a combination of wavefront retardation and superposition effects of the scalar potential field, as Dialect’s Youtube channel explains. This is not the case when the charge is accelerated, in which the field is not synchronized and seems to act against the accelerated motion of the charge. One could think that the gravitational field acts in a similar way as the electromagnetic one, and inertia arises only in the case of accelerated motion. This is a feature of Sciama’s and Treder’s models, in which inertial forces don’t have to be postulated but arise from the gravitational field, and more precisely, from an induction effect of a gravitoelectric field.

In General Relativity, the interpretation of inertial forces is the same as in Newtonian mechanics: pseudo forces from motion in a non-inertial frame, encoded in the Christoffel symbols, which describe how the coordinates change due to spacetime curvature and frame non-inertiality. Masses follow geodesics in spacetime in absence of external forces. Geodesics are the paths of least spacetime distance and the paths of least energy, and extra energy is needed to depart from a geodesic. But why is extra energy required to depart from the path of a geodesic? This question is equivalent to the question of why mass has inertia, or where do inertial forces come from.

All masses have an associated curved spacetime (or according to Newton, an associated gravitational field). When a mass is stationary, the curved spacetime around it counteracts its outward acceleration, maintaining equilibrium with respect to itself.

Following the reasoning of Idea List’s Youtube video “How Time Dilation Causes Gravity, and How Inertia Works“, when a mass is moving at a constant velocity, the object is curving spacetime in front of it at the same rate it is curving spacetime behind it. The mass is attracted to the new curved spacetime, but also attracted in the same way to the space it is leaving, so that its velocity rate does not change due to this balance. In the direction of motion of the mass, it experiences length contraction, which defines its velocity vector: the direction of its momentum. Momentum is the relative rate and magnitude in which the mass curves and uncurves spacetime, and depends on its inertial mass and velocity. It takes more energy to push the mass with increasing velocity until reaching the speed of light, the moment at which it takes infinite energy to curve and uncurve spacetime faster.

From Mach’s perspective, the kinetic energy gained by a mass when a force is applied to it, has an origin in gravity, since the kinetic energy term is built including the gravitational potential and making it relational.

When a particle changes position due to a force applied to it, meaning, an accelerated change in spacetime, its curved spacetime or gravitational field must also adopt this change over time. This change is not instantaneous, but happens at the speed of light. The reaction force exerted by the inertial mass against the cause that it is pulling it from its geodesic must come from curving a new region of spacetime around itself. The force and energy applied tends to separate the source from its gravitational field, and the field reacts in opposition, with an effect we call inertia. That means that the inertial force comes from spacetime resisting a change its curved surroundings. Inertia thus appears when trying to dissociate the inertial mass from its gravitational field or curved spacetime. The higher the inertial mass, the higher the spacetime curvature, and the higher the gravitational field’s force or curved spacetime exerts.

Apart from Idea List’s video, this explanation was also partly described by João Bosco, for which there are links below in the references.

Joao Bosco's explanation of inertia — Joao Bosco’s explanation of inertia

But as in the Newtonian case, it can’t be just space (or in General Relativity, the spacetime of the Minkowski metric), the one that resists acceleration. It must be the gravitational field or curved spacetime, which depends on the rest of the masses of the universe, as Mach conjectured. This could be thought as if the effect of accelerated motion from an inertial mass through the curved spacetime depended on how much spacetime is already curved. Quantitatively, this effect could be formulated as a dependence on the potential of distant masses, and it is exactly how the Reissner’s and Schrödinger’s relationship appears in the modified theories of inertia of Sciama and Treder: the gravitational constant, moved to Newton’s second law, depends on the potential of the rest of masses of the universe.

And what if this effect was nonlinear, as the non-linearity of gravity in General Relativity suggests? The sum of the effect of the curved spacetime from different sources would be nonlinear, and superposition would not hold. This would mean that inertia would depend differently from the linear sum of local and global contributions to the gravitational field. One could think about this as if Mach’s reference frame of the global distribution of matter, for which he substituted Newton’s absolute space, would be shared nonlinearly by local and global gravitational fields, depending on their strength. Not only distant masses would determine which is our Newtonian rest frame, but also local masses would. From the Newtonian perspective, this translates into inertial mass depending nonlinearly on the local and global gravitational field. And this is precisely what MOND suggests, since it can be equivalently reformulated as a correction factor to Newton’s second law based on the local and global accelerations or field intensities through Reissner’s and Sciama’s relationship and the coincidence of MOND’s acceleration parameter with the speed of light and the radius of the observable universe.

In special relativity, velocities are relative and accelerations are absolute, and the Lorentz correction factor contains a fraction of velocities. If acceleration is relativized, as a Machian perspective requires, shouldn’t a factor containing a fraction with accelerations or field intensities arise? Can MOND be derived phenomenologically from Mach’s principle and the relativity of acceleration and inertia, in a relativistic modified theory of inertia?