**4. Is inertia gravitationally induced in general relativity?**

The problem of the origin of inertia, that is, Mach's principle, ceased to be a topic of mainstream interest in general relativity 50 years ago. It was not forgotten by those who lived through the '50s and '60s and were parties to the debates on inertia. For example, John Wheeler, working with Ignazio Ciufolini, made it the center piece of one of his last major books on gravity: *Gravitation and Inertia* [5]. On the otherwise blank page facing the first page of chapter 1 we find, "Inertia here arises from mass there". Exactly as Einstein would have said. In the penultimate chapter (5) of the book we find, "In the next chapter we shall describe in detail dragging of inertial frames and *gravitomagnetism*, which may be thought of as a manifestation of some weak general relativistic interpretation of the Mach principle, their measurement would provide experimental foundation for this general relativistic interpretation of the origin of inertia."

Brans'spectator matter argument does not involve gravitomagnetism. But it is implicitly present in Einstein's Eqs. 118 quoted above. In the vector potential **A**. The sources of **A** are the matter density currents in the universe. As Sciama pointed out in his first paper "On the Origin of Inertia" [6], the integration over cosmic matter currents involved can be vastly simplified by noting that the important currents involved can be singled out by assuming the local accelerating body in question and (instantaneously) has velocity **v**, can be taken as at rest with the universe moving past it rigidly with velocity – **v**. This can be removed from the integration, and the remaining integral just returns the Newtonian gravitational potential for all the stuff in the universe (up to a factor of order unity). If the coefficient of the time derivative of – **v**, **a** that is, is one, then this term in the equation of motion is the inertial reaction force. So, Brans' argument does more than require that the gravitational potential energies conferred on test particles by spectator matter not influence the rest mass of the test particle. *It also demands that if inertial forces are gravitational inductive effects, the coefficient of the acceleration in the equation of motion must be one in all circumstances. This is only possible if the total Newtonian potential is a locally measured invariant equal to the square of the vacuum speed of light (which is also a locally measured invariant in general relativity).*

A series of events, too lengthy to relate in detail here, led to the rejection of the time derivative of the vector potential in the equation of motion in general relativity. This started with an article by Edward Harris in the *American Journal of Physics* in 1991 [7] where he outlined the analog of linearized weak field slow motion equations of general relativity with Maxwell's equations of electrodynamics. In this approximation, it is possible to argue that the *d***A**/*dt* term in the field equation and equation of motion vanish by gauge invariance. Harris, of course, knowing that general relativity is not a gauge theory, allowed as how this could not be generally true. But others took this argument and tried to justify it on other terms. (See references in chapter 4 of Pfister and King. [8]) With the passing of the *d***A**/*dt* term, so too went the question/possibility of the gravitomagnetic origin of inertial forces. This development led Herbert Pfister and Markus King, in their recent book *Inertia and Gravitation*, to remark that, "We hope to give a new synopsis of this theme [that Faraday induction that produces the time derivative of the vector potential is absent in general relativity], with this specific focus not entering most textbook presentations on the foundations of gravitation and general relativity – and in a way amend Cuifolini and Wheeler's view on gravitation and inertia … ".

Cuifolini and Wheeler had opted for an account of inertia based on an initial spacelike hypersurface complemented by elliptic (instantaneous) constraint equations (first discussed by Wheeler and independently Lynden-Bell in the '60s) in preference to integrations over matter currents out the past light-cone to the past particle horizon. Pfister and King, having rejected the existence of Faraday induction effects in general relativity, were left no choice but the hypersurface/constraint equation approach. All of this was, wittingly or unwittingly, motivated by Carl Brans' conclusion that in sufficiently small regions of spacetime one must use the Minkowski metric which makes the local Newtonian potential vanish, eliminating gravity from external sources as an actor at that scale. That is, making the presence of cosmic matter, in Brans' word, "invisible".

Who's right? Cuifolini and Wheeler? Pfister and King? Brans? Arguably, they are, in a sense, all right, and wrong. The source of the confusion and problems regarding inertia in general relativity is the Minkowski metric – which de Sitter showed Einstein to be an acceptable formal solution of his field equations. The Minkowski metric is the metric for flat pseudo-Euclidean spacetime with gravity completely absent. That it is a solution of Einstein's field equations is not surprising. The theory is constructed on the assumption that in sufficiently small regions, spacetime is flat (and special relativity applies). As Einstein explicitly claimed, however, without gravity, there is no spacetime. This makes the Minkowski metric an unphysical pre-general relativistic idealization. Scaffolding to be removed once the construction of the theory is complete. The Maxwellian analog is the "roller bearing" mechanical aether he used to construct his equations of electrodynamics, promptly abandoned once the construction was complete. But spacetime, as a matter of observation, is essentially flat almost everywhere/when. What takes the place of Minkowski spacetime in the completed theory? Spatially flat FLRW spacetime.
