**2. Einstein's conception of gravity and inertia in general relativity**

Einstein started talking about the gravitational induction of inertia as "Mach's principle" shortly after mooting general relativity. Willem de Sitter quickly pointed out that the field equations of general relativity have solutions that are plainly inconsistent with any reasonable interpretation of "Mach's principle". Einstein retreated from full-blown Mach's principle, which seemed to require action at a distance, then deemed inconsistent with the conception of field theory as articulated by Faraday. But he did not abandon the gravitational induction of inertia which is consistent with the tenets of field theory. Einstein advanced his ideas first in an address at Leiden in 1920 where he analogized his evolving view of spacetime to the "aether" of the turn of the century theory of electrodynamics. That is, spacetime is not some pre-existing void in which matter, gravity and the other forces of nature exist. It is a real, substantial entity – not a void – which *is* the gravitational field of matter sources. And then he extended his view in remarks in a series of lectures at Princeton in 1921 [2]. He calculated the action of some nearby, "spectator" matter on a test particle of unit mass (at the origin of coordinates) in the weak field limit of GR. There he found for the equations of motion of the test particle (his Equations 118):

$$
\delta \left( \frac{d}{dl} \right) [(l + \mathbf{\bar{\sigma}}) \mathbf{v}] = \nabla \sigma + \frac{\partial \mathbf{A}}{\partial l} + \nabla \times (\mathbf{A} \times \mathbf{v}), \tag{1}
$$

$$
\overline{\sigma} = (\kappa/8\pi) \int (\sigma/r) \, \, dV\_o,\tag{2}
$$

$$\mathbf{A} = (\kappa/2\pi) \int (\sigma d\mathbf{x}/dl) r^{-1} dV\_o. \tag{3}$$

The second and third of these equations are the expressions for the scalar (̄*σ*Þ and vector (A) potentials of the gravitational action of the spectator masses with density σ on the test particle. *l* is coordinate time and v is coordinate velocity of the test particle. The first equation is just Newton's second law. After writing down these equations, Einstein noted approvingly that,

*The equations of motion, (118), show now, in fact, that. The inert mass [of the test particle of unit mass] is proportional to 1 + σ, and therefore increases when ponderable masses approach the test body.*

*There is an inductive action of accelerated masses, of the same sign, upon the test body. This is the term dA/dl …*

*Although these effects are inaccessible to experiment, because κ [Newton's constant of universal gravitation] is so small, nevertheless they certainly exist according to the general theory of relativity. We must see in them strong support for Mach's ideas as to the relativity of all inertial interactions. If we think these ideas consistently through to the end we must expect the* whole gμν*-field, to be determined by the matter of the universe, and not mainly by the boundary conditions at infinity.*

Note that the small effects singled out by Einstein – the contribution of gravitational potential energy to the rest mass of the test particle and the inductive action, a force, of accelerating masses – are the two features of gravity that would supposedly account for all of inertia – origin of mass and reaction forces – in a properly constituted cosmology.

The above quote was not Einstein's last explicit word on gravity, inertia, and spacetime. In 1924, he again addressed these topics in a paper, "Concerning the Aether" [3]. In it he quickly asserted that by "aether" he did not mean the material aether of turn of the century electromagnetism. Rather, he meant a real, substantial, but not material entity that *is* spacetime, and that spacetime *is* the gravitational field of material sources. No material sources, no spacetime. Why did he make this radical break with the conception if space as a pre-existing void in which nature plays out its events in time? Arguably, this was his way of getting rid of the Minkowski and other metrics that de Sitter had shown to be anti-Machian, delimiting acceptable solutions of his field equations to those consistent with inertia as a strictly gravitational interaction. As he put it toward the end of his article:

*The general theory of relativity rectified a mischief of classical dynamics. According to the latter, inertia and gravity appear as quite different, mutually independent phenomena, even though they both depend on the same quantity, mass. The theory of relativity resolved this problem by establishing the behavior of the electrically neutral point-mass by the law of the geodetic line, according to which inertial and gravitational effects are no longer considered as separate. In doing so, it attached characteristics to the aether [spacetime] which vary from point to point, determining the metric and the dynamical behaviour [sic.] of material points, and determined, in their turn, by physical factors, namely the distribution of mass/energy.*

*That the aether of general relativity differs from those of classical mechanics and special relativity in that it is not "absolute" but determined, in its locally variable characteristics, by ponderable matter. This determination is a complete one if the universe is finite and closed …*

One may reasonably ask, if Einstein was convinced that general relativity, correctly interpreted, encompassed the gravitational induction of inertia, why today is it widely believed in the community of relativists and beyond that inertia is not gravitationally induced? That inertia is no better understood now than it was in the absolute systems of Newton and Minkowski? Carl Brans. And his "spectator matter" argument.
