Mach’s principle, asserting that all inertial forces are due to the distribution of matter in the universe, played a pivotal role in Einstein’s theory of general relativity—but for the wrong reason!
Newton, on the other hand, was a firm believer in “absolute” space, existing in its own right devoid of matter or its influences. According to him, if a single object existed in the universe there would be no way of telling how fast it was going or what direction it was moving in. But if it accelerates, it is possible to determine what direction it is moving in. The example used by Newton of a rotating body is valid only in Euclidean space [for its non-Euclidean generalization see: A New Perspective on Relativity for counter-intuitive behavior].
When a body accelerates its direction is parallel to the axis of rotation. Superimposed on the uniform rotation is the centrifugal force which attempts to distort the uniform motion. For a sphere, this would be observed by the flattening at the poles, and the bulging at the equator. All that is necessary then is to measure these distortions in order to determine how fast the body rotates. But, this rotation is not with respect to any other bodies, rather, with respect to “absolute” space itself. In other words, space is not subordinate to matter, and can exist independently of it.
This idea was criticized by many of Newton’s contemporaries including Berkeley and Leibniz. There can be no space where there is matter, and everything is relative to the “sky of fixed stars.”
When Einstein came onto the world stage at the beginning of the twentieth century, he had high hopes of incorporating Mach’s principle into this theory of relativity, for only then would his theory be “relative.” Even as late as 1917, we hear Einstein claiming that “In a consistent theory of relativity, there can be no inertia relative to space, but only an inertia of masses relative to one another. If, therefore, I have a mass at a sufficient distance from other masses in the universe, its inertia must fall to zero.”
Yet, the theory he came out with is the exact antithesis. There can be absolute space with its own geometry (gravitation) even in the absence of matter. According to Dirac, the “emptiness” of the universe is not disturbed by the gravitational field; all other fields do disturb it. Thus, the vanishing of the eigenvalues of the Ricci tensor became Einstein’s condition of emptiness and from which Schwarzschild was able to solve Einstein’s field equations for its “vacuum” solution. If the universe were really empty, the components of the Ricci tensor would vanish identically, and there would be nothing to solve.
Yet, mass does appear in two forms: a constant central mass that arises by identifying an arbitrary constant of integration in the asymptotic weak field limit, and a “test” mass that is associated with proper time, as distinct from laboratory time. So Newtonian gravity was used as an asymptotic limit, in a theory that negated inertial forces, and relative motion could be accounted for even in the absence of inertial forces. These are two contradictory statements that are fused into a single theory!
So we have a universe without mass, although mass makes its appearance—a bewildering predicament. And inertia exists without reference of the other fixed masses in the universe, since there are none. The motion of the “test” particle follows geodesics (constant velocity curves where the component of acceleration vanishes in the plane of the motion) which are consequence of the field equations themselves, and not a separate law like Lorentz’s force which unites particle motion to Maxwell’s field equations.
By definition of geodesic motion, the tangential component of acceleration must vanish, leaving possibly a component of acceleration normal to the plane of motion. It thus seems absurd that "the equations of general relativity not only predict the usual radial Newtonian gravitational force behavior of a stationary mass on a stationary test body, but they also predict that a moving mass can create forces on a test body which are similar to the usual centrifugal and Coriolis forces, although much smaller." [R L Forward, "Guidelines to Anitgravity" AJP 31 (1963) 166-170] But, we have just finished say that centrifugal forces lie in the plane of the motion and tend to distort the uniformity of rotation. This is contrary to geodesic motion that Einstein insists his "test" particle follows!
Contrary to common belief, there are no “unusual” forces that create accelerations that are independent of the “test” body and the forces are indistinguishable from the usual Newtonian gravitational force. In Newton’s world the law of superposition reigns supreme, whereas it Einstein’s universe it is non-existent due to the inherent nonlinear character of the theory. Mass exists independently of all other masses, and the gravitational field can exist even in the absence of matter. General relativity is anything but relative!
Forward continues by saying that "In addition, when the general field theory equations are linearized, they result in a set of dynamical gravitational field equations similar to Maxwell's relations [sic]. Thus one can use intuitive pictures from electromagnetic theory to design theoretical models. Whether the effects predicted by the linearized theory really exist, will, of course, have to be checked by repeating the calculation with the nonlinearized field equations."
One case in point is the gravitational wave phenomenon, which arises from linearizing the Ricci tensor. No one has "checked" if the nonlinearized field equations bear out the linearized predictions. And to make matters even worse, there are no known periodic solutions to the Einstein field equations. So where does this leave gravitational waves?
What Forward and other were mistaken in, is that even though the linearized field equations bear a striking resemblance to Maxwell's equations, there is one stark difference: Whereas Maxwell needs Lorentz to make contact with particle motion, Einstein does not. The geodesic nature of the motion has already been incorporated into his field equations, and these clearly outlaw fictitious forces like centrifugal and Coriolis forces.