The mathematics of gravitation theory is remarkable for its expansibility and physical ambiguity. To a large extent it applies equally well to an interpretation of gravitation as a force and as a geometric distortion of spacetime. But given the pre-relativistic association of gravitation with force, that ambiguity, combined with the current primacy of mathematics in the interpretation of physical phenomena, has led to an overextension of the mathematics and resulted in theoretical misdirection.
Einstein’s heuristic insight leading to the General Theory derived from a consideration that the ratio of circumference to diameter of a rotating disk will deviate from pi with relativistic accelerations at the radius. In his original pursuit of a generalization of relativity, where he hypothesized the equivalence of inertial acceleration and gravitation, the similarity of the inertial effect on the rotating disk and the gravitational pressure we experience at the earth’s surface suggested that gravitation might be explicable as a geometric principle. Experimentation has confirmed the validity of that seminal insight, and the service of the mathematical analogy. But in the kinematical similarity between objects on a rotating disk and in a gravitational orbit there is a distinct physical difference. A test particle in a box fixed at the edge of a rotating disk presses against the radial wall of the box, manifesting a centrifugal “force”, derivative of the actual force that is rotating the disk; in contrast, a test particle in a box orbiting a massive body floats freely, following its geodesic in spacetime, and gives no indication of the presence of a force or acceleration. There is thus a mathematical analogy due to the similar kinetics of the rotating disk and the orbiting body, but not a physical equivalence.
The subsequent development of the Field Equations was based on another mathematical analogy, formalizing the behavior of bodies being accelerated or pressured toward an attractive or determinant center, as in a field of force or field of gravity. The analogy holds in this case because gravity, like a field of force, produces a typically curved, actually concentric form to the relative motion of affected bodies. But again, the mathematical analogy is not a physical equivalence. A neutral test particle inside a charged box accelerating toward the vertex of a field of force presses against the wall opposite the direction of force, and a non-neutral particle of different mass than the box accelerates at a different rate than the box; in contrast, a particle in a box falling in a gravitational field floats freely, following its geodesic in spacetime in parallel with the box, and gives no indication of the presence of a force or acceleration.
In each case, the rotating disk or orbiting body and the attractive or determinant field, there is a discernable difference in the empirical behavior of test particles being acted upon by a force and those moving in a gravitational field. The mathematical analogy between gravitation and force is limited to the trajectories of idealized, dimensionless particles.
Empirical gravitational phenomena consist in the distortion or compression of spacetime in the presence of mass, the relative curvature of geodesic motion in the presence of a gravitational distortion, and the static acceleration of bodies when their geodesics are resisted at the surfaces of large masses. In these descriptions there is no indication that gravitation might somehow produce energy or manifest a force.
We are left to question how the predictions of gravitational waves and “gravitomagnetic” effects can be justified, as both are based on the supposition of a gravitational energy. They are mathematical extrapolations from the Field Equations, drawn from electromagnetic analogies. It is here that the physical ambiguity and indifference of mathematics has been misleading gravitation theory, and consequently, it is here that the derivative predictions of General Relativity remain unconfirmed. In the absence of a coherent physical theory that could somehow link non-energetic phenomena with the production of energy, there is no reason to expect such predictions will ever be confirmed.