**Abstract**

Gravitation is described as a uniquely geometric phenomenon, incompatible with the concepts of force, energy, and inertia, and only analogically comparable by means of mathematical formalizations. Two thought experiments are employed to demonstrate that the association of gravitation with force and inertia is irreconcilable with the geometric interpretation and without theoretical foundation or empirical support.

**Conceptualization of the General Theory**

The mathematics of relativistic gravitation theory is remarkable for its expansibility and physical ambiguity. To a large extent it can apply equally well to an interpretation of gravitation as a geometric distortion of spacetime and as a force. But given the pre-relativistic association of gravitation with force, that ambiguity, fomented by the consolidation and predominance of mathematics in the interpretation of physical phenomena, has resulted in an overextension of the mathematics and persistent theoretical misdirection.

Two principal
mathematical analogies can be identified in the development of relativistic gravitation
theory and implicated in its diversion. One derives from Einstein's heuristic
insight associating gravitation with geometry, apparently due to an idea
suggested by his friend Paul Ehrenfest ( 1909), who was himself inspired by Max Born's
investigation of relativistic rigidity ( 1909). Ehrenfest noted that
the ratio of circumference to diameter of a rotating disk would have to deviate
from *pi* with relativistic accelerations at the radius. In Einstein's subsequent pursuit of a
generalization of relativity the similarity between the inertial effect produced
at the radius of the rotating disk and the gravitational pressure we experience
at the earth's surface suggested that gravitation might be explicable as a
fundamentally geometric
principle. Experimentation has confirmed the validity of that seminal geometric insight and the service
of the mathematical analogy. But in the kinematical similarity between objects
on a rotating disk and in gravitational orbit there is a distinct empirical
difference: A test particle in a box that is 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 in
parallel with the box, 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 development of the field equations of General Relativity was based on another mathematical analogy , formalizing the behavior of bodies being accelerated or pressured toward an attractive or determinant vortex, as in a field of force or a collapsing, concentrating sphere. The analogy holds in this case because gravity, like a field of force, produces a typically concentric form to the motion of affected bodies. But again, the mathematical analogy is not a physical equivalence. A neutral test particle inside a charged box that is accelerating toward the vortex of a field of force presses against the wall of the box opposite the direction of force, and a non-neutral particle of different mass than the box accelerates at a different rate than the box, moving consequently toward one wall or its opposite. In contrast, a test particle in a box falling or spiraling 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.1

In each case --
in the similarities between the rotating disk or orbiting body and between the
attractive or determinant
field -- there is a discernible difference in the *empirical* behavior of
test particles being acted upon by a force and those moving in a gravitational field. In these
pivotal models grounding relativistic gravitation theory, the mathematical analogies between gravitation and force are
limited to descriptions of curvilinear trajectories of idealized, dimensionless
particles.

**Physics and mathematics**

The special and general theories of relativity were conceptual in origin and mathematical only in their corroboration and utilization. The general theory has represented gravitation as a product of the "curvature" or deformation of spacetime in the presence of mass, and both the evidence and the supportive mathematics have been entirely adequate to justify its acceptance. But the field equations of general relativity are indifferent to the basis of gravitational dynamics, and geometry is distinctly non-dynamic. Theorists who have sought to associate gravitation with force have consequently been compelled to develop extensions of the field equations, usually based on electromagnetic analogy. Gravitation has been described in terms of the mathematics of quantum theory as a

*force*and associated with a hypothetical particle, without either an explanation of the relationship between geometry and force or an explicit dissension from the geometric interpretation, and without empirical evidence of a particle.

Conceptual
physics -- which can be considered roughly coextensive with pre-quantum physics
-- involved the initial development of coherent hypotheses and *secondarily* the employment of
mathematics (and/or experiments) to support their plausibility. A mathematical
formalism without conceptual coherence would have been regarded as irremediably
provisional, if not unsatisfactory, in the former methodology. With respect to
the former, two thought-experiments will be employed below, without resort to
mathematics, to demonstrate that the association of gravitation with force is
conceptually flawed and without empirical support.

**Two thought experiments**

The first experiment would be unnecessary except the pre-relativistic association of gravitation with inertia, and of inertia with universal mass, is still maintained on occasion, if only tacitly, and may be the ultimate basis of the continued identification of gravitation with force. The identification may also be a residue of one of our most familiar and persistent experiences on the earth's surface: The pressure we feel between ourselves and the surface is fundamental to our original concept of gravitation; we tend to regard the pressure as a force ("the force of gravity") and our relatively static surface reference as being at rest. The following experiment may therefore be helpful toward more clearly dispelling the identification of gravitation with inertia, and also in prefacing the second experiment, which will illustrate the force-free continuity between astronomical gravitation and gravitation at the surface of a massive body.

Imagine a spacecraft traveling a uniform path relative to the "fixed stars" which comes under the influence of a stellar object nearby and begins to deviate toward it, while continuing in uniform motion by the evidence of free-floating objects inside. In order to maintain the original course a thruster is fired, and inertial effects are experienced onboard as the craft accelerates just enough to counter the influence of the local gravitational field and maintain the intended course.

In this experiment inertial
effects are associated not with gravitation, but with the *counteraction* of gravitational acceleration, and with *uniform* motion relative to the distant
stars, contrary to the pre-relativistic expectation. Aside from the
discrimination of inertia from any influence of the overall mass of the
universe (an association that is seldom explicitly defended now anyway), the
experiment demonstrates what I hold to be most significant, that at least in
the situation just described, force becomes evident in conjunction with
gravitation only when gravitation is being *resisted*.

Now consider an experiment that comprehends the transition from astronomical gravitation to an involvement with force and inertia at the surface of a massive body:

Imagine
two test bodies gravitating toward the earth from some considerable distance.
For the sake of simplicity, consider the earth to be at rest with the test
bodies gravitating toward its center of mass. (They appear to be simply "falling"
from a perspective on the earth's surface.) One body is an immense hollow
sphere of negligible mass, the other is relatively small in size -- an
extra-vehicular scientist, let's say -- and also of negligible mass. Notice
that while the test bodies are falling toward the earth (or more accurately,
while the three bodies are converging) there is among them a purely relative
transformation of potential energy to kinetic energy as each moves uniformly in
its own frame of reference -- there would be, at least as yet, no occasion for
an exchange of mass-energy in the form of the supposed *gravitational* energy.

Let the sphere and the scientist be placed initially close together so that as they approach the earth their geodesics converge enough to bring their surfaces in contact some time before the larger impact. (It is the fantastic size of the hollow sphere that allows the surfaces of the two bodies to meet somewhere above the earth's surface). From the moment the sphere and the scientist come in contact until they reach the surface of the earth a static inertial acceleration between them will intensify as each tries to conform to its own geodesic at an ever greater angle to the normal. The situation will, if viewed in isolation, come to resemble the gravitation of a small body pressing against a planetary surface (although the gravitation between them is actually insignificant due to their negligible masses) and the scientist will even be able to stand upon the sphere. This development of an increasing inertial acceleration between the test bodies is the only aspect of the situation that changes from the moment they meet; the earthward component of their motion continues as before, a relative gravitation.

In a
manner that is similar to the first experiment, force has developed in the *resistance* to what is in this case a
convergent gravitation of two bodies toward a third. And once the two reach the
earth the situation remains essentially the same: Each of them, now in
conjunction with the entire conglomerate of the earth, presses toward the
center of mass with the same sort of conflict of geodesics as was observed
between them when they were gravitating from a distance. Along with the other
components of the earth at and below the surface, they are resisted, and thereby
induced with a static acceleration, by those further below, due to the
coincidence of the common inclination toward the center of mass and the
subordinate obstructions.

This
second experiment demonstrates that it is only in the inertial conflict of geodesics
(or as in the first experiment, in a singular inertial acceleration) that force
can be observed in association with gravitational phenomena. The intersection
of geodesics and the consequent inertial effects constitute the *interruption* of gravitation, and what is
commonly conceived as "the force of gravity" at a surface could be more
accurately described as *anti-gravitation*.