(You may obtain a PDF version of this lab note at Electrodynamic Geodesics) Note: See also Part 3 of this Lab Note, Gravitational and Electrodynamic Potentials, the Electro-Gravitational Lagrangian, and a Possible Approach to Quantum Gravitation, which contains further development.
1. Introduction It has been understood at least since Galileo’s refutation of Aristotle which legend situates at the Leaning Tower of Pisa, that heavier masses and lighter masses similarly-disposed in a gravitational field will accelerate at the same rate and reach the ground after identical times have elapsed. Physicists have come to describe this with the principle that the “gravitational mass” and the “inertial mass” of any material body are “equivalent.” As a material body becomes more massive and so more-susceptible to the pull of a gravitational field (back when gravitation was viewed as action at a distance), so too this increase in massiveness causes the material body in equal measure to resist the gravitational pull. By this equivalence, the result is a “wash,” and so with the neglect of any air resistance, all the bodies accelerate and fall at the same rate. (The other consequence of Galileo’s escapade, is that it strengthened the role of experimental testing, in relation to the “pure thought” upon which Aristotle had relied to make the “obvious” but untested and in fact false argument that heavy objects should fall faster. In this way, it spawned the essence of what we today know as the scientific method which remains a dynamic blend of thought and creativity, with experience and cold, hard numbers derived from measurement of masses, lengths, and times.)
Along his path to developing the General Theory of Relativity (GTR), Albert Einstein made a brief stop in 1911 in an imaginary elevator, to conduct a gedanken in which he concluded that the physical experience of an observer falling freely in a gravitational field before terminally hitting the ground is no different from what was commonly thought of as Newton’s inertial motion in which a body in motion remained in motion unless acted upon by a “force.” (GTR later showed that this was not quite true, the “asterisk” to this insight arising from the so-called tidal forces.) And, he concluded that the force one feels standing on the floor of an elevator in free fall to which a constant force is then applied, is no different from the force one feels when standing on the surface of the earth.
The General Theory of Relativity, in the end, captured inertial motion and its close cousin of free-fall motion in a gravitational field, in the most elegant way, as simple geodesic motion in a curved geometry along geodesic paths which coincide precisely with the paths one observes for bodies moving under gravitational influences. This was a triumph of the highest order, as it placed gravitational theory on the completely-solid footing of Riemannian geometry, and became the “gold standard” against which all other physical theories are invariably measured, even to this day. (“Marble and wood” is another oft-employed analogy.)
However, the question of “absolute acceleration,” that is, of an acceleration which is not simply a geodesic phenomenon of unimpeded free fall through a swathe carved out by geometry, but rather one in which an observer actually “feels” a “force” which can be measured by a “weight scale” in physical contact between the observer and that body which applies the force, is in fact not resolved by GTR. To this day, it is hotly-debated whether or not there is such a thing as “absolute acceleration.” Surely, the forces we feel on our bodies in elevators and cars and standing on the ground are real enough, but the question is whether there is some way to understand these forces — which are impediments to what would otherwise be our own geodesic free fall motion in spacetime under the influence of gravity and nothing more — as geodesic forces in their own right, simply of a different, supplemental, and perhaps more-subtle character than the geodesics of gravitation. That is the central question to be examined in this lab note. (more…)