In Defense of the Principle of Equivalence

I appreciate Mr. Moreira’s insight into the principle of equivalence. I was a little too brief in my explanation, and should have gone on to include the keywords "uniform gravitational field." In other words, the principle of equivalence is a limiting abstraction, applicable only to point objects, which do not exist, or absolutely parallel lines of gravitational force, which also do not exist, though for many practical cases, within experimental limits, they do. For example, we cannot perform the tidal forces experiment on a spherical drop of liquid and obtain measurable results. However, the lack of ability to measure the effect does not change the validity of the concept, and Mr. Moreira is correct in this aspect. If, on the other hand, we do have, in our local vicinity, a uniform (meaning parallel) gravitational field, then the statement holds that we can not distinguish free-fall through this field from free-fall in gravitational-free space, which is also an abstraction. In fact, it is precisely this at-best asymptotically realizable condition posed by the equivalence principle that gave rise to the view of the universe as a patchwork of locally, causally uniform Minkowski frames as originally suggersted by general relativity.

Mr. Moreira goes on to quote Synge, though on an unrelated version of the principle of equivalence. Dr. Synge states that the equivalence of acceleration due to gravity (experienced when one is fighting the gravitational field), and that of artificial or induced acceleration is incorrect since the field presumably exists even in the absence of a test particle. But Dr. Synge is incorrect in the following respect. The principle of equivalence in this case addresses only what the test particle experiences. The mathematical or physical nature of the fields or forces that produce the effect are inconsequential. To dismiss the assumptions that led to the formation of the Riemann tensors in the first place ("In Einstein’s theory, either there is…"), in favor of treating the theory of general relativity as an a priori given, is to commit grave error. General relativity is but one attempt at explaining observable nature. One must not state that nature is an attempt at modeling general relativity.

Again, this form of the principle of equivalence applies only in a uniform field, and only very locally, since moving through the field changes the acceleration as well. In fact, as the companion article in this issue shows, movement through the field actually changes the effect of the field on the test particle. We find that there is a dynamic component to Newton’s gravitational potential, which, when correctly applied, fully accounts for Mercury’s anomalous perihelion advance, and for photon deflection in a gravitational field.

An interesting addendum regarding the so-called "tidal forces" felt by an extended object in free-fall is the following. Since we are not assuming a uniform field, the test object experiences these forces. For example, a ten foot 2x4 in the presence of a strong gravitational field would become concave in shape. The force on the center of the plank, being closer to the center of the gravitating mass, would be greater than that on each end of the plank. To the extent that the body is deformable, the tidal forces would bend the plank. However, if we also do not have uniform acceleration, but instead apply the accelerative force to the center of the plank, the inertia of the ends of the plank would result in a similar deformation. The plank would again be concave in the direction of acceleration. In this way, a non-uniform gravitational field can be replaced by an equivalent, non-uniform acceleration, and we can apply the principle of equivalence even to extended objects (this is what Einstein did when introducing the dynamic, or parameterized, metric in general relativity), though there is neither a theoretical nor a practical reason for doing so.

Questions? Comments?crenshaw@teleinc.com