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## Classical

Gravitation is characterized by the fact that every test particle traces out a world line which is independent of the parameters characterizing the particle, most notably its mass. Thus, cutting the mass of a planet into half has no effect on its motion in a gravitational field. No intrinsic mass parameter is needed to specify a particle's motion. The world line is determined entirely by the particle's local environment, not by the particle's intrinsic structure such as its mass. We shall refer to this structure independence as the ``Eotvos property'' [1].

This property is the key to making gravitation subject to our comprehension. This independence is far reaching, because, with it, the world lines have a remarkable property: They are probes that reveal the nature of gravitation without themselves getting affected by the idiosyncracies (e.g. mass) of the particles. The imprints of gravitation are acquired without the introduction of irrelevant features such as the mass of the system that carries these imprints. Put differently, the classical world lines of particles highlight a basic principle: Gravitation is to be identified by its essentials.

Einstein used this principle to acquire our understanding about gravitation within the framework of classical mechanics. His line of reasoning was as follows:

(1) He identified an ``accelerated frame'' as a one-parameter family of locally inertial (=free-float) frames [2]. The free-float nature of each of these frames he ascertained strictly within the purview of classical mechanics, namely by means of the straightness of all particle world lines (Newton's first law of motion) [3].

(2) Next he observed that the afore-mentioned Eotvos property of the particle world lines implies and is implied by the statement (equivalence principle) that the motion of particles, falling in what was thought to be a non-accelerated frame with gravity present, is physically equivalent to the motion of free particles viewed relative to an accelerated frame [4,5,6].

(3) With this observation as his staring point, he used Lagrangian mechanics to characterize gravitation in terms of the metric tensor [4,5,6], and then proceeded towards his theory of gravitation [7] along what in hindsight is a straight forward mathematical path [8]

However happy we must be about Einstein's gravitation theory, we must not forget that it rests on two approximations, which, although very fruitful, are approximations nevertheless. They are made in the very initial stages of Einstein's line of reasoning, namely in what he considered an ``accelerated frame'' and in how he used it.

First of all, in step (1) above, Einstein approximates an ``accelerated frame'' as one in which its future and past event (Cauchy) horizons are to be ignored. Such indifference is non-trivial. It results in neglecting the fact that (i) a frame accelerating uniformly and linearly always has a twin, which moves into the opposite direction and (ii) that it takes these two frames (``Rindler frames'') to accomodate a Cauchy hypersurface.

Secondly, the geodesic world lines, the carriers of the imprints from which he constructs his theory of gravitation, are only classical approximations to the quantum mechanical Klein-Gordon wave functions. Their domain extends over all (the regions) of spacetime associated with the pair of accelerated frames. The world lines, by contrast, have domains which are strictly limited to the razor sharp classical particle histories.

It would have been difficult to argue with these approximations ninety years ago. However, the knowledge we have gained in the meantime would expose us to intellectual evasion if we were to insist on adhering to them in this day and age. Thus we shall not make them.

Next: Quantum Mechanical Up: THE EOTVOS PROPERTY Previous: THE EOTVOS PROPERTY
Ulrich Gerlach
1998-02-20