the crucial physical question for the satisfaction of the equivalence principle is whether the geodesic represents a physical free fall.
Einstein [28] pointed out, ?s far as the prepositions of mathematics refers to reality, they are not certain; and as far as they are certain, they do not refer to reality." Thus, an application of a mathematical theorem should be carefully examined although ?ne cannot really argue with a mathematical theorem [18]". If, at the earlier stage, Einstein? arguments are not so perfect, he seldom allowed such defects be used in his calculations. This is evident in his book, ?he Meaning of Relativity' which he edited in 1954. According to his book and related papers, Einstein? viewpoints on space-time coordinates are:
1) A physical (space-time) coordinate system must be physically realizable (see also 2) & 3) below).
Einstein [29] made clear in ?hat is the Theory of Relativity? (1919)' that ?n physics, the body to which events are spatially referred is called the coordinate system." Furthermore, Einstein wrote ?f it is necessary for the purpose of describing nature, to make use of a coordinate system arbitrarily introduced by us, then the choice of its state of motion ought to be subject to no restriction; the laws ought to be entirely independent of this choice (general principle of relativity)". Thus, Einstein? coordinate system has a state of motion and is usually referred to a physical body. Since the time coordinate is accordingly fixed, choosing a space-time system is not only a mathematical but also a physical step.


2) A physical coordinate system is a Gaussian system such that the equivalence principle is satisfied.
One might attempt to justify the viewpoint of accepting any Gaussian system as a space-time coordinate system by pointing out that Einstein [3] also wrote in his book that ?n an analogous way (to Gaussian curvilinear coordinates) we shall introduce in the general theory of relativity arbitrary co-ordinates, x1, x2, x3, x4, which shall number uniquely the space-time points, so that neighboring events are associated with neighboring values of the coordinates; otherwise, the choice of co-ordinate is arbitrary." But, Einstein [3] qualified this with a physical statement that ?n the immediate neighbor of an observer, falling freely in a gravitational field, there exists no gravitational field." This statement will be clarified later with a demonstration of the equivalence principle (see eqs. [6] & [7]).
3) The equivalence principle requires not only, at each point, the existence of a local Minkowski space2)

ds2 = c2dT2 - dX2 - dY2 - dZ2, (1)

but a free fall must result in a co-moving local Minkowskian space (see also [10-13]). Note that the equivalence principle requires that such a local coordinate transformation be due to a specific physical action, acceleration in the free fall alone. Einstein [2] wrote, " For this purpose we must choose the acceleration of the infinitely small (?ocal") system of co-ordinates so that no gravitational field occurs; this is possible for an infinitely small region."
Also, for a Lorentz manifold, if a ?ree fall" results in a local constant metric, which is different from Minkowski metric, then the equivalence principle is not satisfied in terms of physics. Einstein [2] wrote, "...in order to be able to carry through the postulate of general relativity, if the special theory of relativity applies to the special case of the absence of a gravitational field."
According to Einstein, the body to which events are spatially referred is called the coordinate system. To be more precise, a spatial coordinate system attached to a body (i.e., no relative motion nor acceleration) is its ?rame of reference" [2,3]. These coordinates together with the time-coordinate form the space-time coordinate system. A frame of reference can be chosen physically and, due to the equivalence principle, the time-coordinate is determined accordingly (壯 5 & 6). Thus, one may call loosely the frame of reference as a coordinate system. In this paper, for the purpose of considering a satisfaction of the equivalence principle, a frame of reference and a related space-time coordinate system, are distinguished as above.
To clarify the theory, Einstein [3] wrote, ?ccording to the principle of equivalence, the metrical relation of the Euclidean geometry are valid relative to a Cartesian system of reference of infinitely small dimensions, and in a suitable state of motion (free falling, and without rotation)." Thus, at any point (x, y, z, t) of space-time, a ?ree falling" observer P must be in a co-moving local Minkowski space L as (1), whose spatial coordinates are attached to P, whose motion is governed by the geodesic,



= 0, where , (2)

ds2 = g((dx(dx( and g(( is the space-time metric. The attachment means that, between P and L, there is no relative motion or acceleration. Thus, when a spaceship is under the influence of gravity only, the local space-time is automatically Minkowski. Note that the free fall implies but is beyond just the existence of ?rthogonal tetrad of arbitrarily accelerated observer" [4].
Einstein? equivalence principle is very different from the version formulated by Pauli [10, p.145], ?or every infinitely small world region (i.e. a world region which is so small that the space- and time-variation of gravity can be neglected in it) there always exists a coordinate system K0 (X1, X2, X3, X4) in which gravitation has no influence either in the motion of particles or any physical process." Note that in Pauli? misinterpretation, gravitational acceleration as a physical cause is not mentioned, and thus Pauli? version3), which is now commonly but mistakenly regarded as Einstein? version of the principle [30], actually is not a physical principle. Based on Pauli? version, it was believed that in general relativity space-time coordinates have no physical meaning. In turn, diffeomorphic coordinate systems are considered as equiva