. 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 ¥Bree 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 ¥Krthogonal tetrad of arbitrarily accelerated observer" [4].
Einstein£r equivalence principle is very different from the version formulated by Pauli [10, p.145], ¤ƒor 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£r misinterpretation, gravitational acceleration as a physical cause is not mentioned, and thus Pauli£r version3), which is now commonly but mistakenly regarded as Einstein£r version of the principle [30], actually is not a physical principle. Based on Pauli£r version, it was believed that in general relativity space-time coordinates have no physical meaning. In turn, diffeomorphic coordinate systems are considered as equivalent in physics [21] not just in certain mathematical calculations. However, according to Einstein£r calculations [2,3], this is simply not true (see section 3).
The initial form of the equivalence principle is a relation between acceleration and gravity. However, in the above clarification, the role of acceleration is not explicitly shown. One may ask if acceleration does not exist for a static object, would the equivalence principle be satisfied? One must be careful because a geodesic may not represent a physical free fall.
There are three physical aspects in Einstein£r equivalence principle as follows [3]:
1) In a physical space, the motion of a free falling observer is a geodesic.
2) The co-moving local space-time of an observer is Minkowski, when 1) is true.
3) A physical transformation transforms the metric to the co-moving local Minkowski space.
Point 3) must indicate that this physical local coordinate transformation is due to the free fall alone. In other words, the physical validity of the geodesic 1) is a prerequisite for the satisfaction of the equivalence principle, and validity of 3) is an indication of such a satisfaction. Thus, a satisfaction of the equivalence principle is beyond the mathematical tangent space (‰Ñ 5-7).
Perhaps, this inadequate understanding is, in part, due to the fact that it is often difficult to see the physical validity of point 2) directly, i.e., how the metric transformed automatically to a local Minkowski space. To this end, examining point 1) and/or point 3) would be useful. Point 1) is a prerequisite of the equivalence principle point 2). For Point 1) to be valid, i.e., the geodesic representing a physical free fall, it is required that the metric of such a manifold should satisfy all physical principles. Needless to say, such a metric must be physically realizable. If point 1) is valid in physics, point 3) should produce valid physical results. Thus, one can check point 3) to determine the validity of

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