l free fall [9,16]. In other words, given the mathematical existence of local Minkowski space co-moving along a time-like geodesic, 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 ¥Kne cannot really argue with a mathematical theorem [18]". If, at the earlier stage, Einstein£r arguments are not so perfect, he seldom allowed such defects be used in his calculations. This is evident in his book, ¡vhe Meaning of Relativity' which he edited in 1954. According to his book and related papers, Einstein£r 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 ¡yhat 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£r 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 (¥Hocal") 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 ¥Bree 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 ¥Brame 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 (‰Ñ 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 ¥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 point 1) or vice versa.
The mathematical existence of a co-moving Local Minkowski space along a ¥Bree fall" geodesic implies only that Riemannian geometry is compatible with the equivalence principle. The physics is whether the existence of a physical local transformation which transforms the metric to the co-moving local Minkowski space. This is possible only if the geodesic represents a physical free fall, i.e., the equivalence principle ensures the existence of such a physical transformation. Thus, one must carefully distinguish mathematical properties of a Lorentz metric from physical requirements. Apparently, a discussion on the possible failure of satisfying the equivalence principle was over-looked by Einstein and others (see ‰Ñ6 & 7).
3. Einstein£r Illustration of the Equivalence principle
Einstein [3] illustrated his equivalence principle in his calculation of the light bending around the sun. (Note, the other method does not have such a benefit.) His 1915 equation for the space-time metric g(( is

G(( ( R(( -R g(( = -KT(m)(( (3)

where K is the coupling constant, T(m)(( is the energy-stress tensor for massive matter, R(( is the Ricci curvature tensor, and R = R((g((, where g((, is the inverse metric of g((. Now, he considered a coordinate system S (x, y, z, t) with the sun attached to the spatial origin. Based on eq. (3), and the notion of weak gravity, Einstein ¥Fustified" the linear equation,

= 2K(T(( - g((T), where ( (( = g(( - ((( , (4a)

((( is the flat metric, and T = T((g((. Then, from eq. (4a), and Ttt = (, otherwise T(( is zero, by using the asymptotically flat of the metric, Einstein obtained, to a sufficiently close approximation, the metric for coordinate system S

ds2 = c2(1 - )dt2 - (1 + )(dx2 + dy2 + dz2), (4b)

where ( is the mass density and r2 = x2 + y2 + z2.
However, since eq. (3) itself is questionable for dynamic problems [7,8,24-26], it is necessary to justify eq. (4a) again. Also, the notion of weak gravity may not be compatible with the principle of general covariance [3]. In the next section, eq. (4a) will be justified directly and is independent of the details of higher order terms of an exact Einstein equation. For this reason and the dynamical incompatibility with eq. (3), eq. (4a) is called the Maxwell-Newton Approximation [7]. In other words, eq. (4a) should be valid for dynamic problems. Also, an implicit assumption of Einstein£r calculation is that the gravitational effects due to the light itself, is negligible. To address this issue theoretically, would be complicated and is beyond the scope of this paper [9]. Here, this negligibility is justified from the viewpoint of practical observations only.
Now, according to the geodesic eq. (2), one has d2x /ds2 = 0 for x( (= x, y, z) since (gtt/(x( ( 0. Thus, the gravitational force is non-zero, and the equivalence principle would be applicable. (For the non-applicable cases, please see ‰Ñ5-7.) Consider an observer P at (x0, y0, z0, t0) in a ¥Bree falling" state,

dx/ds = dy/ds = dz/ds = 0. (5)

According to the equivalence principle and eq. (1), state (5) implies the time dt and dT are related by

c2(1 - )dt2 = ds2 = c2dT2 (6)

since the local coordinate system is attached to the observer P (i.e., dX = dY = dZ = 0 in eq. [1]). This is the time dilation of metric (4b). Eq. (6) shows that the gravitational red shifts are related to gtt, and is compatible with his 1911 derivation [2]. Moreover, since the space coordinates are orthogonal to dt, at (x0, y0, z0, t0), for the same ds2, eq. (6) implies [3]

(1 + )(dx2 + dy2 + dz2) = dX2 + dY2 + dZ2 . (7)

On the other hand, the law of the propagation of light is characterized by t

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