lent in physics [21] not just in certain mathematical calculations. However, according to Einstein? 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? 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 (壯 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 ?ree 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 壯6 & 7).
3. Einstein? 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 ?ustified" 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? 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 壯5-7.) Consider an observer P at (x0, y0, z0, t0) in a ?ree 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 the light-cone condition,

ds2 = 0. (8)

Then, to the first order approximation, the velocity of light is expressed in our selected coordinates S by

= c(1 - ). (9)

It is crucial to note that the light speed (9), for an observer P1 attached to the system S at (x0, y0, z0), is smaller than c; and this condition is required by the coordinate relativistic causality for a physicall