the z-direction. Metric (24b) does not satisfy point 2) of the equivalence principle since there is no physical cause for transformation (27b). In relativity, such a physical transformation happens only when there is relative motion or acceleration. But, P' is rest at K'. Thus, (27b) illustrates also that geodesic (26) does not represent a physical free fall.


A misunderstanding of the equivalence principle, as Yu (p. 42 of [23]) believed, is that at any space-time point, it is always possible to establish a local Minkowski space, which is related to a ?ree fall". However, this is necessary but insufficient. For instance, at any space-time point of manifold (18a), (22b) or (24b), there is a local Minkowski space, which is co-moving with a ?ree falling" observer in the manifold. But, the geodesic does not represent a physical free fall. Note that Yu? interpretation is essentially rephrasing Pauli? misinterpretation [3, p.145].
The Galilean transformation is an unphysical transformation, and it simply takes another unphysical transformation to cancel out the unphysical properties so introduced. In fact, (24a), and (27b) imply

dt = ( (dT - v/c2dZ) , and dZ = (dz' = ( (dz + v dt). (27c)

Transformation (27c) is just a Lorentz-Poincaré transformation. (27b) completes the transformation (27c) starting form (24a).
It has been shown in different approaches that metric (24b) is incompatible with physics and in particular the equivalence principle. Since (24a) is a Galilean transformation, the Galilean transformation is also not physically valid in general relativity. The failure of satisfying the equivalence principle should be expected since the Galilean transformation is experimentally not realizable. This analysis shows also that the Minkowski metric is only valid constant metric in physics. In fact, a general result is that if ((tt = 0 for ( = x, y, or z, then the equivalence principle is satisfied only if the metric is Minkowski.
Another consequence is the reaffirmation of coordinate relativistic causality in vacuum. That the speed of light could be larger than c through a coordinate transformation is inconsistent with the notion that the light speed c is the maximum possible speed. The equivalence principle rules out such a possibility. It thus follows that physically the speed of light cannot be larger than c at the presence of gravity. In fact, observation confirms that gravity only leads to a reduction of the light speed.
It has been illustrated that the Galilean transformation is incompatible with the equivalence principle in the absence of gravity. In fact, the incompatibility is also true even when gravity is present. To illustrates this, let us consider physical metric (4b) and the physical situation that a particle at (0, 0, z0, t0) moving with velocity v at the z-direction. The Galilean transformation (24a) transforms metric (4b) to

ds2 = c2(1 - )dt'2 - (1 + )(dx'2 + dy'2 + [dz' - v dt? 2 (28a)

If metric (28a) had a physical realizable coordinate system S', the particle would be at (0, 0, z'0, t'0) in the state (0, 0, 0, dt') and the local spatial coordinates dx', dy', and dz' would be attached to the particle at the instance t'0. The problem can be reduced to previous case by considering the limits (? 0.
Moreover, according to Einstein [3], the equivalence principle is valid only if ds2 = 0 produces the correct light speeds. Thus, if S' were realizable, the light speeds in the z-direction would be



= c(1- () + v, or = -c(1- () + v, where ( = (28b)

according to metric (28a). Thus, coordinate relativistic causality is violated for sufficiently large r. In other words, point 1) of the equivalence principle cannot be satisfied and metric (28a) is not realizable.
This illustrates that the equivalence principle is a requirement for a valid physical space-time coordinate system.
7. Restriction of Covariance, and Intrinsically Unphysical Lorentz Manifolds
Einstein proposed that the equivalence principle is satisfied in a physical space-time1). Moreover, the equivalence principle is satisfied only in a physical space-time since the existence of a local Minkowski space has been proven by mathematics and a satisfaction of the equivalence principle requires sufficient satisfactions of all physical conditions. For example, when coordinate relativistic causality is not satisfied, the equivalence principle is proven directly to be not valid for this manifold. The current confusion was due to that the equivalence principle has not been understood correctly from the viewpoint of physics.
However, one may still wonder whether a Lorentz manifold is always diffeomorphic to a physical space. If this were true, then the metric signature would be essentially equivalent to the equivalence principle. But, there are Lorentz manifolds any of which cannot be diffeomorphic to a physical space. In view of this, such misunderstanding of relativity must be rectified. Since the belief that a Lorentz manifold were diffeomorphic to a physical space, has never been proven; the burden of proof is on such believers. Nevertheless, it is desirable to give an example of an intrinsically unphysical Lorentz manifold. This can even be a solution of Einstein? equation if it fails a physical requirement, which is independent of a coordinate system [8,9,16].
For instance, an accepted solution of metric for an electromagnetic plane wave [38] is

ds2 = du dv + H du2 - dxi dxj , where H = hij(u)xixj, hii(u) 0, hij = hji , (29)

u = t - z, v = t + z. This is a Lorentz manifold since its eigen values are H ( (H2 + 1)1/2, -1, and -1. However, since the condition 1 ( (1 + H)/(1 - H) may not be valid, metric (29) does not satisfy coordinate relativistic causality and therefore the equivalence principle. Moreover, since H can be arbitrarily large, metric (29) is incompatible with Einstein? notion of weak gravity4) and the correspondence principle. Also, in the light bending experiment, the gravitational effect of the light is implicitly assum