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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 |
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