ed to be negligible. Thus, metric (29) cannot be valid in physics.
Nevertheless, to show that metric (29) cannot be diffeomorphic to a physical space, needs more work. The gravitational force (related to (ztt = (1/2)((hijxixj)/(t has arbitrary parameters (the coordinate origin). This arbitrariness in the metric violates the principle of causality (i.e., the causes of phenomena are identifiable) [8,11]. Thus, the manifold (29) cannot be diffeomorphic to a physical space since a diffeomorphism cannot eliminate the parameters, which violate the principle of causality.


8. Conclusions and Discussions
Einstein [2,3] proposed the equivalence principle for the reality, which he models as a Riemannian physical space-time. However, Pauli? [10, p.145] version implies that the equivalence principle would be satisfied even though the coordinate system is not physically realizable. Now it is clarified that Einstein correctly objected Pauli? version as a misinterpretation [30]. Also, it is proven that the equivalence principle is satisfied if and only if a manifold is physically realizable.
In general relativity, the Minkowski metric in special relativity is obviously a special case. However, it was not clear that the ?rinciples" which lead to general relativity are compatible with each other even in this special case. Some theorists believe incorrectly that the Galilean transformation were valid for general relativity, although Einstein [2] has made clear, ?pecial theory of relativity applies to the special case of the absence of a gravitational field". To rectify this, it is shown directly that, due to the equivalence principle, the Minkowski metric is the only valid constant space-time metric (§6).
To establish special relativity, the Galilean transformation is proven to be unrealizable by experiments. Thus, theoretically a Galilean transformation should be incompatible with the equivalence principle, which is applicable to only a physical space. This means, in contrast of the belief of some theorists [14,15], that the equivalence of all frames of reference is not the same as the physical equivalence of all mathematical coordinate systems. In fact, it is invalid in physics to extend the space-time physical coordinate system to an arbitrary Gaussian system [9]. For instance, the time coordinate is not arbitrary [2,3].
The Galilean transformation implied that there is no limit on the velocity of light. This, in principle, disagrees with the notion of invariant light speed. However, due to entrenched misconceptions on covariance, this problem was not even recognized for further investigations [20,23]. Moreover, some supported such a misconceptions with other errors and misunderstandings. In other words, such current ?heories" are characterized and maintained with a system of errors. Thus, it is necessary to calculate examples that directly demonstrate a violation of the equivalence principle.
Some theorists incorrectly claimed that the equivalence principle is equivalent to the mathematical existence of the tetrad. They over simplified Einstein? principle merely as the mathematical existence of a co-moving local Minkowski space along a time-like geodesic. However, the physics is not only just such an existence, but also the formation of such local space by the free fall alone. For instance, the local space-time of a spaceship under the influence of only gravity is a local Minkowski space. Thus, the real question for the equivalence principle is whether the geodesic represents a physical free fall.
The fact that there is a distinction between the equivalence principle and the proper metric signature would imply also that the covariance principle must be restricted. An important function of the equivalence principle test is to eliminate unphysical Lorentz manifolds (see §7). For example, the fact that metric (29) is intrinsically unphysical resolves its seemingly paradox with the light bending calculation in which the gravity due to the light is implicitly assumed to be negligible [2,3]. This is another example that a misunderstanding of the equivalence principle can leads to disagreements with experiments.


Perhaps, due to confusing mathematical theorems with Einstein? equivalence principle as Pauli did, this principle is often not explained adequately in some books [21-23]. To deal with all the theoretical inconsistence superficially, some theorists claimed that the space-time coordinates have no physical meaning in general relativity. Such a speculation disagrees with the fact that there are non-scalars in physics. The deflection of light is related to the light ray being observed as an almost straight line away from the sun, and gravitational red shifts are related to gtt - the time-time component of space-time metric.
Nevertheless, based on such an absurd claim, Hawking [18] declares, ?n general relativity, there is no real distinction between the space and time coordinates, just as there is no real difference between any two space coordinates." On the other hand, Hawking [18] also believed, ?n arrow of time, something that distinguished past from future, giving a direction of time". Apparently, he did not see that there is an inconsistency between these two statements. Moreover, like others, Hawking accepted the deflection of light. He probably did not realize that the deflection angle could be defined only in a certain type of physical coordinate systems, where the trajectory of a light ray, when far away from the sun, is approximately a straight line. Note that such logical deficiency is a common problem among those so-called ?tandard" relativists.
Theorists such as Synge [19], Fock [39] and more recently, Hawking [18,40], Ohanian, Ruffini, and Wheeler [22], who do not understand Einstein? equivalence principle for various reasons including inadequate understanding of physics or mathematics at the fundamental level or deficiency in logic, advocated essentially that the basis of general relati