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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. |
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