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ble mathematical choice but not a physical result as it should be. This omission inadvertently gives the opportunity of a misinterpretation that any Lorentz manifold could be considered as a physical space because only the signature of the metric of a manifold is examined while other physical requirements are ignored. Nevertheless, Einstein [9] did not point out clearly the relation between the need of a valid space-time coordinate system and a satisfaction of Einstein’s equivalence principle. This seems to have an effect that helps the acceptance of Pauli’s view. An implicit physical requirement for Einstein’s version of infinitesimal equivalence principle (section 6) is that it must be applied to a physical space. In a free fall, due to the effect of physics, a local space must be uniquely Minkowskian. For example, when a space ship is under the influence of gravity only, the local space-time is known to be automatically Minkowskian, as determined by the physics of gravity. Moreover, if the manifold under consideration is a physical space (-time), it satisfies all physical principles sufficiently. Thus, the mathematical existence of a local Minkowski space need not mean a satisfaction of Einstein’s equivalence principle that depends on the physical validity of the local coordinate transformation (see sections 3 & 4). It follows that, if the conditions for a physical space are taken into consideration, Einstein’s equivalence principle is not exactly a local principle as Fock [11] misinterpreted. 3. The Initial Form of Einstein's Equivalence Principle and the Gravitational Red Shifts In 1911, Einstein [8] derived the gravitational red shifts from the initial form of his principle, the equivalence of a uniformly accelerated frame and the uniform gravity. This is independent of the need of a Riemannian space with a Lorentz signature, which is additionally due to the principle of general relativity and special relativity [1]. A known deficiency of his results then is an incorrect formula for light speeds under gravity. Einstein [15,16] corrected this formula in 1915. Nevertheless, an unverified belief advocated by Fock [11] and others [13,17] is that Einstein’s equivalence principle would be incompatible with the notion of a curved space. Based on this belief, Fock [11] also speculated that the metric related to uniform gravity would have only the form as follows: ds2 = (c2- 2U) dt2 – dx2 - dy2 - dz2, (1) where U is the gravitational potential although his own calculation does not support this speculation (section 7). Such a belief must be very absurd to Einstein since his argument for a Riemannian space is based on his equivalence principle [1,15]. Recently, such a belief has been proven to be fundamentally incorrect since Maxwell-Newton Approximation, the linear field equation for weak gravity, that produced a valid light bending, has been derived [17] with Einstein’s equivalence principle (see section 4) together with the notion of a Riemannian space if Newtonian theory is taken as a form of first order approximation. Another mistake of Ohanian and Ruffini [13] is that, instead of the Maxwell-Newton Approximation, their linear equation is based on the linearized conservation law, which has been proven to be invalid for gravity by Wald [18] and Yu [19]. They [13] also claimed, in disagreement with Eddington [2] and Pauli [3], that the covariance prin