] wrote, " For this purpose we must choose the acceleration of the infinitely small (ocal") system of co-ordinates so that no gravitational field occurs; this is possible for an infinitely small region."
Also, for a Lorentz manifold, if a ree fall" results in a local constant metric, which is different from Minkowski metric, then the equivalence principle is not satisfied in terms of physics. Einstein [2] wrote, "...in order to be able to carry through the postulate of general relativity, if the special theory of relativity applies to the special case of the absence of a gravitational field."
According to Einstein, the body to which events are spatially referred is called the coordinate system. To be more precise, a spatial coordinate system attached to a body (i.e., no relative motion nor acceleration) is its rame of reference" [2,3]. These coordinates together with the time-coordinate form the space-time coordinate system. A frame of reference can be chosen physically and, due to the equivalence principle, the time-coordinate is determined accordingly (壯 5 & 6). Thus, one may call loosely the frame of reference as a coordinate system. In this paper, for the purpose of considering a satisfaction of the equivalence principle, a frame of reference and a related space-time coordinate system, are distinguished as above.
To clarify the theory, Einstein [3] wrote, ccording to the principle of equivalence, the metrical relation of the Euclidean geometry are valid relative to a Cartesian system of reference of infinitely small dimensions, and in a suitable state of motion (free falling, and without rotation)." Thus, at any point (x, y, z, t) of space-time, a ree falling" observer P must be in a co-moving local Minkowski space L as (1), whose spatial coordinates are attached to P, whose motion is governed by the geodesic,
= 0, where , (2)
ds2 = g((dx(dx( and g(( is the space-time metric. The attachment means that, between P and L, there is no relative motion or acceleration. Thus, when a spaceship is under the influence of gravity only, the local space-time is automatically Minkowski. Note that the free fall implies but is beyond just the existence of rthogonal tetrad of arbitrarily accelerated observer" [4].
Einstein equivalence principle is very different from the version formulated by Pauli [10, p.145], or every infinitely small world region (i.e. a world region which is so small that the space- and time-variation of gravity
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