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2) A physical coordinate system is a Gaussian system such that the equivalence principle is satisfied.
One might attempt to justify the viewpoint of accepting any Gaussian system as a space-time coordinate system by pointing out that Einstein [3] also wrote in his book that n an analogous way (to Gaussian curvilinear coordinates) we shall introduce in the general theory of relativity arbitrary co-ordinates, x1, x2, x3, x4, which shall number uniquely the space-time points, so that neighboring events are associated with neighboring values of the coordinates; otherwise, the choice of co-ordinate is arbitrary." But, Einstein [3] qualified this with a physical statement that n the immediate neighbor of an observer, falling freely in a gravitational field, there exists no gravitational field." This statement will be clarified later with a demonstration of the equivalence principle (see eqs. [6] & [7]).
3) The equivalence principle requires not only, at each point, the existence of a local Minkowski space2)
ds2 = c2dT2 - dX2 - dY2 - dZ2, (1)
but a free fall must result in a co-moving local Minkowskian space (see also [10-13]). Note that the equivalence principle requires that such a local coordinate transformation be due to a specific physical action, acceleration in the free fall alone. Einstein [2] 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, |
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