关于什么是爱因斯坦的等效原理 [7]
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关键词:general theoryimportanceEinstein’s equivalence principlechallengedunderstanding
ntally involves with any theory of physics not only that a unification of gravitation and electromagnetism is needed due to that all charged particles are massive. This analysis also clarifies that the space-time in reality having a Riemannian geometry is also due to the equivalence principle.
Einstein considered a Galilean (inertial) system of reference K (x, y, z, t) and a system K’ (x’, y’, z’, t’) in uniform rotation W relatively to K. The origins of both systems and their axes of Z permanently coincide. For reason of symmetry, a circle around the origin in the X, Y plane of K may at the same time be regarded as a circle in the X’, Y’ plane of K’. Then, according to special relativity, in the X, Y plane and the X’, Y’ plane, the metrics of K and K’ [22] are respectively the following:
ds2 = c2 dt2 – dr2 - r2 df2 - dz2 (6a)
and
ds2 = (c2 - W2r’2) dt’2 – dr’2 - (1 - W2r’2/c2)-1r’2 df’2 – dz’2 (6b)
Then,
= (1 - W2r’2/c2)-1/2r’ &nb
sp; (6c)
would be the circumstance of a circle of radius r’ (= r) for an observer in K’. Thus, Einstein concluded, “With a measuring rod at rest relatively to K’, the quotient of circumstances over diameter would be greater than p.” and Euclidean geometry therefore breaks down in relation to the system K’. Moreover, Einstein [8] wrote, “An observer at the common origin of co-ordinates, capable of observing the clock at the circumferences by means of light, would therefore see it lagging behind the clock beside him...So, he will be obliged to define time in such a way that the rate of a clock depends upon where the clock may be.”
According to the principle of equivalence, K’ may also be considered as a system at rest, with respect to which there is a gravitational field (field of centrifugal force, and force of Coriolis). Thus, Einstein’s notion of gravity, though has a cause such as W, needs not relate to a source, but just relate to acceleration to a resting massive particle (see also section 6). This example shows also that the equivalence principle enables an extension of the principle of relativity to accelerated motion.
Thus, Einstein concluded, “In general theory of relativity, space and time cannot be defined in such a way measured by the unit measuring-rod, or difference in the time co-ordinate by a stand clock.” Since a physical space-time is Riemannian, covariance must be done in terms of Riemannian geometry. However, metric (6b) is still consistent with the notion of a superficial Euclidean frame of reference, since a measuring rod, which is attached to the system K’, would be under the same influence of gravity. In fact, it is based on this implicit assumption that the cylindrical coordinate system is well defined in K’. In other words, not only is K’ a Riemannian space as indicated by (6b), but also K’ has an implicit Euclidean structure.
Thus, in general relativity, the notion of a Euclidean space is still implicitly included within a Riemannian space although, just as in special relativity, the Euclidean structure is not invariant. In fact, this is the mathematical basis that the cylindrical coordinate system (r’, f’, z’) is well defined in K’. Thus, the Euclidean structure is an integral part of Einstein’s theory. For clarity, a Euclid-Riemannian space with a Euclidean structure shall be called an Einstein space after its creator.
To see the local coordinate transformation between metri
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