valid in physics, point 3) should produce valid physical results. Thus, one can check point 3) to determine the validity of point 1) or vice versa.
The mathematical existence of a co-moving Local Minkowski space along a ree fall" geodesic implies only that Riemannian geometry is compatible with the equivalence principle. The physics is whether the existence of a physical local transformation which transforms the metric to the co-moving local Minkowski space. This is possible only if the geodesic represents a physical free fall, i.e., the equivalence principle ensures the existence of such a physical transformation. Thus, one must carefully distinguish mathematical properties of a Lorentz metric from physical requirements. Apparently, a discussion on the possible failure of satisfying the equivalence principle was over-looked by Einstein and others (see 壯6 & 7).
3. Einstein Illustration of the Equivalence principle
Einstein [3] illustrated his equivalence principle in his calculation of the light bending around the sun. (Note, the other method does not have such a benefit.) His 1915 equation for the space-time metric g(( is
G(( ( R(( -R g(( = -KT(m)(( (3)
where K is the coupling constant, T(m)(( is the energy-stress tensor for massive matter, R(( is the Ricci curvature tensor, and R = R((g((, where g((, is the inverse metric of g((. Now, he considered a coordinate system S (x, y, z, t) with the sun attached to the spatial origin. Based on eq. (3), and the notion of weak gravity, Einstein ustified" the linear equation,
= 2K(T(( - g((T), where ( (( = g(( - ((( , (4a)
((( is the flat metric, and T = T((g((. Then, from eq. (4a), and Ttt = (, otherwise T(( is zero, by using the asymptotically flat of the metric, Einstein obtained, to a sufficiently close approximation, the metric for coordinate system S
ds2 = c2(1 - )dt2 - (1 + )(dx2 + dy2 + dz2), (4b)
where ( is the mass density and r2 = x2 + y2 + z2.
However, since eq. (3) itself is questionable for dynamic problems [7,8,24-26], it is necessary to justify eq. (4a) again. Also, the notion of weak gravity may not be compatible with the principle of general covariance [3]. In the next section, eq. (4a) will be justified directly and is independent of the details of higher order terms of an exact Einstein equation. For this reason and the dynamical incompatibility with eq. (3), eq. (4a) is called the Maxwell-Newton Approximation [7]. In other words, eq.
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