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tives of the fundamental tensor vanish. That this must be so we see directly in the local system of co-ordinates. In case AÃÄ is skew-symmetrical, we obtain from (80), by contraction with respect to Ä and Á, "AÃÄ = Aà (82) "xÄ In the general case, from (79) and (80), by contraction with respect to Ä and Á, we obtain the equations, "A± ± ± ² = Aà - “ A (83) "x± ò ± "Añ à = + Aà “ A±² (84) "x± ±² The Riemann Tensor. If we have given a curve extending from the point P to the point G of the continuum, then a vector Aµ, given at P, may, by a parallel displacement, be moved along the curve to G. If the continuum is Euclidean (more generally, if by a suitable choice of co-ordinates the gµ½ are constants) then the vector obtained at G as a result of this displacement does not depend upon the choice of the curve joining P and G. But the general theory of relativity 77 Figure 4 otherwise, the result depends upon the path of the displacement. In this case, therefore, a vector suffers a change, "Aµ (in its direction, not its magnitude), when it is carried from a point P of a closed curve, along the curve, and back to P. We shall now calculate this vector change: µ = "Aµ ´A As in Stokes s theorem for the line integral of a vector around a closed curve, this problem may be reduced to the integration around a closed curve with infinitely small linear dimensions; we shall limit ourselves to this case. We have, first, by (67), = "Aµ - “µ A± dx² ±² µ In this, “ is the value of this quantity at the variable point G ±² of the path of integration. If we put = ¾µ (xµ)G - (xµ)P 78 the meaning of relativity µ and denote the value of “ at P by “µ , then we have, with ±² ±² sufficient accuracy, µ "“ ±² µ µ = + “±² “±² ¾½ "x½ Let, further, A± be the value obtained from A± by a parallel dis- placement along the curve from P to G. It may now easily be proved by means of (67) that Aµ - Aµ is infinitely small of the first order, while, for a curve of infinitely small dimensions of the first order, "Aµ is infinitely small of the second order. Therefore there is an error of only the second order if we put = A± A± - “± Aà ¾Ä ÃÄ µ If we introduce these values of “ and A± into the integral, ±² we obtain, neglecting all quantities of a higher order than the second, µ ò µ Á à ± = "Aµ - - “ “ d¾² (85) "“ ¾ "x± Á² ñ A The quantity removed from under the sign of integration refers to the point P. Subtracting 1d(¾x¾²) from the integrand, we obtain 2 1 ± 2 (¾ d¾² - ¾² d¾±) This skew-symmetrical tensor of the second rank, f±², character- izes the surface element bounded by the curve in magnitude and position. If the expression in the brackets in (85) were skew- symmetrical with respect to the indices ± and ², we could con- clude its tensor character from (85). We can accomplish this by the general theory of relativity 79 interchanging the summation indices ± and ² in (85) and adding the resulting equation to (85). We obtain ±² = 2"Aµ - Rµ AÃf (86) ñ² in which µ µ "“ "“ ñ ò µ µ Á µ Á = + + R - “ “ - “ “ (87) ñ² "x² "x± Á± ò Á² ñ µ The tensor character of R follows from (86); this is the ñ² Riemann curvature tensor of the fourth rank, whose properties of symmetry we do not need to go into. Its vanishing is a sufficient condition (disregarding the reality of the chosen co- ordinates) that the continuum is Euclidean. By contraction of the Riemann tensor with respect to the indices µ, ², we obtain the symmetrical tensor of the second rank, ± ± "“ "“ µ½ µ± ± ² ± ² = + + Rµ½ - “ “ - “ “ (88) "x± µ² ½± "x½ µ½ ±² The last two terms vanish if the system of co-ordinates is so = chosen that g constant. From Rµ½ we can form the scalar, = R gµ½Rµ½ (89) Straightest (Geodesic) Lines. A line may be constructed in such a way that its successive elements arise from each other by parallel displacements. This is the natural generalization of the straight line of the Euclidean geometry. For such a line, we have "x± µ µ ´ - “ dx² dx = ±² ds ds 80 the meaning of relativity d2xµ The left-hand side is to be replaced by ,* so that we have ds2 d2xµ µ dx± dx² + = “ 0 (90) ds2 ±² ds ds We get the same line if we find the line which gives a stationary value to the integral µ½ ds or g dxµ dx½ between two points (geodesic line). * The direction vector at a neighbouring point of the curve results, by a parallel displacement along the line element (dx²), from the direction vector of each point considered. THE GENERAL THEORY OF RELATIVITY (continued) We are now in possession of the mathematical apparatus which [ Pobierz caÅ‚ość w formacie PDF ] |