Kostik
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- Is parallel transport exactly norm-preserving or only approximately?
Dirac ("General Theory of Relativity", pp. 12-13, see below) shows the following. Let ##A_\nu (x)## be a vector, and let
$$K_\nu (x+dx) = A_\nu (x) + dA_\nu (x)$$ be the vector after parallel transport from ##x## to ##x+dx##. The formula $$dA_\nu = A_\mu \Gamma^\mu_{\nu\sigma}dx^\sigma $$ is given as equation (7.7) below.
Dirac shows (leading up to and following equation (7.8)) that $$d(A^\nu A_\nu) = 0 \,\,$$ This would seem to say that the length of ##A_\nu## remains unchanged under parallel displacement; that is, it is norm-preserving.
Yet, if you calculate:
$$K^\nu K_\nu (x+dx) = (A^\nu + dA^\nu)(A_\nu + dA_\nu) = A^\nu A_\nu + d(A^\nu A_\nu) + dA^\nu dA_\nu $$ $$ = A^\nu A_\nu + O[(dA_\nu)^2]$$ you seem to find that ##||K_\nu||^2## is equal to ##||A_\nu||^2## only up to order ##O[(dx^\nu)^2]##.
This seems strange. Either parallel transport is either exactly norm-preserving, or it isn't?
Note: Dirac says at the bottom of p. 13: "It follows that, to the first order, the length of the whole vector equals that of its tangential part" (i.e., the parallel-transported vector). So, Dirac also seems to be suggesting that parallel transport is norm-preserving only to the first order in the coordinate displacement.
$$K_\nu (x+dx) = A_\nu (x) + dA_\nu (x)$$ be the vector after parallel transport from ##x## to ##x+dx##. The formula $$dA_\nu = A_\mu \Gamma^\mu_{\nu\sigma}dx^\sigma $$ is given as equation (7.7) below.
Dirac shows (leading up to and following equation (7.8)) that $$d(A^\nu A_\nu) = 0 \,\,$$ This would seem to say that the length of ##A_\nu## remains unchanged under parallel displacement; that is, it is norm-preserving.
Yet, if you calculate:
$$K^\nu K_\nu (x+dx) = (A^\nu + dA^\nu)(A_\nu + dA_\nu) = A^\nu A_\nu + d(A^\nu A_\nu) + dA^\nu dA_\nu $$ $$ = A^\nu A_\nu + O[(dA_\nu)^2]$$ you seem to find that ##||K_\nu||^2## is equal to ##||A_\nu||^2## only up to order ##O[(dx^\nu)^2]##.
This seems strange. Either parallel transport is either exactly norm-preserving, or it isn't?
Note: Dirac says at the bottom of p. 13: "It follows that, to the first order, the length of the whole vector equals that of its tangential part" (i.e., the parallel-transported vector). So, Dirac also seems to be suggesting that parallel transport is norm-preserving only to the first order in the coordinate displacement.
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