![]() ![]() The geodesic equation is useful in establishing one of the necessary theoretical foundations of relativity, which is the uniqueness of geodesics for a given set of initial conditions. We will seldom have occasion to resort to this technique, an exception being example 19. The solution to this chicken-and-egg conundrum is to write down the differential equations and try to find a solution, without trying to specify either the affine parameter or the geodesic in advance. Likewise, we can’t do the geodesic first and then the affine parameter, because if we already had a geodesic in hand, we wouldn’t need the differential equation in order to find a geodesic. We can’t start by defining an affine parameter and then use it to find geodesics using this equation, because we can’t define an affine parameter without first specifying a geodesic. Recall that affine parameters are only defined along geodesics, not along arbitrary curves. Robertson and Walker proved for any curvature universe (which is homogeneous and isotropic), the metric is given as. d s 2 a 2 ( t) ( d r 2 + r 2 d 2 + r 2 s i n 2 d 2) We wish to write the metric in any curvature. If this differential equation is satisfied for one affine parameter \(\lambda\), then it is also satisfied for any other affine parameter \(\lambda' = a \lambda + b\), where a and b are constants (problem 5). FRW Christoffel Symbols: (Graded) For the FRW metric, d s 2 c 2 d t 2 + a (t) 2 (1 k r 2 1 d r 2 + r 2 d 2 + r 2 sin 2 d 2) There are 4 rows of Christoffel symbols, 13 in total. The metric for flat, Euclidean, expanding universe is given as. ![]() \) implies that when \(\kappa\) and \(\nu\) are distinct, the same term will appear twice in the summation. ![]()
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