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Philosophaie
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Homework Statement
Using the Schwarzschild Metric and the Contravariant Position Vector 1x4 ##x^k## with 4 vector:
$$x^{k'} = \left[ \begin {array}{c}r \\ \theta \\ \phi \\ t \end {array} \right]$$
where
##x^1## = r = 1 per unit distance
##x^2## = ##\theta## = 50 Degrees
##x^3## = ##\phi## = 30 Degrees
##x^4## = t = 1 per unit time
Find the next Contravariant Position Vector 1x4 ##x'^k##
where ##\Delta##t = t' - t = Period of One Revolution (in per unit time) / 10000
$$x^{k'} = \left[ \begin {array}{c}r' \\ \theta' \\ \phi' \\ t' \end {array} \right]$$
Homework Equations
Covarient Schwarzschild Metric Tensor 4x4
$$g_{ij} = \left[ \begin {array}{cccc}1/(1-2m/r) & 0 & 0 & 0 \\ 0 & r^2 & 0 & 0 \\ 0 & 0 & r^2(sin(h))^2 & 0\\ 0 & 0 & 0 & -(1-2m/r) \end {array} \right]$$
Contravarient Schwarzschild Metric Tensor 4x4
$$g^{ij} = (g_{ij})^{-1}$$
Christoffel Symbol of Affinity 4x4x4
##\Gamma^i_{jk}## = 1/2*##g^{il}## * (##\frac{d g_{lj}}{d x^k}## + ##\frac{d g_{lk}}{d x^j}## - ##\frac{d g_{jk}}{d x^l}##)
Riemann Curvature Tensor 4x4x4x4
##R^i_{jkl}## = ##\frac{d \Gamma^i_{jl}}{d x^k}## - ##\frac{d \Gamma^i_{jk}}{d x^l}## + ##\Gamma^i_{km}## * ##\Gamma^m_{jl}## - ##\Gamma^i_{lm}## * ##\Gamma^m_{jk}##
The Attempt at a Solution
##x^{k'} = \Lambda^{k'}_k * x^k##
##\Lambda^{k'}_k## 4x4 should contain a form of the Riemann Curvature Tensor or the Christoffel Symbol because ##\Delta##t is reasonably small.
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