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Transform from one position to another.

  1. Jan 29, 2014 #1
    1. The problem statement, all variables and given/known data

    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

    $$x^{k'} = \left[ \begin {array}{c}r' \\ \theta' \\ \phi' \\ t' \end {array} \right]$$

    2. Relevant 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}##


    3. 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.
     
    Last edited: Jan 29, 2014
  2. jcsd
  3. Jan 29, 2014 #2

    maajdl

    User Avatar
    Gold Member

    What does that mean?
     
  4. Jan 29, 2014 #3
    The time the object takes to orbit the central body in per unit time divided by 10000. What I really mean is a Deltat small enough to have a homogeneous Riemann and Christoffel.
     
    Last edited: Jan 29, 2014
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