Transform from one position to another.

[Philosophaie]

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.

 

[maajdl]

Period of One Revolution (in per unit time) / 10000

What does that mean?

 

[Philosophaie]

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.