- #1

Philosophaie

- 462

- 0

## 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.

Last edited: