Transform from one position to another.

In summary, the conversation discusses the use of the Schwarzschild Metric and the Contravariant Position Vector 1x4 to find the next Contravariant Position Vector 1x4. The equations used include the Covariant and Contravariant Schwarzschild Metric Tensors, Christoffel Symbol of Affinity, and Riemann Curvature Tensor. The goal is to find a small enough value for ##\Delta##t to ensure homogeneous Riemann and Christoffel.
  • #1
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.
 
Last edited:
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  • #2
Period of One Revolution (in per unit time) / 10000
What does that mean?
 
  • #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:

Related to Transform from one position to another.

1. What is meant by "transforming from one position to another"?

Transformation refers to the process of changing one position or state to another. In science, this can involve changing the physical location, chemical composition, or other properties of an object.

2. What are some examples of transformations in science?

Examples of transformations include physical changes such as melting or boiling, chemical reactions like rusting or combustion, and biological processes such as growth and development.

3. How do scientists study transformations?

Scientists study transformations through observation, experimentation, and data analysis. They may also use mathematical models and simulations to better understand the underlying processes and predict future transformations.

4. What factors can affect the rate of transformation?

The rate of transformation can be affected by various factors such as temperature, pressure, concentration, and the presence of catalysts. These factors can influence the energy required for a transformation to occur and the stability of the resulting state.

5. Why is understanding transformation important in science?

Understanding transformation is crucial in science as it allows us to explain and predict changes in the natural world. This knowledge can be applied in various fields such as medicine, engineering, and environmental science to develop new technologies and solve complex problems.

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