Flat Space - Christoffel symbols and Ricci = 0?

In summary, the homework statement asks for the christoffel symbols and the ricci tensor. I found the metric to be constant and found the transformation to the usual flat space form.
  • #1
unscientific
1,734
13

Homework Statement


[/B]
(a) Find christoffel symbols and ricci tensor
(b) Find the transformation to the usual flat space form ## g_{\mu v} = diag (-1,1,1,1)##.

ricci1.png

Homework Equations

The Attempt at a Solution



Part(a)
[/B]
I have found the metric to be ## g_{tt} = g^{tt} = -1, g_{xt} = g_{tx} = 2c, g_{xx} = g^{xx} = 0, g_{yy} = g^{yy} = 1, g_{zz} = g^{zz} = 1##.

The christoffel symbols can be calculated by:
[tex] \Gamma_{\alpha \beta}^{\mu} = \frac{1}{2} g^{\mu v} \left( \frac{\partial g_{\alpha v}}{\partial x^{\beta}} + \frac{\partial g_{v \beta}}{\partial x^{\alpha}} - \frac{\partial g_{\alpha \beta}}{\partial x^{v}} \right) [/tex]

Since all components of the metric are constants, it means ##\Gamma_{\alpha \beta}^{\mu} = 0## and ##R_{\alpha \beta}^{\mu} = 0##.

Part (b)

I'm not sure how to approach this question. I know I have to find the Jacobian ##\frac{\partial \tilde{x^{\mu}}}{\partial x^{v}}##. I also know the transformation is ##\tilde{g_{\alpha \beta}} = \frac{\partial x^{\mu}}{\partial \tilde{x^{\alpha}}} \frac{\partial x^v}{\partial \tilde{x^{\beta}}} g_{\mu v}##.
 
Physics news on Phys.org
  • #2
Any input for part (b)?
 
  • #3
For both the original line element and the usual flat space line element the metric components are constants. This suggests a linear transformation. So, maybe introduce a new time coordinate ##\tilde{t} = at + bx## where you can try to find values of ##a## and ##b## to do the job.
 
  • #4
TSny said:
For both the original line element and the usual flat space line element the metric components are constants. This suggests a linear transformation. So, maybe introduce a new time coordinate ##\tilde{t} = at + bx## where you can try to find values of ##a## and ##b## to do the job.

Letting ## c \tilde t = c t - x## makes it work because ##c^2 d \tilde t^2 = c^2dt^2 - 2ct ~dt dx + dx^2##.
 
Last edited:
  • #5
Looks right.
 
  • Like
Likes unscientific
  • #6
TSny said:
Looks right.

Thanks a lot for all your help! I'm just starting out in GR so I'm learning as hard as I can.
 

Related to Flat Space - Christoffel symbols and Ricci = 0?

1. What is flat space in physics?

Flat space refers to a type of space in physics that follows the laws of Euclidean geometry. In other words, it is a space that has a constant curvature and no curvature in any direction. This type of space is often used as a simplification in calculations and theoretical models.

2. What are Christoffel symbols?

Christoffel symbols are mathematical quantities used to describe the curvature of a space. They are used in the field of differential geometry and are important in understanding the properties of space and how objects move within it.

3. How are Christoffel symbols related to Ricci = 0?

Ricci = 0 refers to the Ricci tensor, which is a mathematical object used to describe the curvature of a space. The Christoffel symbols are part of the equation used to calculate the Ricci tensor, so they are closely related.

4. What does a Ricci = 0 mean for a space?

A Ricci = 0 means that the Ricci tensor is equal to zero, indicating that the space is flat and has no curvature. This is often a simplification used in theoretical models to make calculations easier.

5. Are there any real-world examples of flat space?

Flat space is a mathematical concept and does not exist in the physical world. However, it can be used as an approximation for certain real-world situations, such as small regions of spacetime and the surface of a sphere with a large radius.

Similar threads

  • Advanced Physics Homework Help
Replies
3
Views
4K
  • Advanced Physics Homework Help
Replies
9
Views
2K
  • Advanced Physics Homework Help
Replies
2
Views
1K
  • Advanced Physics Homework Help
Replies
2
Views
1K
  • Advanced Physics Homework Help
Replies
0
Views
1K
  • Advanced Physics Homework Help
Replies
6
Views
3K
  • Advanced Physics Homework Help
Replies
3
Views
945
  • Advanced Physics Homework Help
Replies
1
Views
1K
  • Advanced Physics Homework Help
Replies
3
Views
894
  • Special and General Relativity
Replies
11
Views
1K
Back
Top