# Flat Space - Christoffel symbols and Ricci = 0?

• unscientific
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.
unscientific

## 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)##.

## 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:
$$\Gamma_{\alpha \beta}^{\mu} = \frac{1}{2} g^{\mu v} \left( \frac{\partial g_{\alpha v}}{\partial x^{\beta}} + \frac{\partial g_{v \beta}}{\partial x^{\alpha}} - \frac{\partial g_{\alpha \beta}}{\partial x^{v}} \right)$$

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}##.

Any input for part (b)?

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.

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:
Looks right.

unscientific
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.

## 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.

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