What is the Riemann Curvature Tensor for Flat and Minkowski Space?

[Elliptic]

Homework Statement

Show that all components of Riemann curvarue tensor are equal to zero for flat and Minkowski space.

Homework Equations

The Attempt at a Solution

$(ds)^2=(dx^1)^2+(dx^2)^2+...+(dx^n)^2 \\ R_{MNB}^A=\partial _{N}\Gamma ^{A}_{MB}-\partial _{M}\Gamma ^{A}_{NB}+\Gamma ^{A}_{NC}\Gamma_{MB}-\Gamma ^{C}_{NB}\Gamma_{MC}$

 

[WannabeNewton]

It's a straightforward problem. What are the Christoffel symbols for Minkowski space-time in the standard coordinates?

 

[Elliptic]

$\Gamma^{K}_{MK}=\frac{1}{2}\left(\partial_Mg_{ML}+\partial_Ng_{ML}-\partial_Lg_{MN} \right )$

 

[WannabeNewton]

Yes but can you tell me why?

 

[WannabeNewton]

$\Gamma^{K}_{MK}=\frac{1}{2}\left(\partial_Mg_{ML}+\partial_Ng_{ML}-\partial_Lg_{MN} \right )$

Ok but you don't need to edit your posts, you can just reply to my subsequent posts (it makes it easier to keep track of who's saying what). Ok so you know the Christoffel symbols vanish identically because the metric components are constant. So what does that say about the Riemann curvature tensor based on the usual formula?

 

[Elliptic]

That Riemann curvarue tensor is equal to zero?

 

[WannabeNewton]

Yeahp that's it! :)

 

[Elliptic]

Thanks. I have another problem, but i must respect the rules of forum and post a new thread.

 

[WannabeNewton]

Alrighty :)