(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Calculate the Riemann curvature for the metric:

ds^{2}= -(1+gx)^{2}dt^{2}+dx^{2}+dy^{2}+dz^{2}showing spacetime is flat

2. Relevant equations

Riemann curvature eqn:

[itex]\Gamma[/itex]^{α}_{β}_{γ}_{δ}=(∂[itex]\Gamma[/itex]^{α}_{β}_{δ})/∂x^{γ})-(∂[itex]\Gamma[/itex]^{α}_{β}_{γ})/∂x^{δ})+([itex]\Gamma[/itex]^{α}_{γ}_{ε})(R^{ε}_{β}_{δ})-([itex]\Gamma[/itex]^{α}_{δ}_{ε})([itex]\Gamma[/itex]^{ε}_{β}_{γ})

3. The attempt at a solution

I know that the non-vanishing Christoffel components are as follows:

[itex]\Gamma[/itex]^{∅}_{∅}_{∅}=sinθcosθ

[itex]\Gamma[/itex]^{∅}_{θ}_{∅}=[itex]\Gamma[/itex]^{∅}_{∅}_{θ}=cotθ

My guess is that the middle terms disappear creating:

-cos^{2}θ+sin^{2}θ-(-sinθcosθ)(cosθ/sinθ)

The sinθ's cos^{2}θ's cancel each other out making the answer sin^{2}θ

Is this answer correct? My confusion is that I received this answer for the curvature for a different metric (namely, ds^{2}=R^{2}dθ^{2}+R^{2}sin^{2}θd[itex]\vartheta[/itex]). Will I always receive the answer sin^{2}θ? I am not understanding fully what the Riemann curvature is...

Any help would be greatly appreciated!!

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# Homework Help: Need to find the riemann curvature for the following metric

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