- #1
choirgurlio
- 9
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Homework Statement
Calculate the Riemann curvature for the metric:
ds2 = -(1+gx)2dt2+dx2+dy2+dz2 showing spacetime is flat
Homework Equations
Riemann curvature eqn:
[itex]\Gamma[/itex]αβγδ=(∂[itex]\Gamma[/itex]αβδ)/∂xγ)-(∂[itex]\Gamma[/itex]αβγ)/∂xδ)+([itex]\Gamma[/itex]αγε)(Rεβδ)-([itex]\Gamma[/itex]αδε)([itex]\Gamma[/itex]εβγ)
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:
-cos2θ+sin2θ-(-sinθcosθ)(cosθ/sinθ)
The sinθ's cos2θ's cancel each other out making the answer sin2θ
Is this answer correct? My confusion is that I received this answer for the curvature for a different metric (namely, ds2=R2dθ2+R2sin2θd[itex]\vartheta[/itex]). Will I always receive the answer sin2θ? I am not understanding fully what the Riemann curvature is...
Any help would be greatly appreciated!