- #1

Whitehole

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## Homework Statement

Given two spaces described by

##ds^2 = (1+u^2)du^2 + (1+4v^2)dv^2 + 2(2v-u)dudv##

##ds^2 = (1+u^2)du^2 + (1+2v^2)dv^2 + 2(2v-u)dudv##

Calculate the Riemann tensor

## Homework Equations

Given the metric and expanding it ##~~~g_{τμ} = η_{τμ} + B_{τμ,λσ}x^λx^σ + ...##

We have the Riemann tensor, ##R_{τρμν} = B_{τν,ρμ} + B_{ρμ,τν} - B_{ρν,τμ} - B_{τμ,ρν}##

## The Attempt at a Solution

I'm having problems calculating the B's for the above spaces.

There is also one space given by ##ds^2 = (1+(ax+cy)^2)dx^2 + (1+(by+cx)^2)dy^2 + 2(ax+cy)(by+cx)dxdy## where I know how to compute the B's. Since in 2D, the Riemann tensor has only one component, that is ##R_{1212}##. For ##g_{11}## we have ##B_{1122} = c^2##, for ##g_{22}## we have ##B_{2211} = c^2##, for ##g_{12} = g_{21}## we have ##B_{1212} = ½(ab + c^2)##

For the above spaces, it seems every B is 0. I can't seem to understand why.

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