As some may know, I have been studying the Morris-Thorne wormhole metric for quite some time now.(adsbygoogle = window.adsbygoogle || []).push({});

ds^{2}= -c^{2}dt^{2}+ dl^{2}+ (b^{2}+ l^{2})(dθ^{2}+ sin^{2}(θ)d∅^{2})

Now, from this space-time interval, it is easy to see how I would deduce the following metric tensor:

g_{00}= -1

g_{11}= 1

g_{22}= (b^{2}+ l^{2})

g_{33}= (b^{2}+ l^{2})sin^{2}(θ)

where

x^{0}= ct

x^{1}= l

x^{2}= θ

x^{3}= ∅

Now with this metric tensor, the Christoffel symbols yield:

Γ^{1}_{22}= - L

Γ^{1}_{33}= - Lsin^{2}(θ)

Γ^{2}_{12}and its counterpart with the lower indices reversed (Γ^{2}_{21}) = L/(b^{2}+ l^{2})

Γ^{2}_{33}= - sin(θ)cos(θ)

Γ^{3}_{13}and its counterpart = L/(b^{2}+ l^{2})

Γ^{3}_{23}and its counterpart= cot(θ)

If you plug these Christoffel symbols into the Riemann tensor formula:

R^{a}_{bmv}= (∂Γ^{a}_{vb}/ ∂x^{m}) - (∂Γ^{a}_{mb}/ ∂x^{v}) + Γ^{a}_{mc}Γ^{c}_{vb}- Γ^{a}_{vc}Γ^{c}_{mb}

you will see that the following Riemann tensor elements equal as follows (I am doing these specific elements for a certain reason):

R^{2}_{323}= (b^{2}sin^{2}(θ))/(b^{2}+ l^{2})

R^{0}_{323}, R^{1}_{323}and R^{3}_{323}= 0

Now, I did these specific elements for the purpose of calculating the purely covariant version of the Riemann tensor element R_{2323}using the formula:

R_{2323}= g_{2f}R^{f}_{323}

Doing this yields the following result:

R_{2323}= b^{2}sin^{2}(θ)

The following sources however, (http://www.spacetimetravel.org/wurmlochflug/wurmlochflug.html) (http://www.physics.uofl.edu/wkomp/teaching/spring2006/589/final/wormholes.pdf [Broken]) (pg. 4 on the PDF) said that R_{θ∅θ∅}(which is in fact R_{2323}) = b^{2}/ (b^{2}+ l^{2})^{2}

This is the part that I do not understand. I have shown you clearly with the use of formulas how I derived

R_{2323}= b^{2}sin^{2}(θ)

They however, did not show any work on how they got

R_{2323}= b^{2}/ (b^{2}+ l^{2})^{2}

Can anyone please tell me how they got that (starting with the metric tensor in matrix form and going forward)?

I know that they worked in an orthonormal basis. My problem with that is this: I know what a basis is and I know what an orthonormal basis is. I know what basis vectors are and I know (or at least I think I know) how to derive basis vectors using a metric tensor. What I do not know, is what to do with said basis vectors after I have derived them or how to take a tensor product (despite the fact that I know what one is).

Essentially, I don't know how you convert from a coordinate basis to orthonormal with regards to these tensors (like the Riemann).

That is why I ask if someone can please tell me what they did differently in the beginning of their calculations to get:

R_{2323}= b^{2}/ (b^{2}+ l^{2})^{2}

instead of

R_{2323}= b^{2}sin^{2}(θ)

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# Help me make this mathematical connection in general relativity

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