General Relativity For Tellytubbys - Riemann Curverture Tensor

General Relativity For Tellytubbys

Geodesic Deviation and Potbellied Mr. Riemann

Sir Kevin Aylward B.Sc., Warden of the Kings Ale

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Overview

This section attempts to give a handle on the Riemann curvature tensor. I have had a bit of bother with this one as I could not find a really decent web site to steal the derivations from. They all missed out the bits, which I consider are crucial. For example, the fundamental point of the Riemann tensor, as far as G.R. is concerned, is that it describes the acceleration of geodesics with respect to one another. Some sites noted this fact, but did not show in their derivations how that particular derivation actually related to this acceleration. Taking vectors on round trips with talks of parallel transportation don’t immediately explain what's happening, although very impressive sounding it is, indeed. Of course it's probably that I'm just too thick to see it. In addition, of course, all derivations left most of the details to one's futile imagination. I am led to believe that many people don’t have a bleeding clue what's going on, although they can apply the formulas in a sleepwalking sense.

Further point. It is what are called, tidal forces that are equivalent to the acceleration of geodesics (geodesic deviation). If you consider the Newtonian, inverse square force law, at different radiuses, there is an effective differential force that tries to pull apart objects.

Consider Tinky-Winky and Dipsy orbiting the earth with some velocity, in what are assumed to be geodesics. Since they are not the same geodesics, Tinky-Winky and Dipsy may or may not move closer or further away from each other. The Riemann curvature tensor is what tells one what that acceleration between the Tellytubbys will be. This is expressed by

a w = D 2 w a D λ 2 = R a bcd d x b dλ d w c dλ d x d dλ MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCyyamaaBaaaleaacaWG3baabeaakiabg2da9maalaaabaGaamiramaaCaaaleqabaGaaGOmaaaakiaadEhadaahaaWcbeqaaiaadggaaaaakeaacaWGebGaeq4UdW2aaWbaaSqabeaacaaIYaaaaaaakiabg2da9iaadkfadaahaaWcbeqaaiaadggaaaGcdaWgaaWcbaGaamOyaiaadogacaWGKbaabeaakmaalaaabaGaamizaiaadIhadaahaaWcbeqaaiaadkgaaaaakeaacaWGKbGaeq4UdWgaamaalaaabaGaamizaiaadEhadaahaaWcbeqaaiaadogaaaaakeaacaWGKbGaeq4UdWgaamaalaaabaGaamizaiaadIhadaahaaWcbeqaaiaadsgaaaaakeaacaWGKbGaeq4UdWgaaaaa@5768@

or, equivalently

a w = D 2 w a D λ 2 = R a bcd v b w c v d MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCyyamaaBaaaleaacaWG3baabeaakiabg2da9maalaaabaGaamiramaaCaaaleqabaGaaGOmaaaakiaadEhadaahaaWcbeqaaiaadggaaaaakeaacaWGebGaeq4UdW2aaWbaaSqabeaacaaIYaaaaaaakiabg2da9iaadkfadaahaaWcbeqaaiaadggaaaGcdaWgaaWcbaGaamOyaiaadogacaWGKbaabeaakiaadAhadaahaaWcbeqaaiaadkgaaaGccaWG3bWaaWbaaSqabeaacaWGJbaaaOGaamODamaaCaaaleqabaGaamizaaaaaaa@4C98@

or equivalently, in posher notation

a= ∇ v ∇ v w= R a bcd v b w c v d MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCyyaiabg2da9iabgEGirpaaBaaaleaacaGG2baabeaakiabgEGirpaaBaaaleaacaGG2baabeaakiaacEhacqGH9aqpcaWGsbWaaWbaaSqabeaacaWGHbaaaOWaaSbaaSqaaiaadkgacaWGJbGaamizaaqabaGccaWG2bWaaWbaaSqabeaacaWGIbaaaOGaam4DamaaCaaaleqabaGaam4yaaaakiaadAhadaahaaWcbeqaaiaadsgaaaaaaa@4A78@

where D is the covariant derivative operator, w is the separation vector between the Tellytubbys geodesic, and V is the parameterized velocity of the Tellytubbys as they travel on their geodesics. The last form is the second covariant derivative of the connecting vector w in the direction of v, the gist of this will be shown

Calculation of Riemann

This section calculates what the Riemann tensor is, it is then shown afterwards how this is related to the concept of acceleration described above.

First, lets note some prior results,

T α ;β = T α , β + Γ α μβ T μ MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaeivamaaCaaaleqabaGaeqySdegaaOWaaSbaaSqaaiaacUdacqaHYoGyaeqaaOGaeyypa0JaaeivamaaCaaaleqabaGaeqySdegaaOGaaiilamaaBaaaleaacqaHYoGyaeqaaOGaey4kaSIaeu4KdC0aaWbaaSqabeaacqaHXoqyaaGcdaWgaaWcbaGaeqiVd0MaeqOSdigabeaakiaabsfadaahaaWcbeqaaiabeY7aTbaaaaa@4BD8@

T α β ;ρ = T α β ,ρ + Γ α μρ T μ β − Γ μ βρ T α μ MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaeivamaaCaaaleqabaGaeqySdegaaOWaaSbaaSqaaiabek7aIbqabaGcdaWgaaWcbaGaai4oaiabeg8aYbqabaGccqGH9aqpcaqGubWaaWbaaSqabeaacqaHXoqyaaGcdaWgaaWcbaGaeqOSdigabeaakmaaBaaaleaacaGGSaGaeqyWdihabeaakiabgUcaRiabfo5ahnaaCaaaleqabaGaeqySdegaaOWaaSbaaSqaaiabeY7aTjabeg8aYbqabaGccaqGubWaaWbaaSqabeaacqaH8oqBaaGcdaWgaaWcbaGaeqOSdigabeaakiabgkHiTiabfo5ahnaaCaaaleqabaGaeqiVd0gaaOWaaSbaaSqaaiabek7aIjabeg8aYbqabaGccaqGubWaaWbaaSqabeaacqaHXoqyaaGcdaWgaaWcbaGaeqiVd0gabeaaaaa@5E2C@

For a normal second order partial derivative, we have

∂ 2 Φ ∂x∂y = ∂ 2 Φ ∂y∂x MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacqGHciITdaahaaWcbeqaaiaaikdaaaGccqqHMoGraeaacqGHciITcaWG4bGaeyOaIyRaamyEaaaacqGH9aqpdaWcaaqaaiabgkGi2oaaCaaaleqabaGaaGOmaaaakiabfA6agbqaaiabgkGi2kaadMhacqGHciITcaWG4baaaaaa@484E@

For the covariant derivative of a vector this is not true in general. i.e.

V ;αβ ≠ V ;βα MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOvamaaBaaaleaacaGG7aGaeqySdeMaeqOSdigabeaakiabgcMi5kaadAfadaWgaaWcbaGaai4oaiabek7aIjabeg7aHbqabaaaaa@41D1@

So, lets calculate what the difference on a vector A is

[ ∇ α , ∇ β ]= ∇ α ∇ β − ∇ β ∇ α = V ;αβ − V ;βα MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaai4waiabgEGirpaaBaaaleaacqaHXoqyaeqaaOGaaiilaiabgEGirpaaBaaaleaacqaHYoGyaeqaaOGaaiyxaiabg2da9iabgEGirpaaBaaaleaacqaHXoqyaeqaaOGaey4bIe9aaSbaaSqaaiabek7aIbqabaGccqGHsislcqGHhis0daWgaaWcbaGaeqOSdigabeaakiabgEGirpaaBaaaleaacqaHXoqyaeqaaOGaeyypa0JaamOvamaaBaaaleaacaGG7aGaeqySdeMaeqOSdigabeaakiabgkHiTiaadAfadaWgaaWcbaGaai4oaiabek7aIjabeg7aHbqabaaaaa@5A88@

oh, and the first term above is called a commentator, and this does get rather messy, but there you go that’s G.R. for you, and I dropped the A on the LHS just to keep things uncluttered.

( A μ ;α ) ;β = ( A μ ,α + Γ μ αζ A ζ ) ;β MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadgeadaahaaWcbeqaaiabeY7aTbaakmaaBaaaleaacaGG7aGaeqySdegabeaakiaacMcadaWgaaWcbaGaai4oaiabek7aIbqabaGccqGH9aqpcaGGOaGaamyqamaaCaaaleqabaGaeqiVd0gaaOWaaSbaaSqaaiaacYcacqaHXoqyaeqaaOGaey4kaSIaeu4KdC0aaWbaaSqabeaacqaH8oqBaaGcdaWgaaWcbaGaeqySdeMaeqOTdOhabeaakiaadgeadaahaaWcbeqaaiabeA7a6baakiaacMcadaWgaaWcbaGaai4oaiabek7aIbqabaaaaa@53D0@

( A μ ;α ) ;β = ( A μ ,α + Γ μ αζ A ζ ) ,β MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadgeadaahaaWcbeqaaiabeY7aTbaakmaaBaaaleaacaGG7aGaeqySdegabeaakiaacMcadaWgaaWcbaGaai4oaiabek7aIbqabaGccqGH9aqpcaGGOaGaamyqamaaCaaaleqabaGaeqiVd0gaaOWaaSbaaSqaaiaacYcacqaHXoqyaeqaaOGaey4kaSIaeu4KdC0aaWbaaSqabeaacqaH8oqBaaGcdaWgaaWcbaGaeqySdeMaeqOTdOhabeaakiaadgeadaahaaWcbeqaaiabeA7a6baakiaacMcadaWgaaWcbaGaaiilaiabek7aIbqabaaaaa@53C1@

+ Γ μ βγ ( A γ ,α + Γ γ αζ A ζ ) − Γ γ βα ( A μ ,γ + Γ μ γζ A ζ ) MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@6773@

You might have to think a bit about the above, but its just treat the first derivative as one big 2^nd rank tensor, contravariant one, covariant one sort of thing.

( A μ ;α ) ;β = ( A μ ,α + Γ μ αζ A ζ ) ,β + Γ μ βγ A γ ;α − Γ γ βα A μ ;γ MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@6DA6@

That one above, I thought quite neat when I first worked it out. Once again, see what dummy index's are swapped here

( A μ ;α ) ;β = ( A μ ,α + Γ μ αζ A ζ ) ,β + Γ μ βζ A ζ ;α − Γ ζ βα A μ ;ζ MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@6DFE@

( A μ ;α ) ;β = A μ ,αβ + Γ μ αζ,β A ζ + Γ μ αζ A ζ ,β + Γ μ βζ A ζ ;α − Γ ζ βα A μ ;ζ MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@7B1A@

Now swap all the alphas and betas, but note that the Christoffel symbols are symmetric, so we can swap those ones back again.

( A μ ;β ) ;α = A μ ,βα + Γ μ βζ,α A ζ + Γ μ βζ A ζ ,α + Γ μ αζ A ζ ;β − Γ ζ βα A μ ;ζ MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@7B1A@

Now to subtract and collect terms, note the first and last term obviously cancels

V ;αβ − V ;βα =( Γ μ αζ,β − Γ μ βζ,α ) A ζ +− Γ μ αζ ( A ζ ;β − A ζ ,β ) −− Γ μ βζ ( A ζ ;α − A ζ ,α ) MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@84A5@

V ;αβ − V ;βα =( Γ μ αζ,β − Γ μ βζ,α ) A ζ − Γ μ αζ Γ ζ ρβ A ρ + Γ μ βζ Γ ζ ρα A ρ MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@7C75@

And with more index swapping

V ;αβ − V ;βα =( Γ μ αζ,β − Γ μ βζ,α ) A ζ − Γ μ αρ Γ ρ ζβ A ζ + Γ μ βρ Γ ρ ζα A ζ MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOvamaaBaaaleaacaGG7aGaeqySdeMaeqOSdigabeaakiabgkHiTiaadAfadaWgaaWcbaGaai4oaiabek7aIjabeg7aHbqabaGccqGH9aqpcaGGOaGaeu4KdC0aaWbaaSqabeaacqaH8oqBaaGcdaWgaaWcbaGaeqySdeMaeqOTdONaaiilaiabek7aIbqabaGccqGHsislcqqHtoWrdaahaaWcbeqaaiabeY7aTbaakmaaBaaaleaacqaHYoGycqaH2oGEcaGGSaGaeqySdegabeaakiaacMcacaWGbbWaaWbaaSqabeaacqaH2oGEaaGccqGHsislcqqHtoWrdaahaaWcbeqaaiabeY7aTbaakmaaBaaaleaacqaHXoqycqaHbpGCaeqaaOGaeu4KdC0aaWbaaSqabeaacqaHbpGCaaGcdaWgaaWcbaGaeqOTdONaeqOSdigabeaakiaadgeadaahaaWcbeqaaiabeA7a6baakiabgUcaRiabfo5ahnaaCaaaleqabaGaeqiVd0gaaOWaaSbaaSqaaiabek7aIjabeg8aYbqabaGccqqHtoWrdaahaaWcbeqaaiabeg8aYbaakmaaBaaaleaacqaH2oGEcqaHXoqyaeqaaOGaamyqamaaCaaaleqabaGaeqOTdOhaaaaa@7C6A@

V ;αβ − V ;βα =( Γ μ αζ,β − Γ μ βζ,α − Γ μ αρ Γ ρ ζβ + Γ μ βρ Γ ρ ζα ) A ζ MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@76F6@

So, now the bit in brackets is, as you might have guessed is the Riemann tensor

[ ∇ α , ∇ β ]= Γ μ αζ,β − Γ μ βζ,α − Γ μ αρ Γ ρ ζβ + Γ μ βρ Γ ρ ζα MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@70FE@

[ ∇ α , ∇ β ]= Γ μ ζα,β − Γ μ ζβ,α − Γ μ αρ Γ ρ ζβ + Γ μ βρ Γ ρ ζα MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@70FE@

and just to make it agree with the top of the page, renaming indexes using the negative Christoffel term as a base gives

R a bcd = Γ a bd,c − Γ a bc,d + Γ a cρ Γ ρ bd − Γ a dρ Γ ρ bc MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@6064@

Right, now a result needs to be derived

By inspection, it can be seen that Riemann is antisymmetric in the last two indexes d and c

R a bcd =− R a bdc MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOuamaaCaaaleqabaGaamyyaaaakmaaBaaaleaacaWGIbGaam4yaiaadsgaaeqaaOGaeyypa0JaeyOeI0IaamOuamaaCaaaleqabaGaamyyaaaakmaaBaaaleaacaWGIbGaamizaiaadogaaeqaaaaa@41A1@

By cyclic rotation of the last 3 indexes of Riemann we get

R a dbc = Γ a dc,b − Γ a db,c + Γ a bρ Γ ρ dc − Γ a cρ Γ ρ db MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOuamaaCaaaleqabaGaamyyaaaakmaaBaaaleaacaWGKbGaamOyaiaadogaaeqaaOGaeyypa0Jaeu4KdC0aaWbaaSqabeaacaWGHbaaaOWaaSbaaSqaaiaadsgacaWGJbGaaiilaiaadkgaaeqaaOGaeyOeI0Iaeu4KdC0aaWbaaSqabeaacaWGHbaaaOWaaSbaaSqaaiaadsgacaWGIbGaaiilaiaadogaaeqaaOGaey4kaSIaeu4KdC0aaWbaaSqabeaacaWGHbaaaOWaaSbaaSqaaiaadkgacqaHbpGCaeqaaOGaeu4KdC0aaWbaaSqabeaacqaHbpGCaaGcdaWgaaWcbaGaamizaiaadogaaeqaaOGaeyOeI0Iaeu4KdC0aaWbaaSqabeaacaWGHbaaaOWaaSbaaSqaaiaadogacqaHbpGCaeqaaOGaeu4KdC0aaWbaaSqabeaacqaHbpGCaaGcdaWgaaWcbaGaamizaiaadkgaaeqaaaaa@6064@

R a cdb = Γ a cb,d − Γ a cd,b + Γ a dρ Γ ρ cb − Γ a bρ Γ ρ cd MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@6064@

swap some index's in the last two, due to symmetry of the Christoffels

R a dbc = Γ a dc,b − Γ a bd,c + Γ a bρ Γ ρ dc − Γ a cρ Γ ρ bd MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@6064@

R a cdb = Γ a bc,d − Γ a cd,b + Γ a dρ Γ ρ bc − Γ a bρ Γ ρ dc MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@6064@

and adding these to our first Riemann gives

R a bcd + R a dbc + R a cdb =0 MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOuamaaCaaaleqabaGaamyyaaaakmaaBaaaleaacaWGIbGaam4yaiaadsgaaeqaaOGaey4kaSIaamOuamaaCaaaleqabaGaamyyaaaakmaaBaaaleaacaWGKbGaamOyaiaadogaaeqaaOGaey4kaSIaamOuamaaCaaaleqabaGaamyyaaaakmaaBaaaleaacaWGJbGaamizaiaadkgaaeqaaOGaeyypa0JaaGimaaaa@481E@

but forget about this just for now

Back to our commentator, with the index names realigned to our Riemann definition

[ ∇ c , ∇ d ] x a =( ∇ c ∇ d − ∇ d ∇ c ) x a =( V ;cd − V ;dc ) x a = R a bcd x b MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@6460@

We need to expand on this formula a bit in order to derive our acceleration of geodesics i.e. geodesic deviation.

Going back to our geodesic page, we noted

∇ v V≡ V α ;β V β MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaey4bIe9aaSbaaSqaaiaahAhaaeqaaOGaaCOvaiabggMi6kaabAfadaahaaWcbeqaaiabeg7aHbaakmaaBaaaleaacaGG7aGaeqOSdigabeaakiaabAfadaahaaWcbeqaaiabek7aIbaaaaa@4343@

So now to work out the commentator of the above directional derivative

[ ∇ w , ∇ v ]V=[ w γ ∇ γ , v β ∇ β ]V MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaai4waiabgEGirpaaBaaaleaacaWH3baabeaakiaahYcacqGHhis0daWgaaWcbaGaaCODaaqabaGccaGGDbGaaCOvaiabg2da9iaacUfacaWG3bWaaWbaaSqabeaacqaHZoWzaaGccqGHhis0daWgaaWcbaGaeq4SdCgabeaakiaacYcacaWG2bWaaWbaaSqabeaacqaHYoGyaaGccqGHhis0daWgaaWcbaGaeqOSdigabeaakiaac2facaWHwbaaaa@4F81@

[ ∇ w , ∇ v ]V= w γ ∇ γ v β ∇ β v α − v β ∇ β w γ ∇ γ v α MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaai4waiabgEGirpaaBaaaleaacaWH3baabeaakiaahYcacqGHhis0daWgaaWcbaGaaCODaaqabaGccaGGDbGaaCOvaiabg2da9iaadEhadaahaaWcbeqaaiabeo7aNbaakiabgEGirpaaBaaaleaacqaHZoWzaeqaaOGaamODamaaCaaaleqabaGaeqOSdigaaOGaey4bIe9aaSbaaSqaaiabek7aIbqabaGccaWG2bWaaWbaaSqabeaacqaHXoqyaaGccqGHsislcaWG2bWaaWbaaSqabeaacqaHYoGyaaGccqGHhis0daWgaaWcbaGaeqOSdigabeaakiaadEhadaahaaWcbeqaaiabeo7aNbaakiabgEGirpaaBaaaleaacqaHZoWzaeqaaOGaamODamaaCaaaleqabaGaeqySdegaaaaa@5F24@

[ ∇ w , ∇ v ]V= w γ v β ∇ γ ∇ β v α − w γ ( ∇ β v α )( ∇ γ v β ) − v β w γ ∇ β ∇ γ v α + v β ( ∇ γ v α )( ∇ β w γ ) MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@84E7@

and swapping gamma with alpha in the 2^nd product term

[ ∇ w , ∇ v ]V= w γ v β ∇ γ ∇ β v α − w β ( ∇ γ v α )( ∇ β v γ ) − v β w γ ∇ β ∇ γ v α + v β ( ∇ γ v α )( ∇ β w γ ) MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@84E7@

[ ∇ w , ∇ v ]V= w γ v β ∇ γ ∇ β v α − v β w γ ∇ β ∇ γ v α −( ∇ γ v α )( w β ∇ β v γ − v β ∇ β w γ ) MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@7C02@

[ ∇ w , ∇ v ]V= w γ v β ( ∇ γ ∇ β v α − ∇ β ∇ γ v α )−( ∇ γ v α )( w β ∇ β v γ − v β ∇ β w γ ) MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@77AE@

[ ∇ w , ∇ v ]V= w γ v β [ ∇ γ , ∇ β ] v α −( ∇ γ v α )( ∇ w V− ∇ v W) MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaai4waiabgEGirpaaBaaaleaacaWH3baabeaakiaahYcacqGHhis0daWgaaWcbaGaaCODaaqabaGccaGGDbGaaCOvaiabg2da9iaadEhadaahaaWcbeqaaiabeo7aNbaakiaadAhadaahaaWcbeqaaiabek7aIbaakiaacUfacqGHhis0daWgaaWcbaGaeq4SdCgabeaakiaacYcacqGHhis0daWgaaWcbaGaeqOSdigabeaakiaac2facaWG2bWaaWbaaSqabeaacqaHXoqyaaGccqGHsislcaGGOaGaey4bIe9aaSbaaSqaaiabeo7aNbqabaGccaWG2bWaaWbaaSqabeaacqaHXoqyaaGccaGGPaGaaiikaiabgEGirpaaBaaaleaacaWH3baabeaakiaahAfacqGHsislcqGHhis0daWgaaWcbaGaaCODaaqabaGccaWHxbGaaiykaaaa@6369@

[ ∇ w , ∇ v ]V= w γ v β v b R α bγβ −(∇V)( ∇ w V− ∇ v W) MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaai4waiabgEGirpaaBaaaleaacaWH3baabeaakiaahYcacqGHhis0daWgaaWcbaGaaCODaaqabaGccaGGDbGaaCOvaiabg2da9iaadEhadaahaaWcbeqaaiabeo7aNbaakiaadAhadaahaaWcbeqaaiabek7aIbaakiaadAhadaahaaWcbeqaaiaadkgaaaGccaWGsbWaaWbaaSqabeaacqaHXoqyaaGcdaWgaaWcbaGaamOyaiabeo7aNjabek7aIbqabaGccqGHsislcaGGOaGaey4bIeTaaCOvaiaacMcacaGGOaGaey4bIe9aaSbaaSqaaiaahEhaaeqaaOGaaCOvaiabgkHiTiabgEGirpaaBaaaleaacaWH2baabeaakiaahEfacaGGPaaaaa@5CC4@

but for

w α = ∂ x α ∂λ and v β = ∂ x β ∂τ MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4DamaaCaaaleqabaGaeqySdegaaOGaeyypa0ZaaSaaaeaacqGHciITcaWG4bWaaWbaaSqabeaacqaHXoqyaaaakeaacqGHciITcqaH7oaBaaGaaeiiaiaabggacaqGUbGaaeizaiaabccacaWG2bWaaWbaaSqabeaacqaHYoGyaaGccqGH9aqpdaWcaaqaaiabgkGi2kaadIhadaahaaWcbeqaaiabek7aIbaaaOqaaiabgkGi2kabes8a0baaaaa@5080@

i.e. w is an affine parametized connecting vector and v is our velocity, the last term is zero, via

∇ w v− ∇ v w= w α v β ;α − v β w α ;β MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaey4bIe9aaSbaaSqaaiaahEhaaeqaaOGaaCODaiabgkHiTiabgEGirpaaBaaaleaacaWH2baabeaakiaahEhacqGH9aqpcaWG3bWaaWbaaSqabeaacqaHXoqyaaGccaWG2bWaaWbaaSqabeaacqaHYoGyaaGcdaWgaaWcbaGaai4oaiabeg7aHbqabaGccqGHsislcaWG2bWaaWbaaSqabeaacqaHYoGyaaGccaWG3bWaaWbaaSqabeaacqaHXoqyaaGcdaWgaaWcbaGaai4oaiabek7aIbqabaaaaa@50B4@

∇ w v− ∇ v w= w α ( ∂ v β ∂ x α + Γ β εα v ε ) − v β ( ∂ w α ∂ x β + Γ α εβ w α ) MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaabbeaacqGHhis0daWgaaWcbaGaaC4DaaqabaGccaWH2bGaeyOeI0Iaey4bIe9aaSbaaSqaaiaahAhaaeqaaOGaaC4Daiabg2da9iaadEhadaahaaWcbeqaaiabeg7aHbaakiaacIcadaWcaaqaaiabgkGi2kaadAhadaahaaWcbeqaaiabek7aIbaaaOqaaiabgkGi2kaadIhadaahaaWcbeqaaiabeg7aHbaaaaGccqGHRaWkcqqHtoWrdaahaaWcbeqaaiabek7aIbaakmaaBaaaleaacqaH1oqzcqaHXoqyaeqaaOGaamODamaaCaaaleqabaGaeqyTdugaaOGaaiykaaqaaiabgkHiTiaadAhadaahaaWcbeqaaiabek7aIbaakiaacIcadaWcaaqaaiabgkGi2kaadEhadaahaaWcbeqaaiabeg7aHbaaaOqaaiabgkGi2kaadIhadaahaaWcbeqaaiabek7aIbaaaaGccqGHRaWkcqqHtoWrdaahaaWcbeqaaiabeg7aHbaakmaaBaaaleaacqaH1oqzcqaHYoGyaeqaaOGaam4DamaaCaaaleqabaGaeqySdegaaOGaaiykaaaaaa@6E97@

∇ w v− ∇ v w= ∂ x α ∂λ ∂ v β ∂ x α + Γ β εα v ε w α − ∂ x β ∂τ ∂ w α ∂ x β − Γ α εβ v β w ε MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@7AC7@

∇ w v− ∇ v w= ∂ v β ∂λ + Γ β εα v ε w α − ∂ w α ∂τ − Γ β εα v α w ε MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@69BD@

where alpha and beta has been swapped in the last r term

∇ w v− ∇ v w= ∂ x β ∂τ∂λ + Γ β εα v ε w α − ∂ w α ∂λ∂τ − Γ β αε v ε w α MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@7004@

where epsilon and alpha has been swapped in the last r term, thus the commentator of w and v is zero, therefor

[ ∇ w , ∇ v ]V= w γ v β v b R α bγβ MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaai4waiabgEGirpaaBaaaleaacaWH3baabeaakiaahYcacqGHhis0daWgaaWcbaGaaCODaaqabaGccaGGDbGaaCOvaiabg2da9iaadEhadaahaaWcbeqaaiabeo7aNbaakiaadAhadaahaaWcbeqaaiabek7aIbaakiaadAhadaahaaWcbeqaaiaadkgaaaGccaWGsbWaaWbaaSqabeaacqaHXoqyaaGcdaWgaaWcbaGaamOyaiabeo7aNjabek7aIbqabaaaaa@4E93@

Acceleration or Geodesic Deviation

The next task, is to show why the Riemann tensor determines the acceleration of the geodesics, i.e. why

a w = D 2 w a D λ 2 = R a bcd v b w c v d MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCyyamaaBaaaleaacaWG3baabeaakiabg2da9maalaaabaGaamiramaaCaaaleqabaGaaGOmaaaakiaadEhadaahaaWcbeqaaiaadggaaaaakeaacaWGebGaeq4UdW2aaWbaaSqabeaacaaIYaaaaaaakiabg2da9iaahkfadaahaaWcbeqaaiaadggaaaGcdaWgaaWcbaGaamOyaiaadogacaWGKbaabeaakiaadAhadaahaaWcbeqaaiaadkgaaaGccaWG3bWaaWbaaSqabeaacaWGJbaaaOGaamODamaaCaaaleqabaGaamizaaaaaaa@4C9C@

or equivalently

To do this we need to show the following results, where D is the covariant derivative operator and λ is a fine parameter indeed, e.g. x=x(λ), t=t(λ).,

If we go back to our geodesic equation for acceleration, which sort of defines what is meant by acceleration in generalized co-ordinates.

d 2 x μ d τ 2 + Γ μ βγ d x β dτ d x γ dτ =0 MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaWGKbWaaWbaaSqabeaacaaIYaaaaOGaamiEamaaCaaaleqabaGaeqiVd0gaaaGcbaGaamizaiabes8a0naaCaaaleqabaGaaGOmaaaaaaGccqGHRaWkcqqHtoWrdaahaaWcbeqaaiabeY7aTbaakmaaBaaaleaacqaHYoGycqaHZoWzaeqaaOWaaSaaaeaacaWGKbGaamiEamaaCaaaleqabaGaeqOSdigaaaGcbaGaamizaiabes8a0baadaWcaaqaaiaadsgacaWG4bWaaWbaaSqabeaacqaHZoWzaaaakeaacaWGKbGaeqiXdqhaaiabg2da9iaaicdaaaa@54DE@

which can be written as

∇ v V≡ V α ;β V β =0 MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaey4bIe9aaSbaaSqaaiaahAhaaeqaaOGaaCOvaiabggMi6kaabAfadaahaaWcbeqaaiabeg7aHbaakmaaBaaaleaacaGG7aGaeqOSdigabeaakiaabAfadaahaaWcbeqaaiabek7aIbaakiabg2da9iaaicdaaaa@450D@

So we can obviously write, actually this seems to pick something out of nothing, almost.

∇ w ∇ v V=0 MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaey4bIe9aaSbaaSqaaiaahEhaaeqaaOGaey4bIe9aaSbaaSqaaiaahAhaaeqaaOGaaCOvaiabg2da9iaaicdaaaa@3E0A@

∇ v ∇ w V+ ∇ w ∇ v V− ∇ v ∇ w V=0 MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaey4bIe9aaSbaaSqaaiaadAhaaeqaaOGaey4bIe9aaSbaaSqaaiaadEhaaeqaaOGaaCOvaiabgUcaRiabgEGirpaaBaaaleaacaWG3baabeaakiabgEGirpaaBaaaleaacaWH2baabeaakiaahAfacqGHsislcqGHhis0daWgaaWcbaGaamODaaqabaGccqGHhis0daWgaaWcbaGaam4DaaqabaGccaWHwbGaeyypa0JaaGimaaaa@4C71@

∇ v ∇ w V+[ ∇ w , ∇ v ]V=0 MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaey4bIe9aaSbaaSqaaiaadAhaaeqaaOGaey4bIe9aaSbaaSqaaiaadEhaaeqaaOGaaCOvaiabgUcaRiaacUfacqGHhis0daWgaaWcbaGaam4DaaqabaGccaGGSaGaey4bIe9aaSbaaSqaaiaadAhaaeqaaOGaaiyxaiaahAfacqGH9aqpcaaIWaaaaa@47A2@

∇ v ∇ v W+[ ∇ w , ∇ v ]V+ ∇ v ∇ w V− ∇ v ∇ v W=0 MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaey4bIe9aaSbaaSqaaiaadAhaaeqaaOGaey4bIe9aaSbaaSqaaiaadAhaaeqaaOGaaC4vaiabgUcaRiaacUfacqGHhis0daWgaaWcbaGaam4DaaqabaGccaGGSaGaey4bIe9aaSbaaSqaaiaadAhaaeqaaOGaaiyxaiaahAfacqGHRaWkcqGHhis0daWgaaWcbaGaamODaaqabaGccqGHhis0daWgaaWcbaGaam4DaaqabaGccaWHwbGaeyOeI0Iaey4bIe9aaSbaaSqaaiaadAhaaeqaaOGaey4bIe9aaSbaaSqaaiaadAhaaeqaaOGaaC4vaiabg2da9iaaicdaaaa@560D@

∇ v ∇ v W+[ ∇ w , ∇ v ]V+ ∇ v ( ∇ w V− ∇ v W)=0 MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaey4bIe9aaSbaaSqaaiaadAhaaeqaaOGaey4bIe9aaSbaaSqaaiaadAhaaeqaaOGaaC4vaiabgUcaRiaacUfacqGHhis0daWgaaWcbaGaam4DaaqabaGccaGGSaGaey4bIe9aaSbaaSqaaiaadAhaaeqaaOGaaiyxaiaahAfacqGHRaWkcqGHhis0daWgaaWcbaGaamODaaqabaGccaGGOaGaey4bIe9aaSbaaSqaaiaadEhaaeqaaOGaaCOvaiabgkHiTiabgEGirpaaBaaaleaacaWG2baabeaakiaahEfacaGGPaGaeyypa0JaaGimaaaa@54AF@

but, from up above, the last term is zero so finally then, using our extended commentator formula

and we seem to have lost a minus sign, so we'll leave that as an exercise for the reader.

Bianchi Identity and the Einstein Tensor

We have from above:

( V ;cd − V ;dc ) x a = R a bcd x b MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadAfadaWgaaWcbaGaai4oaiaadogacaWGKbaabeaakiabgkHiTiaadAfadaWgaaWcbaGaai4oaiaadsgacaWGJbaabeaakiaacMcacaWG4bWaaWbaaSqabeaacaWGHbaaaOGaeyypa0JaamOuamaaCaaaleqabaGaamyyaaaakmaaBaaaleaacaWGIbGaam4yaiaadsgaaeqaaOGaamiEamaaCaaaleqabaGaamOyaaaaaaa@498F@

x a ;cd − x a ;dc = x b R a bcd MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaCaaaleqabaGaamyyaaaakmaaBaaaleaacaGG7aGaam4yaiaadsgaaeqaaOGaeyOeI0IaamiEamaaCaaaleqabaGaamyyaaaakmaaBaaaleaacaGG7aGaamizaiaadogaaeqaaOGaeyypa0JaamiEamaaCaaaleqabaGaamOyaaaakiaadkfadaahaaWcbeqaaiaadggaaaGcdaWgaaWcbaGaamOyaiaadogacaWGKbaabeaaaaa@489A@

Which means that taking the covariant 2^nd derivative, in different orders, does not give the same result, as do ordinary derivatives.

It should be no surprise that, in the same manner as the covariant derivative itself, that

x a;cd − x a;dc =− x a R a bcd MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBaaaleaacaWGHbGaai4oaiaadogacaWGKbaabeaakiabgkHiTiaadIhadaWgaaWcbaGaamyyaiaacUdacaWGKbGaam4yaaqabaGccqGH9aqpcqGHsislcaWG4bWaaSbaaSqaaiaadggaaeqaaOGaamOuamaaCaaaleqabaGaamyyaaaakmaaBaaaleaacaWGIbGaam4yaiaadsgaaeqaaaaa@4917@

This can be seen from inspection from the initial derivation equation, and that again, just as in the covariant derivative case, where each tensor order index generates its own Christoffel symbol term, higher order tensors generate additional Riemann terms thus:

x a e ;cd − x a e ;dc = x b e R a bcd − x a b R b ecd MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaCaaaleqabaGaamyyaaaakmaaBaaaleaacaWGLbaabeaakmaaBaaaleaacaGG7aGaam4yaiaadsgaaeqaaOGaeyOeI0IaamiEamaaCaaaleqabaGaamyyaaaakmaaBaaaleaacaWGLbaabeaakmaaBaaaleaacaGG7aGaamizaiaadogaaeqaaOGaeyypa0JaamiEamaaCaaaleqabaGaamOyaaaakmaaBaaaleaacaWGLbaabeaakiaadkfadaahaaWcbeqaaiaadggaaaGcdaWgaaWcbaGaamOyaiaadogacaWGKbaabeaakiabgkHiTiaadIhadaahaaWcbeqaaiaadggaaaGcdaWgaaWcbaGaamOyaaqabaGccaWGsbWaaWbaaSqabeaacaWGIbaaaOWaaSbaaSqaaiaadwgacaWGJbGaamizaaqabaaaaa@5504@

Now differentiate

( x a ;cd − x a ;dc ) ;e = x b ;e R a bcd + x b R a bcd;e MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadIhadaahaaWcbeqaaiaadggaaaGcdaWgaaWcbaGaai4oaiaadogacaWGKbaabeaakiabgkHiTiaadIhadaahaaWcbeqaaiaadggaaaGcdaWgaaWcbaGaai4oaiaadsgacaWGJbaabeaakiaacMcadaWgaaWcbaGaai4oaiaadwgaaeqaaOGaeyypa0JaamiEamaaCaaaleqabaGaamOyaaaakmaaBaaaleaacaGG7aGaamyzaaqabaGccaWGsbWaaWbaaSqabeaacaWGHbaaaOWaaSbaaSqaaiaadkgacaWGJbGaamizaaqabaGccqGHRaWkcaWG4bWaaWbaaSqabeaacaWGIbaaaOGaamOuamaaCaaaleqabaGaamyyaaaakmaaBaaaleaacaWGIbGaam4yaiaadsgacaGG7aGaamyzaaqabaaaaa@5739@

Now set

x a e −>− x a ;e MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaCaaaleqabaGaamyyaaaakmaaBaaaleaacaWGLbaabeaakiabgkHiTiabg6da+iabgkHiTiaadIhadaahaaWcbeqaaiaadggaaaGcdaWgaaWcbaGaai4oaiaadwgaaeqaaaaa@3FFF@

And replace into our 2^nd term Riemann expression.

− ( x a ;e ) ;cd + ( x a ;e ) ;dc =−( x b ;e ) R a bcd +( x a ;b ) R b ecd MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@5F28@

− x a ;ecd + x a ;edc =− x b ;e R a bcd + x a ;b R b ecd MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0IaamiEamaaCaaaleqabaGaamyyaaaakmaaBaaaleaacaGG7aGaamyzaiaadogacaWGKbaabeaakiabgUcaRiaadIhadaahaaWcbeqaaiaadggaaaGcdaWgaaWcbaGaai4oaiaadwgacaWGKbGaam4yaaqabaGccqGH9aqpcqGHsislcaWG4bWaaWbaaSqabeaacaWGIbaaaOWaaSbaaSqaaiaacUdacaWGLbaabeaakiaadkfadaahaaWcbeqaaiaadggaaaGcdaWgaaWcbaGaamOyaiaadogacaWGKbaabeaakiabgUcaRiaadIhadaahaaWcbeqaaiaadggaaaGcdaWgaaWcbaGaai4oaiaadkgaaeqaaOGaamOuamaaCaaaleqabaGaamOyaaaakmaaBaaaleaacaWGLbGaam4yaiaadsgaaeqaaaaa@57DA@ `

and bringing down from above

x a ;cde − x a ;dce = x b ;e R a bcd + x b R a bcd;e MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaCaaaleqabaGaamyyaaaakmaaBaaaleaacaGG7aGaam4yaiaadsgacaWGLbaabeaakiabgkHiTiaadIhadaahaaWcbeqaaiaadggaaaGcdaWgaaWcbaGaai4oaiaadsgacaWGJbGaamyzaaqabaGccqGH9aqpcaWG4bWaaWbaaSqabeaacaWGIbaaaOWaaSbaaSqaaiaacUdacaWGLbaabeaakiaadkfadaahaaWcbeqaaiaadggaaaGcdaWgaaWcbaGaamOyaiaadogacaWGKbaabeaakiabgUcaRiaadIhadaahaaWcbeqaaiaadkgaaaGccaWGsbWaaWbaaSqabeaacaWGHbaaaOWaaSbaaSqaaiaadkgacaWGJbGaamizaiaacUdacaWGLbaabeaaaaa@55D5@

Now a little bit of index swapping on the above two equations:

− x a ;ecd + x a ;edc =− x a ;e R b acd + x a ;b R b ecd MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0IaamiEamaaCaaaleqabaGaamyyaaaakmaaBaaaleaacaGG7aGaamyzaiaadogacaWGKbaabeaakiabgUcaRiaadIhadaahaaWcbeqaaiaadggaaaGcdaWgaaWcbaGaai4oaiaadwgacaWGKbGaam4yaaqabaGccqGH9aqpcqGHsislcaWG4bWaaWbaaSqabeaacaWGHbaaaOWaaSbaaSqaaiaacUdacaWGLbaabeaakiaadkfadaahaaWcbeqaaiaadkgaaaGcdaWgaaWcbaGaamyyaiaadogacaWGKbaabeaakiabgUcaRiaadIhadaahaaWcbeqaaiaadggaaaGcdaWgaaWcbaGaai4oaiaadkgaaeqaaOGaamOuamaaCaaaleqabaGaamOyaaaakmaaBaaaleaacaWGLbGaam4yaiaadsgaaeqaaaaa@57D9@

x a ;cde − x a ;dce = x a ;e R b acd + x b R a bcd;e MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaCaaaleqabaGaamyyaaaakmaaBaaaleaacaGG7aGaam4yaiaadsgacaWGLbaabeaakiabgkHiTiaadIhadaahaaWcbeqaaiaadggaaaGcdaWgaaWcbaGaai4oaiaadsgacaWGJbGaamyzaaqabaGccqGH9aqpcaWG4bWaaWbaaSqabeaacaWGHbaaaOWaaSbaaSqaaiaacUdacaWGLbaabeaakiaadkfadaahaaWcbeqaaiaadkgaaaGcdaWgaaWcbaGaamyyaiaadogacaWGKbaabeaakiabgUcaRiaadIhadaahaaWcbeqaaiaadkgaaaGccaWGsbWaaWbaaSqabeaacaWGHbaaaOWaaSbaaSqaaiaadkgacaWGJbGaamizaiaacUdacaWGLbaabeaaaaa@55D4@

Now, if these last two equations are cycled in the last 3 index of Riemann and the 6 added together:

− x a ;dec + x a ;dce =− x a ;d R b aec + x a ;b R b dec MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0IaamiEamaaCaaaleqabaGaamyyaaaakmaaBaaaleaacaGG7aGaamizaiaadwgacaWGJbaabeaakiabgUcaRiaadIhadaahaaWcbeqaaiaadggaaaGcdaWgaaWcbaGaai4oaiaadsgacaWGJbGaamyzaaqabaGccqGH9aqpcqGHsislcaWG4bWaaWbaaSqabeaacaWGHbaaaOWaaSbaaSqaaiaacUdacaWGKbaabeaakiaadkfadaahaaWcbeqaaiaadkgaaaGcdaWgaaWcbaGaamyyaiaadwgacaWGJbaabeaakiabgUcaRiaadIhadaahaaWcbeqaaiaadggaaaGcdaWgaaWcbaGaai4oaiaadkgaaeqaaOGaamOuamaaCaaaleqabaGaamOyaaaakmaaBaaaleaacaWGKbGaamyzaiaadogaaeqaaaaa@57D9@

− x a ;cde + x a ;ced =− x a ;c R b ade + x a ;b R b cde MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0IaamiEamaaCaaaleqabaGaamyyaaaakmaaBaaaleaacaGG7aGaam4yaiaadsgacaWGLbaabeaakiabgUcaRiaadIhadaahaaWcbeqaaiaadggaaaGcdaWgaaWcbaGaai4oaiaadogacaWGLbGaamizaaqabaGccqGH9aqpcqGHsislcaWG4bWaaWbaaSqabeaacaWGHbaaaOWaaSbaaSqaaiaacUdacaWGJbaabeaakiaadkfadaahaaWcbeqaaiaadkgaaaGcdaWgaaWcbaGaamyyaiaadsgacaWGLbaabeaakiabgUcaRiaadIhadaahaaWcbeqaaiaadggaaaGcdaWgaaWcbaGaai4oaiaadkgaaeqaaOGaamOuamaaCaaaleqabaGaamOyaaaakmaaBaaaleaacaWGJbGaamizaiaadwgaaeqaaaaa@57D9@

x a ;ecd − x a ;ced = x a ;d R b aec + x b R a bec;d MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaCaaaleqabaGaamyyaaaakmaaBaaaleaacaGG7aGaamyzaiaadogacaWGKbaabeaakiabgkHiTiaadIhadaahaaWcbeqaaiaadggaaaGcdaWgaaWcbaGaai4oaiaadogacaWGLbGaamizaaqabaGccqGH9aqpcaWG4bWaaWbaaSqabeaacaWGHbaaaOWaaSbaaSqaaiaacUdacaWGKbaabeaakiaadkfadaahaaWcbeqaaiaadkgaaaGcdaWgaaWcbaGaamyyaiaadwgacaWGJbaabeaakiabgUcaRiaadIhadaahaaWcbeqaaiaadkgaaaGccaWGsbWaaWbaaSqabeaacaWGHbaaaOWaaSbaaSqaaiaadkgacaWGLbGaam4yaiaacUdacaWGKbaabeaaaaa@55D4@

x a ;dec − x a ;edc = x a ;c R b ade + x b R a bde;c MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaCaaaleqabaGaamyyaaaakmaaBaaaleaacaGG7aGaamizaiaadwgacaWGJbaabeaakiabgkHiTiaadIhadaahaaWcbeqaaiaadggaaaGcdaWgaaWcbaGaai4oaiaadwgacaWGKbGaam4yaaqabaGccqGH9aqpcaWG4bWaaWbaaSqabeaacaWGHbaaaOWaaSbaaSqaaiaacUdacaWGJbaabeaakiaadkfadaahaaWcbeqaaiaadkgaaaGcdaWgaaWcbaGaamyyaiaadsgacaWGLbaabeaakiabgUcaRiaadIhadaahaaWcbeqaaiaadkgaaaGccaWGsbWaaWbaaSqabeaacaWGHbaaaOWaaSbaaSqaaiaadkgacaWGKbGaamyzaiaacUdacaWGJbaabeaaaaa@55D4@

Hence:

x b ( R a bcd;e + R a bec;d + R a bde;c )+( R b ecd + R b dec + R b cde ) x a ;b =0 MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@6734@

But the 2^nd term is zero from the result up the page, somewhere…

Hence:

R a bcd;e + R a bec;d + R a bde;c =0 MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOuamaaCaaaleqabaGaamyyaaaakmaaBaaaleaacaWGIbGaam4yaiaadsgacaGG7aGaamyzaaqabaGccqGHRaWkcaWGsbWaaWbaaSqabeaacaWGHbaaaOWaaSbaaSqaaiaadkgacaWGLbGaam4yaiaacUdacaWGKbaabeaakiabgUcaRiaadkfadaahaaWcbeqaaiaadggaaaGcdaWgaaWcbaGaamOyaiaadsgacaWGLbGaai4oaiaadogaaeqaaOGaeyypa0JaaGimaaaa@4D19@

This is the Bianchi identity that is needed in the construction of the Einstein Tensor.

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providing that this source is acknowledged.

Website last modified 1^st January 2022

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