General Relativity For Tellytubbys

General Relativity For Tellytubbys

Geodesic Equation

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

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Overview

This section follows on from the section on Euler-Langrange equations. The task here is to find the geodesic equation that describes straight lines in general.

Geodesic Equation

I do hope you recall from the other pages that, one form of the Euler-Langrange equation is

∂f ∂ x α − d dλ ( ∂f ∂ x ˙ α )=0 MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacqGHciITcaWGMbaabaGaeyOaIyRaamiEamaaCaaaleqabaGaeqySdegaaaaakiabgkHiTmaalaaabaGaamizaaqaaiaadsgacqaH7oaBaaGaaiikamaalaaabaGaeyOaIyRaamOzaaqaaiabgkGi2kqadIhagaGaamaaCaaaleqabaGaeqySdegaaaaakiaacMcacqGH9aqpcaaIWaaaaa@4ACD@

I= ∫ f( x α , x ˙ α ,λ) dλ , where x ˙ = dx dλ MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysaiabg2da9maapeaabaGaaiOzaiaacIcacaWG4bWaaWbaaSqabeaacqaHXoqyaaGccaGGSaGabmiEayaacaWaaWbaaSqabeaacqaHXoqyaaGccaGGSaGaeq4UdWMaaiykaaWcbeqab0Gaey4kIipakiaadsgacqaH7oaBcaqGGaGaaeilaiaabccacaqG3bGaaeiAaiaabwgacaqGYbGaaeyzaiaabccaceWG4bGbaiaacqGH9aqpdaWcaaqaaiaadsgacaWG4baabaGaamizaiabeU7aSbaaaaa@5551@

are the conditions that finds a local minimum, maximums or inflection point of an integral of f.

because that was indeed a waste of brain power, we're going to ignore that just for now, and first derive the geodesic equation directly. This is so we can get a better handle on what's going on from more then one point of view.

Geodesic Equation Method 1

Consider a Tellytubby playing on a slide chute, i.e. undergoing acceleration

a= dv dτ MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCyyaiabg2da9maalaaabaGaamizaiaahAhaaeaacaWGKbGaeqiXdqhaaaaa@3C8A@

If there are no net forces acting on Po (this is the deeper meaning bit of G.R.) in order to achieve this acceleration then we have, from Newton's laws

ma= dv dτ =0 MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyBaiaahggacqGH9aqpdaWcaaqaaiaadsgacaWH2baabaGaamizaiabes8a0baacqGH9aqpcaaIWaaaaa@3F3C@

In our newly acquired, very impressive tensor notation, this can be written, noting that derivatives go over to covariant derivatives always, as

dv dτ = ∇ v V≡ V α ;β V β MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaWGKbGaaCODaaqaaiaadsgacqaHepaDaaGaeyypa0Jaey4bIe9aaSbaaSqaaiaahAhaaeqaaOGaaCOvaiabggMi6kaabAfadaahaaWcbeqaaiabeg7aHbaakmaaBaaaleaacaGG7aGaeqOSdigabeaakiaabAfadaahaaWcbeqaaiabek7aIbaaaaa@48EF@

because,

V α ,β V β = ∂ V α ∂ x β ∂ x β ∂τ = d V α dτ MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaeOvamaaCaaaleqabaGaeqySdegaaOWaaSbaaSqaaiaacYcacqaHYoGyaeqaaOGaaeOvamaaCaaaleqabaGaeqOSdigaaOGaeyypa0ZaaSaaaeaacqGHciITcaqGwbWaaWbaaSqabeaacqaHXoqyaaaakeaacqGHciITcaWG4bWaaWbaaSqabeaacqaHYoGyaaaaaOWaaSaaaeaacqGHciITcaWG4bWaaWbaaSqabeaacqaHYoGyaaaakeaacqGHciITcqaHepaDaaGaeyypa0ZaaSaaaeaacaWGKbGaaeOvamaaCaaaleqabaGaeqySdegaaaGcbaGaamizaiabes8a0baaaaa@5613@

and noting the obvious extension to the ";" is required

So, to continue with

V α ;β V β =0 MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaeOvamaaCaaaleqabaGaeqySdegaaOWaaSbaaSqaaiaacUdacqaHYoGyaeqaaOGaaeOvamaaCaaaleqabaGaeqOSdigaaOGaeyypa0JaaGimaaaa@3FAA@

V α ,β V β + V β V μ Γ α μβ =0 MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaeOvamaaCaaaleqabaGaeqySdegaaOWaaSbaaSqaaiaacYcacqaHYoGyaeqaaOGaaeOvamaaCaaaleqabaGaeqOSdigaaOGaey4kaSIaaeOvamaaCaaaleqabaGaeqOSdigaaOGaaeOvamaaCaaaleqabaGaeqiVd0gaaOGaeu4KdC0aaWbaaSqabeaacqaHXoqyaaGcdaWgaaWcbaGaeqiVd0MaeqOSdigabeaakiabg2da9iaaicdaaaa@4CBF@

guess what index's we swapped now

V μ ,β V β + V β V α Γ μ αβ =0 MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaeOvamaaCaaaleqabaGaeqiVd0gaaOWaaSbaaSqaaiaacYcacqaHYoGyaeqaaOGaaeOvamaaCaaaleqabaGaeqOSdigaaOGaey4kaSIaaeOvamaaCaaaleqabaGaeqOSdigaaOGaaeOvamaaCaaaleqabaGaeqySdegaaOGaeu4KdC0aaWbaaSqabeaacqaH8oqBaaGcdaWgaaWcbaGaeqySdeMaeqOSdigabeaakiabg2da9iaaicdaaaa@4CBF@

but we have V α ≡ d x α dτ MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaeOyaiaabwhacaqG0bGaaeiiaiaabEhacaqGLbGaaeiiaiaabIgacaqGHbGaaeODaiaabwgacaqGGaGaaeOvamaaCaaaleqabaGaeqySdegaaOGaeyyyIO7aaSaaaeaacaWGKbGaamiEamaaCaaaleqabaGaeqySdegaaaGcbaGaamizaiabes8a0baaaaa@4B35@

and so the first term can be written as

V μ ,β V β = ∂ V μ ∂ x β ∂ x μ ∂τ = d V μ dτ = d 2 x μ d τ 2 by our wonderfull chain rule MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@7984@

and subbing in again to all terms gets us

d 2 x μ d τ 2 + Γ μ αβ d x α dτ d x β dτ =0 MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaqGKbWaaWbaaSqabeaacaqGYaaaaOGaamiEamaaCaaaleqabaGaeqiVd0gaaaGcbaGaaeizaiabes8a0naaCaaaleqabaGaaGOmaaaaaaGccqGHRaWkcqqHtoWrdaahaaWcbeqaaiabeY7aTbaakmaaBaaaleaacqaHXoqycqaHYoGyaeqaaOWaaSaaaeaacaWGKbGaamiEamaaCaaaleqabaGaeqySdegaaaGcbaGaamizaiabes8a0baadaWcaaqaaiaadsgacaWG4bWaaWbaaSqabeaacqaHYoGyaaaakeaacaWGKbGaeqiXdqhaaiabg2da9iaaicdaaaa@54C3@

Which is the geodesic equation that we are after.

So, this gives one a bit of a feel, one hopes, of what is happening dude

Geodesic Equation Method 2

Now to do the difficult bit and show how things all tie up with the variational principle

Consider the path that light takes

c= ds dτ , where s is the distance traveled and τ is defined as the proper time MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@7AAD@

so that, using our prior result for distance, one can write

d τ 2 = 1 c 2 d s 2 = 1 c 2 g βγ d x β d x γ MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamizaiabes8a0naaCaaaleqabaGaaGOmaaaakiabg2da9maalaaabaGaaGymaaqaaiaadogadaahaaWcbeqaaiaaikdaaaaaaOGaamizaiaadohadaahaaWcbeqaaiaaikdaaaGccqGH9aqpdaWcaaqaaiaaigdaaeaacaWGJbWaaWbaaSqabeaacaaIYaaaaaaakiaadEgadaWgaaWcbaGaeqOSdiMaeq4SdCgabeaakiaadsgacaWG4bWaaWbaaSqabeaacqaHYoGyaaGccaWGKbGaamiEamaaCaaaleqabaGaeq4SdCgaaaaa@4FA3@

To make the sums all work out, an "affine parameter " for the time is introduced. This is simply to get rid of all those dx's, bloody annoyance that they are.

So, let τ=τ(λ) then dτ= dτ dλ dτ MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaae4uaiaab+gacaqGSaGaaeiiaiaabccacaqGSbGaaeyzaiaabshacaqGGaGaeqiXdqNaeyypa0JaeqiXdqNaaiikaiabeU7aSjaacMcacaqGGaGaaeiDaiaabIgacaqGLbGaaeOBaiaabccacaWGKbGaeqiXdqNaeyypa0ZaaSaaaeaacaWGKbGaeqiXdqhabaGaamizaiabeU7aSbaacaWGKbGaeqiXdqhaaa@557D@

and dividing out by dλ in our distance formula above gives, well after taking the square root and all

dτ dλ = 1 c ds dλ = 1 c [ g βγ d x β dλ d x γ dλ ] 1 2 MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaWGKbGaeqiXdqhabaGaamizaiabeU7aSbaacqGH9aqpdaWcaaqaaiaaigdaaeaacaWGJbaaamaalaaabaGaamizaiaadohaaeaacaWGKbGaeq4UdWgaaiabg2da9maalaaabaGaaGymaaqaaiaadogaaaWaamWaaeaacaWGNbWaaSbaaSqaaiabek7aIjabeo7aNbqabaGcdaWcaaqaaiaadsgacaWG4bWaaWbaaSqabeaacqaHYoGyaaaakeaacaWGKbGaeq4UdWgaamaalaaabaGaamizaiaadIhadaahaaWcbeqaaiabeo7aNbaaaOqaaiaadsgacqaH7oaBaaaacaGLBbGaayzxaaWaaWbaaSqabeaadaWcaaqaaiaaigdaaeaacaaIYaaaaaaaaaa@5A3B@

Hence:

dτ= 1 c [ g βγ d x β dλ d x γ dλ ] 1 2 dλ MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamizaiabes8a0jabg2da9maalaaabaGaaGymaaqaaiaadogaaaWaamWaaeaacaWGNbWaaSbaaSqaaiabek7aIjabeo7aNbqabaGcdaWcaaqaaiaadsgacaWG4bWaaWbaaSqabeaacqaHYoGyaaaakeaacaWGKbGaeq4UdWgaamaalaaabaGaamizaiaadIhadaahaaWcbeqaaiabeo7aNbaaaOqaaiaadsgacqaH7oaBaaaacaGLBbGaayzxaaWaaWbaaSqabeaadaWcaaqaaiaaigdaaeaacaaIYaaaaaaakiaadsgacqaH7oaBaaa@52EE@

or finding the total time

τ= ∫ 1 c [ g βγ ( x α ) d x β dλ d x γ dλ ] 1 2 dλ MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiXdqNaeyypa0Zaa8qaaeaadaWcaaqaaiaaigdaaeaacaWGJbaaamaadmaabaGaam4zamaaBaaaleaacqaHYoGycqaHZoWzaeqaaOGaaiikaiaadIhadaahaaWcbeqaaiabeg7aHbaakiaacMcadaWcaaqaaiaadsgacaWG4bWaaWbaaSqabeaacqaHYoGyaaaakeaacaWGKbGaeq4UdWgaamaalaaabaGaamizaiaadIhadaahaaWcbeqaaiabeo7aNbaaaOqaaiaadsgacqaH7oaBaaaacaGLBbGaayzxaaWaaWbaaSqabeaadaWcaaqaaiaaigdaaeaacaaIYaaaaaaaaeqabeqdcqGHRiI8aOGaamizaiabeU7aSbaa@5821@

So, now the job is to minimize this integral, ~~Laa Laa~~ oops, I mean ala this is the celebrated least action integral for our geodesic.

When I was ~~plagiarizing~~ researching for this project on the web I found one or two derivations of this result. However, they were all rather more complicated because it is obvious that whatever locally minimizes f^1/2, will also locally minimize plain old f as well, so we'll drop the square root complication and just consider:

τ= ∫ g βγ ( x α ) d x β dλ d x γ dλ dλ MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiXdqNaeyypa0Zaa8qaaeaacaWGNbWaaSbaaSqaaiabek7aIjabeo7aNbqabaGccaGGOaGaamiEamaaCaaaleqabaGaeqySdegaaOGaaiykamaalaaabaGaamizaiaadIhadaahaaWcbeqaaiabek7aIbaaaOqaaiaadsgacqaH7oaBaaWaaSaaaeaacaWGKbGaamiEamaaCaaaleqabaGaeq4SdCgaaaGcbaGaamizaiabeU7aSbaaaSqabeqaniabgUIiYdGccaWGKbGaeq4UdWgaaa@52D3@

First term, and note we have dropped c because we are equating to 0

∂f ∂ x α = g βγ,α d x β dλ d x γ dλ MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacqGHciITcaWGMbaabaGaeyOaIyRaamiEamaaCaaaleqabaGaeqySdegaaaaakiabg2da9iaadEgadaWgaaWcbaGaeqOSdiMaeq4SdCMaaiilaiabeg7aHbqabaGcdaWcaaqaaiaadsgacaWG4bWaaWbaaSqabeaacqaHYoGyaaaakeaacaWGKbGaeq4UdWgaamaalaaabaGaamizaiaadIhadaahaaWcbeqaaiabeo7aNbaaaOqaaiaadsgacqaH7oaBaaaaaa@5129@

Second term

∂f ∂ x ˙ α = g βγ ∂ ∂ x ˙ α ( d x β dλ d x γ dλ ) MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacqGHciITcaWGMbaabaGaeyOaIyRabmiEayaacaWaaWbaaSqabeaacqaHXoqyaaaaaOGaeyypa0Jaam4zamaaBaaaleaacqaHYoGycqaHZoWzaeqaaOWaaSaaaeaacqGHciITaeaacqGHciITceWG4bGbaiaadaahaaWcbeqaaiabeg7aHbaaaaGccaGGOaWaaSaaaeaacaWGKbGaamiEamaaCaaaleqabaGaeqOSdigaaaGcbaGaamizaiabeU7aSbaadaWcaaqaaiaadsgacaWG4bWaaWbaaSqabeaacqaHZoWzaaaakeaacaWGKbGaeq4UdWgaaiaacMcaaaa@55F4@

∂f ∂ x ˙ α = g βγ ∂ ∂ x ˙ α ( x ˙ β x ˙ γ ) MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacqGHciITcaWGMbaabaGaeyOaIyRabmiEayaacaWaaWbaaSqabeaacqaHXoqyaaaaaOGaeyypa0Jaam4zamaaBaaaleaacqaHYoGycqaHZoWzaeqaaOWaaSaaaeaacqGHciITaeaacqGHciITceWG4bGbaiaadaahaaWcbeqaaiabeg7aHbaaaaGccaGGOaGabmiEayaacaWaaWbaaSqabeaacqaHYoGyaaGcceWG4bGbaiaadaahaaWcbeqaaiabeo7aNbaakiaacMcaaaa@4EDA@

where I've changed the notation to make it a bit clearer what's going on. So mentally ignore the dots on the x's when doing the sums. I have filled in all the steps because they were not done in the derivation where I copied the outline of this from. What these poor excuse's for Tellytubby professors don’t realize is that, precisely because the reader is going through these elementary deviations, it inherently implies that the punter is not familiar with these sorts of calculations, and so more guidance is needed. e.g. Note how the delta swap's index's.

∂f ∂ x ˙ α = g βγ [ ∂ x ˙ β ∂ x ˙ α x ˙ γ + x ˙ β ∂ x ˙ γ ∂ x ˙ α ] MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacqGHciITcaWGMbaabaGaeyOaIyRabmiEayaacaWaaWbaaSqabeaacqaHXoqyaaaaaOGaeyypa0Jaam4zamaaBaaaleaacqaHYoGycqaHZoWzaeqaaOWaamWaaeaadaWcaaqaaiabgkGi2kqadIhagaGaamaaCaaaleqabaGaeqOSdigaaaGcbaGaeyOaIyRabmiEayaacaWaaWbaaSqabeaacqaHXoqyaaaaaOGabmiEayaacaWaaWbaaSqabeaacqaHZoWzaaGccqGHRaWkceWG4bGbaiaadaahaaWcbeqaaiabek7aIbaakmaalaaabaGaeyOaIyRabmiEayaacaWaaWbaaSqabeaacqaHZoWzaaaakeaacqGHciITceWG4bGbaiaadaahaaWcbeqaaiabeg7aHbaaaaaakiaawUfacaGLDbaaaaa@5BCF@

∂f ∂ x ˙ α = g βγ [ ∂ x ˙ β ∂ x ˙ γ ∂ x ˙ γ ∂ x ˙ α x ˙ γ + x ˙ β δ α γ ] MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@60F9@

∂f ∂ x ˙ α = g βγ [ δ γ β δ α γ x ˙ γ + x ˙ β δ α γ ] MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacqGHciITcaWGMbaabaGaeyOaIyRabmiEayaacaWaaWbaaSqabeaacqaHXoqyaaaaaOGaeyypa0Jaam4zamaaBaaaleaacqaHYoGycqaHZoWzaeqaaOWaamWaaeaacqaH0oazdaqhaaWcbaGaeq4SdCgabaGaeqOSdigaaOGaeqiTdq2aa0baaSqaaiabeg7aHbqaaiabeo7aNbaakiqadIhagaGaamaaCaaaleqabaGaeq4SdCgaaOGaey4kaSIabmiEayaacaWaaWbaaSqabeaacqaHYoGyaaGccqaH0oazdaqhaaWcbaGaeqySdegabaGaeq4SdCgaaaGccaGLBbGaayzxaaaaaa@5A05@

∂f ∂ x ˙ α = g βγ δ α γ [ δ γ β x ˙ γ + x ˙ β ] MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacqGHciITcaWGMbaabaGaeyOaIyRabmiEayaacaWaaWbaaSqabeaacqaHXoqyaaaaaOGaeyypa0Jaam4zamaaBaaaleaacqaHYoGycqaHZoWzaeqaaOGaeqiTdq2aa0baaSqaaiabeg7aHbqaaiabeo7aNbaakmaadmaabaGaeqiTdq2aa0baaSqaaiabeo7aNbqaaiabek7aIbaakiqadIhagaGaamaaCaaaleqabaGaeq4SdCgaaOGaey4kaSIabmiEayaacaWaaWbaaSqabeaacqaHYoGyaaaakiaawUfacaGLDbaaaaa@54E3@

∂f ∂ x ˙ α = g βγ δ α γ [ x ˙ β + x ˙ β ] MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacqGHciITcaWGMbaabaGaeyOaIyRabmiEayaacaWaaWbaaSqabeaacqaHXoqyaaaaaOGaeyypa0Jaam4zamaaBaaaleaacqaHYoGycqaHZoWzaeqaaOGaeqiTdq2aa0baaSqaaiabeg7aHbqaaiabeo7aNbaakmaadmaabaGabmiEayaacaWaaWbaaSqabeaacqaHYoGyaaGccqGHRaWkceWG4bGbaiaadaahaaWcbeqaaiabek7aIbaaaOGaay5waiaaw2faaaaa@4FB9@

∂f ∂ x ˙ α =2. g βα x ˙ β MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacqGHciITcaWGMbaabaGaeyOaIyRabmiEayaacaWaaWbaaSqabeaacqaHXoqyaaaaaOGaeyypa0JaaGOmaiaac6cacaWGNbWaaSbaaSqaaiabek7aIjabeg7aHbqabaGcceWG4bGbaiaadaahaaWcbeqaaiabek7aIbaaaaa@4641@

We now have then

g βγ,α d x β dλ d x γ dλ − d dλ [ 2. g αβ d x β dλ ]=0 MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4zamaaBaaaleaacqaHYoGycqaHZoWzcaGGSaGaeqySdegabeaakmaalaaabaGaamizaiaadIhadaahaaWcbeqaaiabek7aIbaaaOqaaiaadsgacqaH7oaBaaWaaSaaaeaacaWGKbGaamiEamaaCaaaleqabaGaeq4SdCgaaaGcbaGaamizaiabeU7aSbaacqGHsisldaWcaaqaaiaadsgaaeaacaWGKbGaeq4UdWgaamaadmaabaGaaGOmaiaac6cacaWGNbWaaSbaaSqaaiabeg7aHjabek7aIbqabaGcdaWcaaqaaiaadsgacaWG4bWaaWbaaSqabeaacqaHYoGyaaaakeaacaWGKbGaeq4UdWgaaaGaay5waiaaw2faaiabg2da9iaaicdaaaa@5DF9@

g βγ,α d x β dλ d x γ dλ −2. g αβ d dλ ( d x β dλ )−2. d dλ ( g αβ ). d x β dλ =0 MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@7029@

g βγ,α d x β dλ d x γ dλ −2. g αβ d 2 x β d λ 2 −2. d d x γ ( g αβ ) d x γ dλ d x β dλ =0 MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@7406@

g βγ,α d x β dλ d x γ dλ −2. g αβ d 2 x β d λ 2 −2. g αβ,γ d x γ dλ d x β dλ =0 MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@7047@

g μα g βγ,α d x β dλ d x γ dλ −2. g μα g αβ d 2 x β d λ 2 −2. g μα g αβ,γ d x γ dλ d x β dλ =0 MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@7DAF@

2. d 2 x μ d λ 2 − g μα g βγ,α d x β dλ d x γ dλ +2. g μα g αβ,γ d x γ dλ d x β dλ =0 MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@74DF@

2. d 2 x μ d λ 2 − g μα g βγ,α d x β dλ d x γ dλ + g μα g αβ,γ d x γ dλ d x β dλ + g μα g αβ,γ d x γ dλ d x β dλ =0 MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@8C60@

d 2 x μ d λ 2 + g μα 2 [ g αβ,γ d x γ dλ d x β dλ + g αβ,γ d x γ dλ d x β dλ − g βγ,α d x β dλ d x γ dλ ]=0 MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@84C0@

Ahmm, getting close, seems familiar? Swap one more time

d 2 x μ d λ 2 + g μα 2 [ g αβ,γ d x γ dλ d x β dλ + g αγ,β d x β dλ d x γ dλ − g βγ,α d x β dλ d x γ dλ ]=0 MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@84C0@

d 2 x μ d λ 2 + g μα 2 [ g αβ,γ + g αγ,β − g βγ,α ] d x β dλ d x γ dλ =0 MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@6B08@

which, by referring to our Christoffel page, is

d 2 x μ d λ 2 + Γ μ βγ d x β dλ d x γ dλ =0 MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaWGKbWaaWbaaSqabeaacaaIYaaaaOGaamiEamaaCaaaleqabaGaeqiVd0gaaaGcbaGaamizaiabeU7aSnaaCaaaleqabaGaaGOmaaaaaaGccqGHRaWkcqqHtoWrdaahaaWcbeqaaiabeY7aTbaakmaaBaaaleaacqaHYoGycqaHZoWzaeqaaOWaaSaaaeaacaWGKbGaamiEamaaCaaaleqabaGaeqOSdigaaaGcbaGaamizaiabeU7aSbaadaWcaaqaaiaadsgacaWG4bWaaWbaaSqabeaacqaHZoWzaaaakeaacaWGKbGaeq4UdWgaaiabg2da9iaaicdaaaa@54AB@

and, obviously, we can let tau = lambda

amazing, ain't it. How different methods give the same answer.

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