Geodesic (general relativity)/Proofs

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This mathematics article is devoted entirely to providing mathematical proofs and support for claims and statements made in the article geodesic (general relativity). This article is currently an experimental vehicle to see how well we can provide proofs and details for a math article without cluttering up the main article itself. See Wikipedia:WikiProject Mathematics/Proofs for some current discussion. This article is "experimental" in the sense that it is a test of one way we may be able to incorporate more detailed proofs in Wikipedia.

[edit] Proof 1

\nabla_{\vec U} \vec U = 0,
U^\alpha \nabla_\alpha \vec U = 0,
UαUβ;α = 0,
Uα(Uβ,α + UσΓβσα) = 0,
UαUβ,α + ΓβσαUαUσ = 0,
\ddot x^\beta + \Gamma^\beta {}_{\sigma \alpha} \dot x^\sigma \dot x^\alpha = 0. \

(return to article)

[edit] Proof 2

The goal being to extremize the value of

l = \int d\tau = \int {d\tau \over d\phi} \, d\phi = \int \sqrt{{(d\tau)^2 \over (d\phi)^2}} \, d\phi = \int \sqrt{{-g_{\mu \nu} dx^\mu dx^\nu \over d\phi \, d\phi}} \, d\phi = \int f \, d\phi

where

f = \sqrt{-g_{\mu \nu} \dot x^\mu \dot x^\nu}

such goal can be accomplished by calculating the Euler-Lagrange equation for f, which is

{d \over d\tau} {\partial f \over \partial \dot x^\lambda} = {\partial f \over \partial x^\lambda}.

Substituting the expression of f into the Euler-Lagrange equation (which extremizes the value of the integral l), gives

{d \over d\tau} {\partial \sqrt{-g_{\mu \nu} \dot x^\mu \dot x^\nu} \over \partial \dot x^\lambda} = {\partial \sqrt{-g_{\mu \nu} \dot x^\mu \dot x^\nu} \over \partial x^\lambda}

Now calculate the derivatives: {d \over d\tau} \left( {-g_{\mu \nu} {\partial \dot x^\mu \over \partial \dot x^\lambda} \dot x^\nu - g_{\mu \nu} \dot x^\mu {\partial \dot x^\nu \over \partial \dot x^\lambda} \over 2 \sqrt{-g_{\mu \nu} \dot x^\mu \dot x^\nu}} \right) = {-g_{\mu \nu, \lambda} \dot x^\mu \dot x^\nu \over 2 \sqrt{-g_{\mu \nu} \dot x^\mu \dot x^\nu}} \qquad \qquad (1)

{d \over d\tau} \left( {g_{\mu \nu} \delta^\mu {}_\lambda \dot x^\nu + g_{\mu \nu} \dot x^\mu \delta^\nu {}_\lambda \over 2 \sqrt{-g_{\mu \nu} \dot x^\mu \dot x^\nu}} \right) = {g_{\mu \nu , \lambda} \dot x^\mu \dot x^\nu \over 2 \sqrt{-g_{\mu \nu} \dot x^\mu \dot x^\nu}} \qquad \qquad (2)

{d \over d\tau} \left( {g_{\lambda \nu} \dot x^\nu + g_{\mu \lambda} \dot x^\mu \over \sqrt{-g_{\mu \nu} \dot x^\mu \dot x^\nu}} \right) = {g_{\mu \nu , \lambda} \dot x^\mu \dot x^\nu \over \sqrt{-g_{\mu \nu} \dot x^\mu \dot x^\nu}} \qquad \qquad (3)

{\sqrt{-g_{\mu \nu} \dot x^\mu \dot x^\nu} {d \over d\tau} (g_{\lambda \nu} \dot x^\nu + g_{\mu \lambda} \dot x^\mu) - (g_{\lambda \nu} \dot x^\nu + g_{\mu \lambda} \dot x^\mu) {d \over d\tau} \sqrt{-g_{\mu \nu} \dot x^\mu \dot x^\nu} \over -g_{\mu \nu} \dot x^\mu \dot x^\nu} = {g_{\mu \nu , \lambda} \dot x^\mu \dot x^\nu \over \sqrt{-g_{\mu \nu} \dot x^\mu \dot x^\nu}} \qquad \qquad (4)

{(-g_{\mu \nu} \dot x^\mu \dot x^\nu) {d \over d\tau} (g_{\lambda \nu} \dot x^\nu + g_{\mu \lambda} \dot x^\mu) + {1 \over 2} (g_{\lambda \nu} \dot x^\nu + g_{\mu \lambda} \dot x^\mu) {d \over d\tau} (g_{\mu \nu} \dot x^\mu \dot x^\nu) \over -g_{\mu \nu} \dot x^\mu \dot x^\nu} = g_{\mu \nu ,\lambda} \dot x^\mu \dot x^\nu \qquad \qquad (5)

(g_{\mu \nu} \dot x^\mu \dot x^\nu) (g_{\lambda \nu ,\mu} \dot x^\nu \dot x^\mu + g_{\mu \lambda ,\nu} \dot x^\mu \dot x^\nu + g_{\lambda \nu} \ddot x^\nu + g_{\lambda \mu} \ddot x^\mu)

= (g_{\mu \nu ,\lambda} \dot x^\mu \dot x^\nu) (g_{\alpha \beta} \dot x^\alpha \dot x^\beta) + {1 \over 2} (g_{\lambda \nu} \dot x^\nu + g_{\lambda \mu} \dot x^\mu) {d \over d\tau} (g_{\mu \nu} \dot x^\mu \dot x^\nu) \qquad \qquad (6)

g_{\lambda \nu ,\mu} \dot x^\mu \dot x^\nu + g_{\lambda \mu ,\nu} \dot x^\mu \dot x^\nu - g_{\mu \nu ,\lambda} \dot x^\mu \dot x^\nu + 2 g_{\lambda \mu} \ddot x^\mu = {\dot x_\lambda {d \over d\tau} (g_{\mu \nu} \dot x^\mu \dot x^\nu) \over g_{\alpha \beta} \dot x^\alpha \dot x^\beta} \qquad \qquad (7)

2(\Gamma_{\lambda \mu \nu} \dot x^\mu \dot x^\nu + \ddot x_\lambda) = {\dot x_\lambda {d \over d\tau} (\dot x_\nu \dot x^\nu) \over \dot x_\beta \dot x^\beta} = {U_\lambda {d \over d\tau} (U_\nu U^\nu) \over U_\beta U^\beta} = U_\lambda {d \over d\tau} \ln |U_\nu U^\nu| \qquad \qquad (8)

This is just one step away from the geodesic equation. (return to article)

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