geodesic

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A geodesic on a manifold M\, is a curve for which the arc length is an extremum. Let \gamma(\lambda): [\lambda_0, \lambda_1] \to M\, be a differentiable curve with coordinates x^\mu(\lambda)\,. Its arc length is the functional of x^\mu(\lambda)\, given by

S[x] = \int_{\lambda_0}^{\lambda_1}\!d\lambda\, \sqrt{ \pm g_{\mu\nu}(x(\lambda)) \frac{d x^\mu}{d\lambda} \frac{d x^\nu}{d\lambda}}\,,

where the negative sign is needed if the metric is not positive definite. Using a dot to indicate differentiation with respect to \lambda\,, we can write this as

S[x] = \int_{\lambda_0}^{\lambda_1}\!d\lambda\, \sqrt{ \pm g_{\mu\nu}(x) \dot x^\mu \dot x^\nu}\,.

Geodesic equation

Consider a variation of the functions x^\mu(\lambda)\,, i.e., x^\mu(\lambda) \to x^\mu(\lambda) + \delta x^\mu(\lambda)\, such that \delta x^\mu(\lambda_0) = \delta x^\mu(\lambda_1) = 0\,. For x^\mu(\lambda)\, to extremize the arc length, we require that

\frac{\delta S}{\delta x} = 0\,,

evaluated at the solution x^\mu(\lambda)\,. This could also be written as \left. \tfrac{d}{d\epsilon} S[x + \epsilon \delta x] \right|_{\epsilon = 0} = 0\,, but this notation is cumbersome. Then

\delta S = \pm \int_{\lambda_0}^{\lambda_1}\!d\lambda\, \frac{1}{ 2\sqrt{ \pm g_{\mu\nu}(x) \dot x^\mu \dot x^\nu} } \left( \frac{\partial g_{\alpha\beta}}{\partial x^\gamma} \delta x^\gamma \dot x^\alpha \dot x^\beta + g_{\alpha\beta}  \delta \dot x^\alpha \dot x^\beta + g_{\alpha\beta} \dot x^\alpha  \delta \dot x^\beta   \right)\,.

With the variation performed, we can now use our coordinate freedom to set \lambda = S\,, or equivalently, \sqrt{ \pm g_{\mu\nu}(x) \dot x^\mu \dot x^\nu} = 1\,. Then

\delta S = \pm \frac{1}{ 2} \int_{\lambda_0}^{\lambda_1}\!d\lambda\, \left( \frac{\partial g_{\alpha\beta}}{\partial x^\gamma} \delta x^\gamma \dot x^\alpha \dot x^\beta + g_{\alpha\beta}  \delta \dot x^\alpha \dot x^\beta + g_{\alpha\beta} \dot x^\alpha  \delta \dot x^\beta   \right)\,.

We can integrate by parts:

\delta S = \pm \frac{1}{ 2}  \int_{\lambda_0}^{\lambda_1}\!d\lambda\, \left( \frac{\partial g_{\alpha\beta}}{\partial x^\gamma} \delta x^\gamma \dot x^\alpha \dot x^\beta + \frac{d}{d\lambda}\left[ g_{\alpha\beta}  \delta x^\alpha \dot x^\beta \right]- \frac{d}{d\lambda} \left[ g_{\alpha\beta} \dot x^\beta\right] \delta \dot x^\alpha + \frac{d}{d\lambda}\left[ g_{\alpha\beta} \dot x^\alpha  \delta x^\beta \right] - \frac{d}{d\lambda}\left[g_{\alpha\beta} \dot x^\alpha \right]\delta x^\beta  \right)\,,

but \delta x^\mu(\lambda_0) = \delta x^\mu(\lambda_1) = 0\,, so the contribution from the endpoints vanishes:

\delta S = \pm \frac{1}{ 2}  \int_{\lambda_0}^{\lambda_1}\!d\lambda\, \left( \frac{\partial g_{\alpha\beta}}{\partial x^\gamma} \delta x^\gamma \dot x^\alpha \dot x^\beta - \frac{d}{d\lambda} \left[ g_{\alpha\beta} \dot x^\beta\right] \delta  x^\alpha - \frac{d}{d\lambda}\left[g_{\alpha\beta} \dot x^\alpha \right]\delta x^\beta  \right)\,,

Using \tfrac{d}{d\lambda} g(x) = \tfrac{\partial g(x)}{\partial x^\mu} \tfrac{d x^\mu}{d\lambda}\,, and the fact that g_{\alpha\beta} = g_{\beta\alpha}\,, and relabeling indices

\delta S\,  = \pm \frac{1}{ 2}  \int_{\lambda_0}^{\lambda_1}\!d\lambda\, \left( \frac{\partial g_{\alpha\beta}}{\partial x^\gamma} \delta x^\gamma \dot x^\alpha \dot x^\beta - \frac{\partial  g_{\alpha\beta} }{\partial x^\gamma} \dot x^\gamma \dot x^\beta  \delta x^\alpha  - g_{\alpha\beta} \ddot x^\beta \delta x^\alpha- \frac{\partial g_{\alpha\beta} }{\partial x^\gamma} \dot x^{\gamma} \dot x^\alpha \delta x^\beta  - g_{\alpha\beta} \ddot x^\alpha \delta x^\beta \right)\,,
 = \pm \frac{1}{ 2}  \int_{\lambda_0}^{\lambda_1}\!d\lambda\, \left( \frac{\partial g_{\alpha\beta}}{\partial x^\gamma}  \dot x^\alpha \dot x^\beta - \frac{\partial  g_{\gamma\beta} }{\partial x^\alpha} \dot x^\alpha \dot x^\beta - 2 g_{\alpha\gamma} \ddot x^\alpha - \frac{\partial g_{\alpha\gamma} }{\partial x^\beta} \dot x^{\beta} \dot x^\alpha\right) \delta x^\gamma\,,
 = \mp  \int_{\lambda_0}^{\lambda_1}\!d\lambda\, \left( \frac{1}{ 2}\left[ \frac{\partial  g_{\gamma\beta} }{\partial x^\alpha} + \frac{\partial g_{\alpha\gamma} }{\partial x^\beta} -\frac{\partial g_{\alpha\beta}}{\partial x^\gamma} \right] \dot x^\alpha \dot x^\beta + g_{\alpha\gamma} \ddot x^\alpha\right) \delta x^\gamma\,,
 = \mp  \int_{\lambda_0}^{\lambda_1}\!d\lambda\, g_{\delta\gamma} \left( \frac{1}{2}  g^{\delta\rho}\left[ \frac{\partial  g_{\rho\beta} }{\partial x^\alpha} + \frac{\partial g_{\alpha\rho} }{\partial x^\beta} -\frac{\partial g_{\alpha\beta}}{\partial x^\rho} \right] \dot x^\alpha \dot x^\beta +  \ddot x^\delta\right) \delta x^\gamma\,,
 = \mp  \int_{\lambda_0}^{\lambda_1}\!d\lambda\, g_{\delta\gamma} \left( \Gamma^{\delta}_{\alpha\beta} \dot x^\alpha \dot x^\beta +  \ddot x^\delta\right) \delta x^\gamma\,,

where we have defined the Christoffel symbol of the second kind \Gamma^\delta_{\alpha\beta} = \tfrac{1}{2}g^{\rho\delta}\left(\partial_\alpha g_{\beta\rho} + \partial_\beta g_{\rho\alpha} - \partial_\rho g_{\alpha\beta}\right)\,. In order for \delta S\, to vanish for an arbitrary variation \delta x^{\gamma}\,, by the fundamental lemma of calculus of variations,

\ddot x^\gamma + \Gamma^{\gamma}_{\alpha\beta} \dot x^\alpha \dot x^\beta = 0\,,

since g_{\delta\gamma}\, is generally not zero. This is known as the geodesic equation, and is satisfied by geodesic curves for which \lambda\, is an affine parameter, and is supplemented by our previous condition g_{\alpha\beta} \dot{x}^\alpha \dot{x}^\beta = \pm 1\,. We can write it in another way:

\dot{x}^\alpha \left( \partial_\alpha \dot x^\gamma + \Gamma^{\gamma}_{\alpha\beta}  \dot x^\beta\right) = 0\,.

If we denote the tangent vector \tfrac{d}{d\lambda} = \tfrac{d x^\alpha}{d \lambda}\tfrac{\partial}{\partial x^\alpha}\, by \vec{v} = v^\alpha \tfrac{\partial}{\partial x^\alpha}\,, then this can be written as

v^\alpha \left( \partial_\alpha v^\gamma + \Gamma^{\gamma}_{\alpha\beta}  v^\beta\right) = 0\,.

Written in this form, the geodesic equation makes use of what is known as the affine connecion (specifically, the Levi-Civita connection). Furthermore, the geodesic equation implies that a geodesic parallel transports its own tangent vector.

Reparameterization

Note that above we used the invariance of the integral

S[x] = \int_{\lambda_0}^{\lambda_1}\!d\lambda\, \sqrt{ \pm g_{\mu\nu}(x) \frac{dx^\mu}{d\lambda} \frac{dx^\nu}{d\lambda}}\,

under reparameterizations of \lambda\,, i.e,. \lambda \to \lambda'(\lambda)\, to choose a parameter for which \sqrt{ \pm g_{\mu\nu}(x) \tfrac{dx^\mu}{d\lambda} \tfrac{dx^\nu}{d\lambda}}\, takes a convenient form. Thus, suppose \pm\dot{x}^\mu \dot{x}_\mu = 1\, or \dot{x}^\mu \dot{x}_\mu = 0\,. Then the integrals

S[x] = \int_{\lambda_0}^{\lambda_1}\!d\lambda\, \sqrt{ \pm g_{\mu\nu}(x) \frac{dx^\mu}{d\lambda} \frac{dx^\nu}{d\lambda}}\,

and

\tilde{S}[x] = \int_{\lambda_0}^{\lambda_1}\!d\lambda\, \left( \pm g_{\mu\nu}(x) \frac{dx^\mu}{d\lambda} \frac{dx^\nu}{d\lambda}\right)\,,

are equal, and we can extermize either. The second form is especially useful for dealing with null geodesics, i.e., geodesics with zero length.


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