# geodesic

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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