curvature

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Geodesic Curvature

The Geodesic curvature of a curve \gamma\, is defined to be

k_g = - t^2 t^a n_b \nabla_a t^b\,,

where t^a\, is the unit tangent vector of \gamma\, and n_a\, is the unit normal one-form orthogonal to t^a\,. The factor t^2\, is an alternative to

k_g = \pm t^a n_b \nabla_a t^b\,,

where +\, corresponds to a timelike (t^2 = -1\,) boundary, -\, corresponds to a spacelike (t^2 = 1\,) boundary.

Weyl transformation

Under g_{ab} \to e^{2\Omega} g_{ab}\,. To maintain normalization,

t^a \to e^{-\Omega} t^a\,,
n_a \to e^{\Omega} n_a\,.
\Gamma^a_{b,c}\, = \frac{1}{2}g^{ad}\left(g_{db,c}+g_{dc,b}-g_{bc,d} \right)\,
\to \frac{1}{2}e^{-2\Omega} g^{ad}\left(e^{2\Omega}g_{db,c}+e^{2\Omega}g_{dc,b}-e^{2\Omega}g_{bc,d} + e^{2\Omega}g_{db} 2\Omega_{,c} + e^{2\Omega}g_{dc} 2\Omega_{,b} - e^{2\Omega}g_{bc} 2\Omega_{,d}\right)\,
=\Gamma^a_{b,c} + g^{ad}\left( g_{db} \Omega_{,c} + g_{dc} \Omega_{,b} - g_{bc} \Omega_{,d}\right)\,
=\Gamma^a_{b,c} + \delta^a_b \Omega_{,c} + \delta^a_c \Omega_{,b} - g_{bc} \Omega^{,a}\,
t^2 t^a n_b \nabla_a t^b\, = t^2t^a n_b \left( \partial_a t^b + \Gamma^b_{ac}t^c\right)\,
\to t^2 t^a n_b \nabla_a (e^{-\Omega} t^b) + t^a n_b \left(\delta^b_a \Omega_{,c} + \delta^b_c \Omega_{,a} - g_{ac} \Omega^{,b} \right)t^c e^{-\Omega}\,
= t^2 t^a n_b \nabla_a (e^{-\Omega} t^b) + \left(t^a n_a t^c \Omega_{,c} + t^a n_b t^b \Omega_{,a} - t^a n_b t^c g_{ac} \Omega^{,b}\right)e^{-\Omega}\,
= t^2 t^a n_b \nabla_a (e^{-\Omega} t^b) - t^2 n^b e^{-\Omega}\Omega_{,b}\,
= e^{-\Omega} n_b t^2 \left( t^a \nabla_a t^b - t^a t^b \partial_a \Omega +\Omega^{,b}\right)\,
= e^{-\Omega} \left( k_g +n^b \Omega_{,b}\right)\,
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