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ergospherical

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Introduction

The theory of geodesic congruences is extensively covered in many textbooks (see References); what follows in the introduction is a brief summary. Consider a 1-parameter family of timelike geodesics ##\gamma_s(\lambda)##, where ##s## labels each geodesic in the family whilst ##\lambda## is anaffine parameteralong each ##\gamma_s##. Then the vector field ##\xi \equiv \partial / \partial s## is tangent to curves of constant ##\lambda## and is interpreted as adeviation vectorbetween neighbouring geodesics.

In some neighbourhood of the family, ##(s,\lambda, x^2, x^3)## is a coordinate chart satisfying ##\xi = \partial/\partial s## and ##u = \partial/\partial \lambda##. By the equality of mixed partial derivatives, the commutator of ##u## and ##\xi## is zero (i.e. ##\xi## is Lie transported along ##u##),\begin{align*}

0 = [u, \xi]^a = (L_{u} \xi)^a = \xi^b \nabla_b u^a – u^b \nabla_b \xi^a

\end{align*}which implies that ##\dfrac{D\xi^a}{d\lambda} = u^b...

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