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BACKGROUND ON CONVENTIONAL SINGULAR PERTURBATIONS
INTRODUCTION TO SINGULAR PERTURBATIONS ON MANIFOLDS
Fp C pthorder tensor approximation 37
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1-parameter family 2-forms acceleration asymptotic expansions asymptotic worldline Bianchi identities calculation Cartan equations chapter compact object components connection forms consider conventional singular perturbation coordinate system correspondence maps covariant derivative curvature forms curve defined degrees of freedom depends domains of validity Einstein field equation Einstein tensor equation and Bianchi event horizon exact solution manifolds example external expansion external field external spacetime fixed flat formalism frame perturbations functions gauge invariance gauge transformations geodesic geometric global asymptotic approximation gravitational field gravitational perturbations higher-order identification maps intermediate limit process introduce jurisdictions Lie derivative linear model manifolds model spacetime motion n-dimensional Newtonian nonlinear nonuniform norm notation objects with strong obtain orthonormal basis parameter perturbation equations perturbations on manifolds preceeding section principal limit pull-back quantities radiation reaction region represent satisfy singular perturbation singular perturbation problems strong internal gravity structure equation Suppose tensor fields trial worldline type II terms uniformity vanish vector zeroth-order term