Abstract: We present an extension of Geodesics in Lorentzian Manifolds (Semi-Riemannian Manifolds or pseudo-Riemannian Manifolds ). A geodesic on a Riemannian manifold is, locally, a length minimizing curve. On the other hand, geodesics in Lorentzian manifolds can be viewed as a distance between “events”. They are no longer distance minimizing (instead, some are distance maximizing) and our goal is to illustrate over what time parameter geodesics in Lorentzian manifolds are defined. If all geodesics in timelike or spacelike or lightlike are defined for infinite time, then the manifold is called “geodesically complete”, or simply, “complete”. It is easy to show that g(σ′, σ′) is constant if σ is a geodesic and g is a smooth metric on a manifold M , so one can characterize geodesics in terms of their causal character : if g(σ′, σ′) 0, σ is spacelike. If g(σ′, σ′) = 0, then σ is lightlike or null. Geodesic completeness can be considered by only considering one causal character to produce the notions of spacelike complete, timelike complete, and null or lightlike complete. We illustrate that some of the notions are inequivalent.
Publication Year: 2016
Publication Date: 2016-04-22
Language: en
Type: book
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Cited By Count: 2
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