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 the magnitude of a geodesic is constant, so one can characterize geodesics in terms of their causal character: if this magnitude is negative, the geodesic is called timelike. If this magnitude is positive, then it is spacelike. If this magnitude is 0, then it is called 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.
","abstract_html":"<p>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 the magnitude of a geodesic is constant, so one can characterize geodesics in terms of their causal character: if this magnitude is negative, the geodesic is called timelike. If this magnitude is positive, then it is spacelike. If this magnitude is 0, then it is called 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.</p>","abstract_has_math":false,"creators":["Botros, Amir A"],"institution":null,"degree_name":"Master of Arts in Mathematics","degree_level":"Thesis","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Dr. Corey Dunn"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2016,"date_issued":"2016-03-01T08:00:00Z","date_published":"2016-03-01T08:00:00Z","updated_at":"2026-07-24T01:52:53Z","subjects":["geodesic completeness","Lorentzian manifolds","pseudo-Riemannian manifolds","Geometry and Topology"],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://scholarworks.lib.csusb.edu/etd/275","outbound_label":"Repository record","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Dr. Corey Dunn"]},{"key":"dc:creator","label":"Author","values":["Botros, Amir A"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.available","label":"Dc Date Available","values":["2016-02-24T08:00:00Z"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Thesis"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Master of Arts in Mathematics"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["geodesic completeness","Lorentzian manifolds","pseudo-Riemannian manifolds","Geometry and Topology"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://scholarworks.lib.csusb.edu/etd/275"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["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 the magnitude of a geodesic is constant, so one can characterize geodesics in terms of their causal character: if this magnitude is negative, the geodesic is called timelike. If this magnitude is positive, then it is spacelike. If this magnitude is 0, then it is called 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.
"]},{"key":"dc:title","label":"Title","values":["GEODESICS IN LORENTZIAN MANIFOLDS"]}]}],"canonical_facts":{"dc:contributor":["Dr. Corey Dunn"],"dc:creator":["Botros, Amir A"],"dc:date.available":["2016-02-24T08:00:00Z"],"dc:description.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 the magnitude of a geodesic is constant, so one can characterize geodesics in terms of their causal character: if this magnitude is negative, the geodesic is called timelike. If this magnitude is positive, then it is spacelike. If this magnitude is 0, then it is called 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.
"],"dc:identifier":["https://scholarworks.lib.csusb.edu/etd/275"],"dc:subject":["geodesic completeness","Lorentzian manifolds","pseudo-Riemannian manifolds","Geometry and Topology"],"dc:title":["GEODESICS IN LORENTZIAN MANIFOLDS"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Thesis"],"thesis:degree_name":["Master of Arts in Mathematics"]},"updated_at":"2026-07-24T01:52:53Z"}