{"id":{"repo_id":"brock","oai_identifier":"oai:brocku.scholaris.ca:10464/14535"},"canonical_url":"https://search.dev.ndltd.org/etd/brock/oai:brocku.scholaris.ca:10464/14535","repository":{"repo_id":"brock","name":"Brock University","base_url":"https://brocku.scholaris.ca/server/oai/request"},"display":{"title":"An Analogue of the Laplace-Runge-Lenz Vector for Timelike Geodesics in Schwarzschild Spacetime","abstract":"In Schwarzschild spacetime, the timelike geodesics are the trajectories of free, massive particles, orbiting a singularity at the origin r = 0. In this work we derive four scalar first integrals of timelike geodesics in Schwarzschild spacetime. Two of the first integrals, corresponding to energy and angular momentum, are well-known. The other two first integrals, an angular quantity and a temporal quantity, are not as well-known. Using the freedom to shift first integrals by a constant value we set a ‘zero-point’ for each of the four first integrals. By choosing a natural point on a non-circular trajectory such as a turning point or inertial point to set the zero-point value, the angular and temporal first integrals will correspond respectively to the angle and time of the chosen zero-point. We then take the Newtonian limit of the angular and temporal first integrals, and show that using a natural choice of zero-point they provide a generalization of the classical Laplace-Runge-Lenz (LRL) vector. We then evaluate the angular first integral for each type of timelike geodesic in Schwarzschild spacetime. In most cases we are able to choose a turning or inertial point to set a zero-point. For an unbound or asymptotic trajectory which falls into the singularity of the metric at r = 0, however, we find that we must take a different point, such as the point where the trajectory crosses the horizon at r = 2M, which we call the ‘horizon point.’ For the case of a precessing elliptic orbit we find that the angular first integral is multi-valued, with the zero-point jumping each time the trajectory crosses an apoapsis. It is found that the angular and temporal first integrals provide a relativistic generalization of the classical LRL vector, where we the first integrals correspond to a larger class of physically meaningful points compared to Newtonian orbits and where the LRL vector and angular and temporal first integrals may always correspond to the periapsis.","abstract_html":"In Schwarzschild spacetime, the timelike geodesics are the trajectories of free, massive particles, orbiting a singularity at the origin r = 0. In this work we derive four scalar first integrals of timelike geodesics in Schwarzschild spacetime. Two of the first integrals, corresponding to energy and angular momentum, are well-known. The other two first integrals, an angular quantity and a temporal quantity, are not as well-known. Using the freedom to shift first integrals by a constant value we set a ‘zero-point’ for each of the four first integrals. By choosing a natural point on a non-circular trajectory such as a turning point or inertial point to set the zero-point value, the angular and temporal first integrals will correspond respectively to the angle and time of the chosen zero-point. We then take the Newtonian limit of the angular and temporal first integrals, and show that using a natural choice of zero-point they provide a generalization of the classical Laplace-Runge-Lenz (LRL) vector. We then evaluate the angular first integral for each type of timelike geodesic in Schwarzschild spacetime. In most cases we are able to choose a turning or inertial point to set a zero-point. For an unbound or asymptotic trajectory which falls into the singularity of the metric at r = 0, however, we find that we must take a different point, such as the point where the trajectory crosses the horizon at r = 2M, which we call the ‘horizon point.’ For the case of a precessing elliptic orbit we find that the angular first integral is multi-valued, with the zero-point jumping each time the trajectory crosses an apoapsis. It is found that the angular and temporal first integrals provide a relativistic generalization of the classical LRL vector, where we the first integrals correspond to a larger class of physically meaningful points compared to Newtonian orbits and where the LRL vector and angular and temporal first integrals may always correspond to the periapsis.","abstract_has_math":false,"creators":["Fazio, Jordan"],"institution":"Brock University","degree_name":"M.Sc. Physics","degree_level":"Masters","degree_discipline":"Faculty of Mathematics and Science","degree_department":"Department of Physics","school":null,"contributors":[],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2019,"date_issued":"2019-09-30T14:10:18Z","date_published":"2019-09-30T14:10:18Z","updated_at":"2026-07-24T01:23:12Z","subjects":["Schwarzschild","geodesic","LRL","first integral","General Relativity"],"languages":["eng"],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/10464/14535","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.department","label":"Department","values":["Department of Physics"]},{"key":"dc:creator","label":"Author","values":["Fazio, Jordan"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2019-09-30T14:10:18Z"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2019-09-30T14:10:18Z"]},{"key":"dc:date.issued","label":"Date","values":["2019-09-30T14:10:18Z"]},{"key":"dc:type","label":"Dc Type","values":["Electronic Thesis or Dissertation"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Faculty of Mathematics and Science"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Masters"]},{"key":"thesis:degree_name","label":"Degree Name","values":["M.Sc. Physics"]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["Brock University"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Schwarzschild","geodesic","LRL","first integral","General Relativity"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language.iso","label":"Language (ISO)","values":["eng"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.uri","label":"Identifier URI","values":["http://hdl.handle.net/10464/14535"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["In Schwarzschild spacetime, the timelike geodesics are the trajectories of free, massive particles, orbiting a singularity at the origin r = 0. In this work we derive four scalar first integrals of timelike geodesics in Schwarzschild spacetime. Two of the first integrals, corresponding to energy and angular momentum, are well-known. The other two first integrals, an angular quantity and a temporal quantity, are not as well-known. Using the freedom to shift first integrals by a constant value we set a ‘zero-point’ for each of the four first integrals. By choosing a natural point on a non-circular trajectory such as a turning point or inertial point to set the zero-point value, the angular and temporal first integrals will correspond respectively to the angle and time of the chosen zero-point. We then take the Newtonian limit of the angular and temporal first integrals, and show that using a natural choice of zero-point they provide a generalization of the classical Laplace-Runge-Lenz (LRL) vector. We then evaluate the angular first integral for each type of timelike geodesic in Schwarzschild spacetime. In most cases we are able to choose a turning or inertial point to set a zero-point. For an unbound or asymptotic trajectory which falls into the singularity of the metric at r = 0, however, we find that we must take a different point, such as the point where the trajectory crosses the horizon at r = 2M, which we call the ‘horizon point.’ For the case of a precessing elliptic orbit we find that the angular first integral is multi-valued, with the zero-point jumping each time the trajectory crosses an apoapsis. It is found that the angular and temporal first integrals provide a relativistic generalization of the classical LRL vector, where we the first integrals correspond to a larger class of physically meaningful points compared to Newtonian orbits and where the LRL vector and angular and temporal first integrals may always correspond to the periapsis."]},{"key":"dc:title","label":"Title","values":["An Analogue of the Laplace-Runge-Lenz Vector for Timelike Geodesics in Schwarzschild Spacetime"]}]}],"canonical_facts":{"dc:contributor.department":["Department of Physics"],"dc:creator":["Fazio, Jordan"],"dc:date.accessioned":["2019-09-30T14:10:18Z"],"dc:date.available":["2019-09-30T14:10:18Z"],"dc:date.issued":["2019-09-30T14:10:18Z"],"dc:description.abstract":["In Schwarzschild spacetime, the timelike geodesics are the trajectories of free, massive particles, orbiting a singularity at the origin r = 0. In this work we derive four scalar first integrals of timelike geodesics in Schwarzschild spacetime. Two of the first integrals, corresponding to energy and angular momentum, are well-known. The other two first integrals, an angular quantity and a temporal quantity, are not as well-known. Using the freedom to shift first integrals by a constant value we set a ‘zero-point’ for each of the four first integrals. By choosing a natural point on a non-circular trajectory such as a turning point or inertial point to set the zero-point value, the angular and temporal first integrals will correspond respectively to the angle and time of the chosen zero-point. We then take the Newtonian limit of the angular and temporal first integrals, and show that using a natural choice of zero-point they provide a generalization of the classical Laplace-Runge-Lenz (LRL) vector. We then evaluate the angular first integral for each type of timelike geodesic in Schwarzschild spacetime. In most cases we are able to choose a turning or inertial point to set a zero-point. For an unbound or asymptotic trajectory which falls into the singularity of the metric at r = 0, however, we find that we must take a different point, such as the point where the trajectory crosses the horizon at r = 2M, which we call the ‘horizon point.’ For the case of a precessing elliptic orbit we find that the angular first integral is multi-valued, with the zero-point jumping each time the trajectory crosses an apoapsis. It is found that the angular and temporal first integrals provide a relativistic generalization of the classical LRL vector, where we the first integrals correspond to a larger class of physically meaningful points compared to Newtonian orbits and where the LRL vector and angular and temporal first integrals may always correspond to the periapsis."],"dc:identifier.uri":["http://hdl.handle.net/10464/14535"],"dc:language.iso":["eng"],"dc:subject":["Schwarzschild","geodesic","LRL","first integral","General Relativity"],"dc:title":["An Analogue of the Laplace-Runge-Lenz Vector for Timelike Geodesics in Schwarzschild Spacetime"],"dc:type":["Electronic Thesis or Dissertation"],"thesis:degree_discipline":["Faculty of Mathematics and Science"],"thesis:degree_level":["Masters"],"thesis:degree_name":["M.Sc. Physics"],"thesis:institution_name":["Brock University"]},"updated_at":"2026-07-24T01:23:12Z"}