The motion of a particle in the Kepler problem is characterized by a single frequency, the Kepler frequency. This means that in spherical coordinates radial, azimuthal and polar angles all have the same frequency and there is no precession. Einstein showed that in General Relativity, for the Schwarzschild metric, this degeneracy is broken and the radial and azimuthal frequencies become distinct. We will show how in the Kerr space-time the third frequency also gets a separate value. We will derive analytic expressions for all the frequencies. These frequencies can be written fully in terms of the canonical elliptic functions and the constants of the motion. We study some limiting examples, namely far away, or keplerian approximation, and the low eccentricity case. And we also extend our results to non-bound null geodesics, finding an exact result for the angle of deflection for equatorial trajectories. We also discuss a completely different problem. It is well known that there are very massive compact objects in the Universe. Despite some inconsistencies it is assumed that some of them become what we know as Black Holes. According to Hawking and his colleagues Black Holes radiate like a blackbody with a certain temperature. This result has been derived in different ways, one of them, presented by Wilczek and his colleagues uses gravitational anomalies. But there exist alternatives to the Black Hole model. In one of them, the very massive objects become gravitational vacuum stars or Gravastars, bubbles of dark energy. Turning the argument of Wilczek on its head, we argue that it solves naturally some of the inconsistencies.
","abstract_html":"<p>The motion of a particle in the Kepler problem is characterized by a single frequency, the Kepler frequency. This means that in spherical coordinates radial, azimuthal and polar angles all have the same frequency and there is no precession. Einstein showed that in General Relativity, for the Schwarzschild metric, this degeneracy is broken and the radial and azimuthal frequencies become distinct. We will show how in the Kerr space-time the third frequency also gets a separate value. We will derive analytic expressions for all the frequencies. These frequencies can be written fully in terms of the canonical elliptic functions and the constants of the motion. We study some limiting examples, namely far away, or keplerian approximation, and the low eccentricity case. And we also extend our results to non-bound null geodesics, finding an exact result for the angle of deflection for equatorial trajectories. We also discuss a completely different problem. It is well known that there are very massive compact objects in the Universe. Despite some inconsistencies it is assumed that some of them become what we know as Black Holes. According to Hawking and his colleagues Black Holes radiate like a blackbody with a certain temperature. This result has been derived in different ways, one of them, presented by Wilczek and his colleagues uses gravitational anomalies. But there exist alternatives to the Black Hole model. In one of them, the very massive objects become gravitational vacuum stars or Gravastars, bubbles of dark energy. Turning the argument of Wilczek on its head, we argue that it solves naturally some of the inconsistencies.</p>","abstract_has_math":false,"creators":["Sanabria, Andres"],"institution":null,"degree_name":"Ph.D.","degree_level":"Campus Access Dissertation","degree_discipline":"Physics and Astronomy","degree_department":null,"school":null,"contributors":["Pawel W Mazur"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2012,"date_issued":"2012-01-01T08:00:00Z","date_published":"2012-01-01T08:00:00Z","updated_at":"2026-07-24T04:38:14Z","subjects":["Physical Sciences and Mathematics","Physics","geodesic","gravastar","gravity","Kerr","quantum","relativity"],"languages":[],"rights":["© 2012, Andres Sanabria"],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://scholarcommons.sc.edu/etd/1760","outbound_label":"Repository record","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Pawel W Mazur"]},{"key":"dc:creator","label":"Author","values":["Sanabria, Andres"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"thesis:degree_discipline","label":"Discipline","values":["Physics and Astronomy"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Campus Access Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph.D."]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Physical Sciences and Mathematics","Physics","geodesic","gravastar","gravity","Kerr","quantum","relativity"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:rights","label":"Dc Rights","values":["© 2012, Andres Sanabria"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://scholarcommons.sc.edu/etd/1760"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["The motion of a particle in the Kepler problem is characterized by a single frequency, the Kepler frequency. This means that in spherical coordinates radial, azimuthal and polar angles all have the same frequency and there is no precession. Einstein showed that in General Relativity, for the Schwarzschild metric, this degeneracy is broken and the radial and azimuthal frequencies become distinct. We will show how in the Kerr space-time the third frequency also gets a separate value. We will derive analytic expressions for all the frequencies. These frequencies can be written fully in terms of the canonical elliptic functions and the constants of the motion. We study some limiting examples, namely far away, or keplerian approximation, and the low eccentricity case. And we also extend our results to non-bound null geodesics, finding an exact result for the angle of deflection for equatorial trajectories. We also discuss a completely different problem. It is well known that there are very massive compact objects in the Universe. Despite some inconsistencies it is assumed that some of them become what we know as Black Holes. According to Hawking and his colleagues Black Holes radiate like a blackbody with a certain temperature. This result has been derived in different ways, one of them, presented by Wilczek and his colleagues uses gravitational anomalies. But there exist alternatives to the Black Hole model. In one of them, the very massive objects become gravitational vacuum stars or Gravastars, bubbles of dark energy. Turning the argument of Wilczek on its head, we argue that it solves naturally some of the inconsistencies.
"]},{"key":"dc:title","label":"Title","values":["Frequencies of Bound Geodesics, Angle of Deflection For Null Geodesics In Kerr Space-Time, and Gravitational Anomalies In Gravastars."]}]}],"canonical_facts":{"dc:contributor":["Pawel W Mazur"],"dc:creator":["Sanabria, Andres"],"dc:description.abstract":["The motion of a particle in the Kepler problem is characterized by a single frequency, the Kepler frequency. This means that in spherical coordinates radial, azimuthal and polar angles all have the same frequency and there is no precession. Einstein showed that in General Relativity, for the Schwarzschild metric, this degeneracy is broken and the radial and azimuthal frequencies become distinct. We will show how in the Kerr space-time the third frequency also gets a separate value. We will derive analytic expressions for all the frequencies. These frequencies can be written fully in terms of the canonical elliptic functions and the constants of the motion. We study some limiting examples, namely far away, or keplerian approximation, and the low eccentricity case. And we also extend our results to non-bound null geodesics, finding an exact result for the angle of deflection for equatorial trajectories. We also discuss a completely different problem. It is well known that there are very massive compact objects in the Universe. Despite some inconsistencies it is assumed that some of them become what we know as Black Holes. According to Hawking and his colleagues Black Holes radiate like a blackbody with a certain temperature. This result has been derived in different ways, one of them, presented by Wilczek and his colleagues uses gravitational anomalies. But there exist alternatives to the Black Hole model. In one of them, the very massive objects become gravitational vacuum stars or Gravastars, bubbles of dark energy. Turning the argument of Wilczek on its head, we argue that it solves naturally some of the inconsistencies.
"],"dc:identifier":["https://scholarcommons.sc.edu/etd/1760"],"dc:rights":["© 2012, Andres Sanabria"],"dc:subject":["Physical Sciences and Mathematics","Physics","geodesic","gravastar","gravity","Kerr","quantum","relativity"],"dc:title":["Frequencies of Bound Geodesics, Angle of Deflection For Null Geodesics In Kerr Space-Time, and Gravitational Anomalies In Gravastars."],"thesis:degree_discipline":["Physics and Astronomy"],"thesis:degree_level":["Campus Access Dissertation"],"thesis:degree_name":["Ph.D."]},"updated_at":"2026-07-24T04:38:14Z"}