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Computing the Arnold Tongue in the Zipoy-Voorhees Space-time
Thesis (MSc)--Stellenbosch University, 2017
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| Format: | Thesis |
| Language: | en_ZA |
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Stellenbosch : Stellenbosch University
2017
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| _version_ | 1867614094770569216 |
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| access_status_str | Open Access |
| author | Sherif, Abbas Mohamed |
| author2 | Brink, Jeandrew |
| author_browse | Brink, Jeandrew Sherif, Abbas Mohamed |
| author_facet | Brink, Jeandrew Sherif, Abbas Mohamed |
| author_sort | Sherif, Abbas Mohamed |
| collection | Thesis |
| dc_rights_str_mv | Stellenbosch University |
| description | Thesis (MSc)--Stellenbosch University, 2017 |
| format | Thesis |
| id | oai:scholar.sun.ac.za:10019.1/101018 |
| institution | Stellenbosch University (South Africa) |
| language | en_ZA |
| last_indexed | 2026-06-10T12:46:35.101Z |
| license_str | Other — see source repository |
| provenance_str_mv | Harvested via OAI-PMH from SUNScholar — Stellenbosch University Repository |
| publishDate | 2017 |
| publishDateRange | 2017 |
| publishDateSort | 2017 |
| publisher | Stellenbosch : Stellenbosch University |
| publisherStr | Stellenbosch : Stellenbosch University |
| record_format | dspace |
| source_str | SUNScholar — Stellenbosch University Repository |
| spelling | oai:scholar.sun.ac.za:10019.1/101018 Computing the Arnold Tongue in the Zipoy-Voorhees Space-time Sherif, Abbas Mohamed Brink, Jeandrew Stellenbosch University. Faculty of Science. Dept. of Physics Poincaré map Arnold tongue Curvature Zipoy-Voorhees Metric Rotation Curves Mathematical physics -- Integrability Geodesics (Mathematics) Bumpy black holes UCTD Invariant (Mathematics) Spacetime (Physics) Thesis (MSc)--Stellenbosch University, 2017 ENGLISH ABSTRACT : In this thesis I study the integrability of the geodesic equations of the ZipoyVoorhees metric. The Zipoy-Voorhees spacetime is a one parameter family of Stationary Axisymmetric Vacuum spacetimes (SAV’s) that is an exact solution to the vacuum Einstein Field Equations (EFE’s). It has been conjectured that the end state of any asymptotically flat black hole formed by astrophysical mechanisms, such as for example, gravitational collapse of a star, merger of two black holes etc will be a characterised by the Kerr metric. The black hole will thus be a possibly rotating, stationary axisymmetric vacuum spacetime characterised by its mass and spin and will possess no closed time-like curves. Investigating orbits in the Zipoy-Voorhees spacetime serves as a concrete example to of how the Kerr hypothesis fails. For this metric, I compute the Poincaré map and then compute the rotation curve. The Poincaré map is a tool to locate the region where chaos occurs in a dynamical system. The rotation curve is used to quantify chaos in the system. I focus my study on the 2/3 resonance for a range of the parameter values δ ∈ [1, 2]. The value δ = 1 corresponds to the Schwarzschild solution where the system is integrable. I then compute the Arnold tongue by plotting the size of the resonant regions against the parameter values to quantify the departure from integrability. I find that the shape of the tongue of instability is nonlinear and the Arnold tongue pinches off at δ = 1.6. AFRIKAANSE OPSOMMING : In hierdie tesis bestudeer ek die integreerbaarheid van die geodesiese vergelykings van die Zipoy-Voorhees metrieke. Die Zipoy-Voorhees ruimtetyd is ’n familie van stilstaande axisimmetriese vakuum ruimtetye (SAV’s) wat ’n presiese oplossing vir die vakuum Einstein veldvergelykings (EFE se). Dit is veronderstel dat die einde toestand van enige asimptotiese plat gravitasiekolk wat gevorm word deur astrofisiese meganismes, soos byvoorbeeld, gravitasie ineenstorting van ’n ster, samesmelting van twee swart gate ens sal ’n gekenmerk word deur die Kerr metrieke. Die gravitasiekolk sal dus ’n moontlik roterende, stilstaande axisimmetriese vakuum ruimtetyd gekenmerk deur die massa en spin en sal geen geslote tyd-agtige kurwes besit nie. Die studie van trajekte in die Zipoy-Voorhees ruimtetyd dien as ’n konkrete voorbeeld van hoe die Kerr hipotese versuim. Vir hierdie metrieke, ek bereken die Poincaré kaart en dan bereken die rotasie kurwe. Die Poincaré kaart is ’n instrument om die streek op te spoor waar chaos plaasvind in ’n dinamiese stelsel. Die rotasie kurwe word gebruik om chaos in die stelsel te kwantifiseer. Ek fokus my studie op die 2/3 resonansie vir ’n verskeidenheid van die parameterwaardes δ ∈ [1, 2]. Die waarde δ = 1 stem ooreen met die Schwarzschild oplossing waar die stelsel integreerbaar is. Ek bereken die Arnold tong deur die grootte van die resonante streke te plot teen die parameterwaardes om die afwyking van integreerbaarheid te kwantifiseer. Ek vind dat die vorm van die tong van onstabiliteit nielineêre is en dat die Arnold tong onverwags by ’n parameter waarde van δ = 1.6 afsluit. 2017-02-26T20:08:57Z 2017-03-29T11:58:29Z 2017-02-26T20:08:57Z 2017-03-29T11:58:29Z 2017-03 Thesis http://hdl.handle.net/10019.1/101018 en_ZA Stellenbosch University xii, 86 pages : illustrations (some colour) application/pdf Stellenbosch : Stellenbosch University |
| spellingShingle | Poincaré map Arnold tongue Curvature Zipoy-Voorhees Metric Rotation Curves Mathematical physics -- Integrability Geodesics (Mathematics) Bumpy black holes UCTD Invariant (Mathematics) Spacetime (Physics) Sherif, Abbas Mohamed Computing the Arnold Tongue in the Zipoy-Voorhees Space-time |
| title | Computing the Arnold Tongue in the Zipoy-Voorhees Space-time |
| title_full | Computing the Arnold Tongue in the Zipoy-Voorhees Space-time |
| title_fullStr | Computing the Arnold Tongue in the Zipoy-Voorhees Space-time |
| title_full_unstemmed | Computing the Arnold Tongue in the Zipoy-Voorhees Space-time |
| title_short | Computing the Arnold Tongue in the Zipoy-Voorhees Space-time |
| title_sort | computing the arnold tongue in the zipoy voorhees space time |
| topic | Poincaré map Arnold tongue Curvature Zipoy-Voorhees Metric Rotation Curves Mathematical physics -- Integrability Geodesics (Mathematics) Bumpy black holes UCTD Invariant (Mathematics) Spacetime (Physics) |
| url | http://hdl.handle.net/10019.1/101018 |
| work_keys_str_mv | AT sherifabbasmohamed computingthearnoldtongueinthezipoyvoorheesspacetime |