The Hawking-Penrose singularity theorem: a self-contained review
In this work, the Hawking-Penrose singularity theorem is studied from the perspective of the problem of geodesic inextendibility in Lorentzian manifolds, a problem that provides a formal definition of the notion of space-time singularity in general relativity. First, geodesic curves were studied in...
- Autores:
-
Ballén Méndez, Iván Camilo
- Tipo de recurso:
- Trabajo de grado de pregrado
- Fecha de publicación:
- 2024
- Institución:
- Universidad de los Andes
- Repositorio:
- Séneca: repositorio Uniandes
- Idioma:
- eng
- OAI Identifier:
- oai:repositorio.uniandes.edu.co:1992/75951
- Acceso en línea:
- https://hdl.handle.net/1992/75951
- Palabra clave:
- Differential Geometry
Lorentzian Geometry
Jacobi Fields
Spacetime Singularities
Conjugate Points
Causal Structure
Matemáticas
- Rights
- openAccess
- License
- Attribution-NonCommercial-NoDerivatives 4.0 International
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The Hawking-Penrose singularity theorem: a self-contained review |
| dc.title.alternative.spa.fl_str_mv |
El teorema de singularidad de Hawking-Penrose: una revisión autocontenida |
| title |
The Hawking-Penrose singularity theorem: a self-contained review |
| spellingShingle |
The Hawking-Penrose singularity theorem: a self-contained review Differential Geometry Lorentzian Geometry Jacobi Fields Spacetime Singularities Conjugate Points Causal Structure Matemáticas |
| title_short |
The Hawking-Penrose singularity theorem: a self-contained review |
| title_full |
The Hawking-Penrose singularity theorem: a self-contained review |
| title_fullStr |
The Hawking-Penrose singularity theorem: a self-contained review |
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The Hawking-Penrose singularity theorem: a self-contained review |
| title_sort |
The Hawking-Penrose singularity theorem: a self-contained review |
| dc.creator.fl_str_mv |
Ballén Méndez, Iván Camilo |
| dc.contributor.advisor.none.fl_str_mv |
Cortissoz Iriarte, Jean Carlos |
| dc.contributor.author.none.fl_str_mv |
Ballén Méndez, Iván Camilo |
| dc.contributor.jury.none.fl_str_mv |
Cardona Guio, Alexander |
| dc.contributor.researchgroup.none.fl_str_mv |
Facultad de Ciencias |
| dc.subject.keyword.eng.fl_str_mv |
Differential Geometry Lorentzian Geometry Jacobi Fields Spacetime Singularities Conjugate Points Causal Structure |
| topic |
Differential Geometry Lorentzian Geometry Jacobi Fields Spacetime Singularities Conjugate Points Causal Structure Matemáticas |
| dc.subject.themes.spa.fl_str_mv |
Matemáticas |
| description |
In this work, the Hawking-Penrose singularity theorem is studied from the perspective of the problem of geodesic inextendibility in Lorentzian manifolds, a problem that provides a formal definition of the notion of space-time singularity in general relativity. First, geodesic curves were studied in the Riemannian sense as solutions to the curve length optimization problem, and it was shown that every geodesic ceases to be a solution to this optimization problem in finite parametrization. Subsequently, the previous result was extended to families of geodesics whose initial point lies on a Lorentzian submanifold and that share their final point. Finally, the causal structure of space-times and the topology of spaces of causal curves were studied to demonstrate that every co-spacelike geodesic whose initial point lies on a trapped surface halts in finite parametrization. That is, if the space-time satisfies certain geometric properties, this result guarantees the existence of singularities. |
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2024 |
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2024-12-06 |
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2025-01-31T21:07:50Z |
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2025-01-31T21:07:50Z |
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Trabajo de grado - Pregrado |
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info:eu-repo/semantics/bachelorThesis |
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Ballén, I. C. Un acercamiento termodinámico a las ecuaciones de campo de Einstein. https://hdl.handle.net/1992/74683 (2024). Boothby, W. M. An introduction to differentiable manifolds and Riemannian geometry (Academic press, 1986). Carroll, S. M. Spacetime and geometry (Cambridge University Press, 2019). Carter, B. Causal structure in space-time. General Relativity and Gravitation 1, 349–391 (1971). Cohn, D. L. Measure theory (Springer, 2013). Eling, C., Guedens, R. & Jacobson, T. Nonequilibrium thermodynamics of spacetime. Physical Review Letters 96, 121301 (2006). Gelfand, I. M., Silverman, R. A., et al. Calculus of variations (Courier Corporation, 2000). Geroch, R. Domain of dependence. Journal of Mathematical Physics 11, 437–449 (1970). Hall, G. Energy conditions and stability in general relativity. General Relativity and Gravitation 14, 1035–1041 (1982). Hawking, S. W. & Ellis, G. F. The large scale structure of space-time (Cambridge university press, 2023). Hawking, S. W. & Penrose, R. The singularities of gravitational collapse and cosmology. Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences 314, 529–548 (1970). Jacobson, T. Thermodynamics of spacetime: the Einstein equation of state. Physical Review Letters 75, 1260 (1995). Jost, J. & Jost, J. Riemannian geometry and geometric analysis (Springer, 2008). Klingenberg, W. Riemannian geometry (Walter de Gruyter, 1995). Lee, J. M. Riemannian manifolds: an introduction to curvature (Springer Science & Business Media, 1997). Lee, J. M. Introduction to Smooth manifolds (Springer, 2012). Leray, J. Hyperbolic differential equations. https://albert.ias.edu/entities/publication/10f4a37d-ea7d-459e-9d20-a4ca313db55f (1953). Loring, W. An introduction to manifolds (Springer, 2008).10 Manasse, F. K. & Misner, C. W. Fermi normal coordinates and some basic concepts in differential geometry. Journal of mathematical physics 4, 735–745 (1963). O’Neill, B. Semi-Riemannian geometry with applications to relativity (Academic press, 1983). Oppenheimer, J. R. & Snyder, H. On Continued Gravitational Contraction. Phys. Rev. 56, 455–459. https://link.aps.org/doi/10.1103/PhysRev.56.455 (5 Sept. 1939). Pfäffle, F. in Quantum Field Theory on Curved Spacetimes: Concepts and Mathematical Foundations 39–58 (Springer, 2009). Wald, R. M. General relativity (University of Chicago press, 2010). Warner, F. W. Foundations of differentiable manifolds and Lie groups (Springer Science & Business Media, 1983). |
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66 páginas |
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Cortissoz Iriarte, Jean Carlosvirtual::22998-1Ballén Méndez, Iván CamiloCardona Guio, AlexanderFacultad de Ciencias2025-01-31T21:07:50Z2025-01-31T21:07:50Z2024-12-06https://hdl.handle.net/1992/75951instname:Universidad de los Andesreponame:Repositorio Institucional Sénecarepourl:https://repositorio.uniandes.edu.co/In this work, the Hawking-Penrose singularity theorem is studied from the perspective of the problem of geodesic inextendibility in Lorentzian manifolds, a problem that provides a formal definition of the notion of space-time singularity in general relativity. First, geodesic curves were studied in the Riemannian sense as solutions to the curve length optimization problem, and it was shown that every geodesic ceases to be a solution to this optimization problem in finite parametrization. Subsequently, the previous result was extended to families of geodesics whose initial point lies on a Lorentzian submanifold and that share their final point. Finally, the causal structure of space-times and the topology of spaces of causal curves were studied to demonstrate that every co-spacelike geodesic whose initial point lies on a trapped surface halts in finite parametrization. That is, if the space-time satisfies certain geometric properties, this result guarantees the existence of singularities.En el presente trabajo se estudia el teorema de singularidad de Hawking-Penrose desde el problema de la no extendibilidad de geodésicas en variedades Lorentzianas, problema que brinda una definición formal de la noción de singularidad espacio-temporal en relatividad general. En primer lugar, se estudiaron curvas geodésicas en el sentido Riemanniano como soluciones al problema de optimización de longitud de curva, y se demostró que toda geodésica deja de ser una solución a dicho problema de optimización en parametrización finita. Posteriormente, se extendió el resultado anterior para familias de geodésicas cuyo punto inicial se encuentra sobre una subvariedad Lorentziana y tal que comparten su punto final. Finalmente, se estudió la estructura causal de espacio-tiempos y la topología de espacios de curvas causales con el objetivo de demostrar que toda geodésica co-spacelike cuyo punto inicial se encuentra sobre una superficie atrapada se detiene en parametrización finita, es decir, si el espacio-tiempo cumple ciertas propiedades geométricas, dicho resultado se garantiza la existencia de singularidades.PregradoMethods of global Riemannian geometry, including PDE methods; curvature restrictions.Space-time singularities, cosmic censorship, etc.66 páginasapplication/pdfengUniversidad de los AndesMatemáticasFacultad de CienciasDepartamento de MatemáticasAttribution-NonCommercial-NoDerivatives 4.0 Internationalhttp://creativecommons.org/licenses/by-nc-nd/4.0/info:eu-repo/semantics/openAccesshttp://purl.org/coar/access_right/c_abf2The Hawking-Penrose singularity theorem: a self-contained reviewEl teorema de singularidad de Hawking-Penrose: una revisión autocontenidaTrabajo de grado - Pregradoinfo:eu-repo/semantics/bachelorThesisinfo:eu-repo/semantics/acceptedVersionhttp://purl.org/coar/resource_type/c_7a1fTexthttp://purl.org/redcol/resource_type/TPDifferential GeometryLorentzian GeometryJacobi FieldsSpacetime SingularitiesConjugate PointsCausal StructureMatemáticasBallén, I. C. Un acercamiento termodinámico a las ecuaciones de campo de Einstein. https://hdl.handle.net/1992/74683 (2024).Boothby, W. M. An introduction to differentiable manifolds and Riemannian geometry (Academic press, 1986).Carroll, S. M. Spacetime and geometry (Cambridge University Press, 2019).Carter, B. Causal structure in space-time. General Relativity and Gravitation 1, 349–391 (1971).Cohn, D. L. Measure theory (Springer, 2013).Eling, C., Guedens, R. & Jacobson, T. Nonequilibrium thermodynamics of spacetime. Physical Review Letters 96, 121301 (2006).Gelfand, I. M., Silverman, R. A., et al. Calculus of variations (Courier Corporation, 2000).Geroch, R. Domain of dependence. Journal of Mathematical Physics 11, 437–449 (1970).Hall, G. Energy conditions and stability in general relativity. General Relativity and Gravitation 14, 1035–1041 (1982).Hawking, S. W. & Ellis, G. F. The large scale structure of space-time (Cambridge university press, 2023).Hawking, S. W. & Penrose, R. The singularities of gravitational collapse and cosmology. Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences 314, 529–548 (1970).Jacobson, T. Thermodynamics of spacetime: the Einstein equation of state. Physical Review Letters 75, 1260 (1995).Jost, J. & Jost, J. Riemannian geometry and geometric analysis (Springer, 2008).Klingenberg, W. Riemannian geometry (Walter de Gruyter, 1995).Lee, J. M. Riemannian manifolds: an introduction to curvature (Springer Science & BusinessMedia, 1997).Lee, J. M. Introduction to Smooth manifolds (Springer, 2012).Leray, J. Hyperbolic differential equations. https://albert.ias.edu/entities/publication/10f4a37d-ea7d-459e-9d20-a4ca313db55f (1953).Loring, W. An introduction to manifolds (Springer, 2008).10Manasse, F. K. & Misner, C. W. Fermi normal coordinates and some basic concepts in differential geometry. Journal of mathematical physics 4, 735–745 (1963).O’Neill, B. Semi-Riemannian geometry with applications to relativity (Academic press, 1983).Oppenheimer, J. R. & Snyder, H. On Continued Gravitational Contraction. Phys. Rev. 56, 455–459. https://link.aps.org/doi/10.1103/PhysRev.56.455 (5 Sept. 1939).Pfäffle, F. in Quantum Field Theory on Curved Spacetimes: Concepts and Mathematical Foundations 39–58 (Springer, 2009).Wald, R. M. General relativity (University of Chicago press, 2010).Warner, F. W. Foundations of differentiable manifolds and Lie groups (Springer Science & Business Media, 1983).202011440Publicationhttps://scholar.google.es/citations?user=44Ujs4QAAAAJvirtual::22998-10000-0002-7440-4425virtual::22998-1https://scienti.minciencias.gov.co/cvlac/visualizador/generarCurriculoCv.do?cod_rh=0000821411virtual::22998-109606ca2-87c9-4df9-b557-b65295156fdfvirtual::22998-109606ca2-87c9-4df9-b557-b65295156fdfvirtual::22998-1ORIGINALThe Hawking-Penrose singularity theorem: a self-contained review.pdfThe Hawking-Penrose singularity theorem: a self-contained review.pdfapplication/pdf486265https://repositorio.uniandes.edu.co/bitstreams/c74ad72c-55d8-4e3d-84d1-f57d57d26e6b/downloadc21ac574ffde5a538b337a0e68060147MD51autorizacion tesis matemáticas.pdfautorizacion tesis matemáticas.pdfHIDEapplication/pdf332085https://repositorio.uniandes.edu.co/bitstreams/dc5e42ec-3b90-4049-a19a-633c47900f20/download1a466bf95cc99718d643798d323da2deMD52LICENSElicense.txtlicense.txttext/plain; charset=utf-82535https://repositorio.uniandes.edu.co/bitstreams/4511825b-c4d0-4ff2-90e5-d67bd49f7c4a/downloadae9e573a68e7f92501b6913cc846c39fMD53CC-LICENSElicense_rdflicense_rdfapplication/rdf+xml; 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