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Manifold joins and jump conditions in general relativity
This thesis has as its aim the analysis of a possible manifold structure on V, a join of two individual manifolds V⁺ and V⁻, and analysing the physics across the join, as implied by Einstein's theory of General Relativity. There are several reasons why one might want to study such a situation. First...
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| Format: | Thesis |
| Language: | English |
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Department of Mathematics and Applied Mathematics
2016
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| _version_ | 1867613202900058112 |
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| access_status_str | Open Access |
| author | Baston, Robert James |
| author2 | Ellis, George F R |
| author_browse | Baston, Robert James Ellis, George F R |
| author_facet | Ellis, George F R Baston, Robert James |
| author_sort | Baston, Robert James |
| collection | Thesis |
| description | This thesis has as its aim the analysis of a possible manifold structure on V, a join of two individual manifolds V⁺ and V⁻, and analysing the physics across the join, as implied by Einstein's theory of General Relativity. There are several reasons why one might want to study such a situation. Firstly, the joining of manifolds is useful in the study of shock waves, be they of gravitational or other origin - we will be able to characterise the propagation of energy in the join. Secondly piecing together manifolds is a potentially fruitful way of obtaining exact solutions of Einstein's equations which do not exhibit any symmetries in the large, and are yet sufficiently homogeneous (in some sense) to enable one to model the apparent Universe - the prototype of this is the Swiss-Cheese model, used to study light transmission in an inhomogeneous Universe. Thirdly, discontinuities in the fundamental quantities in Relativity are of prime importance in the study of singularities and in particular, it is of prime importance to single out the contributions of the differential geometry and metric structure of the Universe to the existence and nature of such singularities. Closely linked to these problems is the problem of linking the small scale structure of the Universe (which is manifestly complicated and potentially full of singularities) and the large scale structure, which seems so well modelled by assumptions of homogeneity and isotropy. In this regard, the techniques of Regge (1961), originally proposed to provide approximate solutions to the Einstein equations, assume a new theoretical importance, for the delta-type singularities in the curvature he used, in a smoothing process, to represent the (assumed) continuous curvature of space, could in themselves play a distinguished role representing the small scale structure of the Universe. Furthermore, the matching together of blocks of space-time with sharp edges and corners may enable to develop a manifold like structure in which the tangent spaces of some points had a surfeit or deficiency of vectors, so that the differential geometry of the resulting space-time forced discontinuities and singularities in the metric structure of the Universe. Although this may be aphysical, it may be a reasonable way of seeking further understanding of the Universe. |
| format | Thesis |
| id | oai:open.uct.ac.za:11427/22232 |
| institution | University of Cape Town (South Africa) |
| language | eng |
| last_indexed | 2026-06-10T12:32:24.523Z |
| license_str | Not specified — see source repository |
| provenance_str_mv | Harvested via OAI-PMH from UCTD — University of Cape Town Open Access Repository |
| publishDate | 2016 |
| publishDateRange | 2016 |
| publishDateSort | 2016 |
| publisher | Department of Mathematics and Applied Mathematics |
| publisherStr | Department of Mathematics and Applied Mathematics |
| record_format | dspace |
| source_str | UCTD — University of Cape Town Open Access Repository |
| spelling | oai:open.uct.ac.za:11427/22232 Manifold joins and jump conditions in general relativity Baston, Robert James Ellis, George F R Applied Mathematics This thesis has as its aim the analysis of a possible manifold structure on V, a join of two individual manifolds V⁺ and V⁻, and analysing the physics across the join, as implied by Einstein's theory of General Relativity. There are several reasons why one might want to study such a situation. Firstly, the joining of manifolds is useful in the study of shock waves, be they of gravitational or other origin - we will be able to characterise the propagation of energy in the join. Secondly piecing together manifolds is a potentially fruitful way of obtaining exact solutions of Einstein's equations which do not exhibit any symmetries in the large, and are yet sufficiently homogeneous (in some sense) to enable one to model the apparent Universe - the prototype of this is the Swiss-Cheese model, used to study light transmission in an inhomogeneous Universe. Thirdly, discontinuities in the fundamental quantities in Relativity are of prime importance in the study of singularities and in particular, it is of prime importance to single out the contributions of the differential geometry and metric structure of the Universe to the existence and nature of such singularities. Closely linked to these problems is the problem of linking the small scale structure of the Universe (which is manifestly complicated and potentially full of singularities) and the large scale structure, which seems so well modelled by assumptions of homogeneity and isotropy. In this regard, the techniques of Regge (1961), originally proposed to provide approximate solutions to the Einstein equations, assume a new theoretical importance, for the delta-type singularities in the curvature he used, in a smoothing process, to represent the (assumed) continuous curvature of space, could in themselves play a distinguished role representing the small scale structure of the Universe. Furthermore, the matching together of blocks of space-time with sharp edges and corners may enable to develop a manifold like structure in which the tangent spaces of some points had a surfeit or deficiency of vectors, so that the differential geometry of the resulting space-time forced discontinuities and singularities in the metric structure of the Universe. Although this may be aphysical, it may be a reasonable way of seeking further understanding of the Universe. 2016-10-21T07:32:47Z 2016-10-21T07:32:47Z 1982 Master Thesis Masters MSc http://hdl.handle.net/11427/22232 eng application/pdf Department of Mathematics and Applied Mathematics Faculty of Science University of Cape Town |
| spellingShingle | Applied Mathematics Baston, Robert James Manifold joins and jump conditions in general relativity |
| thesis_degree_str | Master's |
| title | Manifold joins and jump conditions in general relativity |
| title_full | Manifold joins and jump conditions in general relativity |
| title_fullStr | Manifold joins and jump conditions in general relativity |
| title_full_unstemmed | Manifold joins and jump conditions in general relativity |
| title_short | Manifold joins and jump conditions in general relativity |
| title_sort | manifold joins and jump conditions in general relativity |
| topic | Applied Mathematics |
| url | http://hdl.handle.net/11427/22232 |
| work_keys_str_mv | AT bastonrobertjames manifoldjoinsandjumpconditionsingeneralrelativity |