Spacetime as a Hamiltonian Orbit and Geroch's Theorem on the Existence of Fermions

Over a century since its inception, general relativity continues to lie at the heart of some of the most researched topics in theoretical physics. It seems likely that the coveted solutions to problems like quantum gravity are to be found in an extension of general relativity, one which may only be...

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Main Author:	Bergstedt, Viktor
Format:	Others
Language:	English
Published:	Uppsala universitet, Institutionen för fysik och astronomi 2020
Subjects:	3+1 ADM black hole canonical quantization Cauchy surface causality differential geometry Einstein-Hilbert action general relativity Geroch globally hyperbolic Hamilton's equations numerical relativity parallelizable SL(2 C) spacetime spinor spin structure SO(1 3) Stiefel-Whitney class tetrad topology Physical Sciences Fysik
Online Access:	http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-432488

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spelling	ndltd-UPSALLA1-oai-DiVA.org-uu-4324882021-02-16T05:30:26ZSpacetime as a Hamiltonian Orbit and Geroch's Theorem on the Existence of FermionsengBergstedt, ViktorUppsala universitet, Institutionen för fysik och astronomi20203+1ADMblack holecanonical quantizationCauchy surfacecausalitydifferential geometryEinstein-Hilbert actiongeneral relativityGerochglobally hyperbolicHamilton's equationsnumerical relativityparallelizableSL(2C)spacetimespinorspin structureSO(13)Stiefel-Whitney classtetradtopologyPhysical SciencesFysikOver a century since its inception, general relativity continues to lie at the heart of some of the most researched topics in theoretical physics. It seems likely that the coveted solutions to problems like quantum gravity are to be found in an extension of general relativity, one which may only be visible in an alternate formulation of the theory. In this thesis we consider the possibility of casting general relativity in the form of an initial value problem where spacetime is seen as the evolution of space. This evolution is shown to be constrained and of Hamiltonian type. Not all spacetimes are physically acceptable. To be compatible with particle physics, one would like spacetime to accommodate fermions. Here we can take comfort in Geroch’s theorem, which implies that any spacetime that admits a Hamiltonian formulation automatically supports the existence of fermions. We review the elements that go into the proof of this theorem. Allmän relativitetsteori har i över hundra år legat i teoretiska fysikens framkant. Det är möjligt att lösningarna på öppna problem som kvantiseringen av gravitation går att finna i en utvidgning av allmän relativitetsteori – och kanske uppenbarar sig denna utvidgning bara ur en alternativ formulering av teorin. I den här uppsatsen formuleras allmän relativitetsteori och dess Einsteinekvationer som ett begynnelsevärdesproblem, genom vilket rumtiden kan betraktas som rummets historia. Vi visar att rummets rörelseekvationer är Hamiltons ekvationer med tvångsvillkor. Enligt partikelfysiken bör fermioner kunna finnas till i rumtiden. Härom kan vi åberopa Gerochs sats, enligt vilken rumtider som har en Hamiltonsk formulering också medger fermioner. Vi redogör för huvuddragen i beviset av Gerochs sats. Student thesisinfo:eu-repo/semantics/bachelorThesistexthttp://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-432488FYSAST ; FYSKAND1131application/pdfinfo:eu-repo/semantics/openAccess
collection	NDLTD
language	English
format	Others
sources	NDLTD
topic	3+1 ADM black hole canonical quantization Cauchy surface causality differential geometry Einstein-Hilbert action general relativity Geroch globally hyperbolic Hamilton's equations numerical relativity parallelizable SL(2 C) spacetime spinor spin structure SO(1 3) Stiefel-Whitney class tetrad topology Physical Sciences Fysik
spellingShingle	3+1 ADM black hole canonical quantization Cauchy surface causality differential geometry Einstein-Hilbert action general relativity Geroch globally hyperbolic Hamilton's equations numerical relativity parallelizable SL(2 C) spacetime spinor spin structure SO(1 3) Stiefel-Whitney class tetrad topology Physical Sciences Fysik Bergstedt, Viktor Spacetime as a Hamiltonian Orbit and Geroch's Theorem on the Existence of Fermions
description	Over a century since its inception, general relativity continues to lie at the heart of some of the most researched topics in theoretical physics. It seems likely that the coveted solutions to problems like quantum gravity are to be found in an extension of general relativity, one which may only be visible in an alternate formulation of the theory. In this thesis we consider the possibility of casting general relativity in the form of an initial value problem where spacetime is seen as the evolution of space. This evolution is shown to be constrained and of Hamiltonian type. Not all spacetimes are physically acceptable. To be compatible with particle physics, one would like spacetime to accommodate fermions. Here we can take comfort in Geroch’s theorem, which implies that any spacetime that admits a Hamiltonian formulation automatically supports the existence of fermions. We review the elements that go into the proof of this theorem. === Allmän relativitetsteori har i över hundra år legat i teoretiska fysikens framkant. Det är möjligt att lösningarna på öppna problem som kvantiseringen av gravitation går att finna i en utvidgning av allmän relativitetsteori – och kanske uppenbarar sig denna utvidgning bara ur en alternativ formulering av teorin. I den här uppsatsen formuleras allmän relativitetsteori och dess Einsteinekvationer som ett begynnelsevärdesproblem, genom vilket rumtiden kan betraktas som rummets historia. Vi visar att rummets rörelseekvationer är Hamiltons ekvationer med tvångsvillkor. Enligt partikelfysiken bör fermioner kunna finnas till i rumtiden. Härom kan vi åberopa Gerochs sats, enligt vilken rumtider som har en Hamiltonsk formulering också medger fermioner. Vi redogör för huvuddragen i beviset av Gerochs sats.
author	Bergstedt, Viktor
author_facet	Bergstedt, Viktor
author_sort	Bergstedt, Viktor
title	Spacetime as a Hamiltonian Orbit and Geroch's Theorem on the Existence of Fermions
title_short	Spacetime as a Hamiltonian Orbit and Geroch's Theorem on the Existence of Fermions
title_full	Spacetime as a Hamiltonian Orbit and Geroch's Theorem on the Existence of Fermions
title_fullStr	Spacetime as a Hamiltonian Orbit and Geroch's Theorem on the Existence of Fermions
title_full_unstemmed	Spacetime as a Hamiltonian Orbit and Geroch's Theorem on the Existence of Fermions
title_sort	spacetime as a hamiltonian orbit and geroch's theorem on the existence of fermions
publisher	Uppsala universitet, Institutionen för fysik och astronomi
publishDate	2020
url	http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-432488
work_keys_str_mv	AT bergstedtviktor spacetimeasahamiltonianorbitandgerochstheoremontheexistenceoffermions
_version_	1719377353299197952

Spacetime as a Hamiltonian Orbit and Geroch's Theorem on the Existence of Fermions

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