Dissertation
Projective gauge gravity
University of Iowa
Doctor of Philosophy (PhD), University of Iowa
Summer 2020
DOI: 10.17077/etd.005525
Abstract
We describe a theory of gravitation that modifies general relativity by exploiting projective gauge symmetry. Two affine connections \(\Gamma^a_{\ bc}\) and \(\hat{\Gamma}^a_{\ bc}\) on a manifold \(M\) are considered projectively equivalent when they determine the same space of geodesic paths on \(M\) up to reparameterization. In the context of general relativity, objects move along geodesics determined by the spacetime connection, which is a dynamical field. Therefore, the theory is physically invariant under projective transformations of the connection. However, general relativity fails to mathematically reflect this symmetry when we use the Einstein-Hilbert action as a starting point.
Our theory utilizes a Thomas-Whitehead connection \(\tilde{\Gamma}^{\mu}_{\ \nu \rho}\) on the volume bundle \(VM\) over the spacetime manifold \(M\). We form a dynamical action using the curvature tensor \(K^{\mu}_{\ \nu \rho \sigma}\) associated with \(\tilde{\Gamma}^{\mu}_{\ \nu \rho}\). This produces a gravitational theory that incorporates the usual spacetime connection \(\Gamma^{a}_{\ b c}\) and metric tensor \(g_{ab}\) along with a gauge field \(\mathcal{D}_{ab}\) that recovers projective invariance of the theory. We demonstrate that the resulting theory, Thomas-Whitehead gravity, is indeed projectively invariant and reduces to the ordinary theory of general relativity in a certain limit. We offer projective gauge symmetry as a potential explanation for phenomena such as dark matter and dark energy, and as a starting point for a theory of quantum gravity.
Furthermore, we show that the projective gauge field \(\mathcal{D}_{ab}\) has the transformation law of a Virasoro coadjoint element, i.e. the \textit{diffeomorphism field} \(D(\theta)\), in the one-dimensional setting. This field arises in the context of string theory in the same manner as the Yang-Mills gauge potential \(A_{\mu}\). Finally, we examine the potential role of \(\mathcal{D}_{ab}\) in the presence of fermions.
Details
- Title: Subtitle
- Projective gauge gravity
- Creators
- Samuel J Brensinger
- Contributors
- Vincent G.J. Rodgers (Advisor)Oguz Durumeric (Committee Member)Palle Jorgensen (Committee Member)Yannick Meurice (Committee Member)Wayne Polyzou (Committee Member)
- Resource Type
- Dissertation
- Degree Awarded
- Doctor of Philosophy (PhD), University of Iowa
- Degree in
- Applied Mathematical and Computational Sciences
- Date degree season
- Summer 2020
- DOI
- 10.17077/etd.005525
- Publisher
- University of Iowa
- Number of pages
- viii, 127 pages
- Copyright
- Copyright 2020 Samuel J Brensinger
- Language
- English
- Description illustrations
- illustrations
- Description bibliographic
- Includes bibliographical references (pages 120-127).
- Public Abstract (ETD)
Einstein’s theory of general relativity is a rich and powerful description of gravitation. However, this theory does not seem to be harmonious with the standard model of physics, which successfully describes the other known forces of nature. General relativity predicts the existence of black holes, which have been verified experimentally, but none of the matter in the standard model can act as the source of a black hole. General relativity allows us to mathematically describe the accelerated expansion of the universe, but the standard model does not predict a rate of expansion that matches observation. These issues give rise to the concepts of dark matter and dark energy, entities that appear to be necessary based on our observations, but which currently have no widely accepted theoretical explanation. Furthermore, attempts to apply our understanding of quantum mechanics to the gravitational field have proven mostly unsuccessful. In short, there is ample opportunity to improve our understanding of gravity.
Projective gauge gravity builds on general relativity by taking advantage of a physical symmetry in the theory. Any theory of gravity should be able to predict the motion of objects in spacetime. When we describe the path of a given object mathematically, we must choose some way to parameterize the path, which gives rise to a notion of speed. In this context, speed is just a mathematical construct; we can always change the way we parameterize a path without changing the underlying physics. In other words, the only relevant information is the direction the object is moving in spacetime, not the mathematically arbitrary notion of speed. A theory with this type of symmetry is best described using projective geometry. Usually, physicists choose a particular parameterization so that objects can be thought of as moving at constant speed in spacetime. Shortly after Einstein published his theory, an Italian mathematician named Attilio Palatini showed that this choice is sound, assuming that general relativity does in fact give a complete description of gravitation.
Instead of sidestepping projective symmetry by assuming that general relativity is the end of the story, our theory uses projective symmetry to introduce a new dynamical field called the projective gauge field. Since the projective gauge field would interact with the gravitational field and other fields in nature, it offers a possible explanation for the concepts of dark matter and dark energy discussed earlier. We can also show that the projective gauge field underlies theories of quantum gravity in two dimensions. Furthermore, since projective gauge gravity is a purely geometric theory, it can be extended to any number of spacetime dimensions, in particular the four-dimensional spacetime that we usually study.
In summary, this theory provides possible answers to lingering questions about the nature of the gravitational force.
- Academic Unit
- Interdisciplinary Graduate Program in Applied Mathematical & Computational Sciences
- Record Identifier
- 9983987895902771
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