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The Foundations Of Differential Geometry

A unified gauge approach to both, dynamics and thermodynamics involving gravity, is developed from the local realization of the Poincar\'e group as a particular instance of a spacetime group including translations. The formalism is applied to study the physical features of the spherical non-rotating matter source of an inner Schwarzschild metric.

Matter sources of gauge thermodynamics.

Gauge theories of internal symmetries, e.g.\ the strong and electroweak forces of the Standard Model, have a geometric description in terms of standard fiber bundles. It is tempting then to ask if the remaining force, gravitation, has a description as a gauge theory. The answer is yes, however unlike the internal symmetries of the Standard Model, the story is not so simple. There are dozens of renditions of gravitational gauge theory and no standard fiber bundle description. The main issue in the construction of gravitational gauge theory is the inclusion of translational symmetry. While the Lorentz group, like internal symmetries, acts only at each point, the translational symmetry shifts points in spacetime. For this reason a gauge theory of gravity requires a somewhat more sophisticated fiber bundle known as a composite fiber bundle. When constructing gauge theories of internal symmetries it is easy to take certain topological conditions for granted, like orientability or the ability to define spinors. However it is known that there exist spaces which do not have the properties required to define sensible field theories. Although we may take these topological properties for granted when constructing gauge theories of internal symmetries we haven't had evidence yet to expect we can do the same for gravitational gauge theory. By studying the geometry of the composite bundle formalism underlying viable gauge theories of gravity we have found previously unappreciated subbundles of the primary bundle. We were able to identify these subbundles as the spacetime bundles we would expect to be created by a gauge theory of gravity. Remarkably, the origin of these subbundles leads to the natural inclusion of expected, and unexpected, topological conditions. While the overall bundle used for gravitation is $P(M,ISO(1,3))$, i.e.\ a principal Poincar\'e bundle over a space $M$, the Poincar\'e group ($ISO(1,3)$) can be viewed as a bundle in its own right $ISO(\reals^4,SO(1,3))$. Thinking of the fiber space itself as yet another bundle leads to consideration of two primary bundles $P(E,SO(1,3))$ and $E(M,\reals^4,ISO(1,3),P(M,ISO(1,3)))$. The split of the total bundle $P(M,ISO(1,3))$ into the two bundles $E$ and $P(E,SO(1,3))$ however requires the existence of a global section of the bundle $E$. Such a global section is guaranteed to exist by a theorem of Kobayashi and Nomizu. However it is interesting to investigate the topology of the bundle space $E$ and hence of $P(E,SO(1,3))$. The requirement of the global section leads to the definition of a bundle $Q(M,SO(1,3))\subset P(M,ISO(1,3))$ which can be identified as the frame bundle of spacetime. Its associated bundle, $Q(M,SO(1,3))\times_{SO(1,3)} \reals^4$ where $\times_{SO(1,3)}$ denotes a specific quotient of the product space $Q(M,SO(1,3))\times\reals^4$ by the group $SO(1,3)$, can then be identified as the tangent bundle. The existence of a global section of $E$ leads to topological conditions on the induced spacetime bundles. Using cohomology with compact support one can…

Building a connection through obstruction; Relating gauge gravity and string theory

We consider topological constraints that must be satisfied by formulations of gravitation as a gauge theory. To facilitate the analysis we review and further justify the composite bundle formalism of Tresguerres as a consistent underlying structure capable of incorporating both the local Lorentz and translational degrees of freedom. Identifying an important global structure required by the composite construction, we translate this into conditions on the underlying manifold. We find that in addition to admitting the expected orientability, causality and spin structures, the underlying manifold must also admit a string structure. We take this to imply that even before considerations of quantum consistency, topological considerations of gauge gravity provide a classical motivation for extended degrees of freedom.

Connecting Through Obstruction; Relating Gauge Gravity and String Theory

We consider two concepts often discussed as significant features of general relativity (particularly when contrasted with the other forces of the Standard Model): background independence and diffeomorphism invariance. We remind the reader of the role of backgrounds both as calculational tools and as part of the formulation of theories. Examining familiar gauge theory constructions, we are able to pinpoint when in the formulation of these theories they become background independent. We then discuss extending the gauge formulation to gravity. In doing so we are able to identify what makes general relativity a background independent theory. We also discuss/dispel suggestions that ”active” diffeomorphism invariance is a feature unique to general relativity and we go on to argue against the claim that this symmetry is the origin of background independence of the theory.

Alex Flournoy

Papers

Connecting Through Obstruction; Relating Gauge Gravity and String Theory

Background-Independence from the Perspective of Gauge Theory