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The Erlangen Program and General Relativity: Spacetime, Geometrodynamics, and Observers.
Mathematical Physics| Speaker: | Derek Wise, University of Erlangen |
| Location: | 3106 MSB |
| Start time: | Thu, Nov 20 2014, 4:10PM |
Description
Using recent results, I will discuss what insight can be gained in
general relativity by taking seriously Klein's Erlangen Program and
Cartan's extension of it to differential geometry. The talk will be
divided into three main parts. First, I will review how Cartan
geometry gives us the geometry of spacetime by "breaking" SO(4,1)
symmetry to SO(3,1), as in the MacDowell-Mansouri-Stelle-West
formulation of gravity. Second, I will describe how a "field of
observers" breaks the symmetry further to SO(3), yielding a system of
evolving spatial Cartan geometries, or "Cartan geometrodynamics". This
gives a manifestly Lorentz-covariant counterpart of the
Ashtekar-Barbero formulation, which is the classical starting point
for loop quantum gravity. Third, I will combine the first two parts,
and consider breaking SO(4,1) to SO(3) directly. This describes the
geometry of "observer space", the bundle of future-timelike unit
vectors in spacetime, giving a perspective with key features of both
the spacetime and geometrodynamic approaches. I will also discuss how
observer space geometry provides a unified geometric setting for
relating various alternative theories of gravity.