Temporal insights from the end of space
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This dissertation concerns the Weiss variation, its application in both classical and quantum General Relativity, and the role of spatial boundary conditions in characterizing time evolution. I review the Weiss variation in mechanics and classical field theory, and present a novel geometric derivation of the Weiss variation for the gravitational action: the Einstein-Hilbert action plus the Gibbons-Hawking-York boundary term. In particular, I use the first and second variation of area formulas (I include a derivation accessible to physicists in an appendix) to interpret and vary the Gibbons-Hawking-York boundary term. Though the Weiss variation of the gravitational action is in principle known to the relativity community, the variation of area approach formalizes the derivation, and facilitates the discussion of time evolution in General Relativity. I demonstrate the utility of the Weiss variation in quantum General Relativity by presenting a formal derivation of the Wheeler-DeWitt equation from the functional integral of quantum General Relativity by way of boundary variations. One feature of this approach is that it does not require an explicit 3+1 splitting of spacetime in the bulk. For spacetimes with spatial boundary, I show that variations in the induced metric at the spatial boundary can be used to describe time evolution--time evolution in quantum General Relativity is therefore governed by boundary conditions on the gravitational field at the spatial boundary.