Complex actions in two-dimensional topology change
Nov, 1995
31 pages
Published in:
- Class.Quant.Grav. 14 (1997) 179-204
e-Print:
- gr-qc/9511023 [gr-qc]
Report number:
- SU-GP-95-5-1,
- WISC-MILW-95-TH-16,
- MDDP-PP-96-40
View in:
Citations per year
Abstract: (arXiv)
We investigate topology change in (1+1)--dimensions by analyzing the scalar-curvature Action at the points of metric-degeneration which (with minor exceptions) any nontrivial Lorentzian cobordism possesses. In two dimensions any cobordism can be built up as a combination of only two elementary types, the ``yarmulke'' and the ``trousers.'' For each of these elementary cobordisms, we consider a family of Morse-theory inspired Lorentzian metrics which vanish smoothly at a single point. In the yarmulke case, the distinguished point is analogous to a cosmological initial (or final) singularity, and the spacetime as a whole is obtained from one causal region of Misner space by adjoining a single point. In the trousers case, the distinguished point is a ``crotch singularity'' that signals a change in the spatial topology. We regularize the metrics by adding a small imaginary part whose sign is fixed to be positive by the condition that it lead to a convergent scalar field path integral on the regularized spacetime. As the regulator is removed, the Ricci scalar density approaches a delta-function whose strength is complex: for the yarmulke family it is , where is the rapidity parameter of the Misner space; for the trousers family it is . This implies that in the path integral over spacetime metrics for Einstein gravity in three or more spacetime dimensions, topology change via a crotch singularity is exponentially suppressed, whereas appearance or disappearance of a universe via a yarmulke singularity is exponentially enhanced. We also contrast these results with the situation in a vielbein-cum-connection formulation of Einstein gravity.- gravitation
- dimension: 2
- boundary condition
- topology: effect
- regularization
- bibliography
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