The Random Discrete Action for 2-Dimensional Spacetime

Details The Random Discrete Action for 2-Dimensional Spacetime The Random Discrete Action for 2-Dimensional Spacetime

2010-11-23

arxiv.org/abs/1011.5191v2

Class.Quant.Grav.28:105018,2011

Astronomy and Astrophysics

Astrophysics

General Relativity and Quantum Cosmology

20 pages, 10 figures, Typos corrected

Scientific paper

10.1088/0264-9381/28/10/105018

A one-parameter family of random variables, called the Discrete Action, is defined for a 2-dimensional Lorentzian spacetime of finite volume. The single parameter is a discreteness scale. The expectation value of this Discrete Action is calculated for various regions of 2D Minkowski spacetime. When a causally convex region of 2D Minkowski spacetime is divided into subregions using null lines the mean of the Discrete Action is equal to the alternating sum of the numbers of vertices, edges and faces of the null tiling, up to corrections that tend to zero as the discreteness scale is taken to zero. This result is used to predict that the mean of the Discrete Action of the flat Lorentzian cylinder is zero up to corrections, which is verified. The ``topological'' character of the Discrete Action breaks down for causally convex regions of the flat trousers spacetime that contain the singularity and for non-causally convex rectangles.

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