Scalar Curvature of a Causal Set
Abstract
A one parameter family of retarded linear operators on scalar fields on causal sets is introduced. When the causal set is well approximated by 4 dimensional Minkowski spacetime, the operators are Lorentz invariant but nonlocal, are parametrized by the scale of the nonlocality, and approximate the continuum scalar D’Alembertian □ when acting on fields that vary slowly on the nonlocality scale. The same operators can be applied to scalar fields on causal sets which are well approximated by curved spacetimes in which case they approximate □(1)/(2)R where R is the Ricci scalar curvature. This can used to define an approximately local action functional for causal sets.
 Publication:

Physical Review Letters
 Pub Date:
 May 2010
 DOI:
 10.1103/PhysRevLett.104.181301
 arXiv:
 arXiv:1001.2725
 Bibcode:
 2010PhRvL.104r1301B
 Keywords:

 04.60.Nc;
 02.40.k;
 11.30.Cp;
 Lattice and discrete methods;
 Geometry differential geometry and topology;
 Lorentz and Poincare invariance;
 General Relativity and Quantum Cosmology;
 High Energy Physics  Theory
 EPrint:
 Typo in definition of equation (3) and definition of n(x,y) corrected. Note: published version still contains typo