Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 6 (2010), 064, 11 pages      arXiv:1003.5651
Contribution to the Special Issue “Noncommutative Spaces and Fields”

Global Eikonal Condition for Lorentzian Distance Function in Noncommutative Geometry

Nicolas Franco
GAMASCO, Department of Mathematics, University of Namur FUNDP, 8 Rempart de la Vierge, B-5000 Namur, Belgium

Received March 30, 2010, in final form August 06, 2010; Published online August 17, 2010

Connes' noncommutative Riemannian distance formula is constructed in two steps, the first one being the construction of a path-independent geometrical functional using a global constraint on continuous functions. This paper generalizes this first step to Lorentzian geometry. We show that, in a globally hyperbolic spacetime, a single global timelike eikonal condition is sufficient to construct a path-independent Lorentzian distance function.

Key words: noncommutative geometry; Lorentzian distance; eikonal inequality.

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  1. Beem J.K., Eherlich P.E., Easley K.L., Global Lorentzian geometry, 2nd ed., Monographs and Textbooks in Pure and Applied Mathematics, Vol. 202, Marcel Dekker, Inc., New York, 1996.
  2. Bernal A.N., Sánchez M., On smooth Cauchy hypersurfaces and Geroch's splitting theorem, Comm. Math. Phys. 243 (2003), 461-470, gr-qc/0306108.
  3. Bernal A.N., Sánchez M., Smoothness of time functions and the metric splitting of globally hyperbolic spacetimes, Comm. Math. Phys. 257 (2005), 43-50, gr-qc/0401112.
  4. Connes A., Noncommutative geometry, Academic Press, Inc., San Diego, CA, 1994.
  5. Connes A., Gravity coupled with matter and the foundations of non-commutative geometry, Comm. Math. Phys. 182 (1996), 155-176, hep-th/9603053.
  6. Connes A., Marcolli M., Noncommutative geometry, quantum fields and motives, American Mathematical Society Colloquium Publications, Vol. 55, American Mathematical Society, Providence, RI, 2008.
  7. Erkekoglu F., García-Río E., Kupeli D.N., On level sets of Lorentzian distance function, Gen. Relativity Gravitation 35 (2003), 1597-1615.
  8. Gracia-Bondía J.M., Várilly J.C., Figueroa H., Elements of noncommutative geometry, Birkhäuser Boston, Inc., Boston, MA, 2001.
  9. Hencl S., On the notions of absolute continuity for functions of several variables, Fund. Math. 173 2002, 175-189.
  10. Malý J., Absolutely continuous functions of several variables, J. Math. Anal. Appl. 231 (1999), 492-508.
  11. Moretti V., Aspects of noncommutative Lorentzian geometry for globally hyperbolic spacetimes, Rev. Math. Phys. 15 (2003), 1171-1217, gr-qc/0203095.
  12. O'Neill B., Semi-Riemannian geometry. With applications to relativity, Pure and Applied Mathematics, Vol. 103, Academic Press, Inc., New York, 1983.
  13. Paschke M., Sitarz A., Equivariant Lorentzian spectral triples, math-ph/0611029.
  14. Strohmaier A., On noncommutative and pseudo-Riemannian geometry, J. Geom. Phys. 56 (2006), 175-195, math-ph/0110001.
  15. Wald R.M., General relativity, University of Chicago Press, Chicago, IL, 1984.

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