Remarks on four dimensional Euclidean gravity without Wick rotation

Abstract

In reference to S. W. Hawking's article "Information Loss in Black Holes" [S. W. Hawking, Phys. Rev. D 72 (2005) 084013], where a four dimensional Euclidean spacetime without Wick rotation is adopted for quantum gravity, an arithmetic with multiplicative modulus is mentioned here which incorporates both a hyperbolic (Minkowski) and circular (Euclidean) metric: The 16 dimensional conic sedenion number system is built on nonreal square roots of +1 and -1, and describes the Dirac equation through its 8 dimensional hyperbolic octonion subalgebra [J. K\"oplinger, Appl. Math. Comput. (2006), in print, doi: 10.1016/j.amc.2006.04.005]. The corresponding circular octonion subalgebra exhibits Euclidean metric, and its applicability in physics is being proposed for validation. In addition to anti-de Sitter (AdS) spacetimes suggested by Hawking, these conic sedenions are offered as computational tool to potentially aid a description of quantum gravity on genuine four-dimensional Euclidean spacetime (without Wick rotation of the time element), while being consistent with canonical spacetime metrics.