Remarks on four dimensional Euclidean gravity without Wick rotation

TL;DR

This paper explores a novel mathematical framework using conic sedenions to model four-dimensional Euclidean quantum gravity without Wick rotation, aiming to unify hyperbolic and circular metrics in a consistent algebraic system.

Contribution

It introduces conic sedenions as a new algebraic tool to describe quantum gravity directly in Euclidean spacetime, avoiding Wick rotation and incorporating both Minkowski and Euclidean metrics.

Findings

Conic sedenions unify hyperbolic and circular metrics.

Potential application of conic sedenions in quantum gravity modeling.

Proposes validation of this algebraic approach in physics.

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.