  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.
