Imaginary Time as a Complex Representation of Relativity

Abstract

We develop a systematic complex representation of Lorentzian relativity based on the imaginary temporal coordinate $X^0=ict$. Rather than treating this substitution as a Wick rotation to Euclidean spacetime, we formulate it as an invertible change of representation in which physical spacetime is identified with a real subspace of a complex vector space. This construction introduces two related metric structures: a complex-bilinear form and a Hermitian form, both of which reproduce the original Lorentzian interval when restricted to physical spacetime. We extend the representation to relativistic dynamics, including proper time, four-momentum, and the Hamilton-Jacobi formulation, and show that the metric, connection, curvature tensors, Einstein equations, and conservation laws retain their conventional content in curved spacetime. Explicit applications to the Kerr and Friedmann-Lemaître-Robertson-Walker geometries demonstrate consistency for rotating and time-dependent spacetimes. We further examine relativistic wave equations and their nonrelativistic limit, showing how the Schrödinger equation and its relation to Euclidean-time evolution emerge within the same complex temporal framework. The resulting formulation introduces no new physical predictions, but clarifies the geometric role of imaginary temporal coordinates and reveals structural connections among relativistic geometry, action, wave propagation, and analytic continuation.