Time vector defined in imaginary space of spatial coordinate

TL;DR

This paper proposes modeling space with three complex coordinates, where time is represented as an imaginary component, and explores how Lorentz invariance and gravity can be formulated within this complex space framework.

Contribution

It introduces a novel complex coordinate model of space-time, integrating time as an imaginary component and demonstrating how Lorentz transformations and gravity can be described in this setting.

Findings

Lorentz invariance is realized as holomorphic transformations in complex space.

Scalar fields in complex space can satisfy Klein-Gordon equations in spacetime.

A framework for gravity using the equivalence principle in complex coordinates is outlined.

Abstract

Empirical understanding teaches us that space is three dimensional while relativity merges space with time. We tried to show that it is possible to model space as three complex coordinates. In our construction, the usual spatial coordinate are the real part while time is considered as parameter of a path attached to each spatial point in imaginary parts. For flat spacetime, Lorentz invariant is realized on induced metric. We first consider the space as a six dimensional real manifold corresponding to the three complex coordinates. The complex structure is then implemented in a reference frame dependent manner. With this complex structure, Lorentz transformation induces a holomorphic transformation in the complex manifold so that spatial coordinate remained to be viewed as real axis. Scalar field in complex space could induce a field that satisfying Klein-Gordon equation on spacetime. Finally we described possible construction of gravity by using equivalence principle. Assuming the coordinate transformation to an accelerating observer also induces a holomorphic transformation, we outlined how to recover general relativity in Hamiltonian formalism.