TITLE:
Extra Time Dimension: Deriving Relativistic Space-Time Transformations, Kinematics, and Example of Dimensional Compactification Using Time-Dependent Non-Relativistic Quantum Mechanics
AUTHORS:
Sajjad Zahir
KEYWORDS:
Two-Time Physics, Special Theory of Relativity, Kaluza-Klein Theory, Time-Dependent Schrödinger Equation, Compactification
JOURNAL NAME:
Journal of Modern Physics,
Vol.14 No.10,
September
25,
2023
ABSTRACT: We consider a five-dimensional Minkowski space with two time dimensions characterized by distinct speeds of causality and three space dimensions. Formulas for relativistic coordinate and velocity transformations are derived, leading to a new expression for the speed limit. Extending the ideas of Einstein’s Theory of Special Relativity, concepts of five-velocity and five-momenta are introduced. We get a new formula for the rest energy of a massive object. Based on a non-relativistic limit, a two-time dependent Schr?dinger-like equation for infinite square-well potential is developed and solved. The extra time dimension is compactified on a closed loop topology with a period matching the Planck time. It generates interference of additional quantum states with an ultra-small period of oscillation. Some cosmological implications of the concept of four-dimensional versus five-dimensional masses are briefly discussed, too.