Arrow of Time

Abstract

In most of physics, time is a real number, and the fundamental laws work equally well forward and backward. The arrow of time enters through thermodynamics, as entropy and the second law. Yet we feel time flow at every moment, from the past to the future. We take this direction as self-evident, make it a postulate, and describe time by an arrow number: a real part, the time of a stationary clock, and an imaginary part, the time a signal needs to cross the distance. The measured time is the arrow’s Euclidean length. From this single idea we answer twelve problems. Time gains an algebra, a Euclidean geometry, and a calculus of its own: arrowtimes add and divide, Euclid’s postulates become statements about past, present, and future, and a turning arrow produces the Coriolis, centrifugal, and Euler terms without a rotating frame. A measurement gives a scalar arrow and can reach only the past: the scientific method in geometric form. Frequency, energy, momentum, and mass form one arrow whose length is the energy. The electric and magnetic fields are the two derivatives of one potential, and the square of the arrow gradient gives Maxwell’s equations and electromagnetic waves. A third, scalar electric field comes with them; it drives the conduction current, and where it is harmonic, charge is conserved. The arrows exist in two, four, and eight dimensions only: the quaternion holds one time and three directions of space, and the octonion adds the energy, so the world of arrowtime is eight-dimensional. The uncertainty of time and energy follows from the arrow Fourier transform, with no appeal to probability. Under Boltzmann’s assumption of molecular chaos, the entropy of a single particle grows from its measured past to its uncertain future, and adds up to the entropy of a body.