Relativity in three dimensions

Abstract

We present a new formulation of relativistic kinematics and field dynamics in an entirely three-dimensional Riemannian space. Here the basic idea is to represent local physical observers by three-dimensional vector fields of bounded length and the dynamics of physical fields by the flows of vector fields, which are one-parameter groups of local smooth transformations of the 3-space. The flow parameter plays the role of local time for each physical observer, and the Lie derivative replaces the time derivative. In the kinematics part we show how the introduction of a relative velocity function in terms of the 3-metric of the space leads immediately to the Lorentz time-dilation formula for two inertially equivalent physical observers. A treatment of equivalence class of inertial observers leads to the definition of generalized Lorentz matrix functions with corresponding generalized properties, which reduce to the usual Lorentz matrices when the underlying space is Euclidean. In the dynamics part we treat the Einstein field equations in Gaussian normal coordinates by reformulating them in a three-dimensional set-up and show how certain solutions, including the de Sitter solution, can be derived from a flat-space 3-metric by subjecting it to a transformation by the flow of a three-dimensional vector field. In this way, we obtain a new solution of the vacuum Einstein field equations which is related to the complex form of (real) Kasner-type solutions. Our formalism can also be applied to other field equations, such as the Maxwell equations, and provide a corresponding set of three-dimensional equations to generate four-dimensional solutions.