Finslerian Fields

Abstract

This work begins to address a problem posed at the A MS Joint Summer Research Conference on Finsler Geometry [1]: to “construct classical and quantum field theories defined covariantly on a Finslerian spacetime tangent bundle.” There have been strong indications in recent years that the spacetime-metric field g _μν must in general depend not only on where it is in spacetime, but also on where it is in four-velocity space, namely [1–3],

$${g_{\mu \nu }} = {g_{\mu \nu }}(x,v)$$

((1))

where x denotes the spacetime coordinates x ^μ, v denotes the four-velocity coordinates,

$${\upsilon ^\mu } = \frac{{d{x^\mu }}}{{ds}}$$

((2))

and ds is the spacetime interval. The fact that the spacetime metric may depend explicitly on four-velocity as well as on the spacetime coordinates follows from the fact that proper acceleration must always be less than the maximal proper acceleration a ₀, in order that the spacetime structure remain topologically stable [2–4]. A proper acceleration exceeding a ₀ would give rise to copious production of black hole pairs from the vacuum and to the topological breakdown of the spacetime structure. It also follows that the natural arena for the description of the spacetime-metric field is the tangent bundle of spacetime [1–3].