Dispersion Relations and Relativistic Causality

TL;DR

This paper argues that adherence to dispersion relations implies the refractive index approaches unity at high frequencies, supporting relativistic causality and preventing faster-than-light signals.

Contribution

It provides a compelling argument linking dispersion relations to the impossibility of superluminal communication, emphasizing the physical implications of Titchmarsh's theorem.

Findings

Refractive index n(w) approaches 1 as frequency w approaches infinity.

Dispersion relations are consistent with relativistic causality.

Faster-than-c signals would violate causality constraints.

Abstract

In this paper we show that if the refractive index, or rather (n(w) -1) satisfies the dispersion relations then, it is implied by Titchmarsh's theorem that n(w) -> 1 as w -> infinity. Any other limiting value for n(w) would violate relativistic causality, by which we mean not only that cause must precede effect but also that signals cannot travel faster-than-c (the velocity of light) in a vacuum. This paper does not claim to be a mathematically rigorous proof, but the authors hope to have succeeded in supplying a very convincing argument against faster-than-c light signals. Keywords: Dispersion relations, Kramers Kronig relations, Causality, faster-than-c signals.