# Reexamination of energy flow velocities of non-diffracting localized waves

## TL;DR

This paper establishes a universal relation between energy transport velocity and group velocity for localized electromagnetic waves, showing that energy velocity does not exceed the speed of light, regardless of superluminal or subluminal group velocities.

## Contribution

It introduces a universal relation linking energy flow velocity and group velocity for localized waves, applicable to both superluminal and subluminal cases, under specific physical restrictions.

## Key findings

- Energy transport velocity is universally related to group velocity.
- The energy velocity does not surpass the speed of light in vacuum.
- The relation holds for both scalar and vector electromagnetic waves.

## Abstract

A universal relation has been established between the local energy transport velocity along the direction of propagation and the group velocity of scalar and vector-valued propagation-invariant spatiotemporally localized superluminal and subluminal electromagnetic waves in free space. Under specific restrictions, this relationship is very closely valid for physically realizable almost propagation-invariant spatiotemporally confined subluminal and superluminal electromagnetic fields. In both cases, although the group velocity may be either superluminal or subluminal, the universal relation is in accord with the well-established result that the upper limit of the energy transport velocity is c, the speed of light in vacuum.