Rate-Differential Relations and General Integrals for Power-Law Energies

The purpose of this paper is to generalise the rate-differential relations and general integrals, originally derived for Einstein’s particle energy expression, to a family of Lorentz-invariant power-law particle energies characterised by an arbitrary constant κ such that the Einstein energy is included through the special case κ=0. By Lorentz invariance, we refer to invariance under the combined special relativistic space–time and energy–momentum transformations. The power-law particle energy expressions are important and potentially fundamental since they share the same relationship with an extension of the Planck–de Broglie energy–momentum relations, as does Einstein’s energy expression with the conventional Planck–de Broglie relations. The particle–wave approach is again adopted to determine the structure of the general integrals that apply to the power-law energies, except that here an independent derivation is also provided and from which the necessary conditions are apparent for the validity of the integral. A simple similarity solution is given that demonstrates the correctness of the general integrals, and the case of non-constant initial rest energy (non-constant rest mass) is incorporated in the analysis. An appendix provides an independent derivation of the Lorentz force expressions, from which the precise conditions of their validity become apparent.