## Description

When a particle is in motion, its momentum (p) needs to be considered in the energy equation. When mass isn’t considered, the energy is simply momentum times the speed of light (E=pc). However, the long form of Einstein’s **energy-momentum equation **includes mass and is the foundation of E=mc^{2} (when momentum is zero). This equation is E^{2} = (mc^{2})^{2} + (pc)^{2}.

Since particles are formed from waves, a change in wave frequency needs to be considered when a particle is in motion. Its wavelength will be shorter in the direction of travel on its leading edge, and longer on its trailing edge, relative to the particle when it is at rest. To an observer, the particle experiences the Doppler effect and thus Doppler equations are used to find the leading edge and trailing frequencies. The particle’s frequency while in motion is the geometric mean of the lead and lag frequencies, which explains the use of the Lorentz factor and relativity in the equation. At **relativistic speeds the Lorentz factor needs to be considered**.

## Derivation

The energy-momentum equation is simply a change in wave frequency due to motion and it can derived from the base wave energy equation. Due to particle motion (velocity – v), it requires the complete form of the Longitudinal In- and Out-Wave Energy.

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## Proof

When velocity is zero, the sum of the in-wave and out-waves is the Longitudinal Energy Equation that accurately calculates rest energy and rest mass of the electron (the first part of the complete energy-momentum equation). See E=mc^{2}. The addition of velocity into the equation correctly derives the Lorentz factor.

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