Energy–momentum relation

The energy-momentum relation, also called the relativistic dispersion relation, is a foundational concept in physics, rooted in Albert Einstein's work on special relativity. 
This article will comprehensively overview the Energy-Momentum Relation and present different equations for considering specific cases for this relation. 
It establishes an important connection between an object's energy and momentum, specifically relating its rest (intrinsic) mass, total energy and momentum.
This relation reveals the integral link between these physical properties. As an object's velocity approaches the speed of light, the interdependence of momentum and energy becomes evident, leading to consequences like time dilation and relativistic mass increase. 
ExplanationThe relation assumes the special relativity case of flat spacetime and that the particles are free. The total energy is the sum of the rest energy and kinetic energy. The invariant mass is the mass measured in the centre of the momentum frame, the inertial frame, whereby the total momentum of a system vanishes.
The energy-momentum equation is as follows:
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E2=(pc)2+(m0c2)2E^2= (pc)^2+(m_0c^2)^2E2=(pc)2+(m0​c2)2
Here are the variables of the equation
E = the energy.
c = the speed of light.
m = the rest mass.
p = the momentum.
Input﻿﻿Mass at rest, m(0)
:0.01kg​
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﻿﻿Speed of light, c
:299,792,458.00m / s​
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﻿﻿Momentum, p
:0.01kg*m/s​
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Output﻿﻿Energy squared, E^2
:807,760,871,306,249,000,000,000,000,000.00J​
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E2=(pc)2+(m0c2)2E^2= (pc)^2+(m_0c^2)^2E2=(pc)2+(m0​c2)2
Special CasesThere are three special cases
If the body is a massless particle (m0 = 0), the relation reduces to E = pc. This is the relation, discovered in 19th-century classical electromagnetism, for photons, between radiant momentum, which causes radiation pressure, and radiant energy.
If the body's velocity, v, is much less than c, the speed of light, the relation reduces to the following:
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E=12m0v2+m0c2E= \frac{1}{2}m_0v^2+m_0c^2E=21​m0​v2+m0​c2
that is, the body's total energy is its classical kinetic energy
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12m0v2\frac{1}{2}m_0v^221​m0​v2
plus its rest energy.
If the body is at rest (v = 0), i.e. in its centre-of-momentum frame (p = 0), we have E = E0 and m = m0; thus, the energy–momentum relation and both forms of the mass–energy relation (mentioned above) all become the same.
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Use this calculator to solve these equations!
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Energy–momentum Relation Calculator
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The energy-momentum relation is essential for understanding the behaviour of particles at high speeds, providing insights into the nuanced relationship between mass, energy, and motion in the framework of the universe.
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Referencesfx​Solver. 2023. Energy – Momentum relation. [ONLINE] Available at: https://www.fxsolver.com/browse/formulas/Energy+%E2%80%93+Momentum+relation. [Accessed 13 November 2023].
Wikipedia. 2022. Energy–momentum relation. [ONLINE] Available at: https://en.wikipedia.org/wiki/Energy%E2%80%93momentum_relation. [Accessed 13 November 2023].
Wikipedia. 2023. Mass–energy equivalence. [ONLINE] Available at: https://en.wikipedia.org/wiki/Mass%E2%80%93energy_equivalence#Applicability_of_the_strict_formula. [Accessed 13 November 2023].
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