E = mc² is one of the famous formulas, created by Einstein, to represent the relationship between mass and energy. E² = (mc²)² + (pc)² is the extended version of E = mc², establishing the relationship between energy and momentum. It is created by Paul Dirac in 1928.

p - Momentum

E - Energy

m - Mass

c - Speed of light

To learn the derivation of this formulae, we have to learn about Lorentz Factor.

Lorentz Factor

The Lorentz factor γ is defined as :

- 𝛾 = 1 √1 − 𝑣2 𝑐2

The Lorentz factor or Lorentz term (also known as the gamma factor) is a dimensionless quantity expressing how much the measurements of time, length, and other physical properties change for an object while it moves.

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Derivation of the Formulae

Relativistic energy-momentum relationship :

E² = (γmc²)²

p² = (γmv)²

Square and add the momentum term:

E² = (γmc²)²

p² = (γmv)² = (γ²m²v²)

Add (pc)² to both sides:

E² + (pc)² = (γmc²)² + (γ²m²v² * c²)

- Since (pc)² = p²c² = (γ²m²v²)c² = (γ²m²v²)

E² + (pc)² = (γmc²)² + (γ²m²v²c²)

E² + (pc)² = (γ²m²c⁴) + (γ²m²v²c²)

- Factor out (γ²m²c²):

E² + (pc)² = (γ²m²c⁴)(1 + v²/c²)

- Since γ² = 1/(1 - v²/c²) and v²/c² < 1:

E² + (pc)² = (γ²m²c⁴)(γ²)

E² + (pc)² = γ²m²c⁴ * γ²

E² + (pc)² = (γ²m²c⁴)

Simplify:

E² = (mc²)² + (pc)²

Conclusion

This is just 1 Way, there are lot of other ways to derive this equation. But E² = (mc²)² + (pc)² is what supports that massless particles - like photon - can have energy.