In special relativity (SR), momentum (p) and energy (E) are related through the following equation:
E^2 = (pc)^2 + (mc^2)^2
Where: E = Total energy of the object (including both rest energy and kinetic energy) p = Momentum of the object c = Speed of light in a vacuum m = Rest mass of the object
This equation is known as the relativistic energy-momentum relation or the mass-energy equivalence equation. It shows how energy and momentum are interconnected in the relativistic regime. When an object is at rest (v = 0), the equation simplifies to the famous mass-energy equivalence formula:
E = mc^2
This equation demonstrates that mass (m) and energy (E) are equivalent and interchangeable, as famously stated by Einstein's equation E=mc^2. It means that even an object at rest possesses energy due to its mass.
As an object's speed approaches the speed of light (v ≈ c), its momentum and relativistic energy increase significantly. The equation shows that the kinetic energy contribution becomes negligible compared to the mass-energy term at high speeds, preventing an object with mass from ever reaching or exceeding the speed of light.
So, in special relativity, momentum and energy are tightly related by the relativistic energy-momentum relation, which incorporates the mass-energy equivalence principle and describes the behavior of objects moving at relativistic speeds.