The Full E = mc² Formula
Einstein's famous equation is the rest-energy version of a more complete relativistic energy formula that also includes momentum.
The famous version
The equation E = mc² describes the rest energy of an object: the energy an object has simply because it has mass.
This applies when the object is at rest, meaning its momentum is zero.
The full energy-momentum relation
The more complete formula is:
- E
- Total energy
- m
- Rest mass
- c
- Speed of light in a vacuum, approximately 299,792,458 meters per second
- p
- Momentum
How E = mc² comes from the full formula
If an object is not moving, its momentum is zero: p = 0.
Starting with the full equation:
Substitute p = 0:
Taking the positive square root gives:
Moving objects
For an object moving at velocity v, its total energy can also be written as:
where:
The symbol γ is called the Lorentz factor. As an object's velocity gets closer to the speed of light, γ increases, so the object's total energy increases.
Massless particles
For a massless particle such as a photon, the rest mass is zero: m = 0.
The full equation becomes:
This means a massless particle can still have energy if it has momentum.
What it means
Mass and energy are two forms of the same underlying physical quantity. Because the speed of light squared is enormous, even a small amount of mass corresponds to a tremendous amount of energy.
For example, one kilogram of mass has a rest energy of approximately: