Prior to Einstein, classical physics treated mass and energy as fundamentally different quantities. Mass measured inertia, while energy described the ability to do work.
In 1905, Albert Einstein demonstrated that mass and energy are two forms of the same physical entity. This principle is known as mass–energy equivalence.
“Mass and energy are both but different manifestations of the same thing.” — Albert Einstein
This relationship is summarized by the equation:
E = mc².
Source: Encyclopaedia Britannica, “E = mc²” (Britannica)
Einstein introduced mass–energy equivalence in a short 1905 paper titled:
“Does the Inertia of a Body Depend Upon Its Energy Content?”
Rather than writing E = mc² explicitly, Einstein showed that
when a body emits energy L in the form of radiation, its mass
decreases by:
Δm = L / c²
This implies that mass is a direct measure of a body’s energy content.
Source: Einstein, A. (1905), Annalen der Physik (Einstein Papers Project)
Einstein’s reasoning relied on a thought experiment involving the emission of light and the conservation of energy and momentum.
When this same process is observed from a moving reference frame, relativity predicts that the energies of the two light pulses are no longer equal.
To preserve conservation laws in all frames of reference, the mass of the emitting body must decrease.
Source: Stanford Encyclopedia of Philosophy, “The Equivalence of Mass and Energy” (SEP)
From this argument, Einstein concluded that energy itself must possess inertia. In other words, energy behaves as if it has mass.
Energy contributes to inertia in exactly the same way as mass.
Adding energy to a system increases its mass; removing energy decreases its mass.
Source: American Institute of Physics, “Einstein and Mass–Energy Equivalence” (AIP)
The factor c² arises from the structure of special relativity,
where the speed of light connects space and time.
It also ensures dimensional consistency:
c² converts mass units into energy units
Because c is extremely large, even a small amount of mass
corresponds to an enormous amount of energy.
Source: Encyclopaedia Britannica, “Mass–energy equivalence” (Britannica)
Mass–energy equivalence explains numerous physical phenomena:
It is a foundational principle of modern physics.
Source: Stanford Encyclopedia of Philosophy (SEP)
Special relativity is based on two postulates:
c, is the same in all inertial frames.These imply that physical quantities must transform under Lorentz transformations, not Galilean ones.
Source: Einstein (1905); Rindler, Introduction to Special Relativity
In special relativity, energy and momentum form a four-vector called the four-momentum:
\[ p^\mu = \left( \frac{E}{c}, \vec{p} \right) \]
where:
The relativistic momentum of a particle with rest mass \(m\) and velocity \(v\) is:
\[ \vec{p} = \gamma m \vec{v} \]
where:
\[ \gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}} \]
Source: Jackson, Classical Electrodynamics
The Minkowski norm of the four-momentum is Lorentz invariant:
\[ p^\mu p_\mu = \left(\frac{E}{c}\right)^2 - |\vec{p}|^2 \]
This invariant must have the same value in all inertial frames. In the particle’s rest frame, \( \vec{p} = 0 \), so:
\[ p^\mu p_\mu = \left(\frac{E_0}{c}\right)^2 \]
where \(E_0\) is the energy of the particle at rest.
By definition, the invariant is:
\[ p^\mu p_\mu = m^2 c^2 \]
Source: Landau & Lifshitz, The Classical Theory of Fields
Equating the expressions for the invariant norm gives:
\[ \left(\frac{E}{c}\right)^2 - |\vec{p}|^2 = m^2 c^2 \]
Rearranging:
\[ E^2 = p^2 c^2 + m^2 c^4 \]
This is the exact relativistic energy–momentum relation.
Source: Weinberg, The Quantum Theory of Fields
Consider the particle in its rest frame, where:
\[ \vec{p} = 0 \]
Substituting into the energy–momentum relation:
\[ E^2 = m^2 c^4 \]
Taking the positive root (energy is positive):
\[ E = mc^2 \]
This is the rest energy of a particle.
The equation
\[ E = mc^2 \]
states that mass itself is a form of energy. A particle possesses energy even when its momentum is zero.
This is not an approximation; it is an exact consequence of Lorentz invariance and the definition of four-momentum.
Einstein’s original 1905 derivation used electromagnetic radiation and conservation laws. The four-vector derivation presented here is a modern reformulation that makes the equivalence mathematically transparent.
Source: Stanford Encyclopedia of Philosophy, “The Equivalence of Mass and Energy”