Mass-Energy Equivalence

Equivalent forms

E^2 = (pc)^2 + (mc^2)^2

m = E/c^2

Three symbols connect matter and energy across all of physics.

Symbol	Name	SI unit	Dimension	Range
$E$	Energyoutput Rest energy equivalent of the mass	J	M·L²·T⁻²	1e-15 – 100000000000000000000
$m$	Rest mass Invariant mass of the object	kg	M	1e-30 – 1000
$c$	Speed of light Speed of light in vacuum (constant)	m/s	L·T⁻¹	299792458 – 299792458

Unit systems

Symbol	SI	CGS	Imperial
$E$	J	erg	ft·lbf
$m$	kg	g	lb
$c$	m/s	cm/s	ft/s

Where it holds

Universally valid in special and general relativity for any closed system. Rest-mass form applies to objects in their rest frame; for moving particles use the full relation

E^{2} = (pc)^{2} + (mc^{2})^{2}

.

Dimensional analysis

$[E] = [m]\cdot [c]^{2} \to M\cdot (L\cdot T^{-1})^{2} = M\cdot L^{2}\cdot T^{-2} \checkmark$ (energy)

Discovery

Albert Einstein · 1905

Derived in Einstein's 'Annus Mirabilis' paper on special relativity, showing that mass and energy are interchangeable.

Try this

How much energy is locked inside a single paperclip?

A paperclip has a mass of about 1 gram. If all of its mass were converted to energy, how much energy would be released?

Research status: stable

Real-world applications

Nuclear power plants — uranium fission releases $\sim 0.1$ % of fuel mass as energy.
Stellar fusion — the Sun converts $\sim 4$ million tons of mass to energy every second.
PET medical imaging — positron–electron annihilation produces 511 keV gamma rays.
Particle accelerators — LHC creates new massive particles from collision kinetic energy.

Common misconceptions

$E = mc^{2}$ describes only rest energy; the full energy of a moving particle includes kinetic energy via the Lorentz factor.
Mass is not literally 'converted' to energy — rather, the system's invariant mass changes when energy leaves (e.g., a hot object weighs slightly more than a cold one).
It does not say 'matter and energy are the same thing'; mass is one specific property and energy is another, related by $c^{2}$ .

Experimental verification

Confirmed countless times: nuclear binding energies match the mass defect

\Delta m\cdot c^{2} to

<0.01%. Cockcroft–Walton (1932) split lithium-7 with protons and verified the energy released equals the mass deficit. Pair production

e^{+}e^{-}

from photons (Anderson 1932) and PET scans rely on this daily.

Derivation

Starting from the relativistic momentum

p = \gamma mv

and total energy

E = \gamma mc^{2}

, compute

E^{2} - (pc)^{2} = \gamma ^{2}m^{2}c^{4}(1 - v^{2}/c^{2}) = m^{2}c^{4}

, giving

E^{2} = (pc)^{2} + (mc^{2})^{2}

.

Setting

p = 0

(rest frame) yields

E = mc^{2}

.

Limiting cases

m \to 0

⟶

E \to 0

A massless object has no rest energy (photons carry only kinetic/momentum energy via

E = pc)

.

v \to c

(relativistic)⟶

E \to \infty

Total energy

E = \gamma mc^{2}

diverges as a massive object approaches the speed of light — that is why c is unreachable.

v ≪ c⟶

E \approx mc^{2} + \tfrac{1}{2}mv^{2}

Low-velocity expansion

of \gamma

recovers Newtonian kinetic energy on top of the rest energy.

What if…

What if we could convert all of a 70 kg human's mass to energy?

$E = 70 \times c^{2} \approx 6.3\times 10^{18} J$ — about 1500 megatons of TNT, roughly $30\times the$ largest nuclear weapon ever detonated.

What if c were only 1000 m/s?

Mass-energy equivalence would be $9\times 10^{10}$ times weaker. Stars couldn't fuse, atoms couldn't bind, and chemistry would dwarf nuclear energy — the universe would be unrecognizable.

What if a hot cup of coffee weighed itself?

It would be heavier $by \Delta m = \Delta E/c^{2}$ . Heating 250 g of water by 80 K adds $\sim 84\,\mathrm{kJ}$ , increasing mass $by \sim 9\times 10^{-13} kg$ — far below any scale's resolution but real.

1

Energy in 1 gram of mass

Given ·

m:: 0.001
c:: 299792458

Find · E

Steps

$E = mc^{2}$ with $m = 1\,\mathrm{g} = 10^{-3} kg$
$c^{2} = (2.998\times 10^{8})^{2} \approx 8.99\times 10^{16} m^{2}/s^{2}$
$E = 10^{-3} \times 8.99\times 10^{16} = 8.99\times 10^{13} J$
Equivalent $to \sim 21$ kilotons of TNT — roughly the Hiroshima yield.

Result ·

E = mc^{2} = 0.001 \times (3\times 10^{8})^{2} \approx 9\times 10^{13} J \approx 21

kilotons of TNT

2

Rest energy of an electron

Given ·

m:: 9.1093837e-31
c:: 299792458

Find · E (in MeV)

Steps

$E = (9.109\times 10^{-31})(8.988\times 10^{16}) = 8.187\times 10^{-14} J$
Convert: $1\,\mathrm{eV} = 1.602\times 10^{-19} J \to E \approx 5.11\times 10^{5} eV = 0.511\,\mathrm{MeV}$
This is the famous electron rest-mass energy used everywhere in particle physics.

Result ·

E = mc^{2} \approx 8.187\times 10^{-14} J \approx 0.511\,\mathrm{MeV}

Lorentz Factor→De Broglie Wavelength→