Theory of Relativity Formula: E = mc², Time Dilation, γ

The best known theory of relativity formula is $E = mc^2$ : a mass m at rest carries an energy equal to that mass times the square of the speed of light. Because c² is about 9.0 × 10¹⁶ m²/s², a tiny mass stands for a huge energy, which is the source of energy in nuclear reactions.

Special relativity adds a second group of results built on the Lorentz factor $\gamma = \frac{1}{\sqrt{1 - v^2/c^2}}$ . Moving clocks run slow, moving rods are shorter along their motion, and momentum grows faster than mv as speed nears c. The mass energy relation is part of the Class 12 nuclei chapter, while γ and its effects appear in college modern physics and some advanced school courses. The modern physics formula sheet lists them next to the photon and atom formulas.

Quick facts

Mass energy
$E = mc^2$
Lorentz factor
$\gamma = \frac{1}{\sqrt{1 - v^2/c^2}}$ , always 1 or more
Time dilation
$\Delta t = \gamma\,\Delta t_0$
Length contraction
$L = \frac{L_0}{\gamma}$
Energy and momentum
$E^2 = (pc)^2 + (mc^2)^2$
Nuclear shortcut
1 u of mass is equivalent to 931.5 MeV
SI unit of E
joule (J), dimensional formula [M¹ L² T⁻²]

Key theory of relativity formulas in one table

Special relativity results
Effect	Formula	What it says
Rest energy	$E_0 = mc^2$	Mass at rest stores energy
Lorentz factor	$\gamma = \frac{1}{\sqrt{1 - v^2/c^2}}$	Sets the size of every effect
Time dilation	$\Delta t = \gamma\,\Delta t_0$	A moving clock ticks slower
Length contraction	$L = L_0/\gamma$	A moving object is shorter along v
Momentum	$p = \gamma m v$	Grows without limit as v nears c
Total energy	$E = \gamma mc^2$	Rest energy plus kinetic energy
Kinetic energy	$K = (\gamma - 1)mc^2$	Reduces to ½mv² at low speed

Two starting rules sit under all of these: the laws of physics are the same in every inertial frame, and light in vacuum has the same speed c for every such observer. Δt₀ and L₀ are the proper time and proper length, measured in the frame where the clock or rod is at rest. The value of c used here is 3.00 × 10⁸ m/s; see the speed of light formula page for how it relates to ε₀ and μ₀.

Mass energy equivalence and nuclear binding

In a nucleus, the mass of the nucleus is less than the total mass of its separate protons and neutrons. The missing mass Δm, the mass defect, is the binding energy divided by c²:

$E_b = \Delta m\,c^2$

Nuclear masses are given in atomic mass units, and 1 u × c² = 931.5 MeV, so multiply Δm in u by 931.5 to get the energy in MeV directly. Fission of heavy nuclei and fusion of light ones both release energy because the products are more tightly bound, with a smaller total mass. The same relation sets the energy of a photon pair made from an electron and a positron, and it links to the general energy formula and mass formula pages.

Worked examples

Example 1: energy in one gram

How much energy is equivalent to 1.0 g of mass?

Convert: m = 1.0 g = 1.0 × 10⁻³ kg.

$E = mc^2 = 1.0 \times 10^{-3} \times (3.00 \times 10^8)^2 = 9.0 \times 10^{13}\ \text{J}$

Example 2: binding energy of helium 4

The mass defect of a helium 4 nucleus is 0.0304 u. Find its binding energy.

$E_b = 0.0304 \times 931.5 = 28.3\ \text{MeV}$

Per nucleon that is 28.3 / 4 ≈ 7.1 MeV.

Example 3: time dilation for a fast particle

A particle has a lifetime of 2.2 μs in its own rest frame and moves at 0.98c in the lab. What lifetime does the lab measure?

$\gamma = \frac{1}{\sqrt{1 - 0.98^2}} = \frac{1}{\sqrt{0.0396}} = 5.03$

$\Delta t = \gamma\,\Delta t_0 = 5.03 \times 2.2 = 11\ \mu\text{s}$

The lab sees the particle live about five times longer.

Example 4: length contraction

A rod 100 m long at rest moves past you at 0.60c. What length do you measure?

$\gamma = \frac{1}{\sqrt{1 - 0.36}} = \frac{1}{0.80} = 1.25$ , so $L = \frac{100}{1.25} = 80\ \text{m}$ .

How to use the formulas step by step

Name the frames. Decide which frame the object is at rest in. That frame measures proper time and proper length.
Find v/c first. Write the speed as a fraction of c, since γ depends only on v²/c².
Compute γ. It is 1 or larger. If you get a value below 1, a step has gone wrong.
Multiply or divide. Times get longer (multiply by γ); lengths along the motion get shorter (divide by γ).
For E = mc², use kilograms. Convert grams or atomic mass units before you multiply, or use 931.5 MeV per u.

The calculator above handles the mass energy relation in SI units, using c² = 9.00 × 10¹⁶ m²/s².

Low speeds and the everyday limit

At everyday speeds v/c is tiny and γ is almost exactly 1. A jet at 300 m/s has v/c = 10⁻⁶, so γ differs from 1 by about 5 parts in 10¹³. That is why Newton's laws work for cars, cricket balls and satellites. Expand $K = (\gamma - 1)mc^2$ for small v and the first term is $\frac{1}{2}mv^2$ , the ordinary kinetic energy formula. Relativity matters for particles in accelerators, cosmic rays, and the timing corrections used in satellite clocks. General relativity, the theory of gravity, goes beyond these special relativity formulas and is outside school syllabi.

Common misunderstandings

Treating E = mc² as kinetic energy. It is the rest energy. A moving object has total energy γmc².
Using c = 3 × 10⁸ but forgetting to square it. c² is 9 × 10¹⁶, not 3 × 10¹⁶ or 9 × 10⁸.
Swapping proper and dilated time. The proper time Δt₀ is always the shortest measured interval.
Contracting the wrong length. Only lengths parallel to the motion shrink; widths and heights stay the same.
Letting v exceed c. No object with mass reaches c, since γ would become infinite.

Keep the dimensions in view: mc² has units kg m²/s², which is a joule.

Frequently asked questions

What is the formula of the theory of relativity?

The best known one is E = mc², the rest energy of a mass. Special relativity also gives time dilation Δt = γΔt₀ and length contraction L = L₀/γ, with γ = 1/√(1 − v²/c²).

What does E = mc² mean?

It means mass and energy are two forms of the same thing. A mass m at rest has energy mc², and losing a mass Δm in a nuclear reaction releases energy Δm c².

What is the Lorentz factor?

It is γ = 1/√(1 − v²/c²). It equals 1 at rest, about 1.25 at 0.6c and grows without limit as v approaches c.

Which unit does E take in E = mc²?

With mass in kilograms and c in m/s, E comes out in joules. In nuclear physics it is often converted to MeV, using 1 u = 931.5 MeV.

Is time dilation real or only apparent?

It is measured directly. Fast moving unstable particles live longer in the lab by the factor γ, exactly as the formula predicts.