Intuition The ONE core ideaA moving object carries energy in two independent ways at once : energy just for existing (its rest energy) and energy for moving (its momentum). These two combine like the two legs of a right triangle, and the hypotenuse is the total energy — that is the entire meaning of E 2 = ( p c ) 2 + ( m c 2 ) 2
This page assumes nothing . If the parent note (topic note) used a symbol, we build it here from the ground up.
Before any physics, one piece of geometry powers the whole topic.
Definition Right triangleA right triangle is a triangle with one square corner (a 9 0 ∘ legs . The long slanted side opposite the corner is the hypotenuse .
Picture two arrows starting from the same point: one pointing straight up, one pointing straight right. They are perpendicular (at 9 0 ∘
Intuition What "perpendicular" buys usThe reason the two energies add as squares (not just add) is that they are perpendicular — they point in different directions , so one growing does not shrink the other. The rest-energy leg stays fixed while the motion leg grows. That is why m c 2 E
v v v
c c 3 × 1 0 8 speed limit — a wall no massive object can cross. Every relativity formula compares v c Why the topic needs it: c conversion factor between mass and energy. It turns a mass (kilograms) into an energy (joules) through m c 2
m m measured when it sits still . Picture the reading on a scale when the object is at rest next to you. We call it rest mass or invariant mass because everyone, no matter how they move, agrees on this number.
Why "rest" matters: older books talked about mass "increasing" with speed — the topic deliberately avoids that. Here m
p p = m v quantity of motion .
Every relativistic effect depends on how close your speed is to the light speed , i.e. on the fraction v / c
v / c v / c 0 1 v / c = 0 v / c = 1
Now a tool enters. Why do we need a new symbol at all? Because at high speed, time, length, energy and momentum all get stretched by the same factor, and it saves us writing 1 − v 2 / c 2 1 γ
Let's read this dial in every case — the reader must never meet a value we didn't show:
γ Why it feels right: stretching sounds like it could shrink too.
The fix: v 2 / c 2 ≥ 0 1 − v 2 / c 2 ≤ 1 ≥ 1 γ never dips below 1
Now we assemble the two quantities the triangle needs.
E E all the energy an object has: the energy of existing plus the energy of moving, in one number. Units: joules (J). Relativity writes
E = γ m c 2 . m c 2 γ γ = 1 m c 2
m c 2 fixed vertical leg of our triangle — it never changes because m c
p c p c energy units (joules), so it can sit on the same triangle as E m c 2 growing horizontal leg : at rest p = 0 Why multiply by c Momentum alone (kg·m/s) can't be compared to energy (joules). One factor of c
Here is the master picture that all these symbols were built for:
Intuition Putting it together
Vertical leg m c 2 fixed (rest energy).
Horizontal leg p c grows with speed (motion energy).
Hypotenuse E ( p c ) 2 + ( m c 2 ) 2 E m c 2
The worked examples switch between two energy units, so build both now.
Definition electron-volt (eV)A tiny energy unit natural for particles: 1 eV = 1.6 × 1 0 − 19 = 0.511 = 938 Convert J → eV: divide by 1.6 × 1 0 − 19 Convert eV → J: multiply by 1.6 × 1 0 − 19 1 MeV = 1 0 6
total energy E = gamma m c squared
Pythagoras h squared = a squared + b squared
E squared = pc squared + mc squared squared
Test yourself — reveal only after you answer.
What are the two legs and the hypotenuse of the energy triangle? legs p c m c 2 E
State Pythagoras' theorem. h 2 = a 2 + b 2 a , b h
What is c the speed of light, the cosmic speed limit, ≈ 3 × 1 0 8
Why is m it is measured at rest and every observer agrees on it; it never changes with speed
Write the Lorentz factor γ What is γ v = 0 v → c γ = 1 γ → ∞ v → c
Why can γ v 2 / c 2 ≥ 0 1 − v 2 / c 2 ≤ 1 ≥ 1
Show the identity γ 2 = c 2 / ( c 2 − v 2 ) from γ 2 = 1/ ( 1 − v 2 / c 2 ) c 2
Why multiply momentum p c to convert kg·m/s into energy units (joules) so it fits on the triangle
Write total energy two equivalent ways. E = γ m c 2 and
E = ( p c ) 2 + ( m c 2 ) 2 How many joules is one electron-volt? 1 eV = 1.6 × 1 0 − 19
Convert an energy in joules to eV. divide by 1.6 × 1 0 − 19
Special Relativity — where c Lorentz Factor — the stretch dial γ Photon Momentum — the m = 0 Relativistic Kinetic Energy — the leftover ( γ − 1 ) m c 2 Four-Momentum — the deeper object whose "length" is m c 2 Nuclear Binding Energy — where mass turning into energy has real consequences 
