Intuition The ONE core idea"Amount of motion" — how hard a moving thing is to stop — grows faster than ordinary physics predicts when you push toward the speed of light. All of that extra growth is captured by a single number, the Lorentz factor γ
This page assumes nothing . Before you touch the parent formula, every letter in it — and every hidden idea behind it — must have a plain meaning and a picture. We build them one at a time, each resting on the one before.
v v v v v
Definition The speed of lightc c ≈ 3 × 1 0 8 wall no object with mass can reach. It is the same for every observer, no matter how they move. That last sentence is the strange fact all of relativity is built on.
Look at the number line below.
What to notice in the figure: the horizontal axis is speed v left of the magenta wall at c c Extract this: real speeds fill the open interval 0 ≤ v < c c
Why the topic needs it: the whole story of relativistic momentum is what happens as v c
Speed alone is not enough for momentum. A ball thrown left and one thrown right at the same speed have opposite momenta, and in a collision they can cancel.
Definition Velocity as a signed quantityAlong one line, pick a positive direction (say, "to the right"). Then velocity carries a sign : v > 0 v < 0 ≥ 0 speed . In full 3-D we write it as a vector v
Intuition Why sign matters for momentumMomentum p v p > 0 p < 0 p p magnitude ∣ v ∣ γ γ v 2 direction travels along with v p = γ m v
Why the topic needs it: γ v
Definition Beta — speed as a fraction of lightβ = c ∣ v ∣ β in units of light-speed , ignoring direction. If ∣ v ∣ = c β = 1 β = 0.5
Intuition Why bother withβ v / c fraction of light-speed matters for the relativistic effects, we give that fraction its own short name. β 0 1 1
Why the topic needs it: every relativistic formula is cleaner written with β v / c
Before the star of the show, meet the small quantity underneath it.
What to notice in the figure: the orange curve starts pinned at height 1 0 β → 1 Extract this: in the next step we divide 1 ; dividing by a number heading to zero makes the result blow up — that is the seed of every dramatic effect.
Why the topic needs it: it is the raw material of γ γ
Definition Gamma — the "relativistic stubbornness" numberγ = 1 − β 2 1 = 1 − v 2 / c 2 1 γ one divided by the shrinking factor from Step 4. Because we divide 1 1 0 ≥ 1 β → 1
γ 1 − β 2 down to 0 γ reciprocal , so it goes up to ∞ γ 1
What to notice in the figure: the magenta curve hugs the dotted line γ = 1 β β = 0.8 , γ = 1.667 β → 1 Extract this: below β ≈ 0.1
Why the topic needs it: γ is the entire relativistic correction. Momentum = γ × ( Newton’s m v )
The very same γ Lorentz factor and time dilation , and we borrow one result from it in Step 8.
Definition Rest mass (invariant mass)m m the same number for every observer , no matter how fast the particle moves. It is a fixed label, like a serial number.
Common mistake "Mass grows with speed"Old books wrote γ m m γ motion instead. The particle does not get more stuff; it gets harder to accelerate because γ
Why the topic needs it: m γ m
Definition Momentum — amount of motionp = m v ( Newton ) m v ⇒ p sign of v (Step 2), so it points the way the object moves. Units: kg·m/s.
Intuition Why it deserves a name at allIn collisions the total momentum before equals the total after — it is conserved . That conservation (positives and negatives adding up the same before and after) is what makes p p = m v stops being conserved for everyone once speeds get near c γ p = m v Newtonian momentum p = mv .
This is the one genuinely new idea the derivation needs, so we justify it rather than assert it.
Definition Two different clocks
t lab time : time read on a clock sitting still in your laboratory.τ proper time (Greek "tau"): time read on a clock riding along with the moving particle .
d τ light travels at c . Build a clock from that rule: a light pulse bouncing between two mirrors — each round trip is one "tick". For the astronaut carrying it, the pulse goes straight up and down. But to the lab, the mirrors are also drifting sideways, so the same pulse must travel a longer, slanted zig-zag. Since light's speed is fixed at c more lab-seconds per tick . Both observers are watching the identical pulse, so they agree on how many ticks happened — that shared tick-count is the proper time τ τ invariant .
Plain reading: for each tick d τ γ d t
Here is the logical bridge to the parent formula — no hand-waving.
Intuition The covariance argumentA particle's motion through space is a list of positions x t x and t x / t mixed-up quantity that changes shape from frame to frame.
Now the key: proper time τ — it is the invariant we just built. So if we divide the (scrambling) displacement d x d τ m d x / d τ kind of object in everyone's coordinates. When a rule ("total momentum is unchanged") is written with such objects, and it holds in one frame, the Lorentz transformation carries it unchanged into every other frame. That is exactly what we needed: conservation valid for all observers.
Newton's p = m d x / d t d t d t d τ γ Energy-momentum four-vector .)
Now every symbol is earned, so the parent formula reads as plain English:
Mnemonic Gamma Mounts the Velocityp = γ m v γ γ ≥ 1 top of the fraction 1/ …
Node labels use full names (matching the sections above).
Speed v metres per second
Direction sign of velocity
Shrink factor root of 1 minus beta squared
Newton momentum p equals m v
Proper time tau moving clock invariant
Time dilation dt equals gamma d-tau
Covariance clean transform
Relativistic momentum gamma m v
Cover the right side and test yourself.
What is c The universal speed limit, ≈ 3 × 1 0 8
Why does momentum need a sign / direction? So opposite motions can cancel — that is how total momentum is conserved in a head-on collision.
Define β β = ∣ v ∣/ c 0 ≤ β < 1
What does 1 − β 2 v 0 c It slides from 1 0
Write the Lorentz factor γ Value of γ v → c γ = 1 γ → ∞
Is γ = 1 − β 2 No — that is the shrinking factor; γ
What is rest mass m The invariant amount of stuff; it is the same in every frame (does not grow).
Write Newtonian momentum and say what it measures. p = m v
Why is proper time τ It counts the particle's own light-clock ticks, and everyone watches the identical pulse, so they agree on the tick-count.
Derive d t = γ d τ Pythagoras on the slanted light-pulse path
( c d t ) 2 = ( c d τ ) 2 + ( v d t ) 2 gives
d t / d τ = 1/ 1 − β 2 = γ .
Why does dividing by d τ d t d τ is invariant, so
m d x / d τ transforms as one clean object — a conservation law true in one frame stays true in all.
In p = γ m v γ m