The Galilean equations tell you how one observer’s measurements turn into another’s — but they quietly assume everyone shares the same clock and ruler. Once we accept that light travels at c for everyone, that assumption has to go. The Lorentz transformations are the upgraded, high-speed version: same idea as before, but with one extra ingredient — the gamma factor — that stretches time and shrinks space.
📘 What you need to know
Lorentz transformations correct the Galilean ones for speeds close to c
The correction is a scaling factor, the Lorentz factorγ = 1 ÷ √(1 − v2/c2)
Because v < c always, γ is always greater than 1
Position: x′ = γ(x − vt) and x = γ(x′ + vt′)
Time: t′ = γ(t − vx/c2) and t = γ(t′ + vx′/c2)
Time is now different between frames (t ≠ t′) — the big break from Galilean physics
Sideways coordinates are unchanged: y = y′, z = z′
The Lorentz factor
Everything on this page hangs off one quantity: the Lorentz factor, written γ (gamma). It’s the scaling factor you multiply the old Galilean equations by to make them correct at high speed.
The Lorentz factorγ = 1 ÷ √(1 − v2/c2)
Look at what it does. Since nothing travels faster than light, v is always less than c, so v2/c2 is a fraction less than 1, the square root is less than 1, and dividing 1 by it always gives a number bigger than 1. At everyday speeds γ is a whisker above 1 (so nothing seems to change), but as v climbs toward c it rockets upward.
The Lorentz factor barely moves for most speeds, then shoots toward infinity as v nears c. This steep rise is why relativity only bites at very high speeds.
WE 1
A spacecraft moves at 0.80c relative to Earth. Calculate the Lorentz factor γ for this speed.
Step 1 — use γ = 1 ÷ √(1 − v²/c²)
with v = 0.80c, the c² cancels:
γ = 1 ÷ √(1 − 0.80²) = 1 ÷ √(1 − 0.64)Step 2 — evaluateγ = 1 ÷ √0.36 = 1 ÷ 0.60γ = 1.67Because v is given as a fraction of c, you never need to plug in 3×10⁸ — the c² cancels. Always check γ > 1.
The transformation equations
Now we simply take the Galilean equations and multiply by γ — with one crucial addition. Time is no longer the same in both frames: it picks up a term too. Picture Person E at rest on Earth (frame S) and Person F flying past in a rocket at speed v close to c (frame S′), both timing a distant flash.
Person E (Earth, frame S) and Person F (rocket, frame S′) both record the same flash. In Lorentz physics they disagree not only on where it happened but also on when.
The full set is just the Galilean transformations wearing a γ and, for time, an extra vx/c2 term. Both directions are on your data booklet:
To go the other way (from the moving frame back to the stationary one), you flip the sign of v — because from F’s point of view, E is the one moving, in the opposite direction:
In Earth’s frame S, an event happens at x = 6.0 × 108 m and t = 1.5 s. A ship (frame S′) moves at 0.60c. Find the position x′ of the event in the ship’s frame. (Take c = 3.0 × 108 m s−1.)
Step 1 — find γ for 0.60cγ = 1 ÷ √(1 − 0.60²) = 1 ÷ √0.64 = 1.25Step 2 — use x′ = γ(x − vt), with v = 0.60c = 1.8×10⁸x′ = 1.25 × (6.0×10⁸ − 1.8×10⁸ × 1.5)x′ = 1.25 × (6.0×10⁸ − 2.7×10⁸) = 1.25 × 3.3×10⁸x′ = 4.1 × 10⁸ mThe ship sees the event at a different distance than Earth does — that’s the γ correction at work.
WE 3
For the same event (x = 6.0 × 108 m, t = 1.5 s, ship at 0.60c), find the timet′ the ship records for it.
Step 1 — use t′ = γ(t − vx/c²), γ = 1.25vx/c² = (1.8×10⁸ × 6.0×10⁸) ÷ (3.0×10⁸)²= 1.08×10¹⁷ ÷ 9.0×10¹⁶ = 1.2 sStep 2 — substitutet′ = 1.25 × (1.5 − 1.2) = 1.25 × 0.3t′ = 0.38 sEarth says 1.5 s, the ship says 0.38 s — they genuinely disagree on when. This is what “time is relative” means.
Two things to burn in. First: when v is given as a multiple of c, the c2 inside γ always cancels — never type 3×10⁸ into the gamma formula. Second: the equations are on your data booklet, but the sign flip for going S′→S is not spelled out, so you must remember it yourself. Get the sign wrong and everything after it is wrong.
Galilean vs Lorentz
It’s worth seeing the two side by side. The Lorentz column is just the Galilean column with a γ in front — and, for time, that extra term. As v shrinks far below c, γ → 1 and the whole Lorentz column collapses back into the familiar Galilean one.
Quantity
Lorentz (v ≈ c)
Galilean (v « c)
position
x′ = γ(x − vt)
x′ = x − vt
time
t′ = γ(t − vx/c2)
t′ = t
sideways
y′ = y, z′ = z
y′ = y, z′ = z
The headline difference is time. In Galilean physics t′ = t — everyone’s clocks agree. In Lorentz physics they don’t: time is woven together with space, and two observers moving relative to each other will read different times for the same event. That single change is the doorway to time dilation and length contraction.
WE 4
A rocket of proper length 120 m moves at 0.80c relative to the ground. A probe is fired from the back of the rocket at 0.50c relative to the rocket. Find the time for the probe to reach the front, as measured by (a) an observer at rest in the rocket, and (b) an observer at rest on the ground.
(a) In the rocket (its own frame, S′)
the probe crosses the full 120 m at 0.50c = 1.5×10⁸ m/s
t′ = L ÷ u′ = 120 ÷ 1.5×10⁸t′ = 8.0 × 10⁻⁷ s(b) On the ground (S) — need γ for 0.80c
γ = 1.67, and the event spans x′ = 120 m in the rocket
t = γ(t′ + vx′/c²)= 1.67 × (8.0×10⁻⁷ + [2.4×10⁸ × 120]/9.0×10¹⁶)= 1.67 × (8.0×10⁻⁷ + 3.2×10⁻⁷) = 1.67 × 1.12×10⁻⁶t = 1.9 × 10⁻⁶ sThe ground observer measures a longer time than the rocket observer — exactly what you’d expect once time is relative.
Which way? Going S→S′ uses minus signs; S′→S flips them to plus.
Pick the right equation — position or time — and match primed/unprimed quantities.
Keep v as a fraction of c. The c2 cancels in γ; only put numbers for c where the equation needs them (the vx/c2 term).
Sanity-check. Moving-frame time should come out longer, distances shorter.
💡 Top tips
Always check γ > 1. If you get less than 1, you’ve slipped up.
The c2 cancels. With v in terms of c, never enter 3×10⁸ into the gamma factor.
Remember the sign flip. The booklet gives the equations, but you must switch −/+ for the reverse direction.
Never write c′. There’s no such thing — c is the same in every frame.
Low speed = Galilean. When v « c, γ → 1 and Lorentz becomes Galilean again.
Quick recap: Lorentz transformations are the Galilean ones multiplied by γ = 1÷√(1−v2/c2), which is always > 1. Position uses x′ = γ(x − vt) and time uses t′ = γ(t − vx/c2). Unlike Galilean physics, time now differs between frames — the key to what comes next.
⚠ Common mistakes
Putting 3×108 into γ when v is already given as a fraction of c
Forgetting the sign flip when transforming from the moving frame back to the stationary one
Dropping the vx/c2 term and treating time as if t′ = t (that’s Galilean, not Lorentz)
Getting γ < 1 and not spotting the error — it must always exceed 1
Writing c′ for the speed of light in the moving frame — c never changes
You’ve now got the master key to high-speed relativity. Everything strange and famous flows straight out of these equations. Next we use them to transform velocities properly — the relativistic fix for that broken 1.2c answer — and then unpack the headline effects: time dilation, length contraction, and the space-time interval. The γ factor you just met will be in every one of them.
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