Point a telescope at a galaxy and measure its spectral lines. Red-shifted — it’s running away. Fine. Now point it somewhere else. Red-shifted. And again. And again. If galaxies were milling about at random you would expect roughly half to be coming towards us. Almost none are. Something is very wrong with the picture of a static universe — and putting it right meant accepting that space itself is stretching.
📘 What you need to know
Redshift: the object is moving away — wavelength increases, frequency decreases
Blueshift: the object is moving towards us — wavelength decreases, frequency increases
Redshift is the fractional increase in wavelength (or decrease in frequency) as source and observer recede from each other
A positive velocity relative to Earth means approaching (f0 > f); a negative velocity means receding (f0 < f)
Nearly all galaxies are red-shifted → the universe is expanding
Space between galaxies stretches. The galaxies themselves are not flying through space
The greater the distance to a galaxy, the greater its redshift — so the faster it recedes
Redshift and blueshift
You met this last page for a single star. Now apply it to the whole sky.
Red-shifted — the light has been stretched to longer wavelengths, towards the red end. The galaxy is receding
Blue-shifted — squashed to shorter wavelengths, towards the blue end. The galaxy is approaching
Redshift, definedthe fractional increase in wavelength(equivalently, the fractional decrease in frequency)caused by the source and observer receding from each other
It is a fraction, so it has no units. Numerically it is the same Δλ/λ you used last page, and for slow galaxies it equals v/c.
Positive and negative velocities
Exam questions ask for a velocity, not just a speed, so the sign carries meaning. Using the frequency form Δf = f0 − f:
Observation
Δf
Velocity relative to Earth
Shift
Motion
f0 > f
Positive
Positive
Blue-shift
Approaching Earth
f0 < f
Negative
Negative
Red-shift
Receding from Earth
Careful — and this catches almost everyone. Work in frequencies and a red-shift gives you a negative velocity. Work in wavelengths and the very same red-shift gives a positive Δλ. The two forms carry opposite signs, because longer wavelength means lower frequency. So never decide the direction from the sign alone. Ask instead: did the wavelength get longer? Then it is receding. Say it in words, and you cannot go wrong.
The evidence: spectra of ever more distant galaxies
Line up the spectra of a lab source, a nearby star, and galaxies at increasing distances. The pattern of lines is identical in every one — same elements, same fingerprints. But the further away the object, the further the whole pattern has slid towards red.
Same three lines every time — that is how we know it is the same element. Only the amount of the shift changes, and it grows with distance.
An expanding universe
Almost every galaxy is receding from us. That sounds suspiciously as though we are at the centre of some cosmic explosion. We aren’t. The resolution is stranger and better: the space between the galaxies is expanding.
The galaxies are not flying through space away from us
Space itself is stretching out between them, carrying them apart
Light travelling through that stretching space gets stretched too, arriving at longer wavelengths
The balloon
Draw dots on a balloon and inflate it. Every dot moves away from every other dot. No dot crawls across the rubber — the rubber grows between them. Stand on any dot you like and all the others recede from you.
The dots (galaxies) never move across the rubber. The rubber (space) stretches between them. Pick any dot as home and every other dot recedes.
The balloon is a two-dimensional universe: everything lives on the surface, and there is no centre on that surface to point to. That is the honest answer to “where did the Big Bang happen?” — everywhere at once. Don’t look for the middle of the balloon; you’d have to leave the surface, and there is nowhere to go.
Distance and redshift go together
Measure the redshift of lots of galaxies and also work out how far away each one is. Plot one against the other and the points fall on a straight line through the origin.
The most distant galaxies are the most red-shifted, and so are running away the fastest. Exactly what a uniformly stretching space predicts.
Space expands
stretches the light
Longer λ redshift
and more so if further
Distant galaxies recede fastest
Why almost, not all? A few nearby galaxies — Andromeda, for one — are blue-shifted. They are close enough that their own motion under gravity beats the gentle expansion of the small amount of space between us. Expansion wins only over very large distances.
🌌 Reading a galaxy’s motion
Compare λ0 with λ (or f0 with f). Longer wavelength, or lower frequency → red-shift.
Get the speed from v = cΔλ/λ, or v = cΔf/f.
Check v/c is small — otherwise the equation doesn’t apply.
State the direction in words: red-shift → receding, blue-shift → approaching.
Bigger redshift? Then it is further away and receding faster.
WE 1
A spectral line has a frequency of 6.170 × 10¹⁴ Hz measured in a laboratory. The same line, in light from a distant galaxy, is measured at 6.152 × 10¹⁴ Hz. Calculate the velocity of the galaxy relative to Earth and state whether it is approaching or receding.
Step 1 — find the change in frequencyΔf = f₀ − f = (6.152 − 6.170) × 10¹⁴Δf = −1.8 × 10¹² HzStep 2 — use Δf/f ≈ v/cv = cΔf/f = (3.00 × 10⁸) × (−1.8 × 10¹²) / (6.170 × 10¹⁴)v = −8.8 × 10⁵ m s⁻¹Step 3 — interpret the sign
Negative velocity, and f₀ < f, so the light is red-shifted.
receding at 8.8 × 10⁵ m s⁻¹Step 4 — check the equation was allowed
v/c = 0.0029, safely << 1.
The minus sign is doing real work here: it tells you the direction. Quote the speed as a positive number, then say “receding”.
WE 2
Galaxy A has a redshift of 4.0 × 10⁻³. Galaxy B has a redshift of 1.2 × 10⁻². Calculate the recession speed of each, and state with a reason which galaxy is further from Earth.
Step 1 — redshift is Δλ/λ, so v = c × redshiftvA = (3.00 × 10⁸) × (4.0 × 10⁻³)vA = 1.2 × 10⁶ m s⁻¹Step 2 — repeat for BvB = (3.00 × 10⁸) × (1.2 × 10⁻²)vB = 3.6 × 10⁶ m s⁻¹Step 3 — which is further?
B has the greater redshift, so it recedes faster. Redshift increases with distance, so
galaxy B is further awayB’s redshift is 3× A’s, so it recedes 3× as fast — and on a straight-line graph through the origin, that puts it 3× as far away.
WE 3
A student says: “Nearly all galaxies are red-shifted, so the Earth must be at the centre of the universe, with everything flying away from us.” Explain what is wrong with this reasoning, using the balloon model.
Step 1 — what redshift really tells us
Every galaxy is receding from every other galaxy, not just from Earth.
Step 2 — the balloon
Dots on an inflating balloon all separate. Stand on any dot and all the others recede from you.
Step 3 — the key idea
The dots do not move across the rubber. The rubber stretches between them — space itself is expanding.
Step 4 — concludeno special centre; every observer sees the same thingAn observer in any galaxy would also record almost everything red-shifted, and would be just as tempted to think they were at the middle.
💡 Top tips
Define redshift as the fractional increase in wavelength due to source and observer receding.
Decide direction from longer or shorter wavelength, in words. Never from a stray minus sign.
Say space expands between galaxies. Saying “galaxies fly through space” loses the mark.
Greater distance → greater redshift → faster recession. That is one fact, stated three ways.
The redshift–distance graph is a straight line through the origin. Draw it that way.
⚠ Common mistakes
Concluding that Earth is at the centre of the universe because everything recedes from us
Saying the galaxies move through space — it is the space between them that stretches
Claiming all galaxies are red-shifted. A few nearby ones, such as Andromeda, are blue-shifted
Muddling the signs: a red-shift gives Δf < 0 but Δλ > 0
Thinking a bigger redshift means a bigger galaxy — it means a more distant, faster-receding one
Reporting a negative velocity as the final speed. Give the magnitude, then say “receding”
Quick recap:Redshift is the fractional increase in wavelength (or decrease in frequency) when a source and observer recede from each other; blueshift is the reverse. Comparing galaxy spectra with a lab reference shows almost every galaxy is red-shifted, and the more distant the galaxy, the greater the shift and the faster it recedes. The explanation is not that we sit at a centre, but that space itself expands between the galaxies — the dots on an inflating balloon.
We have squeezed a great deal out of light. Now let’s go back to sound, where the source moves through a medium at a decent fraction of the wave speed — and the neat approximation Δλ/λ ≈ v/c no longer does the job. Sound needs its own exact equations, one for a moving source and one for a moving observer. That’s the last page of this sub-section.
Redshift and the expanding universe not clicking?
Book a free meeting and we’ll work through redshift calculations, the balloon model and past-paper questions together.
