Thought lab · Light & relativity
A world with slow light
Imagine a town where light travels no faster than a bicycle. What Einstein's theory predicts for spaceships near the speed of light would happen there on your way to the bakery. Ride through it and see what changes – and what doesn't.
An old idea: Mr Tompkins rides a bike
The physicist George Gamow invented a bank clerk called Mr Tompkins, who falls asleep during a lecture on relativity and dreams of a town where the speed of light is only about 10 miles per hour (some 16 km/h). There, passing cyclists are noticeably shortened. Gamow's stories first appeared from 1938 in the British magazine Discovery and were published as a book, Mr Tompkins in Wonderland, in 1939/1940 (the sources give both years).
In 2012 the MIT Game Lab turned the same idea into the free game A Slower Speed of Light: you collect glowing orbs, and each one slows light down further until it is barely faster than walking. The game shows not only shortened lengths and slow clocks, but also what you would actually see, such as shifted colours.
This page borrows the idea in a simple 2D form. It keeps two things strictly apart: what you measure and what you see.
- G. Gamow, Mr Tompkins in Wonderland, Cambridge University Press 1939/1940; first as a serial in Discovery (1938). Wikipedia: Mr Tompkins; Wikipedia: George Gamow.
- MIT Game Lab: A Slower Speed of Light (2012); Wikipedia: A Slower Speed of Light.
Ride it yourself
Choose how fast light is in the town, pick who is measuring, and pedal. The town clocks stand on posts every 20 metres; your own clock rides above your head.
What is measured
Lengths shrink in the direction of travel. In your frame of reference the town is moving, so you measure every house shorter by the Lorentz factor γ: at γ = 2, half as long. The height stays the same. The town, in its own frame, measures exactly the same about you and your bike. Neither is wrong; length depends on who is measuring.
Moving clocks run slow. The town measures your clock running slow. You, in turn, measure each town clock running slow. That sounds like a contradiction, but it is not, because of the third effect:
“At the same time” depends on the frame. The town clocks were set to agree in the town's frame. In your frame they don't: the clocks ahead of you are set ahead, those behind you are set back, and the further away, the more. So although each town clock runs slow for you, every clock you meet shows a later time than the one before. Whenever you pass a town clock, it is ahead of your own clock – and both of you agree on that, because it is a comparison in one place.
When you brake, the lengths return to normal. The time difference stays: you have aged less than the town. This is the same effect as in the twin paradox.
Formulas For the curious: the calculation
With speed v and speed of light c, β = v/c and the Lorentz factor is γ = 1/√(1 − β²).
- Length in the direction of travel: L = L₀/γ (L₀ = length at rest).
- Clocks: while your clock advances by Δτ, the town clock beside you advances by γ·Δτ.
- Simultaneity: in your frame, a town clock that, as measured by the town, stands a distance d ahead of you shows v·d/c² more than the town clock beside you.
Example: c = 30 km/h, v = 26 km/h. Then β = 0.867, β² = 0.751, √(1 − 0.751) = 0.499 and γ = 2.00. A 10-metre house is measured at about 5 m. After one minute on your clock, the town clock beside you shows two minutes.
More on this in the chapter Special relativity · the twin journey
- Wikipedia: Length contraction; Wikipedia: Time dilation; Wikipedia: Relativity of simultaneity.
- A. Einstein, „Zur Elektrodynamik bewegter Körper“, Annalen der Physik 17 (1905), § 4.
What you would see is something else
Your eye does not measure; it collects light. Light that reaches your eye at the same moment left different parts of the town at different times. In a town with slow light this travel time is huge, and it distorts the picture in its own way. In 1959 Roger Penrose and James Terrell showed independently of each other that you would not simply see a fast-moving object shortened; instead it appears rotated. Gamow's cyclists are therefore drawn as they are measured, not as you would see them.
Two effects of seeing are easy to calculate, and you can find them in the experiment under “What you would see”: colours shift (Doppler effect) – light from ahead becomes bluer, light from behind redder, at high speed right out into the invisible ultraviolet and infrared. And the world seems to crowd together ahead of you (aberration): a lamp that is exactly beside you appears in front of you.
And in our world?
Real light travels 299,792.458 km per second, or about 1,079 million km/h. If you ride a bike at 25 km/h for an hour, your clock falls behind by roughly one picosecond (a millionth of a millionth of a second). The effect is there, just far too small to notice. It only becomes measurable with very precise atomic clocks or at very high speeds – for example with fast particles from space or with the clocks in GPS satellites.