Length contraction

Suppose that a rod has is length

L

measured on a rest frame

S

(or maybe even better: two identical rulers were manufactured, and one is taken on a spaceship, a bit like the twin paradox).

Question: what is the length

L^{'}

than an observer in frame

S^{'}

moving relative to

S

as speed

v

observe the rod to be?

The key idea is that there are two events to consider in each frame, which we call 1 and 2:

the left end of the rod is an observation event at a given position at a given time: $x_{1}$ and $t_{1}$ for $S$ or $x_{1}^{'}$ and $t_{1}^{'}$ for $S^{'}$
the right end of the rod is an observation event at a given position at a given time : $x_{2}$ and $t_{2}$ for $S$ or $x_{2}^{'}$ and $t_{2}^{'}$ for $S^{'}$

Note that what you visually observe on a photograph is a different measurement to the more precise/easy to calculate two event measurement. On a photograph, it seems you might not even see the contraction in some cases as mentioned at en.wikipedia.org/wiki/Terrell_rotation

Measuring a length means to measure the

x_{2} - x_{1}

difference for a single point in time in your frame (

t 2 = t 1

So what we want to obtain is

x_{2}^{'} - x_{1}^{'}

for any given time

t^{'} 2 = t^{'} 1

In summary, we have:

L L^{'} = x_{2} = x_{2}^{'} - x_{1} - x_{1}^{'} t_{2}^{'} = t_{1}^{'}

(1)

By plugging those values into the Lorentz transformation, we can eliminate

t_{2} an d t_{1}

, and conclude that for any

t_{2}^{'} = t_{1}^{'}

, the length contraction relation holds:

L^{'} = \frac{L}{γ}

(2)

The key question that needs intuitive clarification then is: but how can this be symmetric? How can both observers see each other's rulers shrink?

And the key answer is: because to the second observer, the measurements made by the first observer are not simultaneous. Notably, the two measurement events are obviously spacelike-separated events by looking at the light cone, and therefore can be measured even in different orders by different observers.

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A rod of rest length

L_{0}

moving parallel to its length has simultaneous laboratory length

L = L_{0} / γ

, where

γ

is the Lorentz factor. Here

L_{0}

is the rod's proper length measured in its rest frame. Transverse lengths are unchanged by the corresponding Lorentz transformation.

For a rod at rest in

S^{'}

with endpoints

x^{'} = 0

and

x^{'} = 10 m

, let

S^{'}

move at

v = 0.6 c

relative to the laboratory

S

. Record both endpoint positions at laboratory time

t = 0

. The Lorentz transformation then gives

x^{'} = 1.25 x

, so their recorded positions are

0

and

8 m

: the laboratory length is

10/1.25 = 8 m

. This is an operational definition: synchronized laboratory detectors register the endpoints at the same laboratory time. A photograph collects light emitted at different times and need not show this length. ^[ref].

For those same two endpoint spacetime events,

Δ t^{'} = - 1.25 (0.6 c) (8 m) / c^{2} = - 6 m / c

. Thus the front-end measurement occurs earlier in

S^{'}

by about

20 ns

. Length contraction compares simultaneous endpoint measurements in each frame, which select different pairs of spacetime events; it does not mean that the rod has been compressed in its own rest frame.

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Length contraction

Terrell rotation

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Length contraction

Terrell rotation

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