Length Contraction

Time dilation says a moving clock reads slow. Its mirror image, length contraction, says a moving ruler reads short — and the two turn out to be two sides of exactly the same relativistic coin, so closely linked that the same cosmic-ray muon result can be explained fully by either one on its own.

Time dilation's spatial counterpart

Time dilation shows that a moving clock is measured to run slow. Length contraction is its direct spatial counterpart: a moving object is measured to be shorter, along its direction of travel, than it would be measured at rest. Both effects come from exactly the same underlying cause — the invariance of the speed of light forcing space and time to behave differently from Newtonian expectation — and both are controlled by the same Lorentz factor, $\gamma = 1/\sqrt{1-v^2/c^2}$.

Proper length, $l_0$, is defined as the length of an object measured in the frame in which the object is at rest — its own, unambiguous "true" length, measured by a ruler that isn't moving relative to it. Measured from any other frame, moving relative to the object at speed $v$, the object's length along the direction of motion is found to be shorter:

Since $\gamma \geq 1$ always (exactly as established for time dilation), $l \leq l_0$ always — every moving object is measured as contracted along its direction of travel, relative to its own rest-frame (proper) length, and never expanded. Two important restrictions are worth being precise about: contraction only happens along the direction of motion — a spacecraft flying past you is measured as shorter in its direction of flight, but its height and width (perpendicular to the motion) are completely unaffected — and, just as with time dilation, the effect is utterly negligible at everyday speeds, only becoming significant as $v$ approaches a sizeable fraction of $c$.

The muon result, seen from the muon's own frame

The clearest illustration of how time dilation and length contraction give the same physical answer from two different points of view is the cosmic-ray muon result from the previous lesson. Muons produced roughly $15\,\text{km}$ up in the atmosphere, travelling at $v=0.995c$ ($\gamma\approx10.0$), are detected reaching the ground in far greater numbers than their $2.2\,\mu\text{s}$ rest-frame lifetime should classically allow. The previous lesson explained this from the Earth's frame, using time dilation: the muon's internal decay clock runs ten times slower as measured from Earth, so on average it survives roughly ten times longer than its proper lifetime, long enough for a sizeable fraction to complete the crossing.

Length contraction gives exactly the same physical outcome, explained entirely from the muon's own point of view instead. From the muon's own frame, nothing unusual happens to its own decay clock at all — its lifetime is a perfectly ordinary $2.2\,\mu\text{s}$, just as it would be for a muon at rest in a laboratory. What's different, from the muon's perspective, is the distance it has to cross: the $15\,\text{km}$ depth of atmosphere, which is the proper length $l_0$ measured by an observer at rest on Earth, is measured by the muon as length-contracted:

A mere $1.5\,\text{km}$ can be crossed at nearly the speed of light in a small fraction of the muon's ordinary $2.2\,\mu\text{s}$ lifetime, so a healthy fraction of muons complete the "shortened" journey before decaying.

Mark as complete