Length Contraction

The rest length $L_0$ — also called the proper length — is what you measure with the object sitting still beside you. Anyone moving relative to it measures less, and only along the direction of the motion. The reason goes back to what measuring a length even means. To measure a moving object you must note where its front and its back are at the same moment, and simultaneity is precisely the thing relativity breaks: two observers in relative motion disagree about which distant events happen at the same time. Their disagreement about simultaneity is their disagreement about length. Nothing is being squeezed, and no material property enters — the same factor applies to a steel bar, a cloud of gas and an empty region of space.

The muon is the cleanest illustration because it lets you check both pages against each other. From the ground, the muon lives ten times longer than it should and crosses 15 km of atmosphere. From the muon, its own clock is entirely normal and it is the atmosphere that is rushing past at $\beta = 0.995$: 15 km contracts to $15 \times \sqrt{1 - 0.990} = 1.5$ km, which takes $1500/(0.995 \times 3 \times 10^{8}) = 5.0$ µs to traverse — a couple of lifetimes, so a good fraction survive. Same detector, same count, two irreconcilable-sounding stories that never actually disagree about anything observable. For scale at the other extreme, protons in the LHC at $\gamma \approx 7250$ cross a ring that is 26.7 km around in their own frame's reckoning of about 3.7 m.

This equation is older than relativity and started life as a patch. George FitzGerald in 1889 and Hendrik Lorentz in 1892 both proposed that objects moving through the luminiferous aether physically shrink along their direction of travel, by exactly the amount needed to explain why Michelson and Morley's 1887 interferometer had detected no aether wind. It was an unmotivated fix for one experiment, and it was recognised as such. Einstein's achievement in 1905 was to derive the identical factor from the constancy of the speed of light, with no aether to move through and nothing doing any physical squeezing — turning an ad hoc rescue into an unavoidable consequence. The effect is worked with daily now: heavy-ion collisions at RHIC and the LHC are modelled as collisions between flattened discs rather than spheres, because at those energies the nuclei genuinely are that shape in the laboratory frame.

Three traps. Only the parallel dimension changes — transverse dimensions are untouched, so a passing sphere is shortened in depth and unaltered in width. Second, and this one surprises nearly everybody: a fast object does not look contracted in a photograph. Light reaching the camera from the far side of the object left earlier than light from the near side, and when James Terrell and Roger Penrose worked that through independently in 1959 they found the two effects largely cancel: a sphere photographs as a sphere, apparently rotated rather than flattened. Contraction is what you measure with a set of synchronised clocks, not what an eye or a lens records. Third, no force is involved and no stress appears in the object; it is not being compressed by anything, and the guard on this page — that $L$ can never exceed $L_0$ — is usually tripped by entering the two lengths the wrong way round.