Length contraction

[2][3] Although both FitzGerald and Lorentz alluded to the fact that electrostatic fields in motion were deformed ("Heaviside-Ellipsoid" after Oliver Heaviside, who derived this deformation from electromagnetic theory in 1888), it was considered an ad hoc hypothesis, because at this time there was no sufficient reason to assume that intermolecular forces behave the same way as electromagnetic ones.

So he had to introduce another ad hoc hypothesis: non-electric binding forces (Poincaré stresses) that ensure the electron's stability, give a dynamical explanation for length contraction, and thus hide the motion of the stationary aether.

Such effects would include optical effects in transparent media, such as optical rotation[6] and induction of double refraction,[7] and the induction of torques on charged condensers moving at an angle with respect to the aether.

[8] Lorentz considered local time not to be "real"; rather, it represented an ad hoc change of variable.

[9]: 51, 80 Impressed by Lorentz's "most ingenious idea", Poincaré saw more in local time than a mere mathematical trick.

[11] Hermann Minkowski gave the geometrical interpretation of all relativistic effects by introducing his concept of four-dimensional spacetime.

[12] First it is necessary to carefully consider the methods for measuring the lengths of resting and moving objects.

[13] Using this method, the definition of simultaneity is crucial for measuring the length of moving objects.

[14] In Newtonian mechanics, simultaneity and time duration are absolute and therefore both methods lead to the equality of

The deviation between the measurements in all inertial frames is given by the formulas for Lorentz transformation and time dilation (see Derivation).

This can be vividly illustrated using symmetric Minkowski diagrams, because the Lorentz transformation geometrically corresponds to a rotation in four-dimensional spacetime.

[15][16] Magnetic forces are caused by relativistic contraction when electrons are moving relative to atomic nuclei.

The magnetic force on a moving charge next to a current-carrying wire is a result of relativistic motion between electrons and protons.

[17][18] In 1820, André-Marie Ampère showed that parallel wires having currents in the same direction attract one another.

To the static proton's frame of reference, the electrons are moving and contracted, resulting in the same imbalance.

In addition, even in such a non-co-moving frame, direct experimental confirmations of length contraction are hard to achieve, because (a) at the current state of technology, objects of considerable extension cannot be accelerated to relativistic speeds, and (b) the only objects traveling with the speed required are atomic particles, whose spatial extensions are too small to allow a direct measurement of contraction.

However, there are indirect confirmations of this effect in a non-co-moving frame: In 1911 Vladimir Varićak asserted that one sees the length contraction in an objective way, according to Lorentz, while it is "only an apparent, subjective phenomenon, caused by the manner of our clock-regulation and length-measurement", according to Einstein.

[26][27] Einstein published a rebuttal: The author unjustifiably stated a difference of Lorentz's view and that of mine concerning the physical facts.

He presented the following thought experiment: Let A'B' and A"B" be the endpoints of two rods of the same proper length L0, as measured on x' and x" respectively.

Let them move in opposite directions along the x* axis, considered at rest, at the same speed with respect to it.

Einstein pointed out that length A*B* is shorter than A'B' or A"B", which can also be demonstrated by bringing one of the rods to rest with respect to that axis.

Another famous paradox is the Ehrenfest paradox, which proves that the concept of rigid bodies is not compatible with relativity, reducing the applicability of Born rigidity, and showing that for a co-rotating observer the geometry is in fact non-Euclidean.

Length contraction refers to measurements of position made at simultaneous times according to a coordinate system.

It was shown by several authors such as Roger Penrose and James Terrell that moving objects generally do not appear length contracted on a photograph.

[30] For instance, for a small angular diameter, a moving sphere remains circular and is rotated.

Meanwhile, the proper length of this object, as measured in its rest frame S', can be calculated by using the Lorentz transformation.

: Likewise, the same method gives a symmetric result for an object at rest in S': Length contraction can also be derived from time dilation,[34] according to which the rate of a single "moving" clock (indicating its proper time

, the above procedure would give Additional geometrical considerations show that length contraction can be regarded as a trigonometric phenomenon, with analogy to parallel slices through a cuboid before and after a rotation in E3 (see left half figure at the right).

Right: the world slab of a moving thin plate in Minkowski spacetime (with one spatial dimension suppressed) E1,2, which is a boosted cuboid.

Indeed, special relativity largely comes down to studying a kind of noneuclidean trigonometry in Minkowski spacetime, as suggested by the following table: