Length contraction can be derived in several ways:
Known moving length In an inertial reference frame S, let x_{1} and x_{2} denote the endpoints of an object in motion. In this frame the object's length L is measured, according to the above conventions, by determining the simultaneous positions of its endpoints at t_{1}=t_{2}. Meanwhile, the proper length of this object, as measured in its rest frame S', can be calculated by using the Lorentz transformation. Transforming the time coordinates from S into S' results in different times, but this is not problematic, since the object is at rest in S' where it does not matter when the endpoints are measured. Therefore, the transformation of the spatial coordinates suffices, which gives: :x'_{1}=\gamma\left(x_{1}-vt_{1}\right)\quad\text{and}\quad x'_{2}=\gamma\left(x_{2}-vt_{2}\right) \ \ . Since t_1 = t_2, and by setting L=x_{2}-x_{1} and L_{0}^{'}=x_{2}^{'}-x_{1}^{'}, the proper length in S' is given by {{NumBlk|:|L_{0}^{'}=L\cdot\gamma \ \ . |}} Therefore, the object's length, measured in the frame S, is contracted by a factor \gamma: {{NumBlk|:|L=L_{0}^{'}/\gamma \ \ . |}} Likewise, according to the principle of relativity, an object that is at rest in S will also be contracted in S'. By exchanging the above signs and
primes symmetrically, it follows that {{NumBlk|:|L_{0}=L'\cdot\gamma \ \ . |}} Thus an object at rest in S, when measured in S', will have the contracted length {{NumBlk|:|L'=L_{0}/\gamma \ \ . |}}
Known proper length Conversely, if the object rests in S and its proper length is known, the simultaneity of the measurements at the object's endpoints has to be considered in another frame S', as the object constantly changes its position there. Therefore, both spatial and temporal coordinates must be transformed: :\begin{align} x_{1}^{'} & =\gamma\left(x_{1}-vt_{1}\right) & \quad\mathrm{and}\quad & & x_{2}^{'} & =\gamma\left(x_{2}-vt_{2}\right)\\ t_{1}^{'} & =\gamma\left(t_{1}-vx_{1}/c^{2}\right) & \quad\mathrm{and}\quad & & t_{2}^{'} & =\gamma\left(t_{2}-vx_{2}/c^{2}\right) \end{align} Computing length interval \Delta x'=x_{2}^{\prime}-x_{1}^{\prime} as well as assuming simultaneous time measurement \Delta t'=t_{2}^{\prime}-t_{1}^{\prime}=0, and by plugging in proper length L_{0}=x_{2}-x_{1}, it follows: :\begin{align}\Delta x' & =\gamma\left(L_{0}-v\Delta t\right) & (1)\\ \Delta t' & =\gamma\left(\Delta t-\frac{vL_{0}}{c^{2}}\right)=0 & (2) \end{align} Equation (2) gives :\Delta t=\frac{vL_{0}}{c^{2}} which, when plugged into (1), demonstrates that \Delta x' becomes the contracted length L': :L'=L_{0}/\gamma. Likewise, the same method gives a symmetric result for an object at rest in S': :L=L^{'}_{0}/\gamma.
Using time dilation Length contraction can also be derived from
time dilation, according to which the rate of a single "moving" clock (indicating its
proper time T_0) is lower with respect to two synchronized "resting" clocks (indicating T). Time dilation was experimentally confirmed multiple times, and is represented by the relation: :T=T_{0}\cdot\gamma Suppose a rod of proper length L_0 at rest in S and a clock at rest in S' are moving along each other with speed v. Since, according to the principle of relativity, the magnitude of relative velocity is the same in either reference frame, the respective travel times of the clock between the rod's endpoints are given by T=L_{0}/v in S and T'_{0}=L'/v in S', thus L_{0}=Tv and L'=T'_{0}v. By inserting the time dilation formula, the ratio between those lengths is: :\frac{L'}{L_{0}}=\frac{T'_{0}v}{Tv}=1/\gamma. Therefore, the length measured in S' is given by :L'=L_{0}/\gamma So since the clock's travel time across the rod is longer in S than in S' (time dilation in S), the rod's length is also longer in S than in S' (length contraction in S'). Likewise, if the clock were at rest in S and the rod in S', the above procedure would give :L=L'_{0}/\gamma
Geometrical considerations 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). This is the Euclidean analog of
boosting a cuboid in
E1,2. In the latter case, however, we can interpret the boosted cuboid as the
world slab of a moving plate.
Image: Left: a
rotated cuboid in three-dimensional euclidean space
E3. The cross section is
longer in the direction of the rotation than it was before the rotation. Right: the
world slab of a moving thin plate in Minkowski spacetime (with one spatial dimension suppressed)
E1,2, which is a
boosted cuboid. The cross section is
thinner in the direction of the boost than it was before the boost. In both cases, the transverse directions are unaffected and the three planes meeting at each corner of the cuboids are
mutually orthogonal (in the sense of
E1,2 at right, and in the sense of
E3 at left). In special relativity,
Poincaré transformations are a class of
affine transformations which can be characterized as the transformations between alternative
Cartesian coordinate charts on
Minkowski spacetime corresponding to alternative states of
inertial motion (and different choices of an
origin). Lorentz transformations are Poincaré transformations which are
linear transformations (preserve the origin). Lorentz transformations play the same role in Minkowski geometry (the
Lorentz group forms the
isotropy group of the self-isometries of the spacetime) which are played by
rotations in euclidean geometry. Indeed, special relativity largely comes down to studying a kind of noneuclidean
trigonometry in Minkowski spacetime, as suggested by the following table: ==References==