Consider two observers O and O', each using their own Cartesian coordinate system to measure space and time intervals. O uses K = (t, x, y, z) and O ' uses K' = (t', x', y', z'). Assume further that the coordinate systems are oriented so that the x-axis and the x'-axis are collinear, the y-axis is parallel to the y'-axis, as are the z-axis and the z'-axis. The relative velocity between the two observers is v along the common x-axis. Also assume that the origins of both coordinate systems are the same.
接下來的推導，完全參考 Albert Einstein, Relativity: The Special and General Theory. 1920 的 Appendix I。此外，這邊中英文混用，純粹是懶惰。
對於 O 來說：A light-signal, which is proceeding along the positive axis of x, is transmitted according to the equation
x - ct = 0.因為「光速不變原理」，對於 O' 來說，則是
x' - ct' = 0.把 x 想成光束接收器的位置，根據影片「實驗」，measuring-rod 會縮短，因此
(x' - ct') = λ(x - ct).我們尚未使用「相對性原理」，因此考慮 light rays which are being transmitted along the negative x-axis，應是十分合理的事。因此，
(x' + ct') = μ(x + ct).解一解令一令，可以得到聯立方程式
x' = ax + bct其中 a = (λ + μ)/2, b = (λ - μ)/2. 若觀察者 O' 站在 K' 座標系的原點 (x' = 0)，那麼
ct' = act - bx,
x = (bc/a) t.If we call v the velocity with which the origin of K' is moving relative to K, we then have
v = bc/a.(根據「相對性原理」，the velocity with which the origin of K is moving relative to K' is -v. 因此，假如觀察者 O 站在 (t, x, y, z) 座標系的原點 (x = 0)，那麼 t' = at, x' = -bct = (-bc/a) t' = -vt'，沒有矛盾，但也解不出 a, b)
(推不下去，慘！因為我們尚未使用「相對性原理」及「time」。)
根據相對性原理，
As judged from K, the length of a unit measuring-rod which is at rest with reference to K' must be exactly the same as the length, as judged from K', of a unit measuring-rod which is at rest relative to K.當 t = 0 時，
x' = ax, or Δx' = aΔx.上面意思是 O 看 K' 的 unit measuring-rod，長度是 Δx = 1/a. 此外，當 t' = 0 時，
x' = a(1 - v^2/c^2)x, or Δx' = a(1 - v^2/c^2)Δx.上面意思是 O' 看 K 的 unit measuring-rod，長度是 Δx' = a(1 - v^2/c^2). 又根據相對性原理，O 與 O' 互看的長度要一樣，也就是說：
1/a = a(1 - v^2/c^2).接著就是整理數學式子而已，這邊直接略過，狹義相對論首重創新的洞見，數學只是重現愛因斯坦的洞見。於是乎，冰冷的 Lorentz transformation 有了新生命。
回顧一下「我學到的相對論」：
- Every description of events in space involves the use of a rigid body to which such events have to be referred.
- If, relative to K, K' is a uniformly moving co-ordinate system devoid of rotation, then natural phenomena run their course with respect to K' according to exactly the same general laws as with respect to K. This statement is called the principle of relativity (in the restricted sense).
- The velocity of light c (in vacuum) is constant. This statement is called the law of propagation of light.
- Every reference-body (co-ordinate system) has its own particular time; unless we are told the reference-body to which the statement of time refers, there is no meaning in a statement of the time of an event.
- Lorentz transformation.
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