The Lorentz Transformation

To understand space and time, rely on axioms and measurements rather than intuition.  All space and time measurements must be made with respect to rigid bodies.  Rigid body reference frames in which Newton's first law is true are inertial.  All rigid body reference frames moving with constant velocities relative to an inertial rigid body reference frame (IRBRF) are also inertial. The Lorentz transformation converts space and time measurements between IRBRFs.  A requirement of the Lorentz transformation is that that speed of light in a vacuum must always be the same in all IRBRFs.  Consider the following two IRBRFs in a vacuum:

Assume a photon is emitted and absorbed in the stationary IRBRF with respect to the events (x₁, y₁, z₁, t₁) and (x₂, y₂, z₂, t₂).  Assume the corresponding events in the moving IRBRF are (x'₁, y'₁, z'₁, t'₁) and (x'₂, y'₂, z'₂, t'₂).  Therefore:

 

(x₁ - x₂)² + (y₁ - y₂)² + (z₁ - z₂)² - c(t₁ - t₂)² = 0

(x'₁ - x'₂)² + (y'₁ - y'₂)² + (z'₁ - z'₂)² - c(t'₁ - t'₂)² = 0.     

If y = y' and z = z', then:

(x₁ - x₂)² - c(t₁ - t₂)² = (x'₁ - x'₂)² - c(t'₁ - t'₂)² = 0.

The following transformation meets this requirement for some ξ:

 x' = cosh(ξ)x + sinh(ξ)ct
y' = y                             
z' = z                             
ct' = sinh(ξ)x + cosh(ξ)ct

The velocity v of the moving IRBRF implies a moving origin velocity measurement relative to the stationary IRBRF such that x' = 0 implies x = vt:

0 = cosh(ξ)vt + sinh(ξ)ct

0 = cosh(ξ)v + sinh(ξ)c  
tanh(ξ) = - v / c                                 

Since tanh(ξ) = sinh(ξ) / cosh(ξ) and cosh(ξ) = 1 / √1 - tanh²(ξ):

x' = γ(x - vt)              
y' = y                         
z' = z                         
ct' = γ(ct - vx / c)        

where γ = 1 / √1 - v² / c².  This Lorentz transformation candidate has been confirmed by experiment.

Lorentz transformations do not alter spacetime distances with respect to imaginary time.  Therefore, Lorentz transformations are hyperbolic rotations in spacetime:

The Lorentz transformation implies measurements of rigid body lengths depend on velocities:

The Lorentz transformation also implies measurements of clock rates depend on velocities:

The corresponding relativistic velocity transformation can be found from differentiation.  Let u and u' object velocity measurements relative to the two IRBFs.  Note:

dt / dt' = 1 / (dt' / dt) = 1 / [γ(1 - vu_x / c²)].

Therefore:

u'_x = (u_x - v) / (1 - vu_x / c²)
u'_y = u_y / [γ(1 - vu_x / c²)]   
u'_z = u_z / [γ(1 - vu_x / c²)].