Lorentz Transformation

A graphical representation of t' as a function of t (t' = f(t)) as according to the Lorentz transformation. It is noted that f() is a linear function with a slope and intercept, which can be calculated from the parameters v and c. If the ratio v/c approaches zero, the slope becomes one and the intercept becomes zero: t' approaches t as v/c approaches zero.

A graphical representation of x' as a function of x (x' = f(x)) as according to the Lorentz transformation. It is noted that f() is a linear function with a slope and intercept, which can be calculated from the parameters v and c. If the ratio v/c approaches zero, the slope becomes one and the intercept becomes -vt: x' approaches x-vt as v/c approaches zero.

In physics, the Lorentz transformation describes how, according to the relativity theory of Lorentz the spatial and temporal coordinates (x, y, z, t) of an event referred to one system and the corresponding coordinates (x’, y’, z’, t’) of the same event referred to another system that moves with a uniform velocity with respect to the first system can be converted into each other. It was derived by and is named after the Dutch physicist Hendrik Lorentz. It results into surprising predictions: observers moving at different velocities would measure different distances between two events, different elapsed times between to events, and even different temporal orderings of events. The Lorentz transformation describes the transformations in which the event at the origin (x=0, y=0, z=0, t=0) is left fixed; Transformations that also include translations are said to belong to the "Poincaré group".

Lorentz proposed in 1904 that motion of a material body through the aether produces a contraction in the direction of motion, and a slowing down of all rhythmical processes, both by the factor (1 – v²/c²)^1/2, where v is the velocity of the body and c the velocity of light.

Lorentz showed that if these physical effects were a reality, the Lorentz transformation gives the relation between the coordinates, (x, y, z, t) of an event referred to one system (the first system), and the coordinates, (x', y', z', t'), of the same event referred to a second system moving uniformly in the direction with respect to the first (for simplicity we consider relative motion in the x direction only; and suppose certain initial conditions to be satisfied). The second system is moving with speed v relative to the first. The Lorentz transformation is given by:

t′ =   t - vx/c2 √ 1 - v2/c2
x′ =   x - vt √ 1 - v2/c2
y′ = y
z′ = z

where:

x, y, z are the three spatial coordinates of the event referred to the first system.

t is the time coordinate of the event referred to the first system.

x', y', z' are the three spatial coordinates of the event referred to the second system.

t' is the time coordinate of the event referred to the second system.

v is the velocity of the second system with respect to the first. v may be positive, zero or negative.

c is the speed of light (which is approximately 3 × 10⁸ m/s).

Mathematically the significance of the Lorentz transformation lies in the fact that, in mathematical language, the vacuum equations of electromagnetic theory (i.e. Maxwell's Equations applied to vacuum) are invariant to them; that is to say, if, for x, y, z and t in those equations, we substitute the values given by the Lorentz transformation, we obtain identical equations with x', y', z', t' taking the places of x, y, z, t, and v changing to —v.

Lorentz recognised that the proposed contraction in the direction of motion and slowing down of all rhythmical processes was a pure ad hoc hypothesis, proposed simply because it led to a transformation to which the equations of the electromagnetic theory were invariant. Herbert Dingle, in his 1972 textbook Science at the Crossroads, quotes Lorentz:

"It need hardly be said that the present theory is put forward with all due reserve."

Nevertheless, Lorentz' theory is a physical theory, not a mathematical one; that is to say, the proposal was that motion through the ether produced physical effects on bodies, and the mathematics expressed the physical results produced.

Woldemar Voigt's transformation

Already in 1887, Prof. Dr. Woldemar Voigt published the following transformation that resembles the later developed Lorentz transformation.

x' = x - vt

y' = y (1 –v²/c²)^1/2

z' = z (1 –v²/c²)^1/2

t' = t - vx/c²

Matrix Notation

The Lorentz transformation can be written as an equation involving a matrix-vector product of a 4×4 matrix and a 4×1 vector. The matrix equation can be derived directly from the above shown Lorentz transformation. We obtain:

ct′  =  x′ y′ z′

γ     -γβ     0     0     -γβ     γ     0     0     0     0     1     0     0     0     0     1

ct x y z

where:

β = v/c

γ = (1 – v²/c²)^1/2