New Considerations

As a consequence of what has been seen in Propagation of Electromagnetic Waves in Vacuum, in special relativity it is postulated as a principle that the speed of light c is the same for all inertial frames. But such an assumption does not come without implications:

1. The Galilean transformations must be abandoned as a valid means to transform observations from one inertial frame into another.
2. The intuitive idea that time flows in the same way for all inertial reference frames must be left behind.

It is through these considerations that the Lorentz transformations are obtained, serving as a correction and generalization for the Galilean transformations, which also works for electromagnetic theory.

Derivation of the Lorentz Transformations

Recap on (Linear) Coordinate Transformations

Consider two inertial frames S and S^\prime in standard configuration such that the second origin moves with a constant velocity \vec{v}_0 = v_{x_0}\hat{x} with respect to the first origin. What will be demonstrated next is that if the coordinates of an event seen from two inertial systems S and S^\prime are related by a linear transformation as revised in The Principle of Relativity (specifically, this expression) and it is accepted that the speed of light is the same from all inertial frames, then the coordinate transformation corresponds exactly with the Lorentz transformations that we will obtain later.

Initially, the coordinates (t,x) of an event seen from S, and coordinates (t^\prime, x^\prime) of the same event seen from S^\prime moving with a velocity v_{v}=v_{x_0}\hat{x} relative to S, are related through a linear transformation such that:

\begin{array}{llr} t^\prime &= At + Bx, & [1]\\ x^\prime &= Dt + Ex & [2] \end{array}

where A, B, C and D are constants to be determined and the coordinates y and z have been omitted (without loss of generality) for simplicity.

Introducing the Speed of Light as a Universal Constant

The constants A, B, D and E can be determined from these new considerations by invoking some special cases. First of all, we must consider that the coordinate transformation expressed through [1] and [2] must always work, and as a consequence, it must work in each particular case, and these particular cases are the ones that will be stated below to delve into their form:

• Consider the event as moving at the speed of light: If it has coordinates (t,x) seen from S and (t^\prime, x^\prime) seen from S^\prime, then they must satisfy the relation:

  \displaystyle\frac{x^2}{t^2} = c^2 = \frac{{x^\prime}^2}{{t^\prime}^2}.

  From this, it is inferred that

  c^2 t^2 - x^2 = c^2{t^\prime}^2 - {x^\prime}^2 = 0\;\;\; [3]

• Consider the event as moving along with the inertial frame S^\prime:

  If the event has the same coordinates as the origin of the inertial frame S^\prime, then it will occur that x=v_0 t and x^\prime =0. Consequently, from equation [2] it will be:

  \begin{array}{rl} & 0 = Dt + Ev_0 t \\ \equiv & D = -Ev_0\:\:\;[4] \end{array}

• Finally, consider the event as remaining along with the origin of the inertial frame S:

  In this case, it will have that x=0 and x^\prime = -v_0 t^\prime, so from equation [2] it will be:

  \begin{array}{rl} &-v_0t^\prime = Dt\\ \equiv & t= \displaystyle -\frac{v_0}{D} t^\prime\;\;\;[5] \end{array}

  Then, from [1] and [5] it is concluded that:

  \begin{array}{rl} & t^\prime = A \left(\displaystyle -\frac{v_0}{D}\right) t^\prime + \underbrace{Bx}_{x=0} \\ \\ \equiv & \displaystyle \frac{-Av_0}{D} = 1 \\ \\ \equiv & D = -Av_0\;\;\;[6] \end{array}

Finally, from [4] and [6]: A = E, so the system of equations given by [1] and [2] is reduced to

\begin{array}{rll} t^\prime &= At + Bx & [7]\\ \\ x^\prime &= A(x - v_{x_0} t) & [8] \end{array}

The Velocity Boost and the Lorentz Factor

Now, replacing [7] and [8] into [3] we get

\begin{array}{rl} & c^2 (At +Bx)^2 - A^2 (x - v_{x_0} t)^2 = c^2t^2 - x^2\\ \\ \equiv\; & \color{blue}{(c^2 A^2) t^2} + \color{red}{(2c^2 AB)xt} \color{black} + c^2 B^2 x^2 - A^2 x^2 + \color{red} {(2A^2v_{x_0})xt} \color{black}- \color{blue}{(A^2 v_{x_0}^2) t^2} \color{black}= \color{blue}{(c^2) t^2} \color{black}- x^2. \end{array}

from what has been marked in blue we obtain

\begin{array}{rl} &c^2 A^2 - A^2 v_{x_0}^2 = c^2 \\ \\ \equiv\;& A^2 (c^2 - v_{x_0}^2) = c^2 \\ \\ \equiv\;& \displaystyle A^2 = \frac{c^2}{c^2 - v_{x_0}^2} = \frac{1}{1 - \frac{v_{x_0}^2}{c^2}} \\ \equiv\;& \displaystyle A = \frac{1}{\sqrt{1 - \frac{v_{x_0}^2}{c^2}}} \end{array}

This is generally written replacing A=\gamma_x (Lorentz contraction factor) and \beta_x = v_{x_0}/c (velocity boost), resulting in the form:

\displaystyle A = \gamma_x = \frac{1}{\sqrt{1 - \beta_x^2}},\;\;\;[9]

And replacing [9] into [2] we get:

x^\prime = \gamma_x(x - \beta_x ct)

from what has been marked in red we obtain

\begin{array}{rll} &2c^2 AB + 2A^2v_{x_{x_0}} = 0& \\ \\ \equiv\;& cB^2 + Av_{x_0} = 0 & \\ \\ \equiv\;& B=\displaystyle -\frac{1}{c^2}Av_{x_0} = -\frac{\gamma_x v_{x_0}}{c^2}& \\ \\ \equiv\;& B=\displaystyle -\frac{\gamma_x \beta_x}{c} & [10] \end{array}

thus, replacing [9] and [10] into [7] we get

\begin{array}{rl} &t^\prime =\displaystyle \gamma_x t -\frac{\gamma_x \beta_x}{c} \\ \\ \equiv\; &t^\prime =\displaystyle \gamma_x \left( t -\frac{\beta_x x}{c}\right)\\ \\ \equiv\; &ct^\prime =\displaystyle \gamma_x \left( ct - \beta_x x \right) \end{array}

Synthesis of the Lorentz Transformations

Finally, the linear transformation that models the change of coordinates between the systems S and S^\prime is given by the following expressions.

\begin{array}{rl}ct^\prime &=\gamma_x \left( ct - \beta_x x \right) \\ x^\prime &= \gamma_x(x - \beta_x ct) \end{array}

This system of transformations can be expressed in matrix form as follows

\left(\begin{matrix}ct^\prime \\ x^\prime \\ y^\prime \\ z^\prime \end{matrix}\right) = \left( \begin{matrix}\gamma_x & -\gamma_x\beta_x & 0 & 0 \\ -\gamma_x\beta_x & \gamma_x & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix} \right) \left(\begin{matrix} ct \\ x \\ y \\ z \end{matrix} \right)

This is known as the Lorentz Transformations of special relativity

The Lorentz Transformations Converge and Generalize the Galilean Transformations

The convergence of the Lorentz transformations to the Galilean transformations is observed when seeing what happens with the Lorentz transformations when the velocity between inertial reference frames is much less than the speed of light. When this occurs, we have:

|v_{x_0}| \ll c \longrightarrow \left\{\begin{matrix}\beta_x = \frac{v_{x_0}}{c} \approx 0 \\ \\ \gamma_x = \sqrt{1-\beta_x} \approx 1 \\ \\ \gamma_x \beta_x c = v_{x_0} \gamma_x \approx v_{x_0} \end{matrix}\right.

So:

\left(\begin{matrix}ct^\prime \\ x^\prime \\ y^\prime \\ z^\prime \end{matrix}\right) = \left( \begin{matrix}\gamma_x & -\gamma_x\beta_x & 0 & 0 \\ -\gamma_x\beta_x & \gamma_x & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix} \right) \left(\begin{matrix} ct \\ x \\ y \\ z \end{matrix} \right) = \left(\begin{matrix} \gamma_x ct -\gamma_x \beta_x x \\ -\gamma_x \beta_x c t + \gamma_x x \\ y \\ z \end{matrix} \right) \approx \left(\begin{matrix} ct \\ -v_{x_0}t + x \\ y \\ z \end{matrix}\right)

That is:

\begin{array}{rl} t^\prime &\approx t \\ x^\prime &\approx x - v_{x_0}t \\ y^\prime &\approx y \\ z^\prime &\approx z \end{array}

which exactly matches the Galilean transformations. Through this, it is corroborated that the Lorentz Transformations generalize the Galilean transformations for velocities close to the speed of light and converge to the Galilean transformations when velocities are much lower than the speed of light.