Review of Lorentz Transformations

In special relativity, the idea of absolute time is discarded. Instead, it is established that the speed of light, c, is constant in all inertial frames. This change, combined with the principle of relativity, leads us to Lorentz Transformations. These transformations connect the coordinates of an event observed from two different inertial frames. This topic is explored in detail in the class on Lorentz Transformations in Special Relativity.

Considering inertial frames S and S^\prime in standard configuration, where their axes and origins coincide at t=t^\prime =0, and a photon emitted at t=t^\prime = 0 from the origin, the space and time coordinates of the photon in each frame must satisfy the equation:

c^2t^2 - x^2 - y^2 - z^2 = c^2{t^\prime}^2 - {x^\prime}^2 - {y^\prime}^2 - {z^\prime}^2 = 0.

From this equation and the principle of relativity, we derive the well-known Lorentz transformations:

\begin{array}{rl} ct^\prime &= \gamma_{ss^\prime_x}(ct - \beta_{ss^\prime_x} x), \\ x^\prime &= \gamma_{ss^\prime_x}(x - \beta_{ss^\prime_x} ct), \\ y^\prime &= y, \\ z^\prime &= z. \end{array}

Where \beta_{ss^\prime_x} =v_{ss^\prime_x}/c is the velocity boost acquired by S^\prime when moving relative to S at a speed v_{ss^\prime_x}, and \gamma_{ss^\prime_x} = 1/\sqrt{1-\beta_{ss^\prime_x}^2} is the associated Lorentz factor. This Lorentz transformation in the direction \hat{x} simplifies to the Galilean transformation when v_{ss^\prime_x} \ll c.

Similar to Galilean transformations, there is a symmetry that facilitates calculating the inverse transformation, simply by swapping terms and considering that \beta_{ss^\prime_x} = -\beta_{s^\prime s_x}:

\begin{array}{rl} ct &= \gamma_{ss^\prime_x}(ct^\prime + \beta_{ss^\prime_x} x^\prime),\\ x &= \gamma_{ss^\prime_x}(x^\prime + \beta_{ss^\prime_x} ct^\prime),\\ y &= y^\prime, \\ z &= z^\prime. \end{array}

Minkowski Spacetime

Lorentz transformations reveal that space and time coordinates are intrinsically intertwined. This relationship is particularly evident in the symmetry between ct and x. Considering two events, A and B, with coordinates (ct_A, x_A, y_A, z_A) and (ct_B, x_B, y_B, z_B). In the frame S, we define the quadratic distance as follows:

\begin{array}{rl} \Delta s^2 &= c^2(t_B - t_A)^2 - (x_B - x_A)^2 - (y_B - y_A)^2 - (z_B - z_A)^2 \\ \\ &= c^2\Delta t^2 - \Delta x^2 - \Delta y^2 - \Delta z^2 \\ \\ &= c^2\Delta t^2 - (\Delta x^2 + \Delta y^2 + \Delta z^2) \end{array}

The spacetime distance, \Delta s, is written as \Delta s = \sqrt{c^2\Delta t^2 - (\Delta x^2 + \Delta y^2 + \Delta z^2)}. Here, \Delta t represents a temporal length and \Delta r = \sqrt{\Delta x^2 + \Delta y^2 + \Delta z^2} is a spatial length.

The Minkowski Spacetime, characterized by this notion of spacetime distance \Delta s, is fundamental in special relativity. It was introduced by Hermann Minkowski and is distinguished from spatial and temporal coordinates by being invariant under Lorentz transformations.

\Delta s = \Delta s^\prime

In this model, space and time combine into a four-dimensional continuum. Unlike Euclidean geometry, Minkowski spacetime geometry is pseudo-Euclidean due to the negative signs in its spatial components. However, for a constant time t, Minkowski’s spatial geometry remains Euclidean.

What happens to space, time, and spacetime lengths under Lorentz transformations?

As previously mentioned, spacetime lengths \Delta s are invariant under Lorentz transformations, but in addition to this, space and time lengths change under these transformations. What we will do next is the step-by-step demonstration of these facts.

First, let’s recall the events A and B considered at the beginning with their respective spacetime coordinates concerning the system S:

• Event A: (ct_A,x_A, y_A, z_A)
• Event B: (ct_B,x_B, y_B, z_B)

For these developments, we will use without loss of generality the Lorentz transformations for systems S and S^\prime in standard configuration where S^\prime moves with velocity \vec{v}_{ss^\prime_x}= v_{ss^\prime_x} \hat{x} = \beta_{ss^\prime_x}c \hat{x} relative to S

\begin{array}{rl} ct^\prime &= \gamma_{ss^\prime_x}(ct - \beta_{ss^\prime_x} x), \\ x^\prime &= \gamma_{ss^\prime_x}(x - \beta_{ss^\prime_x} ct), \\ y^\prime &= y, \\ z^\prime &= z. \end{array}

Development for pure time lengths

Suppose the events A and B, observed from the reference frame S, are separated only by time, like the ticks of a clock. In this case, the elapsed time between a tick-tock is calculated as follows:

c\Delta t = c(t_B - t_A)

On the other hand, the temporal separation between the same pair of events observed from S^\prime will be:

c\Delta t^\prime = c(t^\prime_B - t^\prime_A)

These temporal separations are related through Lorentz transformations as follows:

\begin{array}{rl} c\Delta t^\prime &= c(t^\prime_B - t^\prime_A) \\ \\ &= ct^\prime_B - ct^\prime_A \\ \\ &= \gamma_{ss^\prime_x}(ct_B - \beta_{ss^\prime_x} x_B) - \gamma_{ss^\prime_x}(ct_A - \beta_{ss^\prime_x} x_A) \\ \\ &= \gamma_{ss^\prime_x}c \Delta t - \gamma_{ss^\prime_x} \beta_{ss^\prime_x} \Delta x \end{array}

Now, since the events A and B are separated only in time for the observer in S, we have \Delta x = 0. Therefore:

\boxed{\Delta t^\prime = \gamma_{ss^\prime_x} \Delta t}

It is important to note that:

\gamma_{ss^\prime_x} = \dfrac{1}{\sqrt{1 - \beta^2_{ss^\prime_x}}} \in [1, +\infty[

This is because \beta^2_{ss^\prime_x} = \dfrac{v^2_{ss^\prime_x}}{c^2} \in [0,1[.

In simple terms, if an observer in S measures a time interval \Delta t like the tick-tock of a clock, an observer in S^\prime will measure this same interval as \gamma_{ss^\prime_x} \Delta t, which is greater than or equal to \Delta t. This effect, known as time dilation, indicates how time extends between inertial observers experiencing a velocity boost \beta_{ss^\prime_x}. Therefore, the passage of time is not the same for all inertial observers, showing that time lengths are not invariant under Lorentz transformations.

Development for Pure Space Lengths

Suppose that events A and B are separated only in space, like the ends of a ruler. We assume, without loss of generality, that this ruler is oriented along the \hat{x} axis of S. Then, we have:

\Delta x = x_B - x_A

Seen from S^\prime, this spatial separation would be:

\Delta x^\prime = x^\prime_B - x^\prime_A

Applying Lorentz transformations, we can establish the relationship between both observations:

\begin{array}{rl} \Delta x^\prime &= x^\prime_B - x^\prime_A \\ \\ &= \gamma_{ss^\prime}(x_B - \beta_{ss^\prime_x} ct_B) - \gamma_{ss^\prime}(x_A - \beta_{ss^\prime_x} ct_A) \\ \\ &= \gamma_{ss^\prime} \Delta x - \gamma_{ss^\prime}\beta_{ss^\prime_x} c \Delta t \end{array}

Since the events A and B are simultaneous for S, it follows that \Delta t = 0, and therefore:

\boxed{\Delta x^\prime = \gamma_{ss^\prime} \Delta x}

For example, if we place a ruler of length l_0 inside a train car (observer S^\prime), which moves relative to us (observer S), and the ruler is aligned with the direction of motion, the observed length will be:

\begin{array}{rl} & l_0 = \gamma_{ss^\prime} l \\ \\ \equiv & l = \dfrac{l_0}{\gamma_{ss^\prime}} \leq l_0. \end{array}

This means we will perceive the ruler’s length as shorter than it actually is. This phenomenon is known as Lorentz contraction and demonstrates that space intervals are not conserved under Lorentz transformations.

Development for Spacetime Lengths

After analyzing how pure space and pure time lengths transform, let’s now examine the behavior of spacetime lengths under Lorentz transformations. Recall that a spacetime length, observed by the observer S^\prime for two events A and B, is expressed as follows:

\begin{array}{rl} \Delta s^\prime &= \sqrt{c^2\Delta t^{\prime 2} - (\Delta x^{\prime 2} + \Delta y^{\prime 2} + \Delta z^{\prime 2})} \\ \\ &= \sqrt{c^2 (t^{\prime 2}_B - t^{\prime 2}_A) - \left[(x^{\prime 2}_B - x^{2}_A) + (y^{\prime 2}_B - y^{\prime 2}_A) + (z^{2}_B - z^{2}_A) \right]} \end{array}

Next, we will see how these lengths relate after applying Lorentz transformations, in the case that S^\prime has a velocity boost \beta_{ss^\prime_x} concerning S.

\color{black} \begin{array}{rl} \Delta s^{\prime 2} &= (c^2 t^{\prime 2}_B - c^2 t^{\prime 2}_A) - \left[(x^{\prime 2}_B - x^{\prime 2}_A) + (y^{\prime 2}_B - y^{2}_A) + (z^{\prime 2}_B - z^{2}_A) \right] \\ \\ \\ &= \left[\gamma_{ss^\prime_x}(ct_B - \beta_{ss^\prime_x} x_B)\right]^2 - \left[\gamma_{ss^\prime_x}(ct_A - \beta_{ss^\prime_x} x_A)\right]^2 + \cdots \\ \\ & \cdots -\left\{ \left( \left[\gamma_{ss^\prime_x}(x_B - \beta_{ss^\prime_x} ct_B)\right]^2 - \left[\gamma_{ss^\prime_x}(x_A - \beta_{ss^\prime_x} ct_A)\right]^2 \right) + (y^{2}_B - y^{2}_A) + (z^{2}_B - z^{2}_A) \right\} \\ \\ \\ &= \gamma_{ss^\prime_x}^2 (ct_B - \beta_{ss^\prime_x} x_B)^2 - \gamma_{ss^\prime_x}^2(ct_A - \beta_{ss^\prime_x} x_A)^2 + \cdots \\ \\ & \cdots -\left\{ \gamma_{ss^\prime_x}^2(x_B - \beta_{ss^\prime_x} ct_B)^2 - \gamma_{ss^\prime_x}^2(x_A - \beta_{ss^\prime_x} ct_A)^2 + (y^{2}_B - y^{2}_A) + (z^{2}_B - z^{2}_A) \right\} \\ \\ \\ &= \color{red}\gamma_{ss^\prime_x}^2 c^2 t_B^2 \color{black} - \cancel{2 \gamma_{ss^\prime_x}^2 \beta_{ss^\prime_x} c t_B x_B} + \color{green}\gamma_{ss^\prime_x}^2\beta_{ss^\prime_x}^2 x_B^2\color{black} + \cdots \\ \\ & \cdots - \color{blue}\gamma_{ss^\prime_x}^2 c^2 t_A^2\color{black} + 2 \cancel{\gamma_{ss^\prime_x}^2 \beta_{ss^\prime_x} c t_A x_A} - \color{purple}\gamma_{ss^\prime_x}^2\beta_{ss^\prime_x}^2 x_A^2\color{black} + \cdots \\ \\ & \cdots - \color{green} \gamma_{ss^\prime_x}^2x_B^2 \color{black} + \cancel{2 \gamma_{ss^\prime_x}^2 \beta_{ss^\prime_x} ct_B x_B} - \color{red}\gamma_{ss^\prime_x}^2 \beta_{ss^\prime_x}^2 c^2t_B^2 \color{black}+ \cdots \\ \\ & \cdots + \color{purple}\gamma_{ss^\prime_x}^2x_A^2\color{black}- \cancel{2 \gamma_{ss^\prime_x}^2 \beta_{ss^\prime_x} ct_A x_A} + \color{blue}\gamma_{ss^\prime_x}^2 \beta_{ss^\prime_x}^2 c^2t_A^2 \color{black} + \cdots \\ \\ & \cdots - \left\{ (y^{2}_B - y^{2}_A) + (z^{2}_B - z^{2}_A) \right\} \\ \\ \\ &= \color{red}\gamma_{ss^\prime_x}^2 (1- \beta_{ss^\prime_x}^2)c^2 t_B^2\color{black} - \color{blue}\gamma_{ss^\prime_x}^2 (1- \beta_{ss^\prime_x}^2)c^2 t_A^2 \color{black} + \cdots \\ \\ & \cdots - \color{green}\gamma_{ss^\prime_x}^2(1-\beta_{ss^\prime_x}^2)x_B^2\color{black} + \color{purple}\gamma_{ss^\prime_x}^2(1-\beta_{ss^\prime_x}^2)x_A^2 \color{black} + \cdots \\ \\ & \cdots - \left\{ (y^{2}_B - y^{2}_A) + (z^{2}_B - z^{2}_A) \right\} \\ \\ \\ \end{array}

Finally, remembering that \gamma_{ss^\prime_x}^2 = 1/(1-\beta_{ss^\prime_x}^2), we obtain the following:

\begin{array}{rl} \Delta s^{\prime 2} &= c^2 t_B^2 - c^2 t_A^2 - x_B^2 + x_A^2 - \left\{ (y^{2}_B - y^{2}_A) + (z^{2}_B - z^{2}_A) \right\} \\ \\ &= c^2 (t_B^2 - t_A^2) - \left\{ (x_B^2 - x_A^2) + (y^{2}_B - y^{2}_A) + (z^{2}_B - z^{2}_A) \right\} \\ \\ &= c^2 \Delta t^2 - (\Delta x^2 + \Delta y^2 + \Delta z^2) \\ \\ &= \Delta s^2 \end{array}

With this, we have demonstrated that, unlike pure time and space lengths, spacetime lengths remain constant under Lorentz transformations.