In The "Electrodynamics of Moving Bodies" published by Albert Einstein in 1905, in section §2 On the Relativity of Lengths and Times, Einstein's first equation can be found. This equation is easily recognized by any high-school student, which links distance time and velocity. Einstein, clearly states light path, which is distance and not length.

$$velocity = \frac{light\ path}{time\ interval}$$

Einstein then proceeds with an explanation of why events are not simultaneous and follows this with the following two equations which form the basis of his entire paper.

$$t_B - t_A = \frac{r_{AB}}{c-v}\ and\ t_A'-t_B = \frac{r_{AB}}{c+v}$$ "Where \(r_{AB}\) denotes the length of the moving rod" This is the exact point in his paper where Einstein prepares the ground for his erroneous theory of length contraction and time dilation. Indeed, it is not even neccessary to understand any of his equations, only the concept that Einstein is actually proposing. He is stating that the original basic equation of physics, which he himself quoted, is actually wrong and that from this point on, its correct form is actually; $$velocity = \frac{length\ of\ the\ rod}{time\ interval}$$ The length of a rod is not the same as the distance travelled by the rod and it is clear that the two equations shown are mutually exclusive.

Length is not Distance

The length of a rod and the distance traveled are two distinct physical concepts. the length of a Rod is a measure of the rod's physical size from one end to the other. It is a fixed, static measurement that doesn't change unless the physical dimensions of the rod itself is altered. The distance traveled refers to the total length of the path along which an object moves. It is a dynamic measurement that can vary depending on the path taken. The distance traveled can be more than the displacement (straight-line distance from the starting point to the end point) if the path is not straight. In summary, while the length of a rod is a static measurement of the rod's size, the distance traveled is a dynamic measurement of the path an object moves along. These two measurements are fundamentally different depending on the specific context. The original equations from Einsteins paper begin with: $$t_B-t_A=\frac{r_{AB}}{c-v}\ and\ t_A-t_B=\frac{r_{AB}}{c+v}$$ Einstein himself follows these equations with "Where \(r_{AB}\) denotes the length of the moving rod", apparently deeming it necessary to further emphasize his assertion that length and distance are interchangeable.

Therefore:
\(t_B\) and \(t_A\) are the times measured in the stationary system.
\(t_A'\) is the time measured in the moving system.
\(r_{AB}\) is the length of the moving rod, measured in the stationary system.
\(c\) is the speed of light.
\(v\) is the velocity of the moving rod.
The equations can be simplified, inasmuch as \(t_B\) can be eliminated by adding the two equations together: $$ t_B - t_A + t'_A - t_B = \frac{r_{AB}}{c - v} + \frac{r_{AB}}{c + v} $$ Simplifying:

\(t'_A - t_A = r{_{AB}} \left( \frac{1}{c - v} + \frac{1}{c + v} \right)\)

\(t'_A - t_A = r{_{AB}} \left( \frac{(c + v) + (c - v)}{(c - v)(c + v)} \right)\)

\(t'_A - t_A = r{_{AB}} \left( \frac{c + v + c - v}{c^2 - v^2} \right)\)

\(t'_A - t_A = r{_{AB}} \left( \frac{2c}{c^2 - v^2} \right)\)

\(t'_A - t_A = \frac{2c \cdot r{_{AB}}}{c^2 - v^2}\)

The final result being: $$ t'_A - t_A = \frac{2\cdot r{_{AB}}}{c \left(1 - \frac{v^2}{c^2}\right)} $$

At this point \(r_{AB}\) is still a length. $$ \text{time} =\frac{\text{length}}{\text{velocity}} $$

The correct form is of course. $$ \text{time} =\frac{\text{distance}}{\text{velocity}} $$

The next Equation is the beginning of Einstein's derivation of the Lorentz transformation: $$\frac{1}{2} \left[ \tau(0,0,0,t) + \tau\left(0,0,0,t + x_0 \frac{c - v}{c + v} \right) \right] = \tau\left(x_0,0,0,t + x_0 \frac{c - v}{c + v} \right)$$ Following analysis it is found that this equation represents a relationship between two events, one occurring at the origin \((0,0,0)\) and another at the position \((x_0,0,0,0)\) where the left side is the average of two events at the origin with a time interval involving the speed of light \(c\) and \(v\) and the right side is a single event at position \((x_0,0,0,0)\) with a time interval that also depends on the speed of light \(c\) and \(v\). Einstein proceeds with the derivation which results in: $$ \beta = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}} $$

Einstein now emphasizes the need to demonstrate that the speed of light \(c\) remains constant in both the stationary system \(K\) and the moving system. While the principle of the constancy of the speed of light \(c\) is assumed to hold in the stationary system, it hasn't been proven that this principle is compatible with the principle of relativity, which states that the laws of physics are the same in all inertial frames of reference. To prove this, Einstein considers a scenario where a "spherical" light wave is emitted from the common origin of the coordinate systems at time \(t=\tau=0\). He examines how this light wave propagates in the stationary system \(K\) and the moving system to show that the speed of light is indeed constant in both systems. By doing this, he aims to establish that the constancy of the speed of light is consistent with the principle of relativity. This is crucial because it supports the idea that the laws of physics, including the speed of light, are the same in all inertial frames of reference, regardless of their relative motion. Essentially, Einstein then sets up a scenario to explore how a rod's motion and its coordinates transform between the moving system \(k\) and the stationary system \(K\). The function \(\phi(v)\) playing a role in describing these transformations. By examining this setup, Einstein aims to illustrate how the principles of relativity apply to moving objects and their measurements in different reference frames. \[ x_1 = vt, \quad y_1 = \frac{l}{\phi(v)}, \quad z_1 = 0\] And: \[x_2 = vt, \quad y_2 = 0, \quad z_2 = 0\]

Einstein then points out that the function \(\phi\) encapsulates how the "length" of a moving object is affected by its velocity relative to a stationary observer. This relationship is symmetrical, meaning it doesn't matter if the rod moves in one direction or the opposite direction; the measured "length" remains the same, provided the speed is constant.

Clearly, at this juncture it can be seen that it is "distance" that changes relative to velocity and not "length" as suggested by Einstein. The "length" of the moving rod measured in the stationary system does not change, therefore, if \(v\) and \(−v\) are interchanged. Hence follows that

$$ \frac{l}{\phi(v)} = \frac{l}{\phi{-v}}$$ does not change if the velocity is reversed. $$ \phi(v) = \phi(=v)$$

This symmetry indicates that the effect of the velocity on the "length" of the moving rod is the same regardless of whether the rod is moving in the positive or negative direction along its axis. This is important because it suggests the idea that the physical laws, and specifically the way we measure "lengths", should be independent of the direction of motion. This is a key aspect of the principle of relativity, which states that the laws of physics are the same in all inertial frames of reference.

Lorentz Rotation

A Lorentz rotation in three dimensional space can be represented by a unit quaternion. Consider a quaternion \( q = q_0 + q_1 i + q_2 j + q_3 k \), where \( q_0 \) is the scalar part and \( q_1, q_2, q_3 \) are the vector parts, then a rotation by an angle \( \theta \) around an axis defined by the unit vector \( n = (n_x, n_y, n_z) \) can be represented by the quaternion: \[ q = \cos\left(\frac{\theta}{2}\right) + \sin\left(\frac{\theta}{2}\right) (n_x i + n_y j + n_z k) \]

The rotation of a vector \( v = (v_x, v_y, v_z) \) is then given by: \[ v' = q v q^{-1} \] where \( v \) is represented as the pure quaternion: \[ v = 0 + v_x i + v_y j + v_z k \] and \( q^{-1} \) is the conjugate of \( q \), given by: \[ q^{-1} = \cos\left(\frac{\theta}{2}\right) - \sin\left(\frac{\theta}{2}\right) (n_x i + n_y j + n_z k) \]

Boosts

For completeness, a Lorentz boost can be represented using quaternions. A boost in the direction of a unit vector \( n = (n_x, n_y, n_z) \) with a rapidity of \( \phi \) (where \( \phi = \tanh^{-1} \left( \frac{v}{c} \right) \) and \( v \) is the relative velocity) can be represented by the quaternion: \[ q = \cosh\left(\frac{\phi}{2}\right) + \sinh\left(\frac{\phi}{2}\right)(n_x i + n_y j + n_z k) \] The boost of a spacetime event \( x = (ct, x, y, z) \) is then given by representing \( x \) as a quaternion: \[ x = ct + x i + y j + z k \] and transforming it as: \[ x' = q x q^{-1} \]

Combined Lorentz Transformation

A general Lorentz transformation, which includes both a rotation and a boost, can be represented by the product of the corresponding quaternions. If \( q_{r} \) represents the rotation and \( q_{b} \) represents the boost, then the combined transformation is given by: \[ q = q_{r} q_{b} \] The transformation of a spacetime event \( x \) is then: \[ x' = q x q^{-1} \]

A boost in the \( x \)-direction with rapidity \( \phi \) can be considerd. The corresponding quaternion is: \[ q = \cosh\left(\frac{\phi}{2}\right) + \sinh\left(\frac{\phi}{2}\right) i \]

Applying to Einstein's Equations

The quaternion transformation can be applied to a "spacetime event", represented as a quaternion: \[ x = ct + x i + y j + z k \] The transformed event is given by: \[ x' = q x q^{-1} \] Performing the quaternion multiplication and simplifying, we obtain: \[ ct' = \cosh(\phi) ct + \sinh(\phi) x \] \[ x' = \sinh(\phi) ct + \cosh(\phi) x \] \[ y' = y, \quad z' = z \]

This reveals the hyperbolic nature of the Lorentz transformation. The equations for \( ct' \) and \( x' \) can be rewritten as: \[ ct' = \cosh(\phi) ct + \sinh(\phi) x = x' = \sinh(\phi) ct + \cosh(\phi) x \] Since hyperbolic functions satisfy the identity: \[\cosh(\phi) + \sinh(\phi) = e^{\phi} \] it follows that: \[ ct' + x' = e^\phi (ct + x) \] which suggests that this transformation rescales the components \( ct + x \) and \( ct - x \) by a factor of \( e^\phi \). The transformations for \( y \) and \( z \) remain unchanged, meaning that this is only in the \( x \)-direction.

Conclusion

In summary, the quaternion representation of the Lorentz transformation shows that: Distances (coordinate separations) between events change with a boost due to the hyperbolic rescaling effectively mixing \(t\) and \(x\). Lengths (proper lengths or invariant spacetime intervals) remain unchanged because the Lorentz transformation is constructed to preserve the Minkowski metric. Thus, while observers in different inertial frames may measure different coordinate distances (or “distances traveled”), the proper lengths—the intrinsic measures of objects in their own rest frames—are invariant under Lorentz transformations.

References:
1) A. Einstein, On the Electrodynamics of Moving Bodies (1905)