Lorentz Transformations in Inertial Frames

Introduction

Welcome back! In our last lesson, we discovered that simultaneity is not absolute—a direct consequence of Einstein's postulates. We saw that two events happening at the same time for one observer may occur at different times for another observer in relative motion. This realization shatters the framework of Galilean relativity and classical intuition, where time flows identically for everyone.

This brings us to a crucial question: if the old rules for relating coordinates between moving frames (the Galilean transformations) are wrong, what are the correct ones?

Today, we will answer that question by deriving and applying the Lorentz transformations. These equations form the mathematical heart of special relativity, providing the exact rules for how space and time coordinates measured in one inertial frame relate to those in another. Your goal is to learn these equations and apply them to transform coordinates between frames, cementing the link between the postulates and the strange effects of relativity.

Deriving the Lorentz Transformations

The Lorentz transformations can be derived directly from the two postulates of special relativity. The derivation requires some careful algebra, but the physical reasoning is straightforward. The core idea is to find a set of linear equations that ensures the speed of light is measured to be $c$ by all inertial observers.

Since you appreciate the mathematical underpinnings of physics, we'll go through a full derivation. The following video provides an excellent, step-by-step walkthrough.

Deriving the Lorentz Transformations | Special Relativity

This video, titled 'Deriving the Lorentz Transformations' from the channel DeepBean, provides a clear, first-principles derivation based on Einstein's postulates. It carefully explains the crucial assumption of linearity and then uses the invariance of a spherical light wave to find the transformation coefficients.

Please watch the entire video (up to 13:51). Pay close attention to these key steps in the argument: The setup of a spherical light pulse that must appear spherical in both reference frames. The justification for assuming the transformation must be linear (e.g., x' = Ax + Bt). The algebraic steps combining the equations for forward- and backward-propagating light. How the principle of relativity (symmetry between frames) is used to determine the final coefficients.

The Lorentz Transformation Equations

As derived in the video, for a frame S' moving with velocity $v$ along the positive x-axis relative to a frame S, the coordinates of an event $(t, x, y, z)$ in S are related to the coordinates $(t^{'}, x^{'}, y^{'}, z^{'})$ of the same event in S' by the following equations:

Lorentz Transformation Equations and Inertial Frames — The Lorentz transformation equations for a boost in the x-direction. The equations relate the spacetime coordinates of an event in frame S to those in frame S'. Note the definitions of the Lorentz factor, $\gamma$, and the normalized velocity, $\beta$.

Let's write them out clearly. The forward transformation (from S to S') is:

t^{'} x^{'} y^{'} z^{'} = γ (t - \frac{v x}{c ^{2}}) = γ (x - v t) = y = z

where $γ = \frac{1}{1 - v ^{2} / c ^{2}}$ is the Lorentz factor.

A key point from the principle of relativity is symmetry. To get the inverse transformation (from S' to S), we simply swap the primed and unprimed coordinates and replace $v$ with $- v$ :

t x y z = γ (t^{'} + \frac{v x ^{'}}{c ^{2}}) = γ (x^{'} + v t^{'}) = y^{'} = z^{'}

Notice the most striking feature: the new time coordinate $t^{'}$ depends on both the old time $t$ and the old space coordinate $x$ . This "mixing" of space and time is the source of all the strange relativistic effects we've discussed.

Applying the Transformations: The Relativity of Simultaneity, Revisited

Let's use these equations to confirm what we learned in the last lesson. We concluded that if two events are simultaneous in frame S ( $Δ t = 0$ ) but separated by a distance $Δ x \neq = 0$ , they will not be simultaneous in frame S' ( $Δ t^{'} \neq = 0$ ).

Let's see this emerge directly from the math. The resource below contains an example that does exactly this.

5.6: The Lorentz Transformation - Physics LibreTexts

The textbook page 'The Lorentz Transformation' from Physics LibreTexts provides several worked examples. We'll focus on the one that applies the transformation to the concept of simultaneity.

Please read 'Example \PageIndex{3}: Lorentz Transformation and Simultaneity'. Follow the calculation that determines the time separation in the moving frame for events that are simultaneous in the stationary frame.

Let's break down the logic shown in the example:

We start with the Lorentz transformation for a time interval, $Δ t^{'}$ , which we get by considering two events:
$Δ t^{'} = t_{2}^{'} - t_{1}^{'} = γ ((t_{2} - \frac{v x _{2}}{c ^{2}}) - (t_{1} - \frac{v x _{1}}{c ^{2}}))$
Rearranging gives:
$Δ t^{'} = γ ((t_{2} - t_{1}) - \frac{v ( x _{2} - x _{1} )}{c ^{2}}) = γ (Δ t - \frac{v Δ x}{c ^{2}})$
The problem states that the events (the bulbs flashing) are simultaneous in the platform frame S. This is our initial condition: $Δ t = 0$ .
Plugging this into the equation, we get:
$Δ t^{'} = γ (0 - \frac{v Δ x}{c ^{2}}) = - \frac{γ v Δ x}{c ^{2}}$

This is exactly the result we previewed last lesson! The Lorentz transformations mathematically confirm that a time separation $Δ t^{'}$ appears in the moving frame, and its value depends on the spatial separation $Δ x$ in the other frame.

A Deeper Perspective: Transformations as Rotations in Spacetime

Your background in linear algebra allows for a more elegant and powerful way to view the Lorentz transformations. We can express them using matrices. For the (t, x) coordinates (let's use $c t$ for consistent units), the transformation is:

(c t^{'} x^{'}) = (γ - γ β - γ β γ) (c t x)

where $β = v / c$ .

This looks similar to a rotation matrix. However, it's not a rotation in ordinary space. It's a "hyperbolic rotation" in four-dimensional spacetime. In ordinary rotations, the distance $Δ r^{2} = Δ x^{2} + Δ y^{2}$ is invariant. For Lorentz transformations, the invariant quantity is the spacetime interval:

$Δ s^{2} = (c Δ t)^{2} - (Δ x)^{2}$

This invariance is the geometric essence of special relativity. The following video gives a great intuitive explanation of this idea.

The Lorentz Transformations - Intuitive Explanation

This video, 'The Lorentz Transformations - Intuitive Explanation', contrasts Galilean transformations with Lorentz transformations and visualizes the latter as a type of rotation in Minkowski spacetime that preserves the spacetime interval.

Please watch from 09:44 to 13:43. This segment introduces the matrix form of the Lorentz transformation, contrasts it with a normal rotation, and explains the crucial idea of the invariant spacetime interval.

This concept of an invariant interval is profoundly important. It replaces the separate notions of absolute time and absolute distance with a single, unified, and absolute "spacetime distance." We will build on this idea extensively when we introduce four-vectors in the next module.

Conclusion

Today, we've equipped ourselves with the essential mathematical tool of special relativity. You now have the machinery to translate physical reality from one inertial frame to another.

Key Takeaways:

The Lorentz transformations are the correct set of equations for relating spacetime coordinates between inertial frames, replacing the incorrect Galilean transformations.
They are derived from Einstein's two postulates and ensure the speed of light is constant for all observers.
The transformations for time and space are intertwined: $t^{'}$ depends on both $t$ and $x$ , and $x^{'}$ depends on both $x$ and $t$ .
The equations are:
- $t^{'} = γ (t - v x / c^{2})$
- $x^{'} = γ (x - v t)$
These transformations can be viewed as hyperbolic rotations in spacetime that leave the spacetime interval $Δ s^{2} = (c Δ t)^{2} - (Δ x)^{2}$ invariant.

In our next lesson, we will use the Lorentz transformations you've just learned to formally derive time dilation, showing precisely why moving clocks are observed to run slow.