Lorentz transformations

Lorentz transformations are the core of Albert Einstein's theory of special relativity. These transformations redefine the way we understand space and time, especially when we deal with objects moving close to the speed of light. Unlike classical physics, which relied on Newtonian transformations, Lorentz transformations take into account the constant speed of light in all inertial frames of reference. This revelation results in fascinating phenomena such as time dilation, length contraction, and the relativity of simultaneity. This lesson will provide in-depth information on Lorentz transformations using simple English, mathematical formulas in code blocks, and visual representations.

Understanding special relativity

The special theory of relativity was introduced by Albert Einstein in 1905. It is based on two primary principles:

• The laws of physics are the same in all inertial reference frames.
• The speed of light in a vacuum is the same for all observers, regardless of the speed of the light source.

These theories led to revolutionary ideas about the nature of space and time. In classical mechanics, time is considered absolute and independent of the observer. However, Einstein proposed that time and space are intertwined in a single continuum called spacetime, which changes according to the relative motion of the observer.

What are Lorentz transformations?

Lorentz transformations describe how measurements of space and time by two observers are related. When two observers move relative to each other at high speeds, close to the speed of light, they will not agree on measurements of time intervals or distances. Mathematically, Lorentz transformations provide equations to translate these differences between two inertial reference frames.

Basic formula

The Lorentz transformation connects the space and time coordinates of two observers, usually called the "stationary" observer and the "moving" observer. Let us denote the coordinates in the stationary frame as (t, x, y, z) and in the moving frame as (t', x', y', z'). If the relative velocity between these frames is v and the direction is along x axis, the transformations are given by:

t' = γ(t – vx/c²)
x' = γ(x – vt)
y' = y
z' = z
    

where γ (the Lorentz factor) is defined as:

γ = 1 / √(1 - v²/c²)
    

Here, c is the speed of light in vacuum.

Key results of the Lorentz transformation

Time extension

Time dilation means that time passes slower for a moving clock from the point of view of a stationary observer. Consider a spacecraft moving at velocity v relative to Earth. For each tick of the spacecraft's clock that takes t₀ seconds in its own frame (proper time), the Earth observer measures the passage of time as t, where:

t = γt₀
    

This implies that when v is not zero then t > t₀, indicating that time "dilates" or stretches for a moving observer.

Visual example of time dilation

Length contraction

Length contraction is the phenomenon in which the length of a moving object is measured shorter in the direction of motion relative to a stationary observer. If the proper length of an object, measured in its own rest frame, is L₀, then its length L when moving at velocity v is given by:

L = L₀/γ
    

This equation shows that the length of the object decreases as its velocity relative to the observer increases.

Visual example of length contraction

Relativity of simultaneity

With Lorentz transformations the concept of simultaneity becomes relative. Events that are simultaneous in one frame of reference may not be simultaneous in another. Consider two lightning strikes occurring at two different locations along x axis according to a stationary observer. If they are simultaneous in the stationary frame, they are not simultaneous in the moving frame due to the time transformation equation.

The difference in time for a moving observer is evident from the following:

t'₂ - t'₁ = γ((t₂ - vx₂/c²) - (t₁ - vx₁/c²)) 
    

If t₂ = t₁, which means they are simultaneous for the stationary frame, then the difference on the right-hand side becomes:

t'₂ - t'₁ = γv(x₂ - x₁)/c²
    

This shows that from the point of view of a moving observer, these two events are not simultaneous unless they occur at the same location in the stationary frame (x₂ = x₁).

Historical context and mathematical derivation

Before delving deeper into examples and visualizations, let's look at how the Lorentz transformations were derived historically. Dutch physicist Hendrik Lorentz and French scientist Henri Poincaré were among the first to formulate these transformations, aiming to reconcile Maxwell's equations of electromagnetism with the invariance of the speed of light. However, it was Einstein who gave physical significance to these transformations through his theories.

The transformations can be obtained by considering two inertial frames: one stationary and the other moving with constant velocity v along x axis. In order for light to maintain the same speed c in both frames, the coordinates must be transformed to ensure:

c²t² - x² - y² - z² = c²t'² - x'² - y'² - z'²
    

Satisfying these conditions leads to the standard form of the Lorentz transformations mentioned before.

Detailed calculation examples

Example 1: Twin paradox illustration

Imagine twins, Alice and Bob. Alice stays on Earth while Bob travels on a spaceship moving at high velocity relative to Earth. When Bob returns, he discovers that he has aged less than Alice due to time dilation.

Let’s calculate an example where Bob travels at 80% of the speed of light (0.8c) for 10 years according to his onboard clock (proper time, t₀).

Find Alice's time (Earth reference frame, t):

γ = 1 / √(1 - (0.8)²)
  = 1 / √(0.36)
  = 5/3 ≈ 1.667

t = γt₀
  = 1.667 * 10 years
  = 16.67 years
    

Alice is 16.67 years old, which means she will be 6.67 years older than Bob when he returns!

Example 2: Length contraction of the Earth

Suppose an astronaut is passing the Earth at a velocity of 99% of the speed of light (0.99c). Suppose the diameter of the Earth, the proper length, is 12,742 km.

The traveller measures the compressed length of the Earth in his frame as follows:

γ = 1 / √(1 - (0.99)²)
  = 1 / √(0.0199)
  = 7.089

L = L₀/γ
  = 12,742 km / 7.089
  = 1,797 km
    

To the traveller the diameter of the earth appears to have decreased to 1,797 km!

Maximizing understanding through further analysis

Although these transformations seem paradoxical, they highlight the flexible nature of spacetime. Everything from the electronics in GPS satellites to deep-space communications relies on Lorentz transformations and principles derived from special relativity.

The important concept of Lorentz transformations shows that our universe does not follow a rigid sense of space and time, but rather adapts depending on perspective. Whether thinking of time dilation as a slowly moving movie or length contraction as a crushed toy, transformations reveal the dynamic and complex nature of reality.

In conclusion, the Lorentz transformations fundamentally shake up previous notions of an absolute framework for space and time, and outline a flexible spacetime framework where distances shrink, clocks move at different speeds, and simultaneous events vary - all determined by the universal cosmic speed limit, the speed of light.

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