Step-by-Step Derivation and Practical Applications
Lorentz transformation refers to the relationship between two coordinate frames that move at a constant speed and are relative to one another. It is named after a Dutch physicist, Hendrik Lorentz.
We can divide reference frames into two categories:
Frames of inertial motion - motion with a constant velocity
Non-inertial Frames - Rotational motion with constant angular velocity and acceleration in curved paths
Lorentz Transformation in Inertial Frame
A Lorentz transformation can only be used in the context of inertial frames, so it is usually a special relativity transformation. During the linear transformation, a mapping occurs between 2 modules that include vector spaces. The multiplication and addition operations on scalars are preserved when using a linear transformation. As a result of this transformation, the observer who is moving at different speeds will be able to measure different elapsed times, different distances, and order of events, but it is important to follow the condition that the speed of light should be equivalent across all frames of reference.
Lorentz Boost
It is also possible to apply the Lorentz transform to rotate space. A rotation free of this transformation is called Lorentz boost. This transformation preserves the space-time interval between two events.
The Statement of the Principle
The transformation equations of Hendrik Lorentz relate two different coordinate systems in an inertial reference frame. There are two laws behind Lorentz transformations:
Relativity Principle
Light's constant speed
Simplest Derivation of Lorentz Transformation
We will start by scaling Galilean transformations by Lorentz factor (γ) which is-
γ = \[\frac{1}{\sqrt{1 - \frac{v^{2}}{c^{2}}}}\]
γ = \[\frac{1}{\sqrt{1 - β^{2}}}\]
Galilean transformations of Newtonian transformations: -
t’=t
z’=z
y’=y
x’=x- vt
Here, x’, y’ , z’ and ct’ are the new coordinates. We need to transform from x to x’ and ct to ct’.
This implies, x’ = γ(x - βct)
And, ct’ = γ(ct - βx)
Extending it to 4 dimensions,
y’=y
z’=z
Another form of writing the equations, is to substitute β = \[\frac{v}{c}\]
γ = \[\frac{1}{\sqrt{1 - \frac{v^{2}}{c^{2}}}}\] = \[\frac{1}{\sqrt{1 - β^{2}}}\]
x’ = γ(x - ct\[\frac{v}{c}\])
x’ = γ(x - vt)
ct’ = γ(ct - βx)
ct’ = γ(ct - \[\frac{v}{c}\]x)
Dividing by c ,
\[\frac{ct’}{c}\] = γ(\[\frac{ct}{c}\] - \[\frac{vx}{c^{2}}\])
t’ = γ(t - \[\frac{vx}{c^{2}}\])
When , v << c , Then \[\frac{vx}{c^{2}}\] ≈ 0
and when γ is equal to 1,
t’ = γ(t - \[\frac{vx}{c^{2}}\]) becomes t’ ≈ t
x’ = γ(x - vt) becomes x’ = x - vt
Equation of the Lorentz Transformation
Lorentz transformations transform one frame of spacetime coordinates into another frame that moves at a constant speed relative to the other. The four axes of spacetime coordinate systems are x, ct, y, and z.
x’ = γ(x - βct)
ct’ = γ(ct - βx)
Extending it to 4 dimensions,
y’=y
z’=z
Space-Time
The concept of Lorentz transformation requires us to first understand spacetime and its coordinate system.
As opposed to three-dimensional coordinate systems having x, y, and z axes, space-time coordinates specify both space and time (four-dimensional coordinate system). The coordinates of each point in four-dimensional spacetime consist of three spatial and one temporal characteristic.
Need of a Spacetime Coordinate System
Earlier, time was viewed as an absolute quantity. Since space is not an absolute quantity, observers would disagree about the distance (thus, the observers would not agree about the speed of the light) even though they agree on the time it takes for the light to travel.
Consequently, time is no longer considered an absolute quantity due to the Theory of Relativity.
As a result, the distance between events can now be calculated as a function of time.
d = (1/2)c
Where,
d-distance of the event
t-time was taken by a pulse to reach the event and reflect back
c-speed of light
The theory of relativity has changed our understanding of space and time as separate and independent components. Therefore, space and time had to be combined into one continuum.
World-Line
The path that an object follows as it moves through a spacetime diagram is called its world line. Spacetime diagrams are important because world lines may not correspond to paths that objects traverse in space. For example, when a car moves with uniform acceleration, the graph in a velocity-time graph is no longer a straight line. In your reference frame, a world line is a stationary straight line whose x coordinate is always equal to zero.
Fun Facts about Lorentz Transformation
The world line of the speed of light is the only such path that does not change when followed by a series of contraction and expansion.
The world line of the speed of light is always at an angle of 45° to the spacetime coordinate system.
FAQs on Lorentz Transformation Explained for Students
1. What is meant by the Lorentz transformation in Physics?
The Lorentz transformation is a set of fundamental equations in special relativity used to relate the space and time coordinates of an event as measured by two different observers who are in relative motion. These equations replace the older Galilean transformations of classical physics, especially when dealing with velocities approaching the speed of light. They ensure that the speed of light is constant for all observers, a key postulate of relativity.
2. Why were the Lorentz transformation equations originally developed?
The Lorentz transformation was developed to resolve a major conflict between classical mechanics and Maxwell's equations of electromagnetism. Classical physics (using Galilean transformations) predicted that the speed of light would vary for different observers, which contradicted experimental evidence and Maxwell's theory that it is a universal constant. The Lorentz transformations were created to provide a consistent mathematical framework where the speed of light remains constant in all inertial frames of reference.
3. What is the main difference between Galilean and Lorentz transformations?
The primary difference lies in their treatment of time. The Galilean transformation assumes time is absolute and universal (t' = t) for all observers, regardless of their motion. In contrast, the Lorentz transformation treats time as relative, meaning that the measurement of a time interval can change depending on the observer's relative velocity. This leads to the concept of time dilation, where moving clocks run slower.
4. What are the standard equations for the Lorentz transformation?
For an inertial frame S' moving with a constant velocity v along the x-axis relative to another frame S, the coordinates are transformed as follows:
x' = γ(x - vt)
y' = y
z' = z
t' = γ(t - vx/c²)
Here, γ (gamma) is the Lorentz factor, defined as γ = 1 / √(1 - v²/c²), and c is the speed of light.
5. What are the most important physical consequences of the Lorentz transformation?
The Lorentz transformation leads to several profound and counter-intuitive predictions about the nature of space and time. The key consequences are:
Time Dilation: A clock that is in motion relative to an observer will be measured to tick slower than a clock at rest with the observer.
Length Contraction: An object will be measured to be shorter in its direction of motion compared to its length when at rest.
Relativity of Simultaneity: Two events that are simultaneous for one observer may not be simultaneous for another observer in a different inertial frame.
Velocity Addition: Velocities do not simply add up linearly. The relativistic velocity addition formula ensures that the resulting velocity never exceeds the speed of light.
6. Why must the Lorentz transformations be linear equations?
The transformations must be linear to preserve the principle of inertia. If they were non-linear, a particle moving in a straight line at a constant velocity in one inertial frame could appear to be accelerating in another. This would introduce 'fictitious forces' and violate the definition of an inertial frame of reference, which is a frame where an object with no forces acting on it moves in a straight line at a constant speed.
7. What quantity remains unchanged, or 'invariant', under a Lorentz transformation?
The quantity that remains invariant under a Lorentz transformation is the spacetime interval (often denoted as Δs²). While different observers will measure different values for space (Δx) and time (Δt) separations between two events, the combination (cΔt)² - (Δx)² remains the same for all observers in inertial frames. This is a fundamental concept in Minkowski spacetime and is analogous to how the distance between two points remains the same even if you rotate the coordinate system.
8. How do Lorentz transformations explain the concept of 'relativity of simultaneity'?
The relativity of simultaneity is a direct result of the time transformation equation: t' = γ(t - vx/c²). Consider two events that are simultaneous in frame S, meaning they happen at the same time (Δt = 0), but at different locations (Δx ≠ 0). When we calculate the time interval in frame S', the term -vx/c² becomes non-zero. This means Δt' will not be zero, proving that two events considered simultaneous in one frame are not necessarily simultaneous in another moving frame.
9. What happens to the Lorentz transformation equations at very low speeds (v << c)?
When the relative velocity 'v' is much smaller than the speed of light 'c', the ratio v²/c² becomes extremely small and approaches zero. In this case:
The Lorentz factor γ ≈ 1.
The term vx/c² in the time equation becomes negligible.
As a result, the Lorentz equations simplify to x' ≈ x - vt and t' ≈ t. These are precisely the Galilean transformation equations of classical mechanics. This demonstrates that special relativity incorporates classical mechanics as a low-velocity approximation.
