Lorentz Transformation

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

1. 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.

2. The world line of the speed of light is always at an angle of 45° to the spacetime coordinate system.