Lorentz Transformation Equation

The Lorentz transformation equation transforms one spacetime coordinate frame to another frame which moves at a constant velocity relative to the other. The different axes in 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

What is the Lorentz Transformation?

Lorentz transformation helps you to understand the motion of an object from different moving perspectives in an inertial reference frame.

Statement of the Principle

Hendrik Lorentz developed transformation equations relating two different spacetime coordinate systems in an inertial reference frame relative to each other. The basis of Lorentz transformations are two laws:

1. Relativity Principle

2. The constancy of the speed of light

Space-Time

To understand Lorentz transformation, we first need to know what is spacetime and its coordinate system.

Space-time is a mathematical space whose coordinates must specify both space and time (four-dimensional coordinate system), unlike a three- dimensional coordinate system having x, y, and z axes. Thus, each point is specified by four coordinates, three spatial and one temporal in four-dimensional spacetime.

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Need of a Spacetime Coordinate System

Earlier, time was considered as an absolute quantity. Thus, if the observers agree on the time taken by the light to travel from one place to another, they would disagree on the distance (as space is not an absolute quantity) and therefore, would not agree on its speed.

As a result, the Theory of Relativity puts an end to time as an absolute quantity.

Therefore, the distance of an event can now be defined as a function of time.

d = (\[\frac{1}{2}\])c

Where,

d is the distance of the event

t is the time taken by a pulse to reach the event and reflect back

c is the speed of light

This theory has changed our ideas of space and time as different and independent of each other. Thus, there was a need to combine space and time into a single continuum.

World-line

The path of an object moving through a spacetime diagram is called a world line of that object. It is important to note that a world line in a spacetime diagram may not necessarily be the same shape as the path of an object through space.

For example- the graph for a car moving with uniform acceleration will no longer be a straight line in a velocity-time graph.

A world line from your reference frame will be a stationary straight line whose x coordinate will always be equal to zero.

Application of Lorentz Transformation

This topic encompasses the relative contraction and expansion of the spacetime coordinate system, which kept the worldline of the speed of light constant.

Lorentz transformation is an integral part of calculating various attributes of an object in motion observed from a different coordinate system.

Lorentz transformations include various transformations that help us understand the mechanics of a body in motion, and also gives us an insight into the topics of Length Contraction, Time Dilation, and Relative mass.

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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

Fun Facts on Lorentz Transformation

The world line of the speed of light is the only such path which 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.

FAQ (Frequently Asked Questions)

Q1. Why was there a Need to Develop Lorentz Transformation Equations?

A. When objects start approaching the speed of light Newtonian physics starts to collapse. For example, if we think of a hypothetical situation where the sun was to cease to exist at this very moment, according to Newton's law of gravitation, the earth would feel its impact instantaneously. However, we know it is not possible. Therefore, the effect should be felt only after 8 minutes.

In Galilean Transformations, the speed of light is perceived differently depending on the reference frame. We know that nothing ever travels faster than the speed of light. Thus, there was a contradiction which formulated the need to develop a different set of transformational equations. Also, earlier transformations included rotation, scaling, reflection, and shear, which were not sufficient to incorporate spacetime coordinate systems.

Q2. What are the Various Applications of the Lorentz Transformation Equation?

A. Lorentz transformation gives an intuitive insight into: –

Time dilation- The faster you move through spacetime, the more the relative spacetime graph contracts, which are similar to the theory that slower you move through time, the faster you move through space.

Relativity of mass- Mass of an object tends to approach infinity when its speed approaches the speed of light.

If you look at motion in the form of snapshots at various intervals of time, according to Lorentz transformation, the constant time snapshots from your perspectives may not necessarily be constant time snapshots from someone else’s perspective. This visualization is in correlation with our belief that time is not absolute.