What Is Lorentz Transformation?

This section provides an introduction of Lorentz Transformation, which maps coordinates of an event in a stationary frame to a moving frame under the special theory of relativity.

Now we can look at the second contribution from Hermann Minkowski in constructing a geometric model called Minkowski diagram to support the special theory of relativity. Minkowski diagram is closely related to Lorentz Transformation. So let's start with the Lorentz Transformation first.

What Is Lorentz Transformation? Lorentz Transformation is a transformation formula that maps coordinates of an event in a stationary frame to a moving frame under the special theory of relativity.

We can use the moving train thought experiment to present a simplified version of Lorentz Transformation:

For the above example, the Lorentz Transformation for a given event E from Bob's point of view can be expressed as below:

  X' = gamma*(      X - beta*c*T)           #3: Lorentz Transformation
c*T' = gamma*(-beta*X +      c*T)           #4: Lorentz Transformation
   # Observed from Bob's stationary frame

where: 
   (X',cT')                                 #5: Event E in Amy's frame
   (X,cT)                                   #6: Event E in Bob's frame
   gamma = 1/sqrt(1-beta**2)                #7: "gamma" factor
   beta  = v/c                              #8: "beta" factor

For example, assuming the train is moving at the speed of 0.6*c, an event E at (X,cT)=(5.5,6.5) on Bob's light cone would be observed by Bob in Amy's frame as (X',cT')=(2,4) based on Lorentz transformation #3 and #4.

beta = v/c = 0.6*c/c = 0.6
gamma = 1/sqrt(1-beta**2) = 1.25

  X' = 1.25*(     5.5 - 0.6*6.5)            #9: from #3 
c*T' = 1.25*(-0.6*5.5 +     6.5)           #10: from #4

  X' = 2                                   #11: from #9
c*T' = 4                                   #12: from #10
Lorentz Transformation between Two Frames
Lorentz Transformation between Two Frames

Last update: 2014.

Table of Contents

 About This Book

 Introduction of Space

 Introducion of Frame of Reference

 Introducion of Time

 Introduction of Speed

 Newton's Laws of Motion

 Introduction of Special Relativity

 Time Dilation in Special Relativity

 Length Contraction in Special Relativity

 The Relativity of Simultaneity

 Introduction of Spacetime

Minkowski Spacetime and Diagrams

 What Is Minkowski Spacetime?

 Events Interval and Relations

What Is Lorentz Transformation?

 What Is Minkowski Diagram?

 Constancy of Speed of Light in Minkowski Diagram

 Time Dilation in Minkowski Diagram

 Length Contraction in Minkowski Diagram

 Relativity of Simultaneity in Minkowski Diagram

 Invariant Spacetime Interval in Minkowski Diagram

 Multiple Reference Frames in Minkowski Diagram

 References

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