Invariant Spacetime Interval in Minkowski Diagram

This section provides a demonstration of the invariant spacetime interval property using a Minkowski diagram.

Another property of the Minkowski spacetime model is that the interval of two events is a constant in all intertial frames. This is called invariant spacetime interval.

The invariant spacetime interval property can be demonstrated by a simple thought experiment described below:

Now in Bob's frame:

So the interval between events E and O is a constant in both frames.

Minkowski Diagram Showing Spacetime Interval
Minkowski Diagram Showing Spacetime Interval

Of course, the invariant spacetime interval can be derived from the Lorentz transformation:

  X' = gamma*(      X - beta*c*T)       #3: Lorentz Transformation
c*T' = gamma*(-beta*X +      c*T)       #4: Lorentz Transformation
gamma = 1/sqrt(1-beta**2)               #7: "gamma" factor
beta  = v/c                             #8: "beta" factor

s = sqrt(-(c*T)**2+X**2)               #13: Interval: (X,cT) & (0,0)
s' = sqrt(-(c*T')**2+X'**2)            #14: Interval: (X',cT') & (0,0)

s' = sqrt(-(gamma*(-beta*X+c*T))**2+(gamma*(X-beta*c*T))**2)
                                       #15: merging #3 and #4 into #6

s' = sqrt(gamme**2*(-(-beta*X+c*T)**2+(X-beta*c*T)**2))
s' = sqrt((-(-beta*X+c*T)**2+(X-beta*c*T)**2)/(1-beta**2)

s' = sqrt((-((beta*X)**2 - 2*beta*X*c*T + (c*T)**2))
           +(X**2 - 2*X*beta*c*T + (beta*c*T)**2)))/(1-beta**2))

s' = sqrt(-(beta*X)**2 - (c*T)**2 + X**2 + (beta*c*T)**2)/(1-beta**2))
s' = sqrt((1-beta**2)*X**2 - (1-beta**2)*(c*T)**2)/(1-beta**2)
s' = sqrt(-(c*T)**2+X**2)

s' = s                                 #16: Interval is a constant

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

 PDF Printing Version