Physics Notes - Herong's Tutorial Notes - v3.22, by Dr. Herong Yang
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.
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
Table of Contents
Introducion of Frame of Reference
Introduction of Special Relativity
Time Dilation in Special Relativity
Length Contraction in Special Relativity
The Relativity of Simultaneity
►Minkowski Spacetime and Diagrams
What Is Lorentz Transformation?
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