Herong's Notes on Physics - Version 3.21, by Dr. Herong Yang
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
Last update: 2014.
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