Forest Relativity Notes

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Lorentz Transformations

The picture below represents the relative orientation of two different coordinate systems (S,S) . S is at rest (Lab Frame) and S is moving at a velocity v to the right with respect to frame S.

ForestRelativityLorentzFrame.jpg

The relationship between the coordinate(x,y,z,ct) of an object in frame S to the same object described using the coordinates (x,y,z,ct) in frame S is geven by the Lorentz transformation:

xμ=3ν=0Λμνxν

where

x0ct
x1x
x2y
x3z
Λ=[γγβ00γβγ0000100001]
β=vc
γ=11β2
example
x0=2ν=0Λ0νxν=Λ00x0+Λ01x1Λ02x2+Λ03x2
ct=γx0γβx1+0x2+0x3=γctγβx=γ(ctβx)
Or in matrix form the tranformation looks like
(ctxyz)=[γγβ00γβγ0000100001](ctxyz)
Note
Einstein's summation convention drops the symbols and assumes it to exist whenever there is a repeated subscript and uperscript
ie; xμ=Λμνxν
in the example above theν symbol is repeated thereby indicating a summation over ν.