Difference between revisions of "Frame of Reference Transformation"

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<center><math>\underline{\textbf{Navigation}}</math>
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[[4-momenta|<math>\vartriangleleft </math>]]
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[[VanWasshenova_Thesis#Moller_Scattering|<math>\triangle </math>]]
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=Frame of Reference Transformation=
 
Using the Lorentz transformations and the index notation,
 
Using the Lorentz transformations and the index notation,
  
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<center><math>dt^' \equiv  \frac{\partial t'}{\partial t} dt+\frac{\partial t'}{\partial x} dx + \frac{\partial t'}{\partial y} dy+ \frac{\partial t'}{\partial z} dz \Rightarrow dx^{'0} \equiv  \frac{\partial x^{'0}}{\partial x^0} dx^0+\frac{\partial x^{'0}}{\partial x^1} dx^1 + \frac{\partial x^{'0}}{\partial x^2} dx^2+ \frac{\partial x^{'0}}{\partial x^3} dx^3</math></center>
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<center><math>dt^{'} \equiv  \frac{\partial t'}{\partial t} dt+\frac{\partial t'}{\partial x} dx + \frac{\partial t'}{\partial y} dy+ \frac{\partial t'}{\partial z} dz \Rightarrow dx^{'0} \equiv  \frac{\partial x^{'0}}{\partial x^{0}} dx^{0}+\frac{\partial x^{'0}}{\partial x^{1}} dx^{1} + \frac{\partial x^{'0}}{\partial x^{2}} dx^{2}+ \frac{\partial x^{'0}}{\partial x^{3}} dx^{3}</math></center>
  
  
<center><math>dx^' \equiv  \frac{\partial t'}{\partial t} dt+\frac{\partial x'}{\partial x} dx + \frac{\partial x'}{\partial y} dy+ \frac{\partial x'}{\partial z} dz\Rightarrow dx^{'1} \equiv  \frac{\partial x^{'1}}{\partial x^0} dx^0+\frac{\partial x^{'1}}{\partial x^1} dx^1 + \frac{\partial x^{'1}}{\partial x^2} dx^2+ \frac{\partial x^{'1}}{\partial x^3} dx^3</math></center>
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<center><math>dx^{'} \equiv  \frac{\partial t'}{\partial t} dt+\frac{\partial x'}{\partial x} dx + \frac{\partial x'}{\partial y} dy+ \frac{\partial x'}{\partial z} dz\Rightarrow dx^{'1} \equiv  \frac{\partial x^{'1}}{\partial x^{0}} dx^{0}+\frac{\partial x^{'1}}{\partial x^{1}} dx^{1} + \frac{\partial x^{'1}}{\partial x^{2}} dx^{2}+ \frac{\partial x^{'1}}{\partial x^{3}} dx^{3}</math></center>
  
  
<center><math>dy^' \equiv  \frac{\partial y'}{\partial t} dt+\frac{\partial y'}{\partial x} dx + \frac{\partial y'}{\partial y} dy+ \frac{\partial y'}{\partial z} dz\Rightarrow dx^{'2} \equiv  \frac{\partial x^{'2}}{\partial x^0} dx^0+\frac{\partial x^{'2}}{\partial x^1} dx^1 + \frac{\partial x^{'2}}{\partial x^2} dx^2+ \frac{\partial x^{'2}}{\partial x^3} dx^3</math></center>
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<center><math>dy^{'} \equiv  \frac{\partial y'}{\partial t} dt+\frac{\partial y'}{\partial x} dx + \frac{\partial y'}{\partial y} dy+ \frac{\partial y'}{\partial z} dz\Rightarrow dx^{'2} \equiv  \frac{\partial x^{'2}}{\partial x^{0}} dx^{0}+\frac{\partial x^{'2}}{\partial x^{1}} dx^{1} + \frac{\partial x^{'2}}{\partial x^{2}} dx^{2}+ \frac{\partial x^{'2}}{\partial x^{3}} dx^{3}</math></center>
  
  
<center><math>dz^' \equiv  \frac{\partial z'}{\partial t} dt+\frac{\partial z'}{\partial x} dx + \frac{\partial z'}{\partial y} dy+ \frac{\partial z'}{\partial z} dz\Rightarrow dx^{'3} \equiv  \frac{\partial x^{'3}}{\partial x^0} dx^0+\frac{\partial x^{'3}}{\partial x^1} dx^1 + \frac{\partial x^{'3}}{\partial x^2} dx^2+ \frac{\partial x^{'3}}{\partial x^3} dx^3</math></center>
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<center><math>dz^{'} \equiv  \frac{\partial z'}{\partial t} dt+\frac{\partial z'}{\partial x} dx + \frac{\partial z'}{\partial y} dy+ \frac{\partial z'}{\partial z} dz\Rightarrow dx^{'3} \equiv  \frac{\partial x^{'3}}{\partial x^{0}} dx^{0}+\frac{\partial x^{'3}}{\partial x^{1}} dx^{1} + \frac{\partial x^{'3}}{\partial x^{2}} dx^{2}+ \frac{\partial x^{'3}}{\partial x^{3}} dx^{3}</math></center>
  
  
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Again, using a summation over the indicies
 
Again, using a summation over the indicies
  
<center><math>d\mathbf x^{'\mu}=\sum_{\mu=0}^3 \frac{\partial x^{'\mu}}{\partial x^{\mu}}d\mathbf x^{\mu}</math></center>
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<center><math>d\mathbf x^{'\mu}=\sum_{\nu=0}^3 \frac{\partial x^{'\mu}}{\partial x^{\nu}}d\mathbf x^{\nu}</math></center>
  
  
 
Using the Einstein convention
 
Using the Einstein convention
  
<center><math>d\mathbf x^{'\mu}= \frac{\partial x^{'\mu}}{\partial x^{\mu}}d\mathbf x^{\mu}</math></center>
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<center><math>d\mathbf x^{'\mu}= \frac{\partial x^{'\mu}}{\partial x^{\nu}}d\mathbf x^{\nu}</math></center>
  
  
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0 & 0 & 0 & -1
 
0 & 0 & 0 & -1
 
\end{bmatrix}</math></center>
 
\end{bmatrix}</math></center>
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<center><math>\underline{\textbf{Navigation}}</math>
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[[4-momenta|<math>\vartriangleleft </math>]]
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[[VanWasshenova_Thesis#Moller_Scattering|<math>\triangle </math>]]
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[[4-gradient|<math>\vartriangleright </math>]]
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</center>

Latest revision as of 19:04, 1 January 2019

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Frame of Reference Transformation

Using the Lorentz transformations and the index notation,

{t=γ(tvz/c2)x=xy=yz=γ(zvt)


[x0x1x2x3]=[γ(x0vx3/c)x1x2γ(x3vx0)]=[γ(x0βx3)x1x2γ(x3vx0)]


Where βvc

This can be expressed in matrix form as

[x0x1x2x3]=[γ00γβ01000010γβ00γ][x0x1x2x3]


Letting the indices run from 0 to 3, we can write

xμ=3ν=0(Λμν)xν


Where Λ is the Lorentz transformation matrix for motion in the z direction.


Using the Einstein convention, this can be written as

xμ=Λμνxν

If we take the 4-vector quantities to be on an infinitesimally small scale, then there exists a linear relationship between the transformation. Following the rules of partial differentiation,


dtttdt+txdx+tydy+tzdzdx0x0x0dx0+x0x1dx1+x0x2dx2+x0x3dx3


dxttdt+xxdx+xydy+xzdzdx1x1x0dx0+x1x1dx1+x1x2dx2+x1x3dx3


dyytdt+yxdx+yydy+yzdzdx2x2x0dx0+x2x1dx1+x2x2dx2+x2x3dx3


dzztdt+zxdx+zydy+zzdzdx3x3x0dx0+x3x1dx1+x3x2dx2+x3x3dx3


Expressing this in matrix form

[dx0dx1dx2dx3]=[x0x0x0x1x0x2x0x3x1x0x1x1x1x2x1x3x2x0x2x1x2x2x2x3x3x0x3x1x3x2x3x3][dx0dx1dx2dx3]


Again, using a summation over the indicies

dxμ=3ν=0xμxνdxν


Using the Einstein convention

dxμ=xμxνdxν


The Lorentz transformations are also invariant in that they are just a rotation, i.e. Det Λ=1. The inner product is preserved,


ΛμνημνΛνμ=ημν


[γ00γβ01000010γβ00γ][1000010000100001][γ00γβ01000010γβ00γ]T=[1000010000100001]


[γ2β2γ200001000010000γ2+β2γ2]=[1000010000100001]


[γ2(1β2)00001000010000γ2(1β2)]=[1000010000100001]


Where γ11β2


[γ2γ200001000010000γ2γ2]=[1000010000100001]



[1000010000100001]=[1000010000100001]


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