Difference between revisions of "Relativistic Frames of Reference"

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<center><math> \underline{\textbf{Navigation} }</math>
<center><math>\textbf{\underline{Navigation}}</math>
 
  
 
[[Uniform_distribution_in_Energy_and_Theta_LUND_files|<math>\vartriangleleft </math>]]
 
[[Uniform_distribution_in_Energy_and_Theta_LUND_files|<math>\vartriangleleft </math>]]
[[VanWasshenova_Thesis#Weighted_Isotropic_Distribution_in_Lab_Frame|<math>\triangle </math>]]
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[[VanWasshenova_Thesis#Moller_Scattering|<math>\triangle </math>]]
[[Phase_space_Limiting_Particles|<math>\vartriangleright </math>]]
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[[Relativistic_Units|<math>\vartriangleright </math>]]
  
 
</center>
 
</center>
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From the Galilean description of motion for a frame of reference moving relative to another frame considered stationary we know that
 
From the Galilean description of motion for a frame of reference moving relative to another frame considered stationary we know that
  
<center><math>t \equals t'</math></center>
 
<center><math>x=x'</math></center>
 
<center><math>y=y'</math></center>
 
<center><math>z=z'+vt</math></center>
 
  
  
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<center>[[File:GalileanFrames.png|thumb|center|500px|alt=Galilean Frames of Reference|'''Figure 2.1:''' Primed reference frame moving in the z direction with velocity v.  ]]</center>
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In the rest frame of v=0
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<center><math>v=0 \Rightarrow \begin{cases}
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t= t' \\
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x=x' \\
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y=y' \\
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z=z'+vt'
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\end{cases}</math></center>
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While conversely, from the rest frame of v'=0
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<center><math>v'=0 \Rightarrow \begin{cases}
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t'= t \\
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x'=x \\
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y'=y \\
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z'=z-vt
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\end{cases}</math></center>
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Using Einstein's Theory of Relativity, we know that the speed of light is a constant, c, for all reference frames.  In the unprimed frame, from the definition of speed:
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<center><math>speed=\frac{\Delta Distance}{\Delta Time}</math></center>
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<center><math>c=\frac{\Delta d}{\Delta t}</math></center>
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where
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<center><math>c=3\times 10^8\ m/s</math></center>
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Using the distance equation in a Cartesian coordinate system, the equation for the speed of light becomes
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<center><math>c=\frac{\sqrt{\Delta x^2+\Delta y^2+\Delta z^2}}{\Delta t}</math></center>
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Following the postulate of Special Relativity, this implies for the primed frame
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<center><math>c=\frac{\sqrt{\Delta x^{'2}+\Delta y^{'2}+\Delta z^{'2}}}{\Delta t}</math></center>
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We can rewrite this as
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<center><math>\frac{\Delta x^{'2}+\Delta y^{'2}+\Delta z^{'2}}{\Delta t^{'2}}= c^2=\frac{\Delta x^2+\Delta y^2+\Delta z^2}{\Delta t^2}</math></center>
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This is possible since the ratios of distance to time are multiples of the same base, i.e. the square of the speed of light <math>(\frac{3\times 10^8\ m}{s})^2</math>.  Therefore for the relative change in the time in one frame, the distance must change by the same factor to maintain the same constant.  With this we can write
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<center><math>c^2  \Delta t^{'2}=\Delta x^{'2}+\Delta y^{'2}+\Delta z^{'2}\ \ \ \ \ c^2  \Delta t^{2}=\Delta x^2+\Delta y^2+\Delta z^2</math></center>
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<center><math>\Rightarrow c^2 \Delta t^{'2}-\Delta x^{'2}-\Delta y^{'2}-\Delta z^{'2}= c^2 \Delta t^{2}-\Delta x^2-\Delta y^2-\Delta z^2</math></center>
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This quantity is known as the time space interval <math>ds^2</math> when the change is infinitesimal
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<center><math>ds^2\equiv c^2 dt^{'2}-dx^{'2}-dy^{'2}-dz^{'2}= c^2 dt^{2}-dx^2-dy^2-dz^2</math></center>
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Since the speed of light is a constant for all frames of reference, this allows the space time interval to also be invariant for inertial frames.
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<center><math>ds^2\equiv c^2 dt^{'2}-dx^{'2}-dy^{'2}-dz^{'2}= c^2 dt^{2}-dx^2-dy^2-dz^2</math></center>
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<center><math>ds^2\equiv c^2 dt^{'2}-dr^{'2}= c^2 dt^{2}-dr^2</math></center>
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<center><math>ds^2\equiv (c^2 -v^{'2})dt^{'2}= (c^2 -v^2)dt^{2}</math></center>
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From the rest frame of v'=0
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<center><math>ds^2\equiv c^2 dt^{'2}= (c^2 -v^2)dt^{2}</math></center>
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<center><math>\Rightarrow dt^{'2}= (1-\frac{v^2}{c^2 })dt^{2}</math></center>
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<center><math>\Rightarrow \begin{cases}
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dt'= \sqrt{1-\frac{v^2}{c^2 }}dt=\frac{1}{\gamma} dt\\
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\\
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dt= \frac{1}{\sqrt{1-\frac{v^2}{c^2 }}}dt'=\gamma dt'
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\end{cases}</math></center>
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<center><math>\Rightarrow \begin{cases}
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t'=\frac{1}{\gamma} t\\
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\\
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t=\gamma t'
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\end{cases}</math></center>
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Assuming motion is only along the z direction
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<center><math>z \equiv ct\ \ \ \ z'\equiv ct'</math></center>
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Substituting these changes into the Galilean transformations
  
  
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<center><math>\underline{\textbf{Galilean\ Transformations}}\quad \underline{\textbf{Lorentz\ Transformations}}</math></center>
  
<center>[[File:GalileanFrames.png|thumb|center|500px|alt=Galilean Frames of Reference|'''Figure 2.1:''' Primed reference frame moving in the z direction with velocity v.  ]]</center>
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<center><math>
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v=0\Rightarrow
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\begin{cases}
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t= t' \\
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x=x' \\
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y=y' \\
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z=z'+vt'
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\end{cases}
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\qquad \Rightarrow \qquad \quad
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\begin{cases}
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t=\gamma (t'+vz'/c^2) \\
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x=x' \\
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y=y' \\
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z=\gamma (z'+vt')
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\end{cases}
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</math></center>
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<center><math>v'=0\Rightarrow
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\begin{cases}
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t'= t' \\
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x'=x' \\
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y'=y' \\
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z'=z-vt
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\end{cases}
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\qquad \Rightarrow \qquad \quad
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\begin{cases}
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t'=\gamma (t-vz/c^2) \\
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x'=x' \\
  
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y'=y' \\
  
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z'=\gamma (z-vt)
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\end{cases}
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</math></center>
 
----
 
----
  
  
  
<center><math>\textbf{\underline{Navigation}}</math>
+
<center><math> \underline{\textbf{Navigation} }</math>
  
 
[[Uniform_distribution_in_Energy_and_Theta_LUND_files|<math>\vartriangleleft </math>]]
 
[[Uniform_distribution_in_Energy_and_Theta_LUND_files|<math>\vartriangleleft </math>]]
[[VanWasshenova_Thesis#Weighted_Isotropic_Distribution_in_Lab_Frame|<math>\triangle </math>]]
+
[[VanWasshenova_Thesis#Moller_Scattering|<math>\triangle </math>]]
[[Phase_space_Limiting_Particles|<math>\vartriangleright </math>]]
+
[[Relativistic_Units|<math>\vartriangleright </math>]]
  
 
</center>
 
</center>

Latest revision as of 18:45, 15 May 2018

[math] \underline{\textbf{Navigation} }[/math]

[math]\vartriangleleft [/math] [math]\triangle [/math] [math]\vartriangleright [/math]

Relativistic Frames of Reference

From the Galilean description of motion for a frame of reference moving relative to another frame considered stationary we know that


Galilean Frames of Reference
Figure 2.1: Primed reference frame moving in the z direction with velocity v.


In the rest frame of v=0

[math]v=0 \Rightarrow \begin{cases} t= t' \\ x=x' \\ y=y' \\ z=z'+vt' \end{cases}[/math]

While conversely, from the rest frame of v'=0

[math]v'=0 \Rightarrow \begin{cases} t'= t \\ x'=x \\ y'=y \\ z'=z-vt \end{cases}[/math]



Using Einstein's Theory of Relativity, we know that the speed of light is a constant, c, for all reference frames. In the unprimed frame, from the definition of speed:


[math]speed=\frac{\Delta Distance}{\Delta Time}[/math]


[math]c=\frac{\Delta d}{\Delta t}[/math]


where

[math]c=3\times 10^8\ m/s[/math]

Using the distance equation in a Cartesian coordinate system, the equation for the speed of light becomes


[math]c=\frac{\sqrt{\Delta x^2+\Delta y^2+\Delta z^2}}{\Delta t}[/math]


Following the postulate of Special Relativity, this implies for the primed frame


[math]c=\frac{\sqrt{\Delta x^{'2}+\Delta y^{'2}+\Delta z^{'2}}}{\Delta t}[/math]



We can rewrite this as


[math]\frac{\Delta x^{'2}+\Delta y^{'2}+\Delta z^{'2}}{\Delta t^{'2}}= c^2=\frac{\Delta x^2+\Delta y^2+\Delta z^2}{\Delta t^2}[/math]


This is possible since the ratios of distance to time are multiples of the same base, i.e. the square of the speed of light [math](\frac{3\times 10^8\ m}{s})^2[/math]. Therefore for the relative change in the time in one frame, the distance must change by the same factor to maintain the same constant. With this we can write


[math]c^2 \Delta t^{'2}=\Delta x^{'2}+\Delta y^{'2}+\Delta z^{'2}\ \ \ \ \ c^2 \Delta t^{2}=\Delta x^2+\Delta y^2+\Delta z^2[/math]



[math]\Rightarrow c^2 \Delta t^{'2}-\Delta x^{'2}-\Delta y^{'2}-\Delta z^{'2}= c^2 \Delta t^{2}-\Delta x^2-\Delta y^2-\Delta z^2[/math]


This quantity is known as the time space interval [math]ds^2[/math] when the change is infinitesimal


[math]ds^2\equiv c^2 dt^{'2}-dx^{'2}-dy^{'2}-dz^{'2}= c^2 dt^{2}-dx^2-dy^2-dz^2[/math]


Since the speed of light is a constant for all frames of reference, this allows the space time interval to also be invariant for inertial frames.

[math]ds^2\equiv c^2 dt^{'2}-dx^{'2}-dy^{'2}-dz^{'2}= c^2 dt^{2}-dx^2-dy^2-dz^2[/math]


[math]ds^2\equiv c^2 dt^{'2}-dr^{'2}= c^2 dt^{2}-dr^2[/math]


[math]ds^2\equiv (c^2 -v^{'2})dt^{'2}= (c^2 -v^2)dt^{2}[/math]


From the rest frame of v'=0


[math]ds^2\equiv c^2 dt^{'2}= (c^2 -v^2)dt^{2}[/math]


[math]\Rightarrow dt^{'2}= (1-\frac{v^2}{c^2 })dt^{2}[/math]


[math]\Rightarrow \begin{cases} dt'= \sqrt{1-\frac{v^2}{c^2 }}dt=\frac{1}{\gamma} dt\\ \\ dt= \frac{1}{\sqrt{1-\frac{v^2}{c^2 }}}dt'=\gamma dt' \end{cases}[/math]


[math]\Rightarrow \begin{cases} t'=\frac{1}{\gamma} t\\ \\ t=\gamma t' \end{cases}[/math]


Assuming motion is only along the z direction


[math]z \equiv ct\ \ \ \ z'\equiv ct'[/math]



Substituting these changes into the Galilean transformations


[math]\underline{\textbf{Galilean\ Transformations}}\quad \underline{\textbf{Lorentz\ Transformations}}[/math]
[math] v=0\Rightarrow \begin{cases} t= t' \\ x=x' \\ y=y' \\ z=z'+vt' \end{cases} \qquad \Rightarrow \qquad \quad \begin{cases} t=\gamma (t'+vz'/c^2) \\ x=x' \\ y=y' \\ z=\gamma (z'+vt') \end{cases} [/math]


[math]v'=0\Rightarrow \begin{cases} t'= t' \\ x'=x' \\ y'=y' \\ z'=z-vt \end{cases} \qquad \Rightarrow \qquad \quad \begin{cases} t'=\gamma (t-vz/c^2) \\ x'=x' \\ y'=y' \\ z'=\gamma (z-vt) \end{cases} [/math]


[math] \underline{\textbf{Navigation} }[/math]

[math]\vartriangleleft [/math] [math]\triangle [/math] [math]\vartriangleright [/math]