Difference between revisions of "Relativistic Frames of Reference"

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<center><math>speed=\frac{\Delta Distance}{\Delta Time}</math></center>
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<center><math>speed=\frac{Distance}{Time}</math></center>
  
  
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<center><math>c=\frac{\Delta d}{\Delta t}</math></center>
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<center><math>c=\frac{d}{t}</math></center>
  
  
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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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<center><math>c=\frac{\sqrt{x^2+y^2+z^2}}{t}</math></center>
  
  
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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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<center><math>c=\frac{\sqrt{x^{'2}+y^{'2}+z^{'2}}}{t}</math></center>
  
  
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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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<center><math>\frac{x^{'2}+y^{'2}+z^{'2}}{t^{'2}}= c^2=\frac{x^2+y^2+z^2}{t^2}</math></center>
  
  
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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>c^2 t^{'2}=x^{'2}+y^{'2}+z^{'2}\ \ \ \ \ c^2 t^{2}=x^2+y^2+ 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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<center><math>\Rightarrow c^2 t^{'2}- x^{'2}-y^{'2}-z^{'2}= c^2 t^{2}-x^2-y^2-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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This quantity is known as the time space interval <math>s^2</math>  
  
  
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<center><math>ds^2\equiv c^2 d t^{'2}-(d x')^2-(d y')^2-(d z')^2= c^2 d t^{2}-(d x)^2-(d y)^2-(d z)^2</math></center>
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<center><math>s^2\equiv c^2 t^{'2}-x^{'2}-y^{'2}-z^{'2}= c^2 t^{2}-x^2-y^2-z^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.
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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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Revision as of 01:19, 4 June 2017

[math]\textbf{\underline{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.


[math]t= t'[/math]
[math]x=x'[/math]
[math]y=y'[/math]
[math]z=z'+vt[/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{Distance}{Time}[/math]


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


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


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



We can rewrite this as


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



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


This quantity is known as the time space interval [math]s^2[/math]


[math]s^2\equiv c^2 t^{'2}-x^{'2}-y^{'2}-z^{'2}= c^2 t^{2}-x^2-y^2-z^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]\textbf{\underline{Navigation}}[/math]

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

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