A Space-time Coordinate System: The Lorentz Transformations (Space)
The Lorentz Transformation (Part 1)
We have seen how different observers disagree on when and where events happen. We now want one mathematical framework which can relate the observations of different observers. This framework must encompass all the ideas we have seen before (length contraction, time dilation and simultaneity). The Lorentz Transformations do this. They relate coordinate systems of observers moving relative to each other. This means if you know the time and location of one event in space-time; you can say the time and location for another observer moving relative to you.
A coordinate system in space-time has 4 coordinates. We have found that we must specify the location of an event and the time at which it occurs. For the stationary frame we will use the coordinates (x,y,z,t) and for the moving frame we will use the coordinates (x',y',z',t') (a primed coordinate system). To simplify this however we will use one spacial dimension and so the coordinates will be (x,t) and (x',t').
(x' is said as "x prime")
(x' is said as "x prime")
Like in the length contraction post, we will refer to the stationary frame as team platform and the moving frame as team train. We will imagine them as follows:
Team platform (x,t) is a line of people all holding clocks to monitor the time at their point in space.
Team train (x',t') is the same except they are all moving relative to team platform at speed v.
Team platform (x,t) is a line of people all holding clocks to monitor the time at their point in space.
Team train (x',t') is the same except they are all moving relative to team platform at speed v.
Firstly we want to know where the event occurs. Where is x' in relation to x?
Team train consider themselves stationary. They see an event occur at x' and obviously claim it happens at x'.
What does team platform say?
This equation is the first Lorentz Transformation. If we know where an event (x) and when it happens (t); we can work out where it happens for a moving observer (x').
If we consider one moment in time, we get the length contraction formula. This is because length is the measurement of two points at the same point in time. We get
x' = gamma * x.
Gamma (also known as the Lorentz factor) is greater than or equal to 1
From team platform's POV the train is length contracted.
Multiply the length contracted distance that team platform see by gamma and you get the distance according to team train (who consider themselves stationary and so do not see a length contracted train).
If we want to look at many moments in time however this transformation makes it very easy to work out where the event is happening from different frames of reference.
It is also interesting to think about what Newton would have done. Using classical mechanics; the length contraction part of the formula would be missed out. This means
x = vt + x'
x' = x - vt
Classical mechanics misses out the Lorentz factor. At low speeds, the Lorentz factor is very close to 1 and so this approximation is adequate. At significant fractions of the speed of light, relativity is needed for accurate calculations.
Thanks for reading. If you enjoyed this post or any of my others, follow and subscribe to my blog. Feel free to discuss anything related to this post or ask questions in the comments below.
How To Slow Down Time (Time Dilation)
How To Travel To The Future
How Galilean Relativity And The Lorenz Maxwell Equations (The 2 Postulates) Contradict
The Postulates Of Special Relativity
Did you see my previous post? Click the link below to check it out
The Children Problem
Want to see my next post? Click the link below
A Space-time Coordinate System: The Lorentz Transformations (Time)
Team train consider themselves stationary. They see an event occur at x' and obviously claim it happens at x'.
What does team platform say?
This equation is the first Lorentz Transformation. If we know where an event (x) and when it happens (t); we can work out where it happens for a moving observer (x').
If we consider one moment in time, we get the length contraction formula. This is because length is the measurement of two points at the same point in time. We get
x' = gamma * x.
Gamma (also known as the Lorentz factor) is greater than or equal to 1
From team platform's POV the train is length contracted.
Multiply the length contracted distance that team platform see by gamma and you get the distance according to team train (who consider themselves stationary and so do not see a length contracted train).
If we want to look at many moments in time however this transformation makes it very easy to work out where the event is happening from different frames of reference.
It is also interesting to think about what Newton would have done. Using classical mechanics; the length contraction part of the formula would be missed out. This means
x = vt + x'
x' = x - vt
Classical mechanics misses out the Lorentz factor. At low speeds, the Lorentz factor is very close to 1 and so this approximation is adequate. At significant fractions of the speed of light, relativity is needed for accurate calculations.
Thanks for reading. If you enjoyed this post or any of my others, follow and subscribe to my blog. Feel free to discuss anything related to this post or ask questions in the comments below.
Check out my other posts on special relativity! (Link to all the posts)
How To Travel To The Future
How Galilean Relativity And The Lorenz Maxwell Equations (The 2 Postulates) Contradict
The Postulates Of Special Relativity
Did you see my previous post? Click the link below to check it out
The Children Problem
Want to see my next post? Click the link below
A Space-time Coordinate System: The Lorentz Transformations (Time)
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