How To REALLY Draw Yourself: Space-time Diagrams
Space-time Diagrams
In previous posts, we established that we actually live in 4-dimensional space-time and we derived the equations used to describe the location of events in space-time. In this post we will look at how we can show this on a diagram. This will be useful in future posts looking at causality, paradox's and the hyperbolic functions.
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| 4-dimensional space-time |
In the post on space-time, we established that reality is like a movie and we experience reality as the sequence of frames in the movie. A space-time diagram would let us view all the frames of the movie all at once. It would let us show all the places an object would be during the movie on one picture. This is useful because if we were to simply watch the movie, we would only see each object in space at one moment in time. Once that moment in time is gone, we lose where the object previously was. Viewing the movie only allows us to see each object at a single moment in time.
A proper space-time diagram has 3 spacial dimensions and a time dimension. This is impossible to illustrate because we only only can visualize the 3 dimensions we live in. This means we have to simplify the diagram to 2 or 1 spacial dimension(s) with a time dimension. In the examples below, we will use diagrams with 1 axis for space (1 spacial dimension) and 1 for time.
In space-time diagrams, the units make the speed of light = 1. If seconds are used on the time axis then light seconds are used on on the space axis.
This means when 1 second passes, light travels 1 light second and so its speed is 1. This means we can draw light rays 450. This is purely for convenience.
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| A space-time diagram showing how objects moving at different velocities can be represented. We call the line representing an object its world line. |
What About Other Observers?
Now we want to work out how to draw the space-time diagram of a moving observer.Any observer moving at constant velocity can consider themselves to be stationary while everything else moves past them. This means a moving observer can say they are always at their origin. This means from their perspective, they are always at x' = 0.
Now consider that for the stationary observer, they are always at x = 0. The line x = 0 is is the same as their time axis.
This means that for the moving observer, they are always at x' = 0 and this is the same as their time axis. This means on the time axis of a moving observer is the same as their world-line.
It seems like the time axis of a moving observer is given by rotating the time axis of the stationary observer.
You therefore may think that the space axis of a moving observer is given by rotating the space axis of the stationary observer by the same amount the time axis was rotated by.
This would not work however because this would change the speed of light. To see this just rotate your head and view the space-time diagram. You can see the angle representing the light beam has changed. This means it is travelling a different distance in any given time according the moving observer. This means the speed of light changes for the moving observer which contradicts special relativity's second postulate.
How do we draw the x' axis? What is the exact equation for the t' axis?
The x' axis is where t' = 0. This represents a single moment in time for the moving observer. Using the Lorentz transformations we can work out how to draw the x' axis on a space-time diagram.
t' = γ(t - x(v/c2)
0 = γ(t - x(v/c2)
t - xv/c2 = 0
t = vx/c2
But remember the clever thing we did with the units of the speed of light? We made c = 1 so we can simplify this further
t = vx
We know that the speed of the observer (v) is less than light ie it is less than one
This means this line has a gradient less than one and we can easily draw this on the diagram below
Another thing to note is that you can redraw the diagram from the moving frames perspective and you would have the symmetry we often see in relativity. The stationary observer would appear to have axis that are kind of squashed.
The diagram is derived from the Lorentz transformations and so embodies the ideas of time dilation, length contraction and simultaneity.
The strange rotation of the axis is linked to the hyperbolic functions (sinh, cosh, tanh). I am learning about those later in the year in further mathematics and so we will revisit and relate the Lorentz transformations, space-time diagrams and hyperbolic functions. We will find that the Lorentz transformations can actually be written in terms of these functions.
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
t - xv/c2 = 0
t = vx/c2
But remember the clever thing we did with the units of the speed of light? We made c = 1 so we can simplify this further
t = vx
We know that the speed of the observer (v) is less than light ie it is less than one
This means this line has a gradient less than one and we can easily draw this on the diagram below
Things To Note
The axis are not orthogonal (at right angles). This is key because postulate 2 says that c is invariant and if they were at right angles; c would change between reference frames.Another thing to note is that you can redraw the diagram from the moving frames perspective and you would have the symmetry we often see in relativity. The stationary observer would appear to have axis that are kind of squashed.
The diagram is derived from the Lorentz transformations and so embodies the ideas of time dilation, length contraction and simultaneity.
Time Dilation
If you look at the event on the diagram, you can see time dilation visually. The distance between t and the origin is much greater than the distance between t' and the origin. This more time passes in the stationary frame than in the moving frame i.e. time is slow in the moving frame.Simultaneity
The differing notions of simultaneity is shown by the angle of the space axis. The space axis shows all of space a single moment of time. All of space at a single moment in time clearly is different for the two observers. This means occurs in space at each moment in time depends on the frame.Length Contraction
Length contraction is shown when you consider the distance between the origins and the event. The distance between the event and the origin is cleary much greater in the stationary frame. That means the distance appears much shorter in the moving frame i.e. the moving observer sees the distance between the origin and the event as length contracted.The Hyperbolic Functions
The strange rotation of the axis is linked to the hyperbolic functions (sinh, cosh, tanh). I am learning about those later in the year in further mathematics and so we will revisit and relate the Lorentz transformations, space-time diagrams and hyperbolic functions. We will find that the Lorentz transformations can actually be written in terms of these functions.
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
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