2.2 Time as Another Dimension (Space-Time Diagrams)
Description
This article is from the Relativity and FTL Travel FAQ, by Jason W. Hinson jason@physicsguy.com with numerous contributions by others.
2.2 Time as Another Dimension (Space-Time Diagrams)
One of the first points to make as we begin discussing space-time
diagrams is that we are treating time as another dimension along with the
three dimensions of space. Generally, people aren't used to thinking of time
as just another dimension, but doing so allows us to truly understand how
relativity works. So, how do we represent time as just another dimension?
Obviously we can't actually picture four dimensions all at once (three
of space and one of time). Our minds are limited to picturing the three
dimensions of space that we are used to dealing with. However, we can
consider one or two dimensions of space and then use another dimension of
space to represent time.
To see how this can work, consider Diagram 2-1. There you see a film
strip on which each frame represents a moment in time. As you watch a film,
you see each moment in time presented one right after another, and this
gives the impression of seeing time pass. If we cut the film up into frames
then we can stack the frames flat, evenly spaced, and one on top of the
other (as shown in the diagram). Then each frame is a two dimensional
representation of space and as you move through the third dimension you go
up the stack, and each frame you pass represents another point in time.
Thus, we have a three dimensional stack which represents two dimensions of
space and the third dimension represents time.
Diagram 2-1
|o|---------|o|
|o|* |o| ___________
|o| |o| ^ /* /
|o| |o| 5 | / /
|o| |o| | / / 5
|o| |o| | / /
|o|---------|o| | /__________/
|o| |o| | ___________
|o| * |o| | / /
|o| |o| 4 | / * /
|o| |o| T | / / 4
|o| |o| | / /
|o|---------|o| I | /__________/
|o| |o| | ___________
|o| |o| M | / /
|o| * |o| 3 | / /
|o| |o| E | / * / 3
|o| |o| | / /
|o|---------|o| | /__________/
|o| |o| | ___________
|o| |o| | / /
|o| |o| 2 | / /
|o| * |o| | / / 2
|o| |o| | / * /
|o|---------|o| | /__________/
|o| |o| | ___________
|o| |o| | / /
|o| |o| 1 | / /
|o| |o| | / / 1
|o| *|o| | / /
|o|---------|o| /_________*/
Film
Note too that in the diagram the film shows a ball moving from one
corner of the screen to the other. However, in the three dimensional stack,
the ball now follows a three dimensional path through space-time. In four
dimensional space-time, objects which we see moving in time through three
dimensional space are following a four-dimensional path through space-time.
On space-time diagrams, paths you draw represent objects moving through
space as time passes, but we'll see more about that later in the chapter.
Further, consider an event such as "the ball reaches the far corner of
the screen." That is a single event--it occurs at one moment in time and at
one particular place in space. On our diagram, it is a single point (it is a
spot represented by the ball which is on the upper most frame in the stack).
Any single event which occurs is represented by a single point on a
space-time diagram.
And so, a space-time diagram gives us a means of representing events
which occur at different locations and at different times. Every event is
portrayed as a point somewhere on the space-time diagram.
Now, because of relativity, different observers which are moving
relative to one another will have different coordinates for any given event.
However, with space-time diagrams, we can picture these different coordinate
systems on the same diagram, and this allows us to understand how they are
related to one another.
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