If we accept that as a consequence of
Einstein's postulates that time and space become relative then we
might be tempted to ask the question “can we construct a framework
incorporating time and space where we all agree on the 'distance'
between two events?”.
In order to incorporate both time and
space into the same framework we need a way of converting between
them. There is a saying loved by primary school maths teachers “you
can't add apples to oranges”, but maybe they are missing a subtle
point. You can add apples to oranges if you can find some way
of converting between them. If a grocer is selling apples for 30
cents and oranges for 10 cents, then I can buy three oranges for the
same price as 1 apple, or six oranges for the same price as 2 apples,
or 9 oranges for the same price as 3 apples etc. The ratio of the
value of apples to oranges is 3. If I were to construct a graph
relating the number of apples I could exchange for oranges it would
be a straight line, and the slope of this line would be 3. So that
the slope of the apples-oranges curve is the exchange rate or
conversion factor between the two.
What we need to find is an exchange
rate between distance and time. Well the ratio of distance to time
would be measured in metres per second so our exchange rate should be
a velocity. This is promising because the principle of relativity
implies that velocity is something we all agree on. If I see you
moving 10 m s-1 relative to my position, then you see me
moving at 10 m s-1 relative to your position. At the
moment we don't know what the value of this exchange rate velocity is
so we will just call it 'c' (note: at this stage we are making
no assumptions that c is the speed of light). So now to
convert distance, x, to time t we use this exchange rate c.
Actually, the idea of a mashup of space
and time, with velocity as an exchange rate, is not as weird as it
seems. In fact, we do it all the time. If you ask me how 'far' it is
from Coffs Harbour to Sydney I might reply “5 hours”. By which I
mean it takes about 5 hours which, using the average speed of a car
as the exchange rate, is about 400 km. Ask me the distance to Alpha
Centauri and I'll tell you it is about 4 light years. Still not
comfortable? Then take a look at this video from
minutephysics.
So now we have defined an exchange rate
we can start adding apples to oranges, or more specifically, plotting
events in spacetime. Now spacetime consists of four
dimensions: three space dimensions and one time dimension but I can't
draw 4 dimensions on a two dimensional screen (in fact if I am honest
I cant really imagine what four dimensions would look like) but there
is no problem describing it with maths. To represent it in two
dimensions and to keep the maths simple we will roll up the space
dimensions into one axis labelled x. This shouldn't worry you
too much if you remember that our aim is to come up with a way of
measuring the 'distance' between 'events' in spacetime, and if we
know the location of any two points in space we can measure or
calculate the distance between them to come up with a single value,
x, for the space component of their distance. For example, if
I constructed a three dimensional map of my office, the distance
between my chair and the bin can be represented by a single value of
2.1 m.
Enough babble, lets see what a
graphical representation of spacetime might look like. A spacetime
diagram has one axis representing space, x, and another
representing time, ct (remember that c is just the exchange
rate we need to use to convert time into the same units as space). On
this diagram we can plot some 'events' that occur at a particular
locations in space and time. Our first spacetime diagram has three
points, R, G and B. R lies directly to the left G so they both have
the same value for their time component but a different value for the
space component. In other words, events R and G occur at the same
time at different places. Similarly, G lies directly below B so we
can say that these two events occur at different times in the same
place.
We can even start to determine the
spacetime distance between these events. For example, I throw a
screwed up bit of paper into the bin 2.1 m away and at the instance
it lands in the bin (event G) I throw a second piece of paper (event
R) which lands in the bin 1 second later (event B). R and G occur in
different places at the same time so the spacetime distance between
them is simply the spatial distance, 2.1 m. G and B occur at the same
place but at different times so the spacetime distance between them
is simply the time difference, 1 s. But what is the spacetime distance between R and B? That is a much trickier question and to
answer that we need to develop the concept of spacetime in more
detail. That will be the task of our next exciting installment.
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