Ignoring two dimensions and still getting the message...
First of all, read the spacetime notes to get the
necessary background. It is not convenient to plot more than two dimensions on
a flat screen so the spacetime diagram normally plots the x and t dimensions of
spacetime, with x horizontal and t vertical. Spacetime diagrams represent a
snapshot at a particular instant. Below is a basic spacetime diagram
showing two events in spacetime, one marking a commonly used zero on the time
axis, the other perhaps less widely known.
In addition to isolated events we can plot the events encountered by
an object on its journey through spacetime. The resulting path is called the
object's worldline. Shown below are the worldlines of three objects. The red
line represents an object at rest in the
reference frame
of this diagram. The blue object is moving left at a constant speed from the
positive x direction. The green object is moving right at a constant speed from
the negative x direction. Ignore the dashed lines through each object for the
time being.
Notice in the diagram above the horizontal line at a particular time, which
was the age of the universe at time of the snapshot this spacetime diagram
represents. The age of the universe might be a good candidate for
the time we call "now". The Big Bang
marks the beginning of time as we know it and since it happened everywhere at
once we should find the same age for the universe everywhere. This is confirmed
by the
cosmic microwave background radiation. It remains to be demonstrated that
this universal age is the time known to us as "now".
A more complex spacetime diagram would represent a satellite in orbit about
Earth. The satellite would be moving in three spatial dimensions as well as
drifting up the taxis at the speed of light. Our diagram only shows the x
component of the satellite's spatial position. The diagram refers to two clocks
one on the satellite and the other in my office. The range of motion of the
office is so small compared to that of the satellite that the office clock
appears to be moving right up the taxis. Again ignore the dashed lines and
the two star marked events A and B. We will open that particular can of worms
later.
Let's use spacetime diagrams to think about the nature of "now", that being
the only time in which anything can happen. The past is gone and the future has not
arrived so "now" is the only time available to carry out actions. Classically,
before we had to think about relativity effects, we thought of "now" as a slice
through time, separating past from future, sort of as shown below.
With understanding of the relativity principle
the picture got more complicated. With that we have to consider a thing called
a light cone, or null cone. Look at the spacetime diagram below. It is
arranged so that the same units are
applied to both the time and space axes, meters for example.
With time and distance both in meters, what would be the worldline of a
photon of light? Well it would be a line
up and to the right or left at a 45 degree angle. The above image shows a
spacetime diagram including the worldlines of four photons. One is outgoing to
the right from the green object sitting at the uptime terminus of its worldline.
One is outgoing to the left. One is incoming from the left. The last is
incoming from the right. The first two photon worldlines form a 90 degree angle
opening upward. The second two form a 90 degree angle opening downward. The
four photon worldlines divide spacetime into four regions. In the two dimensional
spacetime diagram we are using here, the regions are triangular. If we add a
dimension the regions are cones. Including the third spatial dimension the regions
are spheres. Because we used light photons to establish these regions we get
the name "light cone".
The region labeled "Past" contains all the volume of spacetime from which
events had any chance of causing an effect on the green object at its spacetime
location. Events outside the "Past" region simply did not have time to
communicate with the object, given the fixed speed of light demanded by
relativity. The region labeled "Future" contains the volume of spacetime where
any action by green object at its spacetime location might have caused an event.
Our object is too late to cause any events outside the "Future" area. The
two areas that are neither "Past" nor "Future" contain events that cannot have
an effect on our object, nor be affected by it. Those events are separated from
our object by a spacelike interval.
Because our purpose here is familiarity with spacetime diagrams, I will
leave the discussion of the ramifications of this for a later time... except
to mention that it appears in relativity that "now" is no longer a surface
in spacetime but a separate point for each object. That is the only place where
the object's past and future are joined.
