As we have seen, the Lorentz transformation is not
a conventional rotation of space into time
(and vice versa), even though it seems to come close.
What exactly makes the analogy fail, and how can we make it
seem ``sensible" -- , how can we remodel our common sense
to make room for this new inhabitant?
Since we are trying to reconcile fact with intuition here,
it makes sense to address the part of our brain where
intuition supposedly resides: the right hemisphere.
Since this is also supposedly the part that processes
visual images and ``big pictures," it stands to reason
that a geometrical formulation should work best.
This seems indeed to be the case.
But first we need to carefully define a few more terms.
In plane geometry (and rotations) we identify each point in the plane as just that: a point {x, y} in the x-y plane. When we talk about Lorentz transformations, however, we draw a ``plane" in which one axis is the distance from an origin along some particular direction (usually x) but the other axis is (the speed of light times) the time since some starting instant (t=0). What should we call a ``point" in this ``plane" if we want to have a mnemonically useful terminology? You may have noted that the word ``event" keeps cropping up in any discussion of the . This is exactly the term we need, since it specifies where and when something happens. Our x-ct ``plane" represents a sort of ``event space" in which each ``point" is an EVENT.
Of course, just as in plane geometry, we have left something out:
the other dimensions, in this case two!
As we know from MECHANICS, physics takes place in
three spatial dimensions
{x, y, z}
and a proper description of rotations must take this into account;
this gets complicated, and we usually try to find a way
to describe rotations relative to a fixed axis
whenever we can, to keep the picture simple.
But sometimes we have to visualize things in three dimensions,
which is not too hard (except when we are trying to draw it
on a flat sheet of paper) because we see the world
in three dimensions.
When we include time as a ``fourth dimension,"
visualization is not so easy. So-called spacetime
{ct, x, y, z},
also known as Minkowski space,
is pretty hard to sketch on a flat sheet of paper.
That is why we almost always draw it in the simplest version
(as shown in Fig.
,
for example),
with just one of the spatial dimensions included.
This form still preserves the essential features
of spacetime geometry, but you should always bear in mind
that there are another two spatial dimensions lurking
behind the scenes.