What is a dimension?
When we say that the space we live in has three dimensions,
what does that mean?
When we describe the size of an object, or of a space
like a room, we use three numbers: the height,
the width and the depth.
The height, width and depth of a room are numbers that can vary independently
from one another. That's one way to see that space
is three dimensional. Another way is that we need three numbers
to exactly locate ourselves on the Earth: longitude,
latitude and elevation
above sea level. That's another argument for space being threedimensional.
That's what we see.
When mathematicians or physicists talk about dimensions,
they mean the number of independent coordinates
needed to specify any point in a given space. The tradition is to label
these three coordinates (x,y,z), with z usually denoting the up direction
or the direction of height.
One of the big discoveries of early classical physics
was the similarity between the forces of gravity and electrostatics.
The gravitational force between two planets and the electrostatic force
between two electric charges were both observed to vary as the inverse
square of the distance between the two objects. So if r is your
distance from the center of a planet, then the gravitational force
of that planet on you will vary like r^{2}. If you go twice
as far away, the force will only be one fourth as strong.
But the number of coordinates in a mathematical equation
is easy to increase on paper. When the gravitational and electrostatic
equations are solved in a space with D dimensions,
then the force varies with distance like r^{1D}.
(Notice this gives the right answer when D=3.)
This gives physicists an interesting way to do fine
measurements of the numbers of dimensions of space. They can look at
the gravitational force and put quantitative limits on any funny behavior
that would come from possible extra dimensions.
If three space dimensions is consistent with current
gravitational physics and interior decorating, then why look more closely
at the force law? Because there are ways that extra dimensions of space
can become undetectable or at least very difficult to detect by our
world, so we can eat our cake and hide it, too, so to speak.
Why is time a dimension?
According to Isaac Newton, time was universal for
all objects no matter their motion relative to one another. This point
of view held until Einstein turned it on its head, because he was bothered
that it wasn't consistent with the propagation of light as electromagnetic
radiation.
Einstein's special theory
of relativity, which makes classical mechanics consistent with
classical electromagnetism, treats time
like a coordinate in a unified spacetime geometry.
If time is a coordinate, then instead of three
coordinates to describe a point in space,
we have four coordinates to describe
an event in spacetime. So that's what
is meant by saying that our spacetime has four dimensions. Usually we
label them (t,x,y,z).
Special relativity is an approximate theory that is
a good approximation when we can neglect the force of gravity and the
acceleration of observers in the system. Einstein's full theory of spacetime,
called general relativity, takes the
concept of a four dimensional spacetime and extends it to a curved
spacetime, where time and space make one united fabric that is
curved and stretched and twisted by the distribution of matter and energy
in the fabric.
From a mathematical point of view, both special and
general relativity can be extended easily to higher space dimensions.
If we have D dimensions of space and one time, then we say there are
d = D + 1 dimensions of spacetime. The equations
of motion can be solved and classified in d dimensions just like in
four spacetime dimensions.
Why have more dimensions?
It's not so hard to construct higher dimensional
worlds using the Einstein equations. But the question is then: WHY BOTHER?
It's because physicists dream
of a unified theory: a single mathematical framework in which
all fundamental forces and units of matter can be described together
in a manner that is internally consistent and consistent with current
and future observation.
And it turns out that having extra dimensions of space
makes it possible to build candidates for such a theory.
Extra dimensions in string theory
Superstring theory is a possible unified theory of
all fundamental forces, but superstring theory requires a 10 dimensional
spacetime, or else bad quantum states called ghosts with unphysical
negative probabilities become part of the spectrum.
Now this creates a problem in d=10 string theory:
how to get the d=4 world as we know it out of the theory.
So far there are two main proposals:
1. Roll up the extra dimensions into some very tiny but nonetheless
interesting space of their own. This is called Kaluza
Klein compactification.
2. Make the extra dimensions really big, but constrain all the matter
and gravity to propagate in a three dimensional subspace called the
three brane. (For an analogy, your computer
screen could be said to be a two brane of three dimensional space.)
These types of theories are called braneworlds.
Next:
Superstrings and braneworlds>>
