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Looking for extra dimensions

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 three-dimensional. 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 r1-D. (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.

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