The dimension of a vector space is defined to be the smallest number of linearly independent elements that span the space.  Dimension is thus a measure of the number of independent objects within the space.

When we're talking about the dimensionality of the space we live in, we are referring to the dimension of the vector space of translations - motions in the physical world.

The statement that there are 3 spatial dimensions, then, is simply a statement that there are 3 independent directions of motion in the physical world.  These, of course, can be chosen to be, say, North/South, East/West, and Up/Down.  The physical correspondence to this is the statement that it is impossible to combine motion in the East/West directions and produce motion in the North/South direction.  These directions are therefore (linearly) independent of one another.  Since, from physical experience, we find that there are 3 independent directions of motion, we conclude that the dimension of "ordinary" space is 3.

The reason time gets brought into the picture as a 4th dimension is that the Lorentz transform of special relativity (which is required for electromagnetism to be consistent) mixes the time in a given frame with its spatial coordinates.  This indicates that the 3 spatial dimensions are not enough to describe motion in the physical world - we need to include time in order to produce the correct physics for electromagnetism and for relativistic mechanics.  We then reflect that time is certainly independent of the 3 spatial dimensions, because it is impossible to add up translations in the 3 spatial directions and produce a translation in time.  Therefore the dimension in which relativity operates is 4, and the Lorentz transform defines the algebra under which this 4-dimensional space operates (i.e. relativity).

Nicholas wrote:

As far as the first 3 spatial dimensions are concerned they are all alike. You can't tell one from the other. And these dimensions are expanding.

I'm not sure about the expanding part, but I believe this statement is correct.

What I'm interested in knowing, and maybe Matt can help me out here since he's studied this far more than this materials scientist, is the following: is the temporal dimension fundamentally different from the spatial dimensions in SR? I assume it is, and I am almost certain that I have read why in the past; however, I cannot recall.

Chris wrote:

is the temporal dimension fundamentally different from the spatial dimensions in SR? I assume it is, and I am almost certain that I have read why in the past; however, I cannot recall.

There are two properties in special relativity that I will refer to in answering your question.  The first is the form of the Lorentz transformation.   (As an aside: I'd like to write the transformation as a matrix; can the LaTex engine on this web page handle matrix commands?  If so, can you remind me what they are?  I think I need an \array command, but I think it spans multiple lines, which this engine doesn't like.)

The Lorentz transformation from a frame with coordinates , to an inertial frame with coordinates moving with velocity relative to is






The Lorentz transformation thus couples the timelike coordinates with the spacelike coordinates in the direction of motion.  In this sense, we can say that the timelike dimension is similar to the spacelike coordinates in that they are interchangeable (up to a point).  However, it is also different from the spatial coordinates because Lorentz boosts always transform the timelike coordinate; in this sense, it is a special coordinate.

The second property I will use is the metric in special relativity.  Since light is the obvious relativistic quantity (and its properties are the same in all frames), it is natural to use the trajectories followed by light to define a 4-dimensional length.  Since the speed of light is constant, the distance it travels in any frame (i.e., or ) will be



If we square both sides of this equation, we get



or



And we can write this as a 4-dimensional dot product between the differentials as



where we have defined the 4-dimensional dot product based on this to be



This definition for the dot product can be expressed in terms of a matrix , called the metric tensor, so that



where


  where

and all other components are zero.  This metric tensor, used in special relativity, is called the Minkowski metric.  An important side note is that we could equally well define the Minkowski metric with an overall change in sign, because we could have moved to the other side instead of above.  For whatever historical reasons, particle/high energy physicists tend to use the sign convention I have written above, while relativists tend to use the opposite sign convention.  Also, I have defined the timelike coordinate to be , so that the timelike part of the metric is .  If you insist that the timelike coordinate be , then the factor of appears in the component of the metric.

In defining the metric tensor based on the behavior of light, we have made a subtle but fundamental shift in our description of the 4-dimensional mechanics.  Essentially, our choice is a statement that the behavior of light is a reflection of the geometry of 4-dimensional space.  Equivalently, you could say that we have constructed a geometrical interpretation of the 4-dimensional space based on the behavior of light.  By this definition, we automatically have the result that light merely follows the innate trajectories of the 4-dimensional geometries, called geodesics. 

Our metric tells us that the geometry of the spatial dimensions is ordinary, Euclidean geometry, in which the 3-dimensional distance is



(note that there is an overall difference in minus sign based on my choice of sign convention).  The timelike component of the Minkowski metric differs in sign from the spacelike parts, however, so the timelike coordinates are fundamentally different from the spacelike coordinates in this sense.  Essentially, this says that the coordinates are all perfectly ordinary geometrical coordinates, but that the geometry of our 4-dimensional space is not Euclidean, but Minkowski geometry, which treats the timelike coordinate differently from the spacelike coordinates.

There are all kinds of interesting consequences of interpreting the behavior of light as an underlying 4-dimensional geometry.  One consequence is that, by definition, the 4-dimensional length (called the space-time interval) of any trajectory that light follows is automatically zero.  These are called null, or lightlike, trajectories for this reason.  This means that, in the context of ordinary geometry, the Minkowski metric is somewhat pathological, because it assigns zero length to certain nonzero vectors!  This in turn requires that 4-velocities actually have fixed length.  Thus the 4-momentum, which is mass times the 4-velocity, also has a fixed length :



or



which is the Einstein relation.

Another consequence of our geometrical interpretation is general relativity itself.  Since light is observed to bend around massive objects, our interpretation forces us to conclude that 4-dimensional space itself is distorted by the presence of mass.  Under minimal assumptions about the nature of this distortion, and under the constraints that the results reduce to classical gravitation in the nonrelativistic limit and that the results be expressed in a Lorentz-covariant form, we produce the Einstein equations

,

where is called the Einstein tensor and describes the curvature of spacetime and is the stress-energy tensor, whose elements are either mass, energy density, or energy flow.  (c.f. http://en.wikipedia.org/wiki/Einstein%27s_equations) Thus Einstein's equations tell us that the presence of either mass or energy density cause a curvature in 4-dimensional spacetime.  In the presence of stress-energy, the geodesics (which light follows) are no longer straight lines, but curved trajectories around the stress-energy.  This is a complete description of the gravitational lensing effect and its interpretation in the form of 4-dimensional geometry.

Something Nicolas said in another thread made me go back to my copy of "Exploring Black Holes" by Taylor and Wheeler, which has an excellent first and second chapter with respect to space-time metrics.

Now I just need to convince my department that we should offer a special topics introductory course on GR and that I should teach it. That way, I will be forced to learn all of this intersting stuff.

Black hole theory makes nonsense predictions at event horizons.

You mean General Relativity, since this is essentially "black hole" theory.

Please elaborate. Exactly what nonsense predictions are made?