Scientific Background - Modern Physics
World Geometry for Neuroscientists
Links to Relativity Sites
Lecture Notes on General Relativity - Sean M Carroll
Virginia Polytechnic institute - an excellent primer for beginners
Ohio State University - a more mathematical approach
George Mason University - notes for student revision
Quick tour of relativity theory
Georgia Institute of Technology - excellent "text book" intro
John Gribbins Home page - people as well as things
Chris Hillman's links and articles - from simple to advanced level of treatment
Philosophical musings on relativity:
Interesting analysis of separation formula and curvature
How to get rid of dimensions altogether?
The Internet Encyclopedia of philosophy - try the letter T for time or search for Einstein.
Dimensions
Einstein proposed his Special Theory of Relativity in 1905 but it was the development of the theory into a four-dimensional model of the universe in the succeeding decade or so that made the theory truly comprehensible.
The word "dimension" sounds very important and "sci-fi" but it is fairly straightforward. Dimensions are defined in a simple way. The relativistic method of defining dimensions starts with a small dot or point that is used as a probe. This can be moved and the succession of positions occupied by the point is called a line. A line is defined as a one dimensional object. Lines can be moved to create a plane, the plane is called a two dimensional object and the planes can be moved to create a volume which is a three dimensional object.
The dimensions are related by Pythagoras' theorem. If there are three dimensions, X, Y, and Z then these are related by:
h2 = X2 + Y2
Having calculated the distance of a point in two dimensions this can be used to calculate the distance of the point in three dimensions:
H2 = h2 + Z2
So that:
H2 = X2 + Y2 + Z2
This application of Pythagoras' theorem allows the position of any point to be described in three dimensions.
According to relativity theory time is a fourth dimension which is NEGATIVE. The relativistic time T is related to our normal time by the equation:
T = jct
where j = the square root of -1, c = the speed of light and t is the time interval. (Although the relation is sometimes left in terms of conjugates).
Applying Pythagoras' theorem:
H2 = X2 + Y2 + Z2 - (ct)2
The time subtracts from the other space dimensions. There has been much discussion of whether or not time is a "true" dimension but the issue is clear; any variable that obeys Pythagoras' theorem in a Cartesian coordinate system is a dimension. The only problem is how to interpret the dimensional nature of time.
The essence of the four dimensional nature of space-time lies in the fact that the time coordinate can subtract from the distance of a point from an observer. Space shrinks or expands depending upon the relative velocity of the observer and this is what Pythagoras' theorem is telling us when time is included.