Projective Geometry is the superset of geometry that includes Euclidean and non-Euclidean geometries. It is more general than Affine geometry and is a subset of topology. A good book to review about this subject is "Fundamental Concepts of Geometry" by Bruce Meserve. His book covers Synthetic and Analytic Projective Geometry and how the different types of geometries relate to one another.
The concept I really like is geometry considered as a logical system. The book starts off with an example about student's and their membership into student councils. One can see the example as either that the memberships can be graphed as an abstraction of geometry or what the book states: that geometry is about items and their logical relationship between each other. In Minkowski type space, these would be events and their relations to each other.
My instinct for space time geometry is that their is no physical space. Events and their relations to neighboring events is what we see as space. The idea of drawing a line, marking off units, and then stating the line is real would be like stating that a graph of US population vs year and stating that the line on the graph is the definition of population. We all know that population consists of real people and that the line is an abstraction. My point is that I see the physical line drawn denoting distance is just as much of an abstraction. My reasoning is based on my positivist view in physics.
Events are connected to other events by their light cone. All these events have a proper distance of zero. This may seem philosophical but it is just a matter of quantum physics assumptions, an event is affected by every possible event on its light cone. So when an event occurs, all of history, and all of future, have to be added up to decide the event. In most cases, some events have more influence than others. We see the more influential events as being closer and more immediate in time. Other events seem more remote in time and further away as a neighbor.
The whole idea of Lorentz contraction and time dilation are just cases where the typical less influential contributor becomes a larger contributor due to circumstances.
Last Updated on November 29, 2000 by Bob Rutkiewicz