So i was reading my new book from amazon.com, World Enough And Space-Time: by John Earman, when I came across a chapter that I looked forward too when reading through the contents page. The book is a simple introduction to many of the common philosophical, historical, perspectives or arguments professed in the Substantialist-Relationist debate on the ontological status of space-time (Does it inhere in the objects we use to distinguish space or time or is there a substratum of Spatio-Temporal points independent of objects & their properties). This chapter was dipping into some of the various classical spacetime structures and used few but still enigma rich mathematical notation. The first is Machian Space-Time, as he put it. Here are a few equation questions i'm wondering: In defining the nature of Machian Space-Time he decides to introduce a symmetrical contravariant tensor field, Please Register or Log in to view the hidden image! . What is the purpose of a symmetrical contravariant tensor field? When would you decide to use it and how? What is its purpose? I have seen this kind of device before used in special relativity, Please Register or Log in to view the hidden image! with a signature of either (-,+,+,+) or (+,-,-,-). Than he mixes the simultaneity of time structure with Please Register or Log in to view the hidden image! which forces it to be this: Please Register or Log in to view the hidden image! , this I understand. Later, to make an appropriate metric over instantaneous spaces, he defines a covector Please Register or Log in to view the hidden image! coressponding to a space-like vector Please Register or Log in to view the hidden image! by, Please Register or Log in to view the hidden image! What is the difference between a covector and a regular vector? What makes a vector a space-like vector? Does it mean space-like because its components are only Please Register or Log in to view the hidden image! or is there some other mathematical reason? Then he wrote thatPlease Register or Log in to view the hidden image! was not unique since it solves the above equations so this does also solve it, Please Register or Log in to view the hidden image! for an arbitrary Please Register or Log in to view the hidden image!. What is exactly similar between these two, Please Register or Log in to view the hidden image! and Please Register or Log in to view the hidden image! ? One seems to look like it is a unit vector of sorts with the little arrow hat, but has a few other symbols tagged on for a certain unknown reason.
