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These terms refer to how quantites transform when you move from one coordinate system to another. Einstein realized that gravitation was caused by a warping of space itself, so in relativity we often encounter coordinate systems that do not have straight axes, but axes that curve. To study how objects interact, we have to study quantities in the coordinate systems of different objects, so we need to know how they transform as you move from one system of coordinates (like cartesian coordinates, the 3 perpendicular lines we usually use) to another (perhaps coordinates around a spherically symmetric mass).

Obviously, invariant quantites do not transform when moving from one system to another. Contravariant transformations take tensor (the usual type in relativity) quantites from one system (we'll call the tensor A in the first system) to a second system by using the partial derivatives of system x' (x' represents all coordinates of the new system, where we'll call the transformed tensor A' ) with respect to coordinates x (dx'/dx) and contravariant transformations do the derivative the other way around (A'=dx/dx' A). The mathematics that this leads to is very complicated, and only a few problems in general relativity are actually solvable because of this. Furthermore, non-tensor objects are introduced that do not transform easily.

If you want to read up on the mathematical nature of relativity, I would suggest "Introducing Einstein's Relativity" by D'Inverno. The book is a clear, detailed explanation which can take you through some pretty advanced stuff...don't be fooled by the word "Introduction" in the title. It does use some slightly non-standard notation. Also, be careful of bad relativity books,or ones that are too deep (like "The Large Scale Structure of Spacetime," also called the "yellow terror" because of its yellow cover and how hard it is to understand). These books are everywhere and pretty much useless.

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