During the end of last week I turned towards Sean Carroll's book Spacetime and Geometry: An Introduction to General Relativity. This is not the first time I go through some of it. I started on the first chapter, going through the mathematical details and skipping the part on energy and momentum and classical field theory. I will go through this two sections once I reach the chapter on gravitation. When I finished the first chapter I carried on to the second chapter on manifolds.
The second chapter formalizes ideas familiar ideas and puts them in the context of a manifold. These ideas include vectors, tensors, differentiation and integration. I am currently finishing the third section of the third chapter on curvature. Right now I am exploring the concept of parallel transport and geodesics.
So what have I learned so far? Here is an outline with the basics idea behind the mathematical construction of a nontrivial spacetime (not flat). It also serves as a nice summary of what I got so far...
The motivation behind General Relativity is the matter and energy curve spacetime and this curvature is what we call gravitation. (I have not reach the chapter on gravitation, hopefully by the end of this week, so that line that you just read might be very wrong...). We would like to construct a spacetime that can be curved, and we would like to describe this curvature mathematically. Another motivation behind relativity is the fact that the laws of physics should remain invariant under coordinate transformations. This tells us that we should use objects that have the same form in any coordinate system. We call this objects abstract vectors, dual vectors and higher-rank tensors. We also would like to describe things locally, since relativity tells us that the speed of light is an universal upper bound to the magnitude of the velocity of any object that carries information. From special relativity we know that this means that simultaneity looses validity between different observers. For a given physical object (a tensor, a vector, etc.) different observers will have different components but this components should be related by a coordinate transformation.
Most of the construction comes from making analogies with flat spacetime (a Cartesian product of many real lines R.) The first thing that we want is that given a region of curved spacetime, common sense tells us that locally (a very small patch) will look like flat space. We can imagine then a chunk of curved spacetime that is made up of flat patches of space, all sewn together. This is along the lines of the entity known in mathematics as a manifold. We need to also demand that this patches can communicate through continuous maps. I am not going into the formal details, but the basic idea is this: One construct a generalization of a Cartesian system by connecting neighborhoods of Cartesian systems with continuous and differentiable functions. The manifold is a collection of points, and this point in turn are mapped locally into a flat spacetime.
The next step is to set up the notion of a vector. In Euclidean space (which is the formal name of a space formed by taking a finite number of Cartesian products of real lines R ) a vector is an object that obeys a set of rules that constitute a vector space. This idea works well in Euclidean space, but when one considers the notion of moving a vector along a spacetime that is curved, it is not clear whether the vector changes or not. This is necessary for adding or taking products of vectors. Since each point on a manifold is locally flat, one can define a vector space on each point, this is called the tangent space. The tangent space is formed of all the vectors that originate at the point. A natural choice for basis vectors are the set of partial derivatives.
Other objects that are generalized to curved spacetime include tensors and dual vectors. More on that later.
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