This is a textmap about general relativity, at a level that is meant to be accessible to advanced undergraduates. This is mainly a textmap about general relativity, not special relativity. For someone who has not already learned special relativity, I strongly recommend mastering it first.
- Differential geometry uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra to study problems in geometry.
- We now have enough machinery to be able to calculate quite a bit of interesting physics, and to be sure that the results are actually meaningful in a relativistic context. The strategy is to identify relativistic quantities that behave as Lorentz scalars and Lorentz vectors, and then combine them in various ways. The notion of a tensor has been introduced earlier. A Lorentz scalar is a tensor of rank 0, and a Lorentz vector is a rank-1 tensor.
- General relativity describes gravitation as a curvature of spacetime, with matter acting as the source of the curvature in the same way that electric charge acts as the source of electric fields. Our goal is to arrive at Einstein’s field equations, which relate the local intrinsic curvature to the locally ambient matter in the same way that Gauss’s law relates the local divergence of the electric field to the charge density.
- spacetimes in cases in which the entire universe has no matter. For example, we will be able to calculate general-relativistic effects in the region surrounding the earth, including a full calculation of the geodetic effect.
- Gravitational waves transport energy as gravitational radiation, a form of radiant energy similar to electromagnetic radiation. Newton's law of universal gravitation, part of classical mechanics, does not provide for their existence, since that law is predicated on the assumption that physical interactions propagate at infinite speed—showing one of the ways the methods of classical physics are unable to explain phenomena associated with relativity.