Gravity is geometry. To fully understand this statement, we will need more sophisticated tools and language to describe curved space and, ultimately, curved spacetime. This is the mathematical subject of differential geometry and will be introduced in this section and the next. Armed with these new tools, we will then return to the subject of gravity in Section 4.
Our discussion of differential geometry is not particularly rigorous. We will not prove many big theorems. Furthermore, a number of the statements that we make can be checked straightforwardly but we will often omit this. We will, however, be careful about building up the mathematical structure of curved spaces in the right logical order. As we proceed, we will come across a number of mathematical objects that can live on curved spaces. Many of these are familiar – like vectors, or differential operators – but we’ll see them appear in somewhat unfamiliar guises. The main purpose of this section is to understand what kind of objects can live on curved spaces, and the relationships between them. This will prove useful for both general relativity and other areas of physics.
Moreover, there is a wonderful rigidity to the language of differential geometry. It sometimes feels that any equation that you’re allowed to write down within this rigid structure is more likely than not to be true! This rigidity is going to be of enormous help when we return to discuss theories of gravity in Section 4.
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