Densities

 My new math blog post.

This post introduces "densities". There is a very nice book, available on arxiv, by Sean M. Carroll: "<a href="https://arxiv.org/abs/gr-qc/9712019" target="_blank" rel="noopener">Lecture Notes on General Relativity</a>". There, on p. 52, Carroll writes "... we don't like tensor densities, we like tensors." Well, I do like tensor densities, I consider them important. They exist, and they have their independent life. While it is true that once you have a nondegenerate metric on your manifold, you can avoid using tensor densities. But what if you want to contemplate "pre-metric electrodynamics," or "pre-metric General Relativity"? I see no reason for prohibiting such ideas. In fact, I very much like them. Therefore, here, we do like tensor densities.

[here is some math]

Just while ago I had a Chat with Perplexity AI concerning  a related topic. In particular it tells you how you should never rely on AI.