〈 Berger | Dillon 〉

The µ’s and v’s indicate the indices of a tensor. You’ll notice that some indices are downstairs and some upstairs. This is because the lower indices are in the “dual-space.” These have the property of mapping a vector to ℝ. Don’t worry, we’ll come back to this.pic.twitter.com/NCUHNP2IjO

In the same clip, we can categorize our tensors depending on how many indices are downstairs or upstairs. Rank = (down, up). The first contraction is between a rank (2, 0) tensor and a rank (0, 2) tensor. The second is between a rank (1, 0) tensor and a rank (0, 1) tensor.pic.twitter.com/vdMA7a5Ws5

However, contraction is mediated by something called the 𝑚𝑒𝑡𝑟𝑖𝑐 𝑡𝑒𝑛𝑠𝑜𝑟, which can change the rank of a tensor. Here we see it operate on a four-vector so that: rank(0, 1) ⭢ rank(1, 0) Which lets us perform a contraction with a rank (0, 1) tensor!pic.twitter.com/reyd8TYIXd

Matrices are pretty unwieldily to work with. So we take advantage of tensors and their symmetries (and anti-symmetries). Below we can express the Faraday Tensors for the E&M field as tensors and not matrices.pic.twitter.com/wp3qwYNcvx

By taking advantage of tensors properties, we can express the four Maxwell Equations in two equations. This also has the benefit of translating E&M into relativity much more easily.pic.twitter.com/xJJbuyWsKw

Spot the tensors (and maybe some familiar ones) from this thread in the Einstein field equation, or the Lagrangian of the Standard Model below!pic.twitter.com/PM3Ft78Izf