﻿Erwin Schrodinger's spacetime structure book talks about many different things. The general topic of the book is doing a three way analysis. First step is looking at unconnected manifolds. Next is looking at affine manifolds. And finally looking at affine manifolds that can form a metric. Everything is done in 4d so statements are simplified, which I like.


Anyhow, let’s talk about connections. I’m now able to do some connection calculations to find auto parallelly transported curves. I can do these calculations by choosing a specific chart and doing the calculations of the nose curve following its nose, which basically means that I will need to solve a differential equation to find the curve that satisfies it, which will end up being the auto parallelly transported curve.


Another question to ask is that putting restrictions on the connection can be hard to state in terms of the connection, so we want some tensors to be created out of the connection so that we can indirectly enforce some conditions on the connection. This is exactly what happens with the torsion tensor and the curvature tensor.


What the torsion tensor actually stands for is a bit beyond me, but we want it to vanish, or to be equal to 0. I know the torsion tensor has some effect where basically the gammas are symmetric or something, so you can flip the lower indices a and b and not have an effect.


The torsion tensor actually is a 1,2-tensor field. T(w, X, Y) = w(∇_x Y - ∇_y X - [X, Y]) where [X,Y] is the commutator. You can actually prove that this map is multilinear in each entry, which is the proof required to show it’s a tensor, but in this case you can just trust me.


The curvature tensor is a 1,3-tensor field. T(w, Z, X, Y) = w(∇X ∇Y Z − ∇Y ∇X Z − ∇[X,Y]Z). Again you can prove that this map is multilinear in each entry to show it’s a tensor. The order of the entries is purposefully backwards to make the right side look more organized.


This curvature tensor is called the riemann curvature tensor, and it contains the maximum possible information about curvature. But sometimes you want to talk about more specific types of curvature, and this can be done by talking about other curvature tensors derived from the big daddy riemann tensor, such as the ricci curvature tensor. Where the ricci curvature is the contraction of the first and third index from the riemannian curvature tensor.


The Ricci curvature tensor is a 0,2-tensor field. The letters for riemannian tensor is Riem, whereas for Ricci it's Ric, although most of the time we just use R, and the type of curvature tensor is evident in context because 0,2-tensor fields act differently from 1,3-tensor fields in equations.
  

The lie bracket, kind of like the inner product, measures how well vector fields can form a grid and can act as good coordinate vector fields. If they form a grid, then the lie bracket of the two vector fields is 0.


Newtonian spacetime is curved, when you look at it as three dimensions of space + time.


Let’s do a little group theory.


A group is a set with an operation that satisfies certain rules ANI. Basically that the operation is associative, which means you can switch parentheses around. The group operator has a neutral element, which means it can be “multiplied by 1” or “added by 0”. Essentially there must be an identity element. Finally there needs to be inverses. So each element of the group should have an inverse, that when you operate on an element and its inverse, you get the neutral element.


Now we can hop right into examples.
* The integers Z with addition.
* The integers Z with multiplication.
* The cube with rotations of the cube.
* In fact any field is also a group with either of the addition or multiplication operators. That’s because a field is a stronger statement then a group, but completely encompasses it. So all fields are groups (but not the other way around).
* Finite fields are examples of finite groups.
* Because all the modular fields are fields, they’re also groups.
* Vector spaces are groups, if you forget about the scalar multiplication. But this is quite a dumb example, because then you’re not studying vector spaces anymore. It’s more like “this is a formula to get a group from any vector space”.
* Vector space of matrices is a group with matrix multiplication.


I personally find “symmetry” to be quite a dumb word, because it means that the object structure you’re looking at has an isormorphism to itself, which means a structure preserving map, depending on what category you’re in. I find symmetry to be quite an annoying word because it doesn’t give any clue on the context you’re working in, which can be quite confusing.


An example of a symmetry in the integers Z with addition is negation (and the trivial symmetry of the identity).


OK and multiplication by 2 is not and automorphism because it is either not a bijection, or if you add in the missing elements, it does not respect the additive structure.


Symmetry is the group created from a category.