﻿What is the determinant? Normally the determinant is a rather confusing algorithm you compute on matrices. But this explanation couldn’t be further from the true intuition. The determinant is defined based on a choice of basis, on a linear transformation T. It’s defined to be that first you make a selection of top-forms on the vector space. The top-forms, also called volume forms, measure the “volume” of objects on the top level of the tensors. Essentially it’s a tensor that takes in q vectors and sends them to a volume scalar.


You would’ve thought the notion of volume would need extra structure on the vector space V, but no, it turns out all it needs is a choice of top-forms.


It turns out the determinant is a quotient of two things. On top are the top-forms applied to the linear transformation you’re taking the determinant of, on each basis vector. And below is just directly the top form being applied to the basis vectors. And so in a very real sense you’re measuring the change in volume of the 1 by 1 by 1 cube that the basis vectors form before and after they’ve been passed through the linear transformation.


It turns out that all of the properties you can normally prove about determinants with the weird definition can also be proved, quite easily in fact, using the real definition.


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For bundles, it turns out they’re a category of their own. And that the structure preserving maps is bundle isomorphisms. Basically for two bundles to be identical on a bundle level, there needs to be an invertible map from one total space to the other, and from one base space to the other.


Some more terminology on bundles. It turns out that if a bundle is isomorphic to a product bundle, then it’s called trivial.


If it is locally isomorphic to a product bundle, then it’s called locally trivial.


A crucial idea in bundle theory is the idea of a section of a bundle. Basically you make a selection of a point in E for each point in M, so you make an inverse map from the usual pi. It must also be the case for bundle sections that the pi applied to every point in E goes down to the correct point in M. Pretty much when making a section you can only select points in E above the according point in M.


Cross sections of the bundle are named cross sections because if you imagine a cylinder, then naturally a cross section is an actual example of a section of the circle-cylinder bundle.


When we eventually talk about vector fields, bundle theory will come to be very important because a vector field is a section of the tangent bundle.


Then we have the idea of a fiber bundle, which pretty much just makes the restriction that the fibers attached to each point of the base space are isomorphic to each other.
Then there’s the idea of a pullback. A pullback of a bundle under a map f: M->M’ basically just says “please create a new bundle from the map”.


You can also pull back sections of the original bundle to the new bundle using the map f.


OK so that’s bundle theory, and as I said before, the important part about it is that vector fields are going to end up being sections of the tangent bundle. And vector fields are hugely important in mathematical physics.


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Let’s talk about the classification of sets:


What is the definition of a map between sets? Well a map on a set theoretic level is just defined as an element of the powerset of the cartesian product of the domain and target. If you want to make it a map you also need a few extra restrictions, like every element in the domain is mapped only once into elements of the target.


What does structure preserving mean? It means that first you make a choice of the structure that you equip you set with. Then you ask yourself the question, if you have a random map f from some structure A, and you want to guarantee that the result of the map can be formed into the same structure, then what restrictions must you put on f. And the restrictions you must put on “preserve” the structure. Essentially you want to make all maps of this f closed in the category.


Now that you have a structure preserving map, you can classify the space within groups that are isomorphic to each other. And hopefully even come up with an algorithm using the invariants of the category to determine instantly if there exists an isomorphism. This is useful because humans are lazy.


Invariants are properties that are left unchanged after an isomorphism in a category.


The identity map is always the map that does nothing. That takes every point and sends it to itself.


Metric spaces are a category and they’re structure preserving maps are isometries. A metric space is a set equipped with a metric, which is a way to measure the distance between two points, or a function that takes two points of the space and returns a positive real number. If you take the distance between two equal points (a and a for example), then the distance is 0. You also have the triangle equality rule holding. And also symmetry (duh) so d(a,b)=d(b,a)


An isometry is literally a distance preserving transformation between two metric spaces, and so naturally it’s the structure preserving maps for metric spaces.




The domain is the first part in the inside tuple that defines a map. If a map is an element of the powerset of the cartesian product of A and B, then the domain is the union of all the first parts of that tuple. Or more simply just A.


The target is the union of all the second parts. Or more simply just B.


Image is the application of the map on a desired subset of the domain, and seeing what appears on the other side.
Preimage is the opposite. Reverse the map on the desired subset of the target, and see which parts of the domain actually do things for this subset, and which parts are useless.
Both image and preimage tell you important information about subsets of the domain and target.


A map is surjective if the domain is larger than the target in a sense.
A map is injective if the target is larger than the domain and has some points not being mapped to, in a sense.
A map is bijective if the domain size is equal to the target size, which means they both have the same cardinality. Bijections are the structure preserving maps for sets. If you don’t remember what that means, please recall the section above.


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The directional derivative operator is a derivation on a manifold. It’s the natural “derivative” object that exists at a point.


Algebras are vector spaces with a bilinear product that’s commutative, associative and unital.


If the bilinear product satisfies a few extra properties including antisymmetry and the jacobi identity, then the algebra is a lie algebra. The product in this case is called a lie bracket. Most of the time the lie algebra is a manifold.


IN general, given as associative algebra, you can make a lie algebra by doing the old [a,b] = a*b - b*a trick


On an algebra, a derivation is a linear map satisfying the leibniz rule. Best lookout case is looking at the c-infinity functions. A linear map is very general on a polynomial, but a derivation is pretty much forced to be a derivative. In other words the leibniz rule, or product rule for derivatives, contains a lot of the “derivativeness” of a derivative, and so applied to other instances like a manifold, the resulting thing (directional derivative operator) is also very derivative-like.


Next is that we want a basis on our tangent space to make calculating things easier. A good way to develop this basis is to look at the case of proving that the dimension of the tangent space is equal to the dimension of the manifold. This feels intuitively very easy to do, and it is if you introduce a basis.


To introduce the basis, you’re very smart. You make it so that when making a choice of charts, you get a resulting basis. Algorithmically in English the process is quite simple. You look at the coordinate chart functions, and consider the resulting tangent vector that plops out, and use those as basis.


So the basis is not chart independent, which means we need to care about what a change of charts does about a change of basis, and what that does to our vector components, which are the ultimate thing we care about since most of our calculation will happen with components.


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What is a connection, and what makes it torsion-free?


A connection is the structure equipped with spacetime. More generally outside of spacetime, equipping a smooth manifold with a connection blows up the balloon. It fills up the air inside by making a decision on curvature of the manifold by making decisions on how you move from one point of the manifold to another, specifically considering going from one tangent space to another.


In fact the general idea of a connection is that you want to take directional derivatives of arbitrary sections of the tangent bundles (vector fields), not just the trivial C-infinity functions. And that’s probably the best way to approach the definition of a connection anyways. In theory you could give a geometric approach, which might work, but doing it analytically is much cleaner. You basically make a list of desired properties, and see how much choice you have left. Eventually the final choice will be left up to the Einstein field equations, which are a statement about the metric, and so implicitly about the gammas of the connection (the choice left on the connection).


The connection must eventually be “torsion-free”, which restricts its choice, and also must be compatible with a lorentzian metric, which also further restricts its choice.


But the first thing you say is that the connection is a function that takes in a vector field for direction, and a p,q-tensor field that it is actually differentiating, and returns a p,q-tensor field.


You start off by requiring that the behavior when it takes in a vector field for direction, and a function, is equal to the behavior of regular directional derivatives. In other words that for some vector field X, and function f that it’s differentiating, the result is Xf, or X applied to f, which basically means give me back the new function which corresponds to the directional derivative operator at each point being applied to the function in the direction of the vector field for each point.


That was restriction 1, next is restriction 2 which is that you want everything to be +-linear on the tensor field argument POV. Later on we use this as a property of the connection quite often. Next is restriction 3, which is that you want the leibniz rule to be true. This property ends up being used very often later. Finally restriction 4 is c-infinity linearity on the vector field being used for direction.


Also the leibniz rule is more often expressed using tensor products, which makes it more reasonable to write. You essentially want to assume a generalization of the product rule, expressed with tensor products.


Equipping a manifold with a connection is called an affine manifold. In fact even euclidean space without a connection is quite different from what you intuitively assume it to be. Put another way, when you think of euclidean space, you’re probably already assuming the existence of a connection, or connection-like structure like a metric.


So now we’ve wished out desired properties, and we want to see how much choice we have left. To do this exploration we expand out into charts (which is the land of computation. If you want to compute something, you do it in charts. And exploring the choice remaining in a connection is not an exception).


Badabim badaboom, we see the resulting choice can be focused on a choice of gamma coefficient functions.


But do the gammas stay the same under change of charts? Yes! Magnificent. ANd in fact the new gammas are almost expressible in terms of the old gammas, but there’s a second derivative term blocking the linearity. However it means that if your chart transition map is linear, then the gammas are expressible in terms of the old ones.


How about now when you apply the connection to a covector field? Or an arbitrary tensor field? What happens then? Well all the choices can still be expressed in terms of the same gammas, which is wonderful.


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We now have a connection. And it turns out that a connection intuitively defines a curvature on our object, and that this curvature can be figured out by looking at parallel transport on our manifold.


Intuitively imagining parallel transport is quite easy. Imagine you talk around the room with your nose pointing in some direction. No matter what path you take around the room, you will always return with your nose in the same place as when you started. But now if you imagine walking on the round sphere, and you walk from the pole to the equator, you do a ¼ walk sideways on the equator, and then you return to the pole, you notice that your nose is now pointing in a different direction. The measure by which your nose changes direction when you loop around on the manifold is a measure of the curvature.


For context, round sphere means a sphere smooth manifold equipped with the round connection, so it’s an affine manifold, or manifold with connection.


The connection gives the manifold “shape”.


The vector field that gives direction to a connection doesn’t have to be a full vector field everywhere. I can just be along a curve for example, which comes in handy when talking about parallel transported vector fields.


A parallel transported vector field to a curve is a vector field where the connection is equal to 0 when you consider the vector field caused by the curve (where all the tangent vectors form the vector field). Geometrically you essentially need each tangent vector to be equal at every point when you transport them from one point to the other.


And a (only) parallel vector field is the same but vectors can be a bit larger or smaller from place to place. Mathematically the restriction is that the connection of the curve vector field direction-wise applied to the vector field you’re claiming is parallel is equal to a smooth function times the vector field you’re claiming is parallel.


This image is quite neat and gives a lot of intuition on what is a parallel transported vector field, a (only) parallel vector field, and a nothing vector field:
  

It’s tempting to think of comparing lengths of vectors in the image above, but in reality what the connection is doing is comparing one vector to the one right beside it, and nothing else. Not even comparing length exactly, but comparing the vectors themselves.


Why do we need a connection? To talk about parallel transported curves. Why do we need parallel transported curves? To talk about curvature. Why do we need curvature? Because spacetime is fundamentally a theory about curved space, so you better understand what that means!


Parallel transported curves are the straightest line on the surface of the affine manifold, and so are equal to geodiscs.
The idea of an auto parallel transported curve is that you follow your nose.


When you expand the chart representation of autoparralelly transported curves, it turns out that the acceleration term must be put to 0. Which makes perfect sense.


You can actually do some problems with specific connections and find the auto parallel curve, which is cool. The solutions are actually valid. But yeah all the calculations happen in chart land.


Newton’s first law is on doing an experimentation, you determine the value of the connection.


Why do we care about parallelly transported vector spaces? Because we want to talk about auto parallel curves.


After doing all the theory, we actually need to talk about charts again because we want to do actual computations.


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What is the grassmann algebra? Well it’s the algebra of all d-differential forms united together with a sum, including the wedge product which is the differential that turns a d-differential form into a d+1-differential form. The sum and scalar multiplication is normal, with “scalar” being c-infinity functions.


What’s a d-differential form? It’s a totally antisymmetric contravariant tensor. Which means it takes in d vectors and returns a scalar linearly.


Really it can be a tensor field, so that you have this tensor at every point of the manifold.


Differential forms are quite useful to describe “things” on the manifold, for example eventually matter & energy.


You can use differential forms to talk about Maxwell's equations. This is actually something I studied, but mostly forgot. Will try to explain it another time, especially the specifics.


I remember the intuition is quite cool though: you imagine 1-forms and 2-forms and 3-forms of cubes, tubes, and panes in space, which is quite cool.