There are yet a lot of things to say about manifolds but maybe is a good moment to talk about coordinate systems. Manifolds can have coordinate systems. If not, how the hell it is supposed we can find something in a manifold? We need a point of reference and a way of establishing relationships between dimensions.
Why coordinate systems are important?
Coordinate systems are a way of knowing where things are and/or when they are happening.
If you are familiar with linear algebra (if not, sorry I can't help you) you already know in order to distinguish an element, for instance a vector in a vector space, the element has a set of coordinates that define that particular element. A fast explanation about this: each coordinate corresponds to a dimension and is a number which tell us how far should we go along that dimension in order to "find" the element.
If you start walking from the origin of the system the number of units, along one dimension, the coordinates tell you how much walking do you have left in order to arrive to your destination.
When I studied Linear Algebra, only orthonormal basis were approached. Orthonormal means:
Orthonormal basis are the easier basis we can use -and they're quite useful for physics to a quite high level. Easy to grasp, easy to operate. Cool.
The problem comes when we are talking about manifolds which represent curved space-time: generally speaking it's better idea to use another kind of coordinate systems. And when that happens, sh*t gets real.
If you are familiar with linear algebra (if not, sorry I can't help you) you already know in order to distinguish an element, for instance a vector in a vector space, the element has a set of coordinates that define that particular element. A fast explanation about this: each coordinate corresponds to a dimension and is a number which tell us how far should we go along that dimension in order to "find" the element.
A typical coordinate system and a point represented in coordinates, from here. In Linear Algebra, 3 coordinates are enough... And a reasonable way of describing physics. In order to arrive to point (x,y,z) you have to walk exactly x,y and z in the respective coordinates which, to make things easier, represent only one dimension at a time.
If you start walking from the origin of the system the number of units, along one dimension, the coordinates tell you how much walking do you have left in order to arrive to your destination.
When I studied Linear Algebra, only orthonormal basis were approached. Orthonormal means:
- Each element in the basis is orthogonal (which means is linearly independent from the other elements in the basis, which means it can not be expressed as a combination of coordinates from the other elements in the basis, which means roughly, each coordinate represents only a dimension). Orthogonality is typically used to express perpendicularity.
- Each element is also normalized -which means is a unitary- vector, or versor.
Orthonormal basis are the easier basis we can use -and they're quite useful for physics to a quite high level. Easy to grasp, easy to operate. Cool.
The problem comes when we are talking about manifolds which represent curved space-time: generally speaking it's better idea to use another kind of coordinate systems. And when that happens, sh*t gets real.
What should I know about them
The more you know, the more you'll be able to do. But, for the lazy:
- The difference between curved and non-curved coordinate systems.
- How to use and operate them.
Curved and non-curved. Does it really matters?
Yes. Yes it does. We said before coordinates are a mean to tell us how far an element is from the origin of our coordinate system. Every coordinate is a number (picturing a distance is quite useful) we go for in a particular dimension. But, we are going to work in curved space-time folks: a coordinate no longer applies for just one dimension.
Far more than that, it turns out we can define coordinates projecting in a parallel or an anti-parallel way.
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One step at a time. Holy crap, point one. What is the point in defining such a bizarre coordinate system! It was easier when walking along a coordinate meant walking along one dimension, right? What's the point in mixing things up? What's the point in defining a coordinate which leads us to walk along several dimensions at a time? That's crazy sh*t! It makes everything a lot more complicated!
Well it is, indeed. But, you should remember mathematicians like easy things, just as everybody else. Things should be that way because they turn easy a couple of very difficult things they cannot solve in other way (I suggest you trying to solve some PDE systems in Cartesian orthogonal coordinates and you'll think again where do you want things turn difficult)
And what about the point 2? Parallel and antiparallel? Yeap. When we had orthogonal basis we just use one projection of coordinates, because the other one gave us cero. But when we have a coordinate system not orthogonal, you can define an element in two ways.
The good news: we know how dimension distance is changed and we can define tricks to take into account this. And we can have easy ways of dealing with the covariant and contra-variant issue, so no problem.
So, now I'm supposed to talk about operations, but it's late and I'm sleepy and you must be wishing this post ends, so better we start sudokus in the next post.
Far more than that, it turns out we can define coordinates projecting in a parallel or an anti-parallel way.
A spherical coordinate system from here and another fancy drawing from here. In the second drawing we can understand the use of such a twisted way of defining things: it's easier to place something over a sphere with latitude, longitude and radius.
One step at a time. Holy crap, point one. What is the point in defining such a bizarre coordinate system! It was easier when walking along a coordinate meant walking along one dimension, right? What's the point in mixing things up? What's the point in defining a coordinate which leads us to walk along several dimensions at a time? That's crazy sh*t! It makes everything a lot more complicated!
Well it is, indeed. But, you should remember mathematicians like easy things, just as everybody else. Things should be that way because they turn easy a couple of very difficult things they cannot solve in other way (I suggest you trying to solve some PDE systems in Cartesian orthogonal coordinates and you'll think again where do you want things turn difficult)
And what about the point 2? Parallel and antiparallel? Yeap. When we had orthogonal basis we just use one projection of coordinates, because the other one gave us cero. But when we have a coordinate system not orthogonal, you can define an element in two ways.
Vector P can be defined in X basis projecting along the directions provided by itself in a parallel way (x1 and x2) or antiparallel way (x1 and x2) Not pretty sure about which is covariant and which contra-variant, so better explanation later. The figure is from here.
The good news: we know how dimension distance is changed and we can define tricks to take into account this. And we can have easy ways of dealing with the covariant and contra-variant issue, so no problem.
How to use them and operate with them.
So, now I'm supposed to talk about operations, but it's late and I'm sleepy and you must be wishing this post ends, so better we start sudokus in the next post.
See you later!
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