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.
