After the failure in describing geodesics from Schwarzschild solution I've decided to come back to the origins. Starting from the basics. Manifolds. Yeah. Manifolds rock!
General Relativity -uppercase means it's important- is defined in a 4 dimensional pseudo Riemannian manifold or more specifically, a Lorentzian manifold.
So, before continuing, what is a manifold?
Some people could say mathematicians are weird. I share this appreciation with a deep and sincerely respect (sometimes a little bit of fear) for mathematicians. Level of mental abstraction in Math is something out of my grasp, so I can only understand it by drama.
That means, in 4 steps:
- I've got a problem I need to solve but nothing I understand so far helps, so...
- I find someone's work who already has solved the issue and I try to grasp the Math behind his/her solution...
- I believe I have understood the Math, so I try to solve the problem again.
- I come back to the first point if the problem is not solved yet. More than once, 'cause I can be a real donkey sometimes...
Quite algorithmic. You have to remember I'm an engineer...
So instead of explaining what is a manifold, I'm going to explain what I understand a manifold is...
A manifold in one phrase...
So... Let's assume you are interested in describing a physical property. Like, let's say, a velocity. High school taught you, you need a vector space to do so. There's a trick in that. Vector spaces are based in Euclidean geometry (those explained in High, are) and that means two parallel lines, for instance, never touch each other.
What about if you are in a curved space? What about if you hate Euclid because he was too squared? In this kind of space, two lines initially parallel can touch each other after some time due to the curvature.
Two parallel lines over a spherical surface. (From here)
And the funny thing is, this is not a mathematical abstraction. If you consider Earth as spherical (just, don't antagonize yet, please?) basing a travel plan in Euclidean geometry can be a bad, bad idea.
So, a good definition of manifold oriented to the evil purposes of this blog is:
A manifold is a mathematical set of rules to play in a non-euclidean playground.
Mathematicians reading this must be yelling in pain. Sorry.
Applying this sh*t to General Relativity
So, what do I need to know about manifolds? The answer is: all you can. If you are in a hurry, like me right know, you better start from:
- General Relativity is defined in a Lorentzian Manifold, that is, a psedo Riemannian manifold with signature (1,3) (sometimes, notation changes, and it's (3,1) )
- This signature thing means we can have a clear separation between "space" components in the manifold and temporal ones. I'll go further in this in other moment (when I understand this, I mean).
- Lorentzian manifolds, as any other manifold, can be locally comparable to a R n That's good, OK? Because it allows to define tangent spaces at each point, allowing too, as I have understood, differentiability.
- Differentiability is important, because curves and surfaces in the manifold, can be defined not only with algebraic relationships, but with differential ones. Yes, trust me. These are good news. :)
Summarizing
- So... Now we know a manifold is a set of rules which allow us to define coordinate systems in non-euclidean cool spaces.
- Manifolds allow us to obtain parametric expressions of curves and surfaces from algebraic and differential expressions.
- The set of rules manifolds establish are the reason for the existence of metric tensors, curvature tensors and so on.
- General Relativity is defined in a very general manifold (pseudo Riemannian) but with a very special signature (Lorentzian manifold) which defines a causal structure. Which I'll discuss later.
And I guess that's all for the moment. Lots of things out, like that mysterious causal structure, and the coordinate systems role in all of this. For next posts. I have a life to live :)
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