Hi. I've been thinking about vectors, coordinate systems and all things associated for a long time. I'd like to know if (at least in the context of General Relativity) my interpretation of these subjects is correct. I will try to summarize my thoughts as follows:(adsbygoogle = window.adsbygoogle || []).push({});

- We start with a general manifold.

- We assign values to each point [itex]P[/itex] on the manifold, through a map from a closed numerical interval, by a function [itex]c[/itex]. So [itex]c: [a, b] \longrightarrow \mathbb{R} [/itex]; [itex]t \mapsto c(t)[/itex]. Additionaly, we impose the condition that no two points on the manifold have the same value.

- Now we map c(t) into [itex] \mathbb{R} ^m[/itex], where m stands for the dimension of the manifold, through a function [itex]\Psi: \mathbb{R}\longrightarrow \mathbb{R} ^m[/itex]; [itex] c(t) \mapsto \Psi (c(t)) \equiv x[/itex].

Define a tangent vector V at P by V = [itex]\frac{\partial c(t)}{\partial t}[/itex] and using the chain rule, V = [itex]V^{\mu} \frac{\partial}{\partial x^{\mu}}[/itex], where [itex]V^{\mu} = f(V, x)^{\color{red}*} \frac{\partial x^{\mu}}{\partial t}[/itex]. Define the basis vector to be [itex] \frac{\partial}{\partial x^{\mu}} [/itex].

Now it is clear that the basis vector in this representation depends on the point [itex]x[/itex] of [itex]\mathbb{R} ^m[/itex] (though the vector itself depends only on the point [itex]P[/itex] on the manifold). So when we change the basis, is it like as we were transporting the vector through [itex]\mathbb{R} ^m[/itex]?

* - We know that at [itex]P[/itex] we can construct as many vectors as we desire, by multiplying the basis vectors byscalarsand adding them together. But since we define V = [itex]V^{\mu}\frac{\partial}{\partial x^{\mu}}[/itex] and [itex]V^{\mu} = \frac{\partial x^{\mu}}{\partial t}[/itex], we see that all vectors would lie in a same line! Because [itex]\frac{\partial x^{\mu}}{\partial t}[/itex] is the same for all of them. But we must be able to construct a tangent plane (or hyperplane) of vectors at [itex]P[/itex], so I inserted that function [itex]f(V, x)[/itex] in *, so that each vector component can be independent from each other, and the vectors can fill an entire plane at [itex]P[/itex]. Is it correct? If not, what would be a solution to get rid of this problem?

Now, I'd like to talk about isometries of a metric. My doubt here is: when people talk about rotations in General Relativity, what they are talking about is of rotations as we know from elementary math courses, like "rotation of the coordinate axis through an angle about a point x"?

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# I Vectors and isometries on a manifold

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