# What is Coordinate chart: Definition and 17 Discussions

In topology, a branch of mathematics, a topological manifold is a topological space which locally resembles real n-dimensional Euclidean space. Topological manifolds are an important class of topological spaces, with applications throughout mathematics. All manifolds are topological manifolds by definition. Other types of manifolds are formed by adding structure to a topological manifold (e.g. differentiable manifolds are topological manifolds equipped with a differential structure). Every manifold has an "underlying" topological manifold, obtained by simply "forgetting" the added structure.

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1. ### I Time dilation vs Differential aging vs Redshift

Hi, I would like to ask for a clarification about the terms time dilation vs differential aging vs gravitational redshit. As far as I can tell, time dilation is nothing but the rate of change of an object's proper time ##\tau## w.r.t. the coordinate time ##t## of a given coordinate chart (aka...
2. ### I ##SL(2,\mathbb R)## Lie group as manifold

Hi, consider the set of the following parametrized matrices $$\begin{bmatrix} 1+a & b \\ c & \frac {1 + bc} {1 + a} \\ \end{bmatrix}$$ They are member of the group ##SL(2,\mathbb R)## (indeed their determinant is 1). The group itself is homemorphic to a quadric in ##\mathbb R^4##. I believe...
3. ### I Schwarzschild spacetime in Kruskal coordinates

As explained here in Kruskal coordinates the line element for Schwarzschild spacetime is: $$ds^2 = \frac{32 M^3}{r} \left( – dT^2 + dX^2 \right) + r^2 \left( d\theta^2 + \sin^2 \theta d\phi^2 \right)$$ My simple question is: why in the above line element are involved 5 coordinates and not just...
4. ### I Wald synchronous reference frame proof

Hi, on Wald's book on GR there is a claim at pag. 43 about the construction of synchronous reference frame (i.e. Gaussian coordinate chart) in a finite region of any spacetime. In particular he says: $$n^b\nabla_b (n_aX^a)=n_aX^b\nabla_b \, n^a$$Then he claims from Leibnitz rule the above equals...
5. ### B Further Understanding Simultaneity Conventions

Summary Almost a year ago, I created a post titled “Understanding the phrase 'simultaneity convention'”. The answers included requirements for defining a simultaneity convention. But some simultaneity conventions, while meeting all the requirements, still appear problematic. What am I missing...
6. ### I Einstein Definition of Simultaneity for Langevin Observers

Hi, reading this old thread Second postulate of SR quiz question I'd like to ask for a clarification on the following: Here the Einstein definition of simultaneity to a given event on the Langevin observer's worldline locally means take the events on the 3D spacelike orthogonal complement to...
7. ### I Clarification about submanifold definition in ##\mathbb R^2##

Hi, a clarification about the following: consider a smooth curve ##γ:\mathbb R→\mathbb R^2##. It is a injective smooth map from ##\mathbb R## to ##\mathbb R^2##. The image of ##\gamma## (call it ##\Gamma##) is itself a smooth manifold with dimension 1 and a regular/embedded submanifold of...
8. ### I Coord. Time Vector Field: Schwarzschild vs Gullstrand-Painleve

Hi, I was reading this insight schwarzschild-geometry-part-1 about the transformation employed to rescale the Schwarzschild coordinate time ##t## to reflect the proper time ##T## of radially infalling objects (Gullstrand-Painleve coordinate time ##T##). As far as I understand it, the vector...

15. ### Vectors in Tangent Space to a Manifold Independent of Coordinate Chart

In Nakahara's book, "Geometry, Topology and Physics" he states that it is, by construction, clear from the definition of a vector as a differential operator [itex] X[\itex] acting on some function [itex]f:M\rightarrow\mathbb{R}[\itex] at a point [itex]p\in M[\itex] (where [itex]M[\itex] is an...
16. ### Do Coordinate Charts Alone Identify a Unit Circle?

Hi. I have been looking at the coordinate charts for the unit circle x^2 + y^2 = 1. In the notes I have the circle is split into 4 coordinate charts the first being - ##U_1## : x>0 , ##A_1## = y (PS without the symbols tab I have used A for the letter phi ) There are 3...
17. ### Coordinate Chart on Manifold: What is $\mathbb{R}^{n}$?

In defining a coordinate chart, \left ( U,\phi \right ), U \in M, \phi : U \to \mathbb{R}^{n}, on a manifold M, what exactly is \mathbb{R}^{n}: the set of all n-tuples, a topological space, a metric space, a vector space, Euclidean space conceived of as an inner product space, Euclidean...