Coordinate chart Definition and 17 Threads

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. cianfa72

    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. cianfa72

    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. cianfa72

    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. cianfa72

    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. F

    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. cianfa72

    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. cianfa72

    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. cianfa72

    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...
  9. cianfa72

    I Clarification on Rindler coordinates definition

    Hi, starting from this post Basic introduction to gravitation as curved spacetime I would ask for a clarification about Rindler coordinates. From this wiki entry Rindler coordinates I understand that the following transformation (to take it simple drop ##y,z##) $$T = x\sinh{(\alpha t)} ...
  10. cianfa72

    I Global coordinate chart on a 2-sphere

    Hi, I know there is actually no way to set up a global coordinate chart on a 2-sphere (i.e. we cannot find a family of 2-parameter curves on a 2-sphere such that two nearby points on it have nearby coordinate values on ##\mathbb R^2## and the mapping is one-to-one). So, from a formal...
  11. cianfa72

    I About global inertial frames in GR

    Hi, starting from this thread Principle of relativity for proper accelerating frame of reference I'm convincing myself of some misunderstanding about what a global inertial frame should actually be. In GR we take as definition of inertial frame (aka inertial coordinate system or inertial...
  12. cianfa72

    I GPS clock synchronization in ECI frame

    Hi, starting from this old thread GPS clock synchronization I've a doubt about the physical process employed to synchronize clocks bolted on GPS system satellites. We said that clock synchronization is frame dependent. In other words we must select a coordinate chart (aka reference frame) that...
  13. cianfa72

    I Reference frame vs coordinate chart

    Hello, here on PF I've seen many threads about the concepts of 'reference frame' and 'coordinate system'. In the context of SR my 'envision' about the concept of 'frame of reference' is basically the 'rods & clocks latticework' as introduced in the book Spacetime physics (Taylor, Wheeler)...
  14. A

    I Tangent vector basis and basis of coordinate chart

    I am learning the basics of differential geometry and I came across tangent vectors. Let's say we have a manifold M and we consider a point p in M. A tangent vector ##X## at p is an element of ##T_pM## and if ##\frac{\partial}{\partial x^ \mu}## is a basis of ##T_pM##, then we can write $$X =...
  15. D

    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. D

    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. Rasalhague

    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...
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