Questions tagged [coordinate-systems]

A set of numbers used to quantify location in space.

288 questions with no upvoted or accepted answers
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Is there an equivalent of Rindler coordinates for an object in centripetal motion?

Rindler coordinates are a parametrization of (a subset of) Minkowski space that are "natural" for an object experiencing constant acceleration - more specifically, an object experiencing constant ...
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429 views

On the embedding of the Schwarzschild metric in six dimensions

At every point of the 4-D space-time, it's metric, being a symmetric 2-tensor, has $\frac{D(D+1)}{2}=10$ independent components. From this we can subtract four degrees of freedom according to the four ...
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1answer
272 views

How the Poisson bracket transform when we change coordinates?

I'm studying the book Geometric Mechanics by Darryl D. Holm and there's one exercise in the book I'm not quite getting what has to be done. The same discussion the author makes in the book is made on ...
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27 views

Fixed coordinate frame as limit of rotating coordinate frame

I have a question about a fixed coordinate system as limit of rotating system. Consider for example a pendulum. The Lagrangian in the rotating frame is given by \begin{equation} L(\mathbf{r}, \mathbf{\...
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206 views

Generating function of point transformation

I am asked to show that the generating function corresponding to a point transformation in Lagrangian mechanics can be taken as null. The point transformation consists of $$ Q_i=Q_i(q,t), $$ and ...
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221 views

Does the spin1/2 rotation operator rotate spin in real space?

in quantum mechanics, the rotation operator for spin one half is $R_{\alpha}\left(\hat{\boldsymbol{n}}\right)=\mbox{exp}\left(-i\frac{\alpha}{2}\boldsymbol{\sigma}\cdot\hat{\boldsymbol{n}}\right)$. ...
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162 views

EM Field Quantization in Spherical polars

Is it possible to quantize the electromagnetic field in spherical polar coordinates instead of cartesian ones? Such that creation and annihilation operators correspond to harmonic oscillator modes ...
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364 views

Is classical electromagnetism conformally invariant? (and a bit of general covariance)

The contest is a flat $4d$ Minkowsky space. A conformal transformation is a diffeomorphism $\tilde x(x)$ such that the metric transforms as \begin{equation*} \tilde g_{\tilde \mu \tilde \nu} = w^2(x) ...
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1answer
76 views

General Relativity - Confusion between choosing basis (orthonormal & coordinate) and coordinate transformations

I am reading the book 'Gravity' by Hartle and presently I am at the section discussing orthonormal and coordinate bases. I am confused about a few points I had read previously and can't exactly ...
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55 views

Variations of tensors are tensors?

Recently I posted a question about variation of metric. I thought I understood it and talked with my friend about it. After that he said he's not convinced because he can't prove variation of metric ...
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82 views

How is time “homogeneous”?

My book$^1$ states: Let's consider a clock moving freely over a curve such as: \begin{equation} \frac{dx^i}{dt}=\text{const} \tag{1.20} \end{equation} We define the proper time $\tau$ as the ...
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164 views

Rindler Coordinates Derivation

In my GR lectures we've derived Rindler coordinates by first showing that the four velocity, which we defined as $$u^{\mu} = (\gamma c, 0, 0, \gamma u),$$ as a function of proper time can be written ...
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109 views

Using action-angle variables in non-periodic system

I'm a little confused by the discussion in the last section $\S 50$ of Landau and Lifshitz's (Classical) Mechanics (1960, first English ed.). Here, they consider finite motion of a system whose ...
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103 views

Difficult coordinate transformation

I am trying to introduce a tortoise coordinate for a modified Schwarzschild metric $$\mathrm{d}s^2=\left(1-\frac{2M\mathop{}\!\mathrm{erf}(r)}{r}\right) \mathrm{d}t^2 + \left(1-\frac{2M\mathop{}\!\...
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109 views

When is it possible to diagonalize a metric tensor with a coordinate transformation?

The question is a mathematical one, but I believe more likely to find here interested people, as it is relevant for GR. The problem arose for Kerr space-time and Boyer-Lindquist coordinates, where the ...
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158 views

Curl operator in Schwarzschild metric

I'm trying to write down the curl operator explicitly for a Schwarzschild metric in cylindrical coordinates. I am trying to use the general expression of the curl operator in orthogonal curvilinear ...
3
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1answer
163 views

Gravity assist, Coordinate system transformation

I am currently learning about the physics and the mathmatics behind gravity assist. I have a question regarding the coordinate transformation between the planet frame and the sun frame. I want to ...
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99 views

Interpreting meaning of coordinates given a metric

I was working problem 3.6 in Carroll's GR textbook and was given the following metric, which is a good approximation to the metric outside the surface of the Earth. $ds^2=-(1+2 \Phi(r))dt^2 + (1-...
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1answer
179 views

Is Gauss electric flux law valid in all coordinate systems?

The derivation of Gauss electric flux is as follows : $$\iint{\vec{E}}\cdot{\vec{dS}}=\iint E \, dS \cos\theta \, .$$ The projection of infinitesimal area on the surface $\vec{dS}$ on the radial ...
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172 views

Is a solution to the Klein-Gordon equation homeomorphic (or even diffeomorphic) to a solution of an equation with a different covariance group?

Consider some solution $\psi(x,t)$ to the linear Klein-Gordon equation: $-\partial^2_t \psi + \nabla^2 \psi = m^2 \psi$. Up to homeomorphism, can $\psi$ serve as a solution to some other equation ...
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1answer
97 views

Sagnac Effect and clocks synchronization

I read this from "The Sagnac effect and its interpretation by Paul Langevin", in cylindrical coordinates $$ ds^2=c^2t^2-dr^2-r^2d\theta^2 $$ ...This transformation means that the observer O (...
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22 views

Spherical polar coordinates in a tetrad frame

I am looking at a paper which writes the spatial components of a vector $S_i$ in terms of spherical polar coordinates w.r.t the local tetrad frame as (Eq 33 in the linked paper), $$ S_1 = s \sin \...
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In Bondi gauge/coordinate, how to obtain covariant derivative of the 2-dimemsional sphere using null projection operators?

In Bondi coordinate , the Bondi-Sachs metric is written as $$ ds^2=-\frac{V}{r}e^{2\beta}du^2-2e^{2\beta}dudr+r^2h_{AB}(dx^A-U^Adu)(dx^B-U^Bdu) $$ Before any decomposition, the covariant derivative $...
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1answer
106 views

Simple Pendulum in Cartesian Coordinates

Riffing on the question in Simple Pendulum Why Generalized Coordinate Always Angle? , I'm trying to write down Newton's law for a simple pendulum in Cartesian coordinates. (I'm doing this as an ...
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40 views
+50

To find the probability of the direction of the velocity after impact lying between given limits

I'm trying to understand Maxwell's "Illustrations of dynamical theory of gas" puts forward the argument of equal scattering probability. The area between the spherical zone is not very clear. How does ...
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1answer
65 views

Choosing initial condition for Hamilton-Jacobi PDE

For separable solutions to Hamilton-Jacobi PDE (say in 2D), we treat the Hamilton's principal function $S$ as $$S= W(x) + W(y) - E*t$$ and treat the separate parts as constants and find $W(x)$, $W(y)$...
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Is Gerard 't Hooft's Cellular Automaton Interpretation of Quantum Mechanics background independent?

In Gerard 't Hooft in his Cellular Automata Interpretation of Quantum Mechanics (https://www.researchgate.net/publication/...
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1answer
66 views

How to determine if a tensor is covariant or contravariant?

In special relativity, the coordenates of a event are in general written using a 4-vector: $$x^{\mu} = \binom{ct}{\textbf{x}}$$ where $\textbf{x} = (x,y,z)$ are the spacial coordenates. This is a ...
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67 views

Spherically Symmetric Spacetimes

I have been studying the Schwarzschild metric $g$ and its derivation. The starting point is to assume the spacetime it describes is spherically symmetric. This means that the algebra of its Killing ...
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45 views

Acceleration interpretation in accelerated frames in general relativity

I would like to know whether my physical interpretation of some dynamics in accelerated frames is correct. In a frame with acceleration $a$ we have the metric $$ds^2 = (1+ax)^2 dt^2 - dx^2$$ The ...
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Pictures of Different Coordinate Systems in General Relativity

In General Relativity by Woodhouse there are the three following diagrams in Chapter 9 about Black Holes. Despite a (very brief) description of these diagrams in the book itself, I am struggling to ...
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How to interpret Einstein's development of Euclidean orthogonal transformations in The Meaning of Relativity?

My questions pertain to the development of Euclidean 3-space orthogonal transformations presented in Einstein's The Meaning of Relativity. Also available at https://en.wikisource.org/wiki/...
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2answers
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Definition of velocity in classical mechanics

Let $(r_1,r_2,r_3)$ be the coordinates of a particle $r$ in the coordinate system $\phi$. Let $\{\hat{e_1},\hat{e_2},\hat{e_3}\}$ be the coordinate basis of $\phi$. Why do we define the velocity $v$ ...
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How to understand intuitively the three relations about coordinate transformation?

The question is related to the representation theory of group theory for physics. Generally, one can define the coordinate transformation $T$ by the following relation: $$\vec{r}'=R(T)\vec{r}+\vec{t}(...
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1answer
397 views

Does angular momentum depend on coordinate frame?

I am working through a problem and getting what seems to be different answers in different reference frames. The angular momentum of a point mass with respect to an axis that doesn't go through the ...
2
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1answer
104 views

In special relativity, is there a prescription to find the coordinate transformation to a general (for instance accelerated) observer? And in GR?

Suppose we are in an inertial frame, so that the metric in our coordinates is just the Minkowski metric, \begin{align} \text{d}s^2 = -\text{d}t^2 + \text{d}x^2 + \text{d}y^2 + \text{d}z^2. \end{align} ...
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1answer
170 views

1+1D curved spacetime diagram example

This is a very basic question about General Relativity. I haven't take any GR course yet. Suppose a flat spacetime with one space direction and one time direction, as follows: Now add a mass at rest ...
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381 views

Special relativity - loss of simultaneity - Is that real?

Below is normal example that is generally given for loss of simultaneity in relativity A person (A) is on platform. Another person (B) is travelling in train (moving left to right). When person A ...
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195 views

Circumference of a circle in Schwarzschild metric in different frames

Suppose that we want to measure the circumference of a circle of radius $R+h$ around a spherical star of radius $R$. The metric there is the Schwarzschild metric, $$ds^2=-\big(1-\frac{r_S}{r}\big)dt^...
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2answers
242 views

How to represent a axisymmetric, stationary metric in a coordinate independent way?

A classic example of a stationary, axisymmetric metric in GR is the Kerr metric. In Boyer-Lindquist coordinates $(t,r,\theta,\phi)$ it is obvious that the metric is independent of $t,\phi$ and so is ...
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Does the curvature parameter $k$ change with coordinate choice (de Sitter spacetime)?

The de Sitter spacetime can be derived from the vacuum Friedmann equations given a choice of $k=0$, where $k$ defines the spatial curvature of the spacetime. The resulting metric in $(t,x,y,z)$ is ...
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Spherical Formulation of Quantum Mechanics

I always wondered, during my QM courses, if we don't explore enough of the freedom that the Lagrangian and Hamiltonian Classical Dynamics give us. Classically, we can always make canonical ...
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54 views

Gravitational time dilation and the pace of time

If we are in empty space far from a black hole, at rest relative to the hole, we would look at a clock and a light source inside the gravitational field of the hole, then we would, according to the ...
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56 views

What will happen to a particle in a vortex?

Consider a particle starting at rest at a position $(L, 0)$ in a coordinate system with polar coordinates (R, $\theta$). Act on the particle with a force $(0, C)$ in the same coordinates. I.e. an ...
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349 views

Kruskal-Szekeres spacetime

How does one determine the Kruskal Szekeres regions? I know the Schwarzschild solution in Kruskal-Szekeres coordinates is given by $$(ds)^2 = \frac{32(GM)^3}{r}\exp(-r/2GM)(-dT^2 + dX^2) + r^2d\Omega^...
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55 views

Eight-shaped orbit determination

I was playing a game on my smartphone whose goal is to draw certain orbit in presence of certain central gravitational potential. I noticed that when there are two center of force is possible to have ...
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144 views

Coordinate system for numerical simulation of general relativity

Lets say i want to simulate the differential equations of GR with some numerical method. I can express the Einstein tensor in terms of the christoffel symbols which in turn can be expressed in terms ...
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135 views

Active and passive transformations in coordinate-free relativity

Consider a coordinate-free metric tensor $g$ which can take different coordinate forms, say $g_{ij}(x)$ and $g_{i'j'}(x')$. Also consider a metric tensor $h$ which is related to $g$ by an active ...
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75 views

Noether charge in light-cone coordinates (1+1D)

I have read in this article http://arxiv.org/abs/1107.2917 that the noether charge (in 1+1 D) $$ Q= \int dx \; q_t$$ could be written in terms of lightcone coordinates $x^\pm = t\pm x$ as $$Q=\int dx^...
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186 views

Lapse Function and Shift Vector in Minkowski and de Sitter

I'd like to find the lapse function and shift vector in 1+1 Minkowski as well as 1+1 de Sitter (flat foliation) for a region foliated this way: The $y$-axis represents time while the x-axis ...