Questions tagged [coordinate-systems]
A set of numbers used to quantify location in space.
621
questions with no upvoted or accepted answers
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"Covariant general coordinate transformations" in the context of gauged spacetime translations
Reference:
Chapter 11.3.1 of Freedman and Van Proeyen's Supergravity textbook.
\begin{eqnarray}
\notag
\delta(a,\lambda) \phi(x) &=& \left(a^\mu(x) P_\mu -\frac{1}{2}\lambda^{\mu\nu}(x)M_{\...
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Contradiction in canonical transformation
The problem I'm supposed to solve is finding $Q$, such that $(p,q)\rightarrow(P,Q)$ is a canonical transformation. In this case $\mathcal{H}=\frac{p^{2}+q^{2}}{2}$ and the new hamiltonian $\mathcal{K}$...
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Active and passive transformation
Some active transformations on the system can be seen also as passive transformations for example the rotation of the system can be seen as the rotation of the observer in the opposite direction. ...
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Time-independent Canonical transformation conditions
A solution manual to Goldstein's $9.1$ states that for an explicit time-independent transformation (for a system with two degrees of freedom) to be canonical it must satisfy
$$\frac{\partial Q}{\...
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Hamiltonian dependence of variables
How can one say that in Hamilton mechanics the $q$'s are independent of the $p$'s while if I have the Lagrangian $L = \frac{1}{2}\dot{x}^2 + \frac{1}{2}x^2\dot{y}^2$ then $p_y = \frac{\partial{L}}{\...
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A problem related to metric variation
Under the coordinate transformation $\bar x=x+\varepsilon$, the variation of the metric $g^{\mu\nu}$ is:
$$
\delta g^{\mu\nu}(x)=\bar g^{\mu\nu}(x)-g^{\mu\nu}(x)=-\frac{\partial{ g^{\mu\nu}}}{\...
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How to find the canonical transformation if given the new Hamiltonian we want?
If we know the new Hamiltonian $H^{\prime}$ we want, and $H^{\prime} \neq 0$, how can we find the canonical transformations?
For example, we want to transform the $$H(p,q)=\frac{p^2}{2m}+\frac{1}{2}...
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Canonoid transformation that is NOT canonical
Canonoid transformations are defined here as preserving the hamiltonian structure of the dynamical system for a particular hamiltonian, but not necessariy for every hamiltonian, in such a way that a ...
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In Bondi gauge/coordinate, how to obtain covariant derivative of the 2-dimemsional sphere using null projection operators?
In Bondi coordinate , the Bondi-Sachs metric is written as
$$
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)
$$
Before any decomposition, the covariant derivative $...
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Is Gerard 't Hooft's Cellular Automaton Interpretation of Quantum Mechanics background independent?
In Gerard 't Hooft in his Cellular Automata Interpretation of Quantum Mechanics
(https://www.researchgate.net/publication/...
user234845
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Proper time of an object crossing event horizon in Kruskal coordinates
So I am reading a paper on a certain black hole paradox. The specifics actually don't matter, but if you want context (p16): black hole thought experiment. An object falls into a black hole. The ...
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Spherically Symmetric Spacetimes
I have been studying the Schwarzschild metric $g$ and its derivation.
The starting point is to assume the spacetime it describes is spherically symmetric. This means that the algebra of its Killing ...
- 3,836
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Acceleration interpretation in accelerated frames in general relativity
I would like to know whether my physical interpretation of some dynamics in accelerated frames is correct.
In a frame with acceleration $a$ we have the metric
$$ds^2 = (1+ax)^2 dt^2 - dx^2$$
The ...
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Pictures of Different Coordinate Systems in General Relativity
In General Relativity by Woodhouse there are the three following diagrams in Chapter 9 about Black Holes. Despite a (very brief) description of these diagrams in the book itself, I am struggling to ...
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Definition of velocity in classical mechanics
Let $(r_1,r_2,r_3)$ be the coordinates of a particle $r$ in the coordinate system $\phi$. Let $\{\hat{e_1},\hat{e_2},\hat{e_3}\}$ be the coordinate basis of $\phi$. Why do we define the velocity $v$ ...
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How to understand intuitively the three relations about coordinate transformation?
The question is related to the representation theory of group theory for physics. Generally, one can define the coordinate transformation $T$ by the following relation:
$$\vec{r}'=R(T)\vec{r}+\vec{t}(...
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3
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Sign problem in projectile with drag
So recently, I've been playing around with the variables in projectile motion with drag to see how the trajectory changes. I had my ODE and solved it using separation of variables but the equation ...
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Possible error in Marion and Thornton's Classical Dynamics of Particles and Systems
I was going over my notes on classical mechanics and just started to review rotation matrices which is the first topic the book starts with. On page 3
The rotation matrix associated with 1.2a and 1....
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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,
$$ds^2=-\big(1-\frac{r_S}{r}\big)dt^...
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Spherical Formulation of Quantum Mechanics
I always wondered, during my QM courses, if we don't explore enough of the freedom that the Lagrangian and Hamiltonian Classical Dynamics give us.
Classically, we can always make canonical ...
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Gravitational time dilation and the pace of time
If we are in empty space far from a black hole, at rest relative to the hole, we would look at a clock and a light source inside the gravitational field of the hole, then we would, according to the ...
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Kruskal-Szekeres spacetime
How does one determine the Kruskal Szekeres regions?
I know the Schwarzschild solution in Kruskal-Szekeres coordinates is given by
$$(ds)^2 = \frac{32(GM)^3}{r}\exp(-r/2GM)(-dT^2 + dX^2) + r^2d\Omega^...
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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 ...
- 1,526
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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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2
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284
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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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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
$$Q=\int dx^...
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Lapse Function and Shift Vector in Minkowski and de Sitter
I'd like to find the lapse function and shift vector in 1+1 Minkowski as well as 1+1 de Sitter (flat foliation) for a region foliated this way:
The $y$-axis represents time while the x-axis ...
2
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Gauge invariance in gravitational field
I have read that the linearized equation for the metric fluctuations $h_{\mu\nu}$, namely:
$$ \partial^2h^{\mu\nu}-\partial_{\alpha}(\partial^{\mu}h^{\nu\alpha}+\partial^{\nu}h^{\mu\alpha}) +\partial^...
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Euler angles and curvilinear coordinate systems
If I have a curvilinear coordinate system and supposing I impose the condition that back transformations to Cartesian coordinate system are not permitted. I perform a rotation of the three axes( say ...
user66043
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950
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Euclidean AdS space in Poincaré coordinates
I have read anti-de Sitter (AdS) space and its Euclidean version both in Global and Poincaré coordinates. For Lorentzian case it is clear how one Poincaré patch cover only one half of the whole AdS ...
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433
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Question about Origins in Galilean transformation
I'm just learning about relativity, and every equation I see for a galilean transformation of frame $S'$ (moving with uniform velocity in the $x$-direction with respect to frame $S$) is $x'=x-vt$, $y'=...
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Double dot product in Cylindrical polar coordinates - Strain Energy
I'm working with a problem in linear elasticity, and I have to calculate the strain energy function as follows:
$$2W=σ_{ij}ε_{ij}$$
Where σ and ε are symmetric rank 2 tensors.
For cartesian ...
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How coordinate system shifting is related to similarity transformations?
I know that coordinate system shifting can be represented using matrices. But how exactly are similarity transformations related to coordinate shifts ?
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What do the components of light velocity look like in polar coordinates?
The Schwarzschild solution makes use of polar coordinates, and I'm wondering how the different components of velocity of light change with the position. Might I get some examples of light velocity ...
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1
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Are there functions of the metric that are scalars under spatial diffs up to total derivatives?
Let $g_{\mu\nu}$ be a metric on a manifold with a time direction $x^0$ singled out. I'm wondering if there exists a function $F(g_{\mu\nu},\partial_\rho g_{\mu\nu},\ldots)$ that transforms under ...
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Falling into a black hole emitter vs observer
Let's say we are working with the Schwarzschild metric and we have an emitter of light falling into a Schwarzschild black hole.
Suppose we define the quantity $$u=t- v$$ where $$dv/dr= 1/(1-r_{s}/r)$$...
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Operational Definition of Reference Frame in General Relativity
Most treatments of GR begin with the assumption that spacetime is a pseudo-Riemannian manifold (or, sometimes, that it is a more general manifold). But this entails quite a few tacit assumptions about ...
user59591
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Converting a gravitation potential from spherical to Cartesian and adding arbitrary tilt
I am working on utilising different potentials in my N-body code to do toy simulations (massless particles in a potential. However, I ran into issues converting a certain potential from a paper into ...
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1
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Ricci tensor in locally Lorentz frame
In the book A Relativist's Toolkit by Eric Poisson, section 4.1.4, page $123$, it is written that in a local Lorentz frame at a point $P$:
$$\delta R_{\alpha \beta} \stackrel{*}{=} \delta\left(\Gamma^\...
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Significance of simultaneous transformation of $t$ and $\phi$ in kerr metric
The Kerr metric in Boyer-Lindquist coordinate is invariant under simultaneous transformations $t \rightarrow -t$ and $\phi \rightarrow -\phi$ but not invariant if we apply one transformation only. ...
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Definition of molecular vibrational coordinates
The wavefunction of a molecule is a function $f(x_1,\ldots,x_N)$ taking the coordinates of the $N$ atoms in the molecules (we ignore the electronic motion) and returning a complex number.
...
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Nonvanishing $g_{00}$ metric component in isotropic coordinates
I have obtained a numerical solution to the Einstein's equations using the isotropic metric ansatz
$$ds^2=-e^{f(r)}\,dt^2+e^{g(r)}\left(dr^2+r^2\,d\Omega^2\right).$$
The associated energy density ...
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Is wedge product a tensor or a pseudo tensor?
I'm doing an exercise where $J$ is a 1-form on a manifold $M$ of dimension $N$.
The exercise ask me to calculate $J∧(*J)$ with $J=dx^0+2dx^1$ in a minkowski space with metric =(-1,1,1,1) where $*J$ is ...
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Problem in calculation of spherically symmetric Laplacian in electrodynamics
I have come across the following operation in two electrodynamics textbooks, which I find problematic: When evaluating an integral over a Laplacian in a spherically symmetric function, the radial term ...
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2
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What equation governs the time boost of a distant object caused by local acceleration?
This isn't the time dilation aka rate change, but rather due to perspective change, where is Sam looking on Sally's worldline?
For example, imagine Sam is 1000 lightyears from Sally and Sally is ...
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Question on the transformation from Boyer-Lindquist to Kerr-Schild coordinates, for a modified Kerr metric
From Kerr metric, we do know that there exist a function with the form of:
$$\Delta = r^2 - 2 M r + a^2 \tag{1}.$$
Following $[1]$, I did understand the coordinate transformation from Boyer-Lindquist (...
- 3,075
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Rescaling the null coordinates
Given a $4$-dimensional spacetime described by four coordinates $(t,r,\theta,\phi)$, we usually define the null coordinates by,
\begin{equation}
u = \frac{t-r}{2}, \quad v = \frac{t+r}{2}
\end{...
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Feynman parameters for $n=3$
I proved the general formula for the Feynman parameters:
\begin{equation}
\frac{1}{P_1^{a_1}...P_n^{a_n}}=\frac{\Gamma(a_1+...+a_n)}{\Gamma(a_1)...\Gamma(a_n)}\int_0^1dx_1...dx_n\delta(1-x_1-...-x_n)\...
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Expansion of the universe and superluminal speeds - an analogy
To illustrate, let's imagine the whole universe having a coordinate grid. Essentially a great big mesh grid of interlocking meter sticks.
Now... Einstein's theory of relativity tells us that nothing ...
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Reparameterization invariance in gravity
It's often said that gravity/general relativity has 'reparameterization invariance.' In particular, this comes up when people talk about the duality between the Sachdev-Ye-Kitaev (SYK) model and ...
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