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Questions tagged [coordinate-systems]

A set of numbers used to quantify location in space.

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3 answers
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Choice of Generalized Coordinates

I am working on solving problem 5.24 from the $3^{\mathrm{rd}}$ edition of Goldstein's Classical Mechanics: A wheel rolls down a flat inclined surface that makes an angle $\alpha$ with the horizontal....
Georgy Zhukov's user avatar
2 votes
0 answers
30 views

Hamiltonian is unbounded from below in only one coordinate system

I'm studying a complex scalar field theory in a spatially flat FLRW background. Using the standard conformal time metric $$ds^2 = dt^2 - a^2(dr^2 - r^2 d\Omega_2^2)$$ (where $a$ is the scale factor), ...
Daniel Waters's user avatar
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Transforming normals into a specific coordinate system [migrated]

Let's say I have normals defined from points of latitude and longitude on a sphere (represents the satellite object). The coordinate system we want to transform these normals into is z points to ...
AGJADSGK's user avatar
3 votes
4 answers
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How do I know if a motion is 1 dimensional or 2 dimensional?

If an object is moving in a straight line with an angle with x axis (it may be vertical or horizontal) , is it 1 dimensional or 2 dimensional? The question was asked by my teacher and he himself gave ...
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4 votes
3 answers
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Meaning of general Lorentz transformations

According to Wikipedia, the Lorentz transformations for two inertial frames are written:$$\begin{cases} t'=\gamma(t-\frac{\mathbf {r}_{\parallel }.\mathbf{v} }{c^{2}} )\;\;\;(*)\\\mathbf {r'...
The Tiler's user avatar
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2 votes
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Boundary conditions on transition maps on general relativity

On the initial courses of topology and differential geometry, we learn again and again about charts, and atlas, and transition maps. I feel that transition maps are a very powerful idea, because they ...
UnkemptPanda's user avatar
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Problem with linearized gravity on flat background in spherical coordinates

I am solving the linearized Einstein equations with a flat background in spherical coordinates, i.e $ds^2 = -dt^2 + dr^2 + r^2 d\Omega^2$ and writing $h_{\mu\nu}$ in terms of spherical harmonics. ...
1 vote
2 answers
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What is the correct way to think of position?

How accurate would it be to think of position (along some axis) as the component of radius vector. Example: $$ \textbf{r} = x \hat{\textbf{i}} + y \hat{\textbf{j}} $$ And if that is correct, we could ...
Alexander Djurovich's user avatar
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Changing coordinate system [migrated]

Someone please explain how did we get second term in equation 2.15.
Mr. Wayne's user avatar
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1 answer
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How does this proof of Gauss’ law generalize from $1$ to $n$ charges?

I am having trouble seeing how the proof of Gauss’ law for one charge generalizes to hold for multiple charges in Griffiths’ introduction to electrodynamics. Gauss’ law is proved for one charge (for ...
Joa's user avatar
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Length Contraction: is $t'$ or $t = 0$?

To demonstrate my confusion - let's say there is a rod traveling with velocity +v relative to S, and in S, the length of the rod is measured to be $L$. If I want to go from S to S', the frame where a ...
Emil Sriram's user avatar
1 vote
1 answer
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Invertibility between generalized and actual coordinates

Chapter $1$, page $13$ of Classical Mechanics by Goldstein ($2^{nd}$ edition), he states the following after defining a transformation equation: "It is always assumed that one can transform back ...
Aditya Krishna Panickar's user avatar
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1 answer
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How can I call a transformation where only the $z$-axis is modified by a function in each point?

What is called a transformation that maps $z \rightarrow |f(z)|\, z$, while the other axis stays the same? Is it a conformal transformation?
Aleph12345's user avatar
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2 answers
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Generalized momentum

I am studying Hamiltonian Mechanics and I was questioning about some laws of conservation: in an isolate system, the Lagrangian $\mathcal{L}=\mathcal{L}(q,\dot q)$ is a function of the generalized ...
user1255055's user avatar
1 vote
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How to derive Feffermann-Graham expansion for AdS Vaidya geometries?

Introduction The Feffermann-Graham expansion for an asymptotically AdS spacetime [0] looks like Poincare AdS but with the flat space replaced by a more general metric i.e. $$ds^2=\frac{1}{z^2}(g_{\mu \...
Sanjana's user avatar
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Differential form of Lorentz equations

A Lorentz transformation for a boost in the $x$ direction ($S'$ moves in $+x$, $v>0$) is given by: $$ t'=\gamma\left(t-v\frac{x}{c^2}\right),~x'=\gamma(x-vt)$$ In the derivation of the addition of ...
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1 answer
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How to transform coordinates in the Action equation?

For a particle moving in one dimension, and we have two people describing the motion of that particle one is stationary (let's call him Lenny) and the other (George) is moving relative to Lenny. Lenny'...
zizaaooo's user avatar
6 votes
1 answer
160 views

Radial reparametrization ansatz in Schwarzschild metric derivation

The standard derivation of Schwarzschild solution (and Birkhoff's theorem) seem to begin with the most general spherically symmetric static metric $$ds^2 = -U(\rho) dt^2 + V(\rho) d\rho^2 + W(\rho) \...
UnkemptPanda's user avatar
1 vote
1 answer
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Confusing Goldstein Statement about Magnitude of the Lagrangian

On page 345 of Goldstein's Classical Mechanics 3rd Ed., he writes: ...the Hamiltonian is dependent both in magnitude and in functional form upon the initial choice of generalized coordinates. For the ...
user1247's user avatar
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Removing the cosmic horizon in the de Sitter metric

The metric for de Sitter spacetime in static coordinates is $$ds^2 = \left(1-\frac{r^2}{\ell^2}\right)dt^2 - \frac {1}{1 - \frac{r^2}{\ell^2}}dr^2 - r^2\,d\Omega_2^2.$$ It is evident that there exists ...
Daniel Waters's user avatar
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1 answer
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Birkhoff's theorem and Schwarzschild vacuum solution [duplicate]

Birkhoff's theorem states that any spherically symmetric solution of the vacuum field equations must be static and asymptotically flat, but the well-known Schwarzschild solution satisfies these ...
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3 votes
7 answers
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Physical Quantities Sign Convention

I see that almost all physical quantities carry signs. But the confusion I have is what they really mean. Does negative velocity mean decreasing velocity or velocity in the opposite direction? Does ...
Singing Account's user avatar
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1 answer
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Interior Solution for Black Hole in Particular

This paper seems to suggest that the interior metric for a black hole in particular (a.k.a not a different kind of spherically symmetric non-rotating body) is just the exterior Schwarzschild metric ...
user345249's user avatar
4 votes
3 answers
199 views

Change of variables from FRW metric to Newtonian gauge

My question arises from a physics paper, where they state that if we take the FRW metric as follows, where $t_c$ and $\vec{x}$ are the FRW comoving coordinates: $$ds^2=-dt_c^2+a^2(t_c)d\vec{x}_c^2$$ ...
Wild Feather's user avatar
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3 answers
130 views

Can you tell who is moving through time? [closed]

In relativity, there is no way to tell if you are moving through space. So, if you were inside of a box, there would be no way for you to tell if you were moving or not. However, can you know who is ...
John W's user avatar
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Derivation of measure for summation over surfaces, including the polyakov action

In his 1981 paper "Quantum geometry of bosonic strings" Polyakov defines a measure for the summation over continuous surfaces. This measure must count all surfaces of a given area with the ...
Jens Wagemaker's user avatar
3 votes
1 answer
81 views

Circumference of ellipse in post-Newtonian metric

The post-Newtonian metric, in harmonic coordinates, is: $$\tag{1} \mathrm{d}s^2=-\left(1+\dfrac{2\phi}{c^2}\right)c^2\mathrm{d}t^2 + \left(1-\dfrac{2\phi}{c^2}\right)\mathrm{d}\mathbf{x}^2$$ where $\...
gravitone123's user avatar
1 vote
1 answer
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Number of independent reparametrization gauge invariances of the 'world $(n+1)$-manifold action' of $n$-dimensional objects

As a generalization of point particle dynamics, one can conceive of a theory of $n-$dimensional objects with 'world-manifold' action given by $$ S[X] = -\frac{T}{2} \int d^{n+1}\sigma \sqrt{h} h^{\...
Hyeongmuk LIM's user avatar
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2 answers
112 views

Event horizon is a null surface - what about the angular coordinates?

From the Schwarzschild metric $$ds^2=(1-2m/r)dt^2-(1-2m/r)^{-1}dr^2-r^2(dθ^2+\sin^2⁡θ dϕ^2)$$ on the surface $r=2m$ (setting $dr=0$) we have $$ds^2=-r^2(dθ^2+\sin^2⁡θ dϕ^2).$$ This looks spacelike ($...
Khun Chang's user avatar
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3 answers
104 views

Free particle in spherical coordinates

I'm trying to solve the very simple equation: $$-\frac{\hbar^2}{2m}\nabla^2 \psi = E\psi$$ but in polar coordinates. I used separation of variables to find out that my wave function is of the form: $$\...
Habouz's user avatar
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0 answers
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Position and displacement vector in Arc coordinate system

In Arc coordinate system the position of the particle is given by the length of the path(which is pre-determined and may also be curved) that it has travelled so how can we write it's position vector ...
Manish's user avatar
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-3 votes
1 answer
71 views

Does quantum entanglement arise from perpendicular time vectors? [closed]

From what I understand, "quantum entanglement" is a phenomenon where certain information travels instantly between entangled particles, regardless of distance in space. When thinking of ...
Quantum Wonder's user avatar
1 vote
1 answer
71 views

Generating function condition not satisfied?

We want to find a generating function $S(q_i,P_i,t)$ such that we get the best possible canonical transformations. So it must satisfy the Hamilton-Jacobi equation: $$H(q_i,\frac{\partial S}{\partial ...
Krum Kutsarov's user avatar
4 votes
3 answers
408 views

Confusion over what constitutes a uniform gravitational field in relativity

Suppose we have some observer moving upwards with a constant proper acceleration, by the equivalence principle this is the same as the observer remaining stationary in a gravitational field, like ...
NaiDoeShacks's user avatar
2 votes
2 answers
290 views

A worldline from the perspective of another in special relativity

Suppose we have two worldlines $\mathbf{x}(\alpha)$ and $\mathbf{y}(\alpha)$ parameterised by $\alpha$. What does the first worldline look like in a frame moving along the second? Is the correct ...
Ted Burgess's user avatar
1 vote
1 answer
100 views

Understanding differentials in an equation in general relativity

I have not studied physics but I was browsing Carroll's relativity book and randomly stumbled upon the following which I would like to understand mathematically. It says $$ds^{2} = 0 = - \left( 1 - \...
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4 votes
3 answers
110 views

Complex coordinates $ds^2 = dzdz̄$ in 2d

I have a very elementary question about complex coordinates in two dimensions. When we have a 2D Euclidean space, $$ds^2 = dx^2 +dy^2$$ and we go to complex coordinates: $$z = x + iy$$ $$z̄ = x - iy$$ ...
j_stoney's user avatar
6 votes
4 answers
124 views

How do we interpret measurements of Mercury's position?

When scientists measured the position of Mercury in the 18th century, they interpreted the results assuming a Euclidean background, because they did not know general relativity. So they measured $r$ ...
Giovanni's user avatar
0 votes
1 answer
127 views

Rotating a system

brekely physics book chapter 2 page 30 , a question about rotating a system by $ \frac{\pi}{2} $ around the z axis clockwise direction and writing vectors according to the new axis after rotation ...
dareen's user avatar
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0 answers
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Galilean transformation vs boost matrices

I'm confused about the difference between a Galilean transformation and boost with reference to their matrices. I was given four statements (listed below) but I'm not sure what I should be looking for ...
rose's user avatar
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1 answer
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Schroedinger equation with value of wave function in polar coordinates?

I'm trying to get a better sense of what causes an increase in the magnitude and phase of the wave function at a given point. Is there a way to rewrite the schroedinger equation such that it ...
user56834's user avatar
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1 vote
3 answers
161 views

Whether nonlinear coordinate transformations are symmetries of flat spacetime

I am editing this question after the answers are posted just to present my question a little clearly (without changing the main theme of the question). Moreover, this question is solely about flat ...
Solidification's user avatar
0 votes
0 answers
39 views

Taylor expansion of scalar function for a coordinate infinitesimal transformation (Poincaré group)

For a coordinate infinitesimal transformation of the form $x^{\prime \mu} = x^{\mu} + a^{\mu} + \omega^{\mu}_{ \ \nu}x^{\nu}$, we want to derive its effect on a space of scalar functions $f(x)$. This ...
SweetTomato's user avatar
2 votes
3 answers
246 views

Orbit description in Schwarzschild metric

Suppose to have a restricted 2-body system (BH + star with $M_{BH}\gg M_{\mathrm{star}}$) and you want to describe the orbit of the star relative to the BH, i.e. in the Schwarzschild metric. Usually, ...
gravitone123's user avatar
0 votes
0 answers
35 views

Susskind GR: expansion of one set of coordinates in terms of another coordinate set

In lecture 3 of Leonard Susskind's Theoretical Minimum course on general relativity, he shows the expansion of one set of coordinates, $x^m$, in terms of another set, $y^m$, as $$ \begin{equation} x^...
MattHusz's user avatar
  • 239
1 vote
2 answers
71 views

Why is the overall phase of a wave traveling backward notated as $-kx - \omega t$ instead of $kx + \omega t$?

in textbook i read the equation for the wave equation that travel forward in the $+x$ direction is $$ y(x,t) = Ae^{i(kx -ωt)} $$ and for wave equation that travel backward in the $-x$ direction is $$...
Tulip Lhospita's user avatar
1 vote
1 answer
55 views

Worldsheet action in the presence of background fields in complex coordinates

We will start with the worldsheet action under massless background fields - the graviton $G_{\mu\nu}$ and Kalb-Ramond field $B_{\mu\nu}$ (we choose to exclude the dilaton $\Phi$ that also appears in ...
Daniel Vainshtein's user avatar
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0 answers
62 views

Can you model relativistic interactions without locality?

Assume $c=1$ I've been doing relativity by myself so I may be making some assumptions here that I would not have if my learning had been more extensive. One such assumption is that you can model the ...
NaiDoeShacks's user avatar
0 votes
1 answer
58 views

Non-homogenous Helmholtz equation in 3+1D: Green's function and solution

I've been reading Jackson's Chapter 8.10 and trying to find the Green's function for a non-homogenous Helmholtz equation. The problem is in cylindrical coordinates. I first made a Fourier transform to ...
Rosabella M's user avatar
2 votes
3 answers
155 views

Geodesic in flat space in spherical coordinates

let's consider the expression, where $u^\mu$ is the tangent vector to the geodesic $\theta = \nabla_\mu u^\mu$....scalar $\Rightarrow$ valid in every coordinate system So in flat space in Cartesian ...
Coderboy's user avatar

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