A set of numbers used to quantify location in space.

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1answer
64 views

$\frac{d}{dr}=0$ and $\frac{d}{dz}=0$ (cylindrical coordinates) for a 1D ring

In http://ritchie.chem.ox.ac.uk/Grant%20Teaching/2010/Lecture%204%202010.pdf slide 21 of 26, he says "Radius of ring is fixed and so derivatives in $r$ are 0." Presumably this goes for ...
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1answer
66 views

Coordinate velocity and free-falling past an event horizon

Ws can re-arrange the standard Schwarzschild metric as $$\left(\frac{d\tau}{dt}\right)^2=\frac{r-1}{r}\left[1-\left(\frac{r}{r-1}\frac{dr}{dt}\right)^2\right]$$ Where $\frac{r-1}{r}$ is the local ...
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1answer
51 views

In central-force mechanics, how do we substitute $ξ=\frac{1}{r}$?

I have taken a look at central-force mechanics in the past, but I still cannot understand how $ξ=\frac{1}{r}$ is substituted to find $\frac{d^2r}{dt^2}$ in terms of ξ. So I know from $F=ma$ that: ...
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3answers
69 views

Prove that the spacetime interval is not invariant under Galilean transformations [closed]

The spacetime interval $(\Delta s)^2 = (\Delta x)^2 + (\Delta y)^2 + (\Delta z)^2 - c^2(\Delta t)^2$ is invariant under the Lorentz transformation and this isn't the case for the Galilean ...
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0answers
248 views

Hamiltonian for Electron in Magnetic Field with Symmetric Gauge in Polar Coordinates

I am new on the board and have a question about how to write the Hamiltonian for an electron in a magnetic field rotating at a fixed radius. I would like to write the hamiltonian using the symmetric ...
4
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2answers
943 views

Light-cone coordinates

The light-cone coordinates are defined as $$x^{\pm} ~=~\frac{x^0 \pm x^3}{\sqrt{2}}.$$ Then in the light cone coordinates the position 4-vector becomes: $(x^+, x^-, x^1, x^2)$ . Zwiebach, in his A ...
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1answer
86 views

Schwarzschild metric: Change in coordinates corresponds to change in object?

I have been reading about the Schwarzschild metric in the book "General Relativity: An Introduction for Physicists" by Hobson, Efstathiou and Lasenby and it appears to say something counter intuitive. ...
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0answers
27 views

Evaluating derivatives with respect to certain vector axis

So, I am trying to work in Spherical coordinates. I have a predefined fixed axis, $\hat{v}_0$, so that $\alpha=\vec{r}.\hat{v}_0$ Now, I am interested in the following: \begin{equation} ...
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3answers
402 views

How to prove a symmetric tensor is indeed a tensor?

Our professor defined a rank $(k,l)$ tensor as something that transforms like a tensor as follows: $$T^{\mu_1' \mu_2'...\mu_k'}{}_{\nu_1'\nu_2'...\nu_l'} ~=~ ...
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0answers
133 views

Accelerometer Pitch and Roll Calculation

I am developing an application for a device that needs to know its tilt/orientation, specifically, pitch and roll. Roll is positive if the right side of the device is elevated, and pitch is positive ...
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1answer
40 views

Ergosphere in different coordinates

I am currently working with the concept of an ergosphere and I was wondering if it has any meaning to consider the ergosphere after changing coordinates. I mean if someone looks only on the sign of ...
0
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1answer
123 views

Tension in the simple pendulum (polar coordinates)

Let's consider the simple pendulum as is displayed here or over there (page 10). The analysis of the second Newton's law in polar coordinates goes as follows: $$ \vec{F} = m\frac{d^2\vec{r}}{dt^2}, ...
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1answer
80 views

Confusing concepts in proof of spherical addition theorem

In http://scipp.ucsc.edu/~haber/ph116C/SphericalHarmonics_12.pdf, section 4, pages 6..9 is a proof of the spherical harmonics addition theorem. Page 8 has eq.(25), an application of Laplace series: ...
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2answers
156 views

Why does the Lorentz transformation have to be a linear transformation? [duplicate]

In my textbook, they say the following statements before doing a proof for the Lorentz transformation: We know that the Galilean transformation $x' = x - vt$ is incorrect, but what is the ...
2
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1answer
106 views

Why does a system have to be holonomic?

So I'm doing some work from Taylor's mechanics book. He says for the problems in the book, we require the system to be holonomic - that is the number of generalized coordinates = number of Deg. of ...
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2answers
694 views

What is the physical meaning of the Eddington-Finkelstein coordinates?

What is the physical meaning of the Eddington-Finkelstein coordinates? I want to see a some physical process (experimental) that could explain the many transformations of coordinates into this ...
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1answer
44 views

Eulerian angle understanding

I have alot of confusion with Eulerian angle so first of all I would like to address something I don't understand from the book and maybe that would shed some light on the intuition of eulerian ...
2
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2answers
137 views

What coordinate systems allows the magnitude of the basis vectors to change with position?

I'm familiar with coordinate systems where the direction of the basis vectors changes with position, but I haven't come across any where the relative magnitude of the basis vectors themselves are ...
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0answers
39 views

Is Lorentz Transformation about the difference of coordinates or coordinates of itself?

I have seen different authorities talking about different interpretation of Lorentz transformation. In his book 'Introduction to Classical Mechanics', David Morin states We always talk about eh ...
3
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1answer
317 views

Non-inertial frames in Lagrangian mechanics?

Building on this Phys.SE post I am interested in how non-inertial frames can be considered in Lagrangian mechanics. My understanding is that changing the reference frame causes a transformation of the ...
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0answers
43 views

Transform velocities from one frame to an other within a rigid body

I come from non-physics background but just came to face the following problem. I have a rigid body with two attached frames of reference A and A'. I know: the rotation and translation between A ...
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1answer
32 views

Co-ordinate rotations

I need help transforming a magnetic field vector from one co-ordinate system to another. I have the components of the Earth's magnetic field in a co-ordinate system with z facing radially into the ...
0
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1answer
116 views

Free-fall path into a black hole in Kruskal Coordinates

If an object at t=0 begins to free-fall into a black hole from X in Kruskal coordinates (https://en.wikipedia.org/wiki/Kruskal%E2%80%93Szekeres_coordinates), what does its path on the Kruskal-Szekeres ...
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4answers
499 views

Why is displacement negative during free fall?

I am confused by this question. Displacement is shortest path travelled by an object, but I had seen in my book that during free fall displacement is negative.
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1answer
38 views

$\sqrt{\frac{\omega ^2}{c^2}-k_z^2}$ in cylindrical harmonics

The radial component of the solution of the wave equation in cylindrical coordinates is $$J_\nu \bigg(\rho\sqrt{\frac{\omega ^2}{c^2}-k_z^2}\,\,\bigg).$$ But I always thought that $\frac \omega c$ ...
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1answer
64 views

Non-inertial system [duplicate]

Supposing I am in a non-inertial system and I don't know what forces are acting. How can I test EXPERIMENTALLY and in practice to be in a non inertial system? If I am in a system and I don't know how ...
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2answers
2k views

Coordinate Transformation of Scalar Fields in QFT

By definition scalar fields are independent of coordinate system, thus I would expect a scalar field $\psi [x]$ would not change under the transformation $x^\mu \to x^\mu + \epsilon^\mu $. Correct? ...
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0answers
13 views

Coordinate Transformation across different media

I have an arrangement that is as follows : (1) inner cylinder(radius r1) filled with water (2) outer cylinder(concentric, radius r2>r1) made of glass I have a sensor S1 at radius r(r1) at angle beta. ...
2
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2answers
9k views

Derive vector gradient in spherical coordinates from first principles

Trying to understand where the $\frac{1}{r sin(\theta)}$ and $1/r$ bits come in the definition of gradient. I've derived the spherical unit vectors but now I don't understand how to transform ...
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5answers
2k views

Why do we need coordinate-free descriptions?

I was reading a book on differential geometry in which it said that a problem early physicists such as Einstein faced was coordinates and they realized that physics does not obey man's coordinate ...
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1answer
40 views

Euler Angles with respect to base body when Euler Angles with respect to another body is known

Let's say I have a fixed base body $B_0$ with a reference frame $X_0Y_0Z_0$, and two other bodies, $B_1$ and $B_2$, rotated arbitrarily with respect to this base body. Coordinate systems fixed to ...
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1answer
64 views

Spherical Symmetric Metrics

In the case where all books try to illustrate a spherical metric, the procedure goes this way: First they impose isotropy in terms of polar coordinates so that one can write: $$ds^2=-A(r)dt^2 + ...
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1answer
33 views

E field using cylindrical coordinates

Can someone explain why, when I am going to calculate the $\vec{E}$ or $\vec{B}$ field of a charged ring in its axis (using cylindrical coordinates), the position of source field is $(R,0,0)$ and not ...
2
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1answer
77 views

Do Einstein's equations allow multiple solutions that agree in a neighborhood of a spacelike hypersurface?

This question is an extension of my a question that I have recently asked: Why doesn't a global frame of reference exist for GR?, where it was recommended that I post another question (so I am ...
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2answers
84 views

Stationary v/s Static

Blau, in his GR book, says that a stationary and spherically symmetric metric is automatically static. He says this easily follows from the fact that for a stationary metric, and in spherical ...
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2answers
127 views

Meaning of Proper time

Sorry for a bit of a basic question, but want to clarify things in my head. Is proper time quantified by the amount of physical process that an object, or physical system undergoes, for example the ...
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0answers
47 views

Change of variables for integral operator

One can write the operator $L=(\sqrt{1-i\partial_x^2}-1)$, as an integral, that is $$(\sqrt{1-i\partial_x^2}-1)B(x,t)=\frac{i}{4\pi^2} \int_{-\infty}^{\infty}(\omega(k_o+\kappa)-\omega(k_o))e^{i ...
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2answers
144 views

How are FRW metric and Minkowski metric physically different?

According to GR, matrices are coordinate invariant. Does this mean we can transform FRW metric to Minkowski metric with a coordinate transformation like $$dx'=dx\cdot a(t), dy' = dy\cdot a(t), dz' = ...
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0answers
41 views

Test bodies general relativity

I'm studying section 82 of the Landau & Lifshitz Field Theory vol.2 In this page it's written that the relative position of test bodies can't remain unchanged during time. And ok with this. But ...
4
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1answer
134 views

Invariant equations of motion under Lorentz transformations

My question regards the statement that an equation of motion may be invariant under a Lorentz transformation I just finished watching the Stanford University special relativity lectures on special ...
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6answers
2k views

Minkowski Metric Signature

When I learned about the Minkowski Space and it's coordinates, it was explained such that the metric turns out to be $$ ds^{2} = -(cdx^{0})^{2} +(dx^{1})^{2} + (dx^{2})^{2} + (dx^{3})^{2} $$ where $ ...
0
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2answers
107 views

Are spherical coordinates distances or angles?

I've become confused about spherical coordinates when dealing with electric fields. The way I always understood spherical coordinates is something like the below picture. To define a vector, you give ...
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2answers
85 views

Best coordinate system for Projectile motion [closed]

What is the best coordinate system for describing the projectile motion? Rectangular coordinate system or n-t(normal and tangential) coordinate system.
6
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2answers
287 views

Kerr Metric in Orthogonal form

I've seen the Kerr metric usually presented in the Boyer-Lindquist coordinates where there is a cross term in the $d\phi$ and $dt$ term. I've done a good bit of searching and cannot find any ...
3
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1answer
112 views

Manifolds, unit 2-sphere and stereographic projection

I am always passing through this example while reading about manifolds that I don't quite get. It is when describing the unit 2-sphere $S^2$ as an example of a manifold. They say, initially it may ...
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0answers
20 views

How do I compute the galactic cooridinates of the Earth for a given date?

The question is simple enough, but I wasn't able to find any tools online. Does anyone know of one, or a simple formula?
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3answers
1k views

Is there a quick way of finding the kinetic energy on spherical coordinates?

Assume a particle in 3D euclidean space. Its kinetic energy: $$ T = \frac{1}{2}m\left(\dot x^2 + \dot y^2 + \dot z^2\right) $$ I need to change to spherical coordinates and find its kinetic energy: ...
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0answers
57 views

commutation relation of angular momentum operator in non cartesian coordinates

The angular momentum operator $J$ in quantum mechanics with the commutation relation \begin{equation*} [J_i,J_j]=i\hbar\epsilon_{ijk}J_k \end{equation*} has the structure of a Lie-algebra. It is ...
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0answers
79 views

What exactly are generalized coordinates and how do they differ from regular coordinates?

I'm trying to learn the basics of Hamiltonian mechanics, which typically distinguishes itself from Newtonian mechanics as being described in terms of "generalized coordinates and momenta". What ...
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3answers
201 views

How can we define a frame of reference in general relativity?

I have started reading general relativity. (A First Course in General Relativity, Bernard Schutz). I am finding very hard to understand a frame of reference. When I was reading special relativity ...