A set of numbers used to quantify location in space.

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1answer
390 views

Stokes' theorem in complex coordinates (CFT)

I am studying CFT, where I encounter Stokes' theorem in complex coordinates: $$ \int_R (\partial_zv^z + \partial_{\bar{z}}v^{\bar{z}})dzd\bar{z} = i \int_{\partial R}(v^{z}d\bar{z} - v^{\bar{z}}dz). ...
0
votes
1answer
195 views

Field transformations

I'm reading Maggiore's book "A modern introduction to quantum field theory" and I'm very confused by what he did in chapter 2.6 page 31 eq. (2.80). He basically wants to find the generators of the ...
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2answers
328 views

Locally flat coordinate and Locally inertial frame

I am having some doubts on myself regarding the above concepts in General Relativity. First, I want to point out how I understand them so far. A male observer follows a timelike worldline ($\gamma$) ...
3
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1answer
47 views

contravariant and covariant vectors and their orthogonality in Euclidean space

I am reading this paper Sigma Coordinate - Contravariance and covariance and I understand how covariant and contravariant vectors are defined mathematically Covariance and Contravariance and I had ...
3
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5answers
125 views

Local inertial frame

In general relativity we introduce local inertial frames to be such frames where the laws of special relativity holds. Let $\xi^{\alpha}$ the coordinates in the local inertial frame, so we get ...
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5answers
666 views

Does coordinate time have physical meaning?

I have always been a little confused by the meaning of the "$t$" which appears in spacetime intervals or metrics in general relativity. I concluded that $t$ was just a mathematical thing which allow ...
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2answers
96 views

A manifold question: Why smooth functions and what is a Jacobian?

My question is what does a Jacobian have to do with the change of coordinates (coordinate transformation). Why do we care about this notion to start with? Also, why should it be non-singular?
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0answers
53 views

Feynman lectures, Volume I, chapter 13-4 [migrated]

While reading Feynman lectures on Physics, volume I, Chapter 13-4, I found following assumption, which I don't understand: Then, since $r^2 = \rho^2 + a^2$, $\rho\,d\rho = r\,dr$. Therefore ... ...
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1answer
34 views

Metric components transformation under change of coordinates

I have been studying Lie derivatives and some applications. While searching the web I found a refence with the following statement: For a general Riemannian manifold $M$, take a tangent vector field ...
10
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3answers
484 views

Age of the universe versus absolute time [duplicate]

In Wikipedia, the age of the universe is defined as the "time elapsed since the Big Bang" while "time" links to "the cosmological time parameter of comoving coordinates" which itself links to "the ...
2
votes
5answers
536 views

A reference frame is any coordinate system or just a set of Cartesian axes?

In Physics the idea of a reference frame is one important idea. In many texts I've seem, a reference frame is not defined explicitly, but rather there seems to be one implicit definition that a ...
1
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1answer
225 views

What is the procedure (matrix) for change of basis to go from Cartesian to polar coordinates and vice versa?

I'm following along with these notes, and at a certain point it talks about change of basis to go from polar to Cartesian coordinates and vice versa. It gives the following relations: ...
0
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1answer
46 views

Transformation matrices for basis and coordinate transformation in non-orthonormal coordinates

The transformation matrices for covariant and contravariant vectors are different but in orthonormal coordinate system numerical values in matrices turn out to be same although in mathematical proof ...
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2answers
120 views

Coordinate Singularity in Metric

Suppose I have some metric $$ds^2=g(t)dt^2+\frac{1}{r}dr^2$$ which has a singularity at $r=0$. However, if I make the coordinate transformation $u=\frac{1}{r}$, then I get: $$ds^2=g(t)dt^2+r^3 ...
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2answers
100 views

How can I convert Right Ascension and declination to distances?

I am calculating galaxy rotation curves for various galaxies in Ursa Major cluster and I want distance of those galaxies from the centre of Cluster. The values referred as coordinated are RA and dec ...
0
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1answer
153 views

Coordinate system for crystallographic groups

In the International Tables for Crystallography for each crystallographic group an asymmetric unit is supplied (mathematicians call this a fundamental domain of the group). This region is a bounded ...
2
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0answers
41 views

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 ...
0
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2answers
73 views

What does coordinate invariance mean?

I would like to really understand what the mathematical as well as Physical meaning of coordinate invariance is. I have pretended to know what this means, but upon thinking a little harder today, I am ...
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2answers
79 views

What is the metric tensor for?

I am wondering how to use the metric tensor, in practice? I read the book and done the exercises in A student's guide to vectors and tensors by Dan Fleisch. The concept of a tensor and their ...
0
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1answer
27 views

Inverse gauge transformation in general relativity [closed]

Can someone explain to me how (8.21) follows from (8.20). The Picture comes from A first course in general relativity (Schutz). Thanks and regards, Jens Wagemaker
3
votes
3answers
499 views

Clarifying what metric counts as flat space

In (2D) Cartesian coordinates, the Euclidean metric... $$\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$$ ...is flat space. If the diagonal elements are exchanged for other real numbers ...
2
votes
1answer
123 views

Description of charged sphere with Heaviside function in cylindrical coordinates

I need to describe density of charge of uniformly charged sphere (radius R, total charge Q, position of centre (0,0,0)) with Dirac delta function and Heaviside step function. The hard part is to ...
3
votes
3answers
120 views

Why doesn't $\vec{E} =\frac{1}{4\pi\epsilon_0} \int\frac{\rho \hat{r}\;dxdydz}{r^2}$ blow up at $r=0$, when $\rho$ is finite?

Electric field at $(x,y,z)$ produced by a continuous distribution of charges is given by:$$\mathbf{E}(x,y,z) =\dfrac{1}{4\pi\epsilon_0} \int\dfrac{\rho(x',y',z') \mathbf{\hat{r}} ...
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3answers
128 views

Covariant and contravariant 4-vector in special relativity

I've just learned about contra- and covariant vector in the context of special relativity (in electrodynamic) and I'm struggling with some concept. From what I found, an intuitive definition of ...
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0answers
26 views

Stewart platform formulas [closed]

What kind of formulas/equations are commonly used to implement Stewart Platforms in electronics and mechanics? Using a co-ordinate system, how would you determine the position of each actuator, etc?
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2answers
50 views

Killing field in Minkowski space-time

If we look at the killing equation for a vector field $X$ in $\mathbb{R}^{(p,q)}$ (or on an open subset thereof) in coordinates with constant diagonal pseudo-metric we get: ...
1
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1answer
36 views

Shear stress in cylindrical coordinates?

In cylindrical coordinates the momentum flux is given by (in the $r$ direction): $$ \Pi=-\eta \frac{\partial (r\omega)}{\partial r}$$ Where $\eta$ is the viscosity. Therefore one would expect that the ...
2
votes
3answers
208 views

Why doesn't a global frame of reference exist for GR?

I only have at best a layperson's familiarity with GR, so forgive me if I am asking a basic question, but I have heard that in GR, we cannot have a global frame of reference, that is a frame of ...
0
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2answers
72 views

For a giving metric in GR, how do we learn which observer the metric refer to?

For example, I have been told the Schwarzschild observer is far away from blackhole and events,(namely, I think, the observer is static at infinity of the coordinate.) And the second example,the ...
1
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1answer
56 views

Conversion of satellite coordinates from ITRF to J2000

I have coordinates of various satellites in two coordinate systems: Cartesian coordinates in the international terrestrial reference frame (ITRF) RA / DEC in J2000 epoch, as derived from plate ...
1
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1answer
39 views

Clarification in deriving the radial momentum operator $p_r$

In deriving an expression for $p_r$, a particle's radial momentum, I am unsure what is happening at a certain step. The derivation given in The Physics of Quantum Mechanics by Binney and Skinner is as ...
4
votes
1answer
2k views

How to determine satellite position in J2000 from latitude, longitude and distance from Earth?

Due to my task of writing orbit prediction routines I am trying to understand the reference frames better and how to use them ( particularly for Earth orbits ). I think I get the idea of what ECI ...
4
votes
0answers
31 views

Change of coordinates [duplicate]

this year I finished off my course of Physics I (the first general physics) at my university and we had a lot of exercises to do where in order to complete them I had to change the system ...
0
votes
1answer
50 views

Calculus of Variations - Virtual displacements

I am currently reading "The Variational Principles of Mechanics - Cornelius Lanczos", in which the author talks about the variation of a function $F(q_1, q_2, \dots q_n)$ where $q_1, q_2, \dots q_n$ ...
0
votes
1answer
55 views

Exact meaning of radial coordinate of the Schwarzschild metric

In this answer as well as on Wikipedia the radial coordinate of the Schwarzschild metric is described as follows: ...the r co-ordinate is the value you get by dividing the circumference of the ...
1
vote
0answers
50 views

Are there any universal forces which are cartesian in nature? [closed]

I was recently talking with someone about how I think the whole Cartesian xyz understanding of the universe evolved from animals thinking earth was flat. They could get along fine without having to ...
4
votes
0answers
113 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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1answer
70 views

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 ...
1
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1answer
179 views

Components of acceleration in spherical polar co-ordinate [closed]

I wanted to calculate two component of acceleration in polar co-ordinate. Starting from the lagrangian $$L= \frac{1}{2}m( \dot{r} ^{2}+ r^{2} \dot{ \theta } ^{2} ) -V(r, \theta )$$ I ...
-1
votes
1answer
31 views

Velocity and acceleration (as vectors) in a straight line

A student is trying to determine the acceleration of a feather as she drops it to the ground. If the student is looking to achieve a positive velocity and positive acceleration, what is the most ...
2
votes
1answer
61 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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votes
2answers
1k views

What is considered now over astronomical distances?

For the sake of discussion, let's say that Mars is exactly 5 light-minutes away and that Earth and Mars are moving with the exact same velocity so that special relativistic effects are irrelevant. ...
1
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1answer
47 views

Describing the shape of a singularity

Hawking and Ellis write about the difficulty of describing the shape of a singularity when presented with a manifold that has curves of finite length that don't reach a point in the manifold. ...
0
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0answers
36 views

How can you translate and rotate a inertial reference frame to a reference frame on the earths surface?

Suppose I have a fixed coordinate system $\mathcal C_1=(x'',y'',z'')$ with origin at the earths center. How can I translate/rotate that coordinate system into a tilted moving coordinate system on the ...
0
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0answers
28 views

How many degrees of freedom to diagonalize the metric?

In A. Zee's Einstein Gravity in a Nutshell, he starts with the following expansion of the metric at some point $P$ of a Riemannian manifold, with coordinates $x^\mu$ that have the origin at $P$: $$ ...
0
votes
1answer
22 views

The equation of continuity in isothermal system in spherical axis(transport phenomena)

My homework is about finding the equation of continuity in isothermal systems in spherical axis, I can't imagine a workaround for that since its a little complicated for me to understand velocities ...
0
votes
0answers
79 views

Confusion between primed and unprimed coordinates

While deriving Ampere's law in Magnetostatics we come across a term $\nabla\times J(r') = 0$ and the reasoning we give is because current density is only a function of primed coordinates hence it's ...
4
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3answers
946 views

Centrifugal Force and Polar Coordinates

In Classical Mechanics, both Goldstein and Taylor (authors of different books with the same title) talk about the centrifugal force term when solving the Euler-Lagrange equation for the two body ...
0
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1answer
69 views

Trajectory of a projectile in a three dimensional space [closed]

$g$: the gravitational acceleration—usually taken to be $9.81\:\mathrm{m/s^2}$ near the Earth's surface $θ$: the angle at which the projectile is launched $v$: the speed at which the projectile is ...