Questions tagged [coordinate-systems]
A set of numbers used to quantify location in space.
2
votes
1answer
38 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 ...
-2
votes
2answers
58 views
In the Schwarzschild Metric does $\sin^2\theta{\Delta}\phi^2={\Delta}\phi^2\sin^2\theta$?
In the Schwarzschild Metric as the spacetime interval between two points in spacetime approaches $0$ for any ratio between the length of time and space the spacetime interval between the points in ...
2
votes
1answer
55 views
Do canonical transformations form a group?
In a course on classical mechanics, we barely touched upon canonical transformations via generating functions. Just like Lorentz transformations form a group, I want to know if canonical ...
0
votes
1answer
16 views
Radial component of velocity at extreme distances
Suppose I am given that a planet's position with respect to some star is of the form $\textbf{r} = r\textbf{e}_{r}$. Then of course $\textbf{v} = \dot{r}\textbf{e}_{r} + r\dot{\theta}\textbf{e}_{\...
1
vote
1answer
36 views
How to calculate spherical coordinate components of dipole field?
I understand well enough how to calculate the radial and tangential components in spherical coordinates at a point due to a magnetic dipole field using the magnetic potential gradient ($\...
1
vote
1answer
52 views
Line Element Transformation
This is just something that I've made up to see if I understand the method.
If I have the line element:
$$ds^2 = dr^2 + r^2\,d\phi^2$$
and I want to carry out a transformation with $r = \dfrac{...
0
votes
1answer
44 views
Minkowski diagram
I have not well understood the picture of geogebra regarding the angle of time (t') that is inclined compared to (t) of 26.57°angle .
In the picture we see that the velocity is setted at 0.5c, for ...
4
votes
2answers
78 views
Lorentz-transformation
I don't understand how to derive the matrix representing the Lorentz-transformation given two systems S and S':
$$x' = \Lambda x$$
these transformations do not leave the differences $\Delta x^\mu$ ...
0
votes
0answers
17 views
Rindler observers are at rest with respect to each other?
I'm studying a chapter about Rindler coordinates right now. In this they say that two Rindler observers at $x = 1/a_1$ and $x= 1/a_2$ will both have speed $v =0$ at $\tau = 0$ compared to the inertial ...
0
votes
1answer
89 views
The role of Lorentz transformation
I will assume that spacetime is flat four dimensional manifold equipped with a Lorentzian metric and define,
Physical systems: any object that is capable of causing a response ( measurement) in the ...
0
votes
0answers
31 views
Why can't we rule out other synchronization procedures?
According to what I've read, it is generally regarded that Einstein's choice to have the speed of light equal in every direction is only a convention, and since there is no way to measure the one-way ...
0
votes
0answers
19 views
Is a change of coordinates the same thing as a change of chart on a manifold?
I am familiar with coordinate transformations in the common case. (Say, polar to cartesian and back)
I have recently been introduced to the definition of a differentiable manifold. Is it correct to ...
1
vote
0answers
45 views
Coordinate transformations [migrated]
I have two scalar functions of $x$ and $y$ that I can define:
$$f(x,y)=x^2+y^2\qquad
\text{and}\qquad
g(x,y)=x^2 + \sin^2(x) y^2.$$
Is it true that there is literally no coordinate change that will ...
1
vote
2answers
45 views
Differential geometry: If $\vec v = v^i \vec e_i$, then why is $\vec r = r \vec e_r$ in spherical coordinates?
In differential geometry (and later carried over to GR) any abstract vector $\vec v$, exists on its own vector space.
We can then choose to represent this vector in a coordinate basis $\vec v = v^i ...
1
vote
2answers
85 views
Metric tensor: Why relate it to Cartesian/Minkowski coordinates?
Why does the metric tensor always relate to cartesian coordinates?
Let's take the simple case for the metric tensor in 3D-space without a time dimension,
$g_{ij}=
\begin{bmatrix}
1 & 0 &...
1
vote
2answers
76 views
Do objects move in 2 directions at once?
If a velocity vector of an object can be divided into an x and y component relative to a second object's position, and both objects have gravity that attracts both objects to each other. We then know ...
1
vote
0answers
44 views
Is it possible to derive $2\times 2$ Lorentz transformation matrix from only eigenvectors?
As a preface, I am somewhat familiar with year 1 linear algebra but not too familiar with how one makes the connection to Lorentz transformation matrices so I apologize if the answer is obvious.
One ...
0
votes
1answer
46 views
Coordinate transformation of basis vectors
The question
Let $e_a$ be the coordinate basis vectors in a manifold described by coordinate system $x^a$. The vector displacement between two nearby points is given by
\begin{equation}
ds=dx^ae_a=dx'...
1
vote
0answers
42 views
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 ...
1
vote
1answer
38 views
A paradox about canonical transform preserving Poisson bracket?
Let $q,p$ denote the position and momentum. Consider a transform generated by $g$:
$q' = q + \epsilon \{q,g\}---(1a)$
$p' = p + \epsilon \{p,g\}---(1b)$
Then:
$\{q',p'\} = \{q,p\}+o(\epsilon^2)+\...
16
votes
8answers
3k views
Formal Definition of Dot Product
In most textbooks, dot product between two vectors is defined as:
$$\langle x_1,x_2,x_3\rangle \cdot \langle y_1,y_2,y_3\rangle = x_1 y_1 + x_2 y_2 + x_3 y _3$$
I understand how this definition ...
1
vote
1answer
40 views
Do Bianchi identities hold in all coordinates?
I understand by expanding out the Riemann tensor, that the Bianchi identities can be derived within a local inertial frame (LIF) by taking the partial derivatives of the Riemann tensor relations in a ...
1
vote
1answer
66 views
Confusion Einstein notation polar coordinates
I'm having issues using Einstein notation in polar coordinates in flat space, I must be missing something basic.
Consider the following example. Take the following metric on a 2+1 spacetime; $ds^2 = ...
0
votes
4answers
79 views
Curved space-time and metric tensor
I'm studying about curved spaces and I read that a manifold is flat if there a coordinate system such that the metric tensor is constant everywhere.
Then I also read that when the space-time tensor ...
0
votes
0answers
51 views
How to show the metric is unchanged under an orthogonal transform?
Say we have the transform such that $x^i \rightarrow (x')^{i'}=M^{i'}_i(x)$
where $M$ is an orthogonal rotation matrix
I've been asked to show that a general metric $g_{ij}(x)$ invariant under the ...
0
votes
1answer
54 views
Journey away and back at close to the speed of light $c$
If a person drives away from a clock tower at a speed close to
$c$ for one whole day, and then drives back to it at the same speed (for another day), what would he see during each of the journeys?
I ...
0
votes
2answers
42 views
Gram-Schmidt procedure for generating orthogonal generalised coordinates
For a general natural system, the kinetic energy part of the lagrangian may be written as $$T = \frac{1}{2}\sum_{ij} a_{ij}(q_{1}, q_{2}, ..., q_{n})\dot{q}_{i}\dot{q}_{j}.$$ For $n = 2$, $$T = \frac{...
0
votes
0answers
45 views
What do orbits around a black hole look like doing the calculations in isotropic coordinates?
JPL and others calculating ephemerides in the solar system are using a method based on taking the Schwarzschild solution, not as expressed in the most common Schwarzschild coordinates but in the ...
2
votes
0answers
72 views
Derivation of Lagrange's Equation [duplicate]
So I was going through the derivation of the Lagrange's Equation given
in Goldstein. And I have doubts about some of the steps. I would appreciate some clarifications
on them.
First:
$$ \frac{d}{...
-2
votes
2answers
70 views
To find the equation of Motion of a Simple Pendulum using Lagrangian mechanics [closed]
The Equation of the Trajectory of a particle can be obtained by eliminating the variable $t$ as we are doing in the equation of Trajectory of a parabolic Projectile. So, my question is if the equation ...
1
vote
0answers
24 views
How to properly align my relaxed molecule onto my gold electrode?
the question is described below since it can be considered a chemistry or physics question: How to translate from one plane to another? originally posted on mathematics SE
I am trying to get ...
1
vote
0answers
27 views
1
vote
2answers
80 views
Why the zero-order term in a variational transformation of coordinates should be identically the same as the old coordinates?
In the Ref.[1, page 61] the author proposes that transformations between two coordinate systems can be described by a continuous parameter $\varepsilon$ such that when $\varepsilon=0$ the original ...
0
votes
2answers
89 views
Is the concept “space” actually needed?
I started making my mind around space and time and recently came to a point where I wondered if the concept of "space" is actually needed to describe physical processes at all and not just some ...
77
votes
4answers
7k views
Why does nature favour the Laplacian?
The three-dimensional Laplacian can be defined as $$\nabla^2=\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}+\frac{\partial^2}{\partial z^2}.$$ Expressed in spherical coordinates, it ...
2
votes
1answer
46 views
Thermodynamical conjugate variables
In thermodynamics the potentials are typically only a function of 2 variables, say
$$U=U(S,V)$$
with entropy $S$ and volume $V$. I see that conjugate pairs $S,T$ or $p,V$ always have the unit of ...
-1
votes
1answer
29 views
Angular velocity of a pendulum in Cartesian coordinates
Hello I have a problem how to write down equations for the pendulum correctly. Say I would use Cartesian coordinates $x, y$ representing the position of the mass. Then the velocities would be usually ...
1
vote
1answer
76 views
Length contraction of an accelerating rod
Let us assume a rod of length, $4l$, initially at rest. The coordinates are defined s.t - (assuming uniform density)
$$X_{back-end}(0) = -2l$$
$$X_{front-end}(0) = +2l$$
$$X_{mid-point}(0) = 0$$
...
0
votes
1answer
43 views
What is the apparent radius of the photon sphere in isotropic coordinates?
In Schwarzschild coordinates the photon sphere is located at an "r-parameter" of $r=3GM/c^2$. If we are watching the photon sphere from infinity using a telescope such as the Event Horizon Telescope ...
0
votes
1answer
34 views
How to cast dipole point charge force expression from cylindrical to Cartesian coordinates
Background
I am currently building simulations of molecular dynamics and one thing I want to model is dipole interactions. I recently came across this post about calculating the force between a point ...
0
votes
0answers
21 views
Equation for Simple Harmonic Oscillator with moving base
It is known that the base of a simple harmonic oscillator moves according to a known function $u(t)$. Is the dynamics of this system given by
$$m\ddot{x} = -\nabla V = -\nabla\frac{k}{2}|x - u(t)|^2 =...
0
votes
1answer
75 views
Mass conservation in spherical coordinate
See four velocity $u^\alpha = \gamma(1,\beta,0,0)$ in a spherical coordinates $(ct,r,\theta,\phi)$,
The mass conservation is
\begin{equation}
\nabla_\mu(\rho u^\mu) = 0
\end{equation}
Then how it ...
1
vote
2answers
87 views
What exactly is the meaning of length contraction? [duplicate]
Let's say a train of length $L$ (wrt ground) is standing on rails which have markings for every femtometre (or even smaller units). Now if someone takes a photograph and measures the distance between ...
0
votes
2answers
74 views
Quantum Lorentz Transformations
Now I am reading a Weinberg's book "Quantum theory of field". Vol.1 page: 55
Сould you explain me the following things?
Einstein's principle of relativity states the equivalence of certain '...
0
votes
2answers
41 views
When calculating centripetal force, do we ignore non-radial or tangential forces
Suppose an object moving in circular motion in the vertical plane (ie such that gravity points directly downwards) around a central point attached by a string; the object is constantly accelerating as ...
0
votes
0answers
43 views
The chain rule and velocity transformation in relativity (2)
First of all, these answers (How to derive the law of velocity transformation using chain rule?, The chain rule and velocity transformation in relativity, and other from a quick search on this site.) ...
1
vote
0answers
31 views
Independence of position and velocity vector [duplicate]
Hi I am a mathematics student with an interest in Physics. In our Physics elective our prof. said if $\vec r$ denotes the position vector then the velocity vector $\vec v = \vec {\dot r} $ is ...
0
votes
1answer
39 views
Coordinate axes, convention, and the isotropy of space
Say something has cylindrical symmetry so I align it on an axis to take advantage of that symmetry (this will make things like calculating the volume of a cylinder much easier if the $z$ axis pierces ...
2
votes
1answer
62 views
Magnitude of vector field [closed]
I think this is more of a mathematical question, but since it's for a physics problem I decided to ask it here.
I have this complicated magnetic field in spherical coordinates $(r, \theta,\phi)$,
$$ ...
2
votes
0answers
42 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 ...
