Questions tagged [coordinate-systems]

A set of numbers used to quantify location in space.

Filter by
Sorted by
Tagged with
1 vote
1 answer
19 views

Is the massive Fierz-Pauli action invariant under changes of coordinates?

The action for linearlized gravity on a curved background has the form $S_{LinGrav}=\frac{1}{16 \pi G} \int d^{4} x \;\sqrt{-g}\;\left[-\frac{1}{4} \nabla_{\rho} h_{\mu \nu} \nabla^{\rho} h^{\mu \nu} +...
  • 11
1 vote
0 answers
36 views

Decomposing velocities in the tetrad frame

Background At each point in a manifold we can find an orthonormal set of basis vectors $\hat{\mathbf{e}}_\alpha$ such that $$ g(\mathbf{e}_\hat{\alpha}, \mathbf{e}_\hat{\beta}) = \eta_{\alpha \beta} $...
13 votes
3 answers
1k views

What is the relevance of the Lorentz factor in general relativity?

The Lorentz factor is ubiquitous in Special Relativity and is used to express "how much the measurements of time, length, and other physical properties change for an object while that object is ...
1 vote
2 answers
66 views

Double pendulum damping and spring forces

If I have the equations of motion for a double pendulum (from https://www.phys.lsu.edu/faculty/gonzalez/Teaching/Phys7221/DoublePendulum.pdf), can I include a time dependent damping and spring force ...
1 vote
0 answers
25 views

Are expressions like "x axis," "x dimension," "x direction," "x plane," "x boundary," etc. all hyphenated? [closed]

I'm not a native speaker of English. I'm currently writing a journal article and want to make sure I follow all conventions properly. Are expressions like "x axis," "x dimension," &...
1 vote
0 answers
19 views

Proof of the valence $\lambda$ of a canonical transformation equaling its Jacobian determinant [closed]

Let $Q(q,p),P(q,p)$ be a canonical transformation with valence $\lambda$. The following is intended to be a proof of the following relation: $$\lambda = \frac{\partial(Q,P)}{\partial(q,p)}.$$ Let $F(q,...
  • 183
1 vote
2 answers
45 views

Metric tensor determinant under coordinate transformation

I've been studying GR through Wald's and Carroll's books, and I've been trying to derive one expression. $$g(x^{\mu^\prime}) = \left|\dfrac{\partial x^{\mu^\prime}}{\partial x^{\mu}}\right|^{-2} g(x^\...
1 vote
2 answers
144 views

Where is Schwarzschild solution valid?

The extrinsic Schwarzschild solution is a vacuum solution, meaning that it is valid for regions of spacetime where there is no matter or energy. This seems to imply that the Schwarzschild solution is ...
  • 428
15 votes
2 answers
2k views

Why is clock synchronisation such a big deal in physics?

I was reading Classical Mechanics : The theoretical minimum by Leonard Susskind, and he says Assume that two clocks at different places can be synchronised. I don't understand why one should do that....
0 votes
2 answers
39 views

Net force acting on current loop in $B$ field linearly dependent $z$

Suppose there is a magnetic field $\overline B$, only in the $z$ direction and dependent linearly in $z$ coordinate e.g. $$\overline B=(0,0,B_0*z)$$ $B_0$ is a constant. And a circular current loop ...
3 votes
1 answer
123 views

What is the Schwarzschild metric in cylindrical coordinates?

I was researching online for different metrics of spacetime out of curiosity, and I found one that was said to be Schwarzschild metric in cylindrical coordinates: $$ds^2 = -\left(1-\frac{r_s}{r}\right)...
  • 1,684
1 vote
1 answer
74 views

Tortoise coordinate transformation

The differential form $dr$ can be written $\left(1-\dfrac{2GM}{r}\right)dr^*$ where $r^*$ is the tortoise coordinate. Writing the Schwartzchild metric then gives $ds^2$ = $\left(1-\dfrac{2GM}{r}\right)...
  • 428
2 votes
1 answer
55 views

Derivation of Hamilton-Jacobi (HJ) Equation

In the Derivation of Hamilton Jacobi Equation, I didn't understand the bold parts: we can write (1) formally as, $$ \frac{\partial F\left(q_i, Q_i, t\right)}{\partial t}=-H\left(p_i, q_i, t\right)=-H\...
0 votes
1 answer
24 views

Sagnac effect and interval invariance

In this article there is one thing I don't get: the author considers (page 9) transformation of the coordinates $$(t,r,\theta)\mapsto (t,r,\theta+\omega t)$$ and then apparently uses invariance of the ...
  • 143
6 votes
6 answers
340 views

On state transformations and the requirement of space-time invariance in (non-relativistic) quantum mechanics

I am trying to follow the development in Ballentine's Quantum Mechanics: A Modern Development but am struggling a lot. Please excuse my attaching of a picture of the development, but my question quite ...
  • 464
2 votes
1 answer
67 views

Vector field coordinate transformation

On Carroll's spacetime and geometry book, page 67, the book gives the component form of vector field commutator $$[X,Y]^\mu=X^\lambda\partial_\lambda Y^\mu- Y^\lambda\partial_\lambda x^\mu \tag{2.23}$$...
  • 949
1 vote
0 answers
38 views

Physical interpretation of linear non-Lorentz coordinate transformations

In school we are taught that Lorentz transformations relate coordinates of observers in uniform motion. Later we are taught that nonlinear coordinate transformations are associated with acceleration. ...
  • 43
1 vote
0 answers
25 views

Aren't the virtual work/virtual power principles in mechanics simply solving for the vector differential equation of motion in a preferred direction?

My conceptual understanding of the virtual work/virtual power principles is that, by hypothesizing "virtual displacements"/"virtual velocities", one can solve the equations of ...
  • 165
1 vote
1 answer
48 views

What are good books/chapters of books or articles to study canonical transformations in quantum mechanics at a graduate level?

I am looking for any kind of sources about canonical transformations in quantum mechanics in the operatorial formulation of the theory and its connection with the classical canonical transformation ...
3 votes
5 answers
118 views

Is it meaningful to draw $ct$ and $x$ axes by two lines intersecting at right angles?

In the Cartesian coordinate system, the x-axis is really perpendicular to the y-axis, by construction. Also, under a rotation of the coordinate system, the transformed coordinate axes $x',y'$ remain ...
1 vote
2 answers
50 views

How the equation of a projectile represents a parabola? [closed]

I am not able to prove that equation of motion of a projectile is parabola. The book simply says the given below is the equation of a parabola but does not clarify it $$y= {\tan\theta}x - \frac{g}{2(...
  • 21
2 votes
1 answer
44 views

Why is cosmological time unique?

According to the definition I have encountered for the concept of cosmological time, it is defined in the following way: The cosmological principle states that, at each location in the universe, it ...
0 votes
1 answer
55 views

Understanding generalized coordinates [closed]

I understand that for each of N particles they have a position: $$\vec{r}_i=\vec{r}_i (q_1, q_2, \dots, q_n, t)$$ where $q_1, q_2, \dots, q_n$ are n generalized coordinates. However I don't understand ...
1 vote
0 answers
48 views

What should be the definition of a comoving frame in phase space?

In short, I think there are two types of comoving frame when talking about distribution function, since it is defined in phase space. Which one should be the real one? I suppose this question is ...
  • 51
1 vote
1 answer
75 views

General Lorentz transformation between two four-vectors

I am reading a paper that makes the claim that $$ \Lambda^\mu {}_\nu = g^\mu{}_\nu - 2 \frac{(V + W)^\mu (V + W)_\nu}{(V + W)^2} + 2 \frac{W^\mu V_\nu}{V^2} $$ is a proper Lorentz transformation, ...
  • 244
0 votes
2 answers
58 views

Why is Lorentz Transformation defined with one super and one sub index?

I came across the Lorentz transformation in tensor form, usually written as $$\Lambda ^\mu _{\nu}$$ I understand that the first index usually corresponds to rows and the second to columns, and while I ...
2 votes
1 answer
108 views

Contravariant Components (Susskind's book)

In his book about SR & classical field theory, Susskind generalizes from the differential of $X'$ (function differential) to any 4-vector. I got stuck there trying to figure out why it is ...
1 vote
2 answers
59 views

How to deduce the general boost transformation matrix? [duplicate]

The general matrix for a boost in an arbitrary direction, is given by: $$\Lambda (\vec{v})= \begin{pmatrix} \gamma & -\gamma\beta_x & -\gamma\beta_y & -\gamma\beta_z\\ -\gamma\beta_x & ...
  • 51
1 vote
0 answers
117 views

How are co-ordinate systems built physically in curved space-time?

How do we physically choose a co-ordinate system for making astronomical observations? In a special relativistic system, the definition of relative velocity, clock synchronization is well understood ...
  • 1,224
1 vote
2 answers
67 views

What prevents the line element being Minkowskian in the vicinity of a point mass?

This is probably a naive question and I'm missing something really simple. The Schwarzschild solution has been constructed in consideration of the following requirements: The field equations $ \frac{\...
0 votes
2 answers
35 views

Why is the Centripetal Force Positive, if the acceleration is negative? [closed]

If $ma_r = F $, but F is positive and $a_r$ is negative, wouldn't this mean a negative mass? I understand why centripetal acceleration is or is defined to be negative, since it is in the opposite ...
2 votes
2 answers
132 views

Why can we put these conditions on coordinates of worldsheet?

https://www.asc.ohio-state.edu/mathur.16/classicalstring.pdf At first, I write some notations I need here. $I=[0,1]$, $M$ means $(1,3)$ Minkowski space, smooth map $X:I\times I\to M$ is timelike ...
0 votes
1 answer
50 views

Showing that the contravariant base vector transforms as a vector [closed]

I wanna show that $Z^a$ is indeed a contravariant vector in the same way I showed that $Z_i$ is indeed a covariant vector (see attached image).This is how I define $Z^a$ : $Z^a = \frac{\partial y^a}{\...
0 votes
0 answers
31 views

Vectors and One-forms in Cylindrical Coordinates and the Angular Momentum

The angular component of the velocity of a particle in cylindrical coordinates has different units if we consider the vector component $v^{\phi}$ or the one-form component $v_{\phi}$: $$ v^{\phi} = \...
  • 186
2 votes
0 answers
77 views

Frame of reference of a freely falling observer in General Relativity

Suppose I have an Einstein manifold (a manifold with a metric that solves the Einstein field equations. We can take $\Lambda = 0$ for simplicity in this example). The worldline of a freely falling ...
0 votes
0 answers
21 views

Why do we consider the whole process, taking a single moment? Universality of the vector equation

I have considered a simple model for describing the elastic force vector of a spring. First, I chose a reference frame in an arbitrary way, then I drew the necessary vectors, we get a ratio that ...
0 votes
1 answer
61 views

Deriving Lagrange equation with constraint

I'm having a hard time understanding the derivation of Lagrange equation from Newton's law when there is constraint (I'm ok with the basic case where there is only kinetic energy and potential ...
  • 119
1 vote
1 answer
28 views

Derivatives of the lagrangian of generalized coordinates [closed]

I know that $$U= \frac{1}{2} \sum_{j,k} A_{jk} q_j q_k \quad \quad T= \frac{1}{2} \sum_{j,k} m_{jk} \dot{q}_j \dot{q}_k $$ and the lagrangian is $$ \frac{\partial U}{\partial q_k} - \frac{d}{dt} \...
1 vote
0 answers
46 views

Visualizing the conformal compactification diagram of $G$

I asked a question a year and 3 months ago on mathstackexchange but after 3 bounties and still no answer I've decided to try here. Here's the link: conformal compactification. Construct a conformal ...
  • 157
0 votes
0 answers
18 views

To derivation the 2D flow ${\phi }_{2}-{\phi }_{1}$ and $ {\psi }_{2}-{\psi }_{1} $ in cylindrical coordinates

I try to derivation the 2D flow ${\phi }_{2}-{\phi }_{1}$ and $ {\psi }_{2}-{\psi }_{1} $ in cylindrical coordinates as follows, if there is a wrong concept of physics,please say some thing about it. (...
0 votes
3 answers
92 views

Is frame of reference a point of view? [duplicate]

The definition of frame of reference I found is :"It is just a coordinate". But in solving problem, my teacher always uses frame of reference by considering it at rest (although it is moving)...
0 votes
1 answer
29 views

Rigid body motion in Arnold's book

During the study of the motion of a rigid body, in Arnold's book, two coordinates systems are introduced: one is fixed $k=\{O',\hat e_1',\hat e_2',\hat e_3'\}$ and one is inside the rigid body $K=\{O,\...
  • 137
1 vote
2 answers
62 views

Choosing coordinates in Lagrangian Mechanics

Consider the problem of a hoop rolling down an inclined plane, with the plane sliding (frictionless) in a horizontal motion. I don't know how to choose the generalized coordinates for this system. In ...
0 votes
0 answers
47 views

What comprises of a 'sensible' coordinate transformation?

I am doing the course on general relativity at my university and have been struggling with covariant and contravariant vectors. I understand that components of contravariant vectors transform in a way ...
  • 1
0 votes
1 answer
28 views

Canonical Transformation of Poisson Bracket [closed]

In Goldstein section 9.4(pg 381) it tells us that for a Hamiltonian that is not explicitly time dependent, transformations of $Q = Q(q,p), P = P(q,p)$ are canonical if $$\frac{\partial Q}{\partial q} =...
  • 414
0 votes
1 answer
49 views

Fraction with components of Lorentz transformation

I want to show how partial derivative transforms under a Lorentz transformation. Since the partial derivative has a fixed definition with respect to the $x$-coordinate it stays unchanged: $\partial_\...
  • 415
0 votes
1 answer
77 views

How does one covert from Cartesian coordinates to Boyer-Lindquist coordinates?

I am new to physics stackexchange, but I have a question which I seem to have not been able to find an answer to. I already know that the transformations from Boyer-Lindquist coordinates to Cartesian ...
0 votes
2 answers
73 views

How to understand the quadratic form of kinetic energy with $\dot{q}$ coefficients?

Kinetic energy can be written as: $$ T=\frac{1}{2}\sum_{\alpha=1}^K\sum_{\beta=1}^K a_{\alpha \beta}(q)\dot{q}^\alpha \dot{q}^\beta$$ Where the object $a_{\alpha \beta}$ is a certain tensor. How to ...
  • 1,270
0 votes
0 answers
25 views

Coordinate transformation and gauge choice in gravitational waves [duplicate]

Assuming weak field : $g_{\mu\nu}=\eta_{\mu \nu}+h_{\mu\nu}$ and considering terms only linear in $h$ We get christoffel sybol as $\Gamma^\mu _{\nu \lambda}=\frac{1}{2}\eta^{\mu \alpha}(\partial_\nu ...
0 votes
2 answers
48 views

How is Coriolis acceleration in polar coordinate, different from Coriolis acceleration due to observation in non-inertial frame of reference?

In Kleppner and Kolenkow's book: An Introduction to Mechanics, on page 34 (pasted below) on the topic titled "Acceleration in Polar coordinates", it has been mentioned that: "when $r$ ...

1
2 3 4 5
54