Questions tagged [coordinate-systems]

A set of numbers used to quantify location in space.

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28
votes
8answers
3k views

Interval preserving transformations are linear in special relativity

In almost all proofs I've seen of the Lorentz transformations one starts on the assumption that the required transformations are linear. I'm wondering if there is a way to prove the linearity: Prove ...
82
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8answers
9k views

How can time dilation be symmetric?

Suppose we have two twins travelling away from each other, each twin moving at some speed $v$: Twin $A$ observes twin $B$’s time to be dilated so his clock runs faster than twin $B$’s clock. But twin ...
13
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1answer
1k views

Which transformations are canonical?

Which transformations are canonical? Why do canonical transformations preserve the measure of integration in phase space?
7
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2answers
278 views

How do frames of reference work in general relativity, and are they described by coordinate systems?

In both Newtonian gravity and special relativity, every frame of reference can be described by a coordinate system covering all of time and space. How does this work in general relativity? When an ...
43
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5answers
10k views

Conformal transformation/ Weyl scaling are they two different things? Confused!

I see that the weyl transformation is $g_{ab} \to \Omega(x)g_{ab}$ under which Ricci scalar is not invariant. I am a bit puzzled when conformal transformation is defined as those coordinate ...
13
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6answers
6k 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 $$ ds^{2} = -(c^{2}dx^{0})^{2} +(dx^{1})^{2} + (dx^{2})^{2} + (dx^{3})^{2} $$ ...
6
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3answers
361 views

Velocity in a turning reference frame

I often see the relation that $\vec v=\vec v_0+ \vec \omega \times \vec r$ in a turning reference frame, but where does it actually come from and how do I arrive at the acceleration being $$\vec a=\...
12
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2answers
1k views

Why does cancellation of dots $\frac{\partial \dot{\mathbf{r}}_i}{\partial \dot{q}_j} = \frac{\partial \mathbf{r}_i}{\partial q_j}$ work?

Why is the following equation true? $$\frac{\partial \mathbf{v}_i}{\partial \dot{q}_j} = \frac{\partial \mathbf{r}_i}{\partial q_j}$$ where $\mathbf{v}_i$ is velocity, $\mathbf{r}_i$ is the ...
8
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3answers
4k views

How to understand the definition of vector and tensor?

Physics texts like to define vector as something that transform like a vector and tensor as something that transform like a tensor, which is different from the definition in math books. I am having ...
4
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3answers
1k 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 ...
18
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5answers
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What does a frame of reference mean in terms of manifolds?

Because of my mathematical background, I've been finding it hard to relate the physics-talk I've been reading, with mathematical objects. In (say special) relativity, we have a Lorentzian manifold, $...
12
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2answers
5k views

Why is light described by a null geodesic?

I'm trying to wrap my head around how geodesics describe trajectories at the moment. I get that for events to be causally connected, they must be connected by a timelike curve, so free objects must ...
6
votes
3answers
354 views

In electromagnetism, why does nature prefer the right-hand rule over the left-hand rule? [duplicate]

At school I learnt the Right-hand rule to remember the resulting direction of different phenomena, such as geometrical cross products, mechanical torque, or the direction a screw will move when ...
5
votes
3answers
2k 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 ...
9
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4answers
3k views

Understanding “natural variables” of the thermodynamic potentials using the example of the ideal gas

I'm struggling with the concept of "natural variables" in thermodynamics. Textbooks say that the internal energy is "naturally" expressed as $$ U = U(S,V,N)$$ For an ideal gas, I could take the ...
5
votes
2answers
807 views

Lagrange multiplier and constraint force

The Lagrangian with Lagrange multiplier in the form $$L= T- V + \lambda f(q, \dot{q},t).$$ But there are different ways of writing the constraint $f = 0$. Will that lead to different EOMs? Let me ...
5
votes
2answers
1k views

Curved space-time VS change of coordinates in Minkowski space

I'm looking for a rather intuitive explanation (or some references) of the difference between the metric of a curved space-time and the metric of non-inertial frames. Consider an inertial reference ...
2
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1answer
2k views

General matrix Lorentz transformation

I just finished an introduction course into theory of relativity and am trying to find the general matrix Lorentz transformation. I have already looked into this question, but I could not make much ...
21
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4answers
1k views

How do we know the Schwarzschild solution contains an object of mass $M$?

The Schwarzschild metric is $$ds^2 = - \left( 1 - \frac{2GM}{r} \right) dt^2 + \left(1-\frac{2GM}{r}\right)^{-1} dr^2 + r^2 d\Omega^2.$$ In all GR books, it is stated that $M$ is the mass of the black ...
6
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5answers
2k views

Why are coordinates and velocities sufficient to completely determine the state and determine the subsequent motion of a mechanical system?

I am a Physics undergraduate, so provide references with your responses. Landau & Lifshitz write in page one of their mechanics textbook: If all the co-ordinates and velocities are ...
3
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1answer
277 views

Holonomic constraints and degrees of freedom

Wikipedia and other sources define holonomic constraints as a function $$ f(\vec{r}_1, \ldots, \vec{r}_N, t) \equiv 0, $$ and says the number of degrees of freedom in a system is reduced by the ...
24
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6answers
11k views

What are holonomic and non-holonomic constraints?

I was reading Herbert Goldstein's Classical Mechanics. Its first chapter explains holonomic and non-holonomic constraints, but I still don’t understand the underlying concept. Can anyone explain it to ...
4
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2answers
414 views

Simple conceptual question conformal field theory

I come up with this conclusion after reading some books and review articles on conformal field theory (CFT). CFT is a subset of FT such that the action is invariant under conformal transformation ...
12
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2answers
721 views

Why do we use the Lagrangian and Hamiltonian instead of other related functions?

There are 4 main functions in mechanics. $L(q,\dot{q},t)$, $H(p,q,t)$, $K(\dot{p},\dot{q},t)$, $G(p,\dot{p},t)$. First two are Lagrangian and Hamiltonian. Second two are some kind of analogical to ...
7
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5answers
386 views

Does an expanding event horizon “swallow” nearby objects?

In a view of a remote observer, an object falling into a black hole is "hanging" at the horizon (slowly falling with a deceleration). Around this moment, the event horizon expands for some reason that ...
3
votes
2answers
312 views

Proof of constructing Action-Angle Coordinates on Hamiltonian System

By Liouville-Arnold Theorem, we know we can construct action-angle coordinates such that the Hamiltonian system, when described in these coordinates, will have a form that is integrable by quadratures....
4
votes
2answers
437 views

Under what representation do the Christoffel symbols transform?

I often read the statement, that the Christoffel symbols aren't tensors. But then, under which representation do they transform?
3
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4answers
335 views

Transformation of $d^4x$ under translation disregarded?

Under a translation in spacetime i.e., $$x\mapsto x^\prime=x+a,\tag{a}$$ a scalar field $\phi(x)$ $$\phi(x)\mapsto\phi^\prime(x)=\phi(x-a).\tag{b}$$ My aim is to verify the invariance of an action of ...
4
votes
2answers
1k views

Local inertial coordinates/Fermi normal coordinates

It is said that we can introduce local inertial coordinates/Fermi normal coordinates for any timelike geodesic. But why only for timelike geodesics? What about null geodesics? Perhaps it has to do ...
4
votes
4answers
473 views

What makes a coordinate curved?

Bear with me while I try to explain exactly what the question is. The question Can a curvature in time (and not space) cause acceleration? is imagining a coordinate system in which the curvature is ...
7
votes
2answers
251 views

Group representations and active/passive transformations

Suppose we are in Euclidean 3-space with coordinates $x$ and a scalar function $\phi(x)$ defined on it, and consider the group of rotations $SO(3)$ for simplicity. Take a rotation matrix $R \in SO(3)$;...
4
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2answers
2k views

A few questions on passive vs active Lorentz transformations

1.) How do we physically interpret an active Lorentz transformation? The passive transformation seems simple enough: you view a fixed object from the perspective of a new observer. When we actively ...
5
votes
2answers
1k views

Covariance of Euler-Lagrange equations under change of generalized coordinates

Suppose I have an inertial frame with coordinate $\{q\}$. Now I define another reference frame with coordinate $\{q'(q,\dot q,t)\}$. I obtain the equation of motion in $\{q'\}$ in two different ways: ...
5
votes
2answers
2k views

How do you derive Lagrange's equation of motion from a Routhian?

Given a Routhian $R(r,\dot{r},\phi,p_{\phi})$, how do you derive Lagrange's equation for $r$? Do you just solve the following for $r$? $$\frac{d}{dt}\frac{∂R}{∂\dot{r}}-\frac{∂R}{∂r}=0.$$ And as a ...
4
votes
1answer
534 views

Diffeomorphism invariance and geodesic action

I'm trying to understand the role of diffeomorphism and isometry invariance in the geodesic action in GR: $$ S = \int_{\tau_1}^{\tau_2} \! d\tau~ g_{ab}(x(\tau)) \frac{dx^a}{d\tau} \frac{dx^a}{d\tau} ...
3
votes
3answers
3k 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 ...
0
votes
2answers
210 views

Conformal transformation vs diffeomorphisms

I am reading Di Francesco's "Conformal Field Theory" and in page 95 he defines a conformal transformation as a mapping $x \mapsto x'$ such that the metric is invariant up to scale: $$g'_{\mu \nu}(x'...
9
votes
2answers
799 views

Homogeneity and isotropy and derivation of the Lorentz transformations

In deriving the Lorentz transformations I have found (from reading a few different sets lecture notes) that it is argued that they must be linear and thus there general form must be $$x'=Ax+Bt,\quad t'...
7
votes
2answers
688 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 ...
5
votes
1answer
469 views

Extent of coordinate freedom to set metric components along a spacetime path

If we describe spacetime with a Lorentzian manifold, it is always possible to choose a coordinate system such that at any particular point $x^\alpha$, the components of the metric are: $$ g_{\mu\nu}(x^...
4
votes
1answer
979 views

How to explain the different forms of the Hamilton-Jacobi equation?

In Arnold's Mathematical Methods of Classical Mechanics, he derives the Hamilton-Jacobi equation (HJE) using a generating function $S_1(Q, q)$ to get $$ H\left(\frac{\partial S_1(Q, q)}{\partial q}, ...
1
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1answer
619 views

For an infinitesimal transformation in phase space, what functions are allowed for this to be a canonical transformation?

Consider an infinitesimal transformation: $$(q_{i},p_{j}) \quad\longrightarrow \quad(Q_{i},P_{j}) ~=~ \left(q_{i} + \alpha F_{i}(q,p),~p_{j} + \alpha E_{j}(q,p)\right) $$ where $α$ is considered to ...
63
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5answers
4k views

What is really curved, spacetime, or simply the coordinate lines?

It is often said that, according to general relativity, spacetime is curved by the presence of matter/energy. But isn't it simply the coordinate lines of the coordinate system that are curved?
34
votes
4answers
5k views

Why is this vector field curl-free?

The curl in cylindrical coordinates is defined: $$\nabla \times \vec{A}=\left({\frac {1}{\rho }}{\frac {\partial A_{z}}{\partial \varphi }}-{\frac {\partial A_{\varphi }}{\partial z}}\right){\hat {\...
16
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4answers
2k views

Is simultaneity well-defined in general relativity?

In special relativity for each event and reference frame we can find a plane of simultaneous events. I wonder is it possible to do the same in general case in curved space? Is simultaneity even ...
19
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1answer
2k views

Does light really travel more slowly near a massive body?

It is a routine problem for beginners in general relativity to calculate the coordinate velocity of light for the Schwarzschild metric. Starting from the metric: $$ ds^2 = -\left(1-\frac{r_s}{r}\...
8
votes
5answers
4k 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 ...
16
votes
4answers
2k views

Does spacetime position not form a four-vector?

When one starts learning about physics, vectors are presented as mathematical quantities in space which have a direction and a magnitude. This geometric point of view has encoded in it the idea that ...
14
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18answers
1k views

What is an event in Special Relativity?

Lorentz transformations help us transform coordinates of one frame to that of another. For example, let the coordinates of an event in an inertial frame $S$ be $(x, t)$, then the coordinates in frame ...
9
votes
3answers
10k 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: $$...