A set of numbers used to quantify location in space.

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0
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1answer
540 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: $$\begin{...
23
votes
5answers
4k 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 ...
-1
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0answers
18 views

Velocities from rotating frame of reference to fixed frame of reference [on hold]

I have the following flexible structure, where P1 coordinate system is fixed to the ground and P3 is rotating around the $y_3$ axes. I would like to model this structure as a simple steady state ...
2
votes
2answers
271 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 du^...
11
votes
2answers
3k 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 ...
1
vote
2answers
102 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 symmetry,...
0
votes
3answers
110 views

Why centripetal force does not increase the value of tangential velocity?

I found in a text book that the value does not change because the centripetal force is perpendicular to the tangential velocity. But I am confused, because a vector can have a component, which is ...
2
votes
1answer
212 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 ...
2
votes
0answers
93 views

Determination of Euler angles from known angles between the axes of two coordinate systems

Suppose there is a unprimed coordinate system XYZ, tilted w.r.t. a primed coordinate system but their origins coincide. Let the angle between $X,X^\prime$, $Y,Y^\prime$ and $Z,Z^\prime$ are ...
0
votes
2answers
99 views

Why do Newton's laws hold in a normal and tangential coordinate system?

In my book it says: When applying the equations of motions, it is important that the acceleration of a particle be measured with respect to a reference frame that is either fixed or translates ...
0
votes
1answer
165 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 ...
4
votes
3answers
552 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 ($\gamma$) ...
0
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1answer
39 views

Marsden and Hughes Mathematical Foundations of Elasticity Derivation

in the beginning of the book they discuss transformation between coordinate systems and they use this to show that the velocity of a point transforms into that coordinate system;, in terms of the ...
5
votes
1answer
142 views

What is the difference between active and passive transformations in Quantum Mechanics?

I am trying to understand what each transformation means and what their differences are but many books that don't state which transformation they are referring to make it a bit confusing to understand ...
1
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2answers
243 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 ...
6
votes
1answer
252 views

Geometric formulation of the equivalence principle

Let $(M,g)$ be a $4$-dimensional Lorentzian manifold. It is well know that given $(U,\psi=(x^1,\ldots,x^4))$ local chart around some $p\in M$, it is posible to find a change of coordinates given by $(...
1
vote
1answer
43 views

What is the meaning of a negative Galactic longitude?

What is the meaning of negative longitude in Galactic coordinate system, for instance in this article? Does the longitude $-65^\circ$ equal $295^\circ$?
3
votes
1answer
487 views

Stokes' theorem in complex coordinates (CFT)

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

How is Riemann tensor related to the curvature in the coordinates?

I came across statements such as "the acceleration observed in a weak gravitational field is mainly due to curvature in the time coordinate. " I want to know how we can explicitly find the curvature ...
1
vote
1answer
124 views

Spherical phase space dynamics

I have a hamiltonian of the form $$H(\phi,z) = (1-z^2)\cos(2\phi) + \chi z^2$$ with position $\phi$ and conjugate momentum $z$. It has this form provided that $z \in [-1,1]$ and we have natural ...
3
votes
1answer
95 views

The most general way to write flat space metric [closed]

What is the most general way to write flat space (in d=4 in particular), but still preserving some isometries? In particular I'm interested in the case with 2 isometries, basically by using explicitly ...
5
votes
2answers
197 views

Why do we use orthogonal axes?

I have been asked several times that “why do we use orthogonal axes in coordinate systems?” and I was always replying that “because of simplicity”. But, today morning, someone asked me that question ...
0
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0answers
41 views

Momentum flux through cylinder wall

A fluid flows through a hovercraft, is bended and the freejet has a certain velocity $c$ with which it flows after exiting the hovercraft / beeing bended. There's a pressure difference between under ...
1
vote
1answer
44 views

Notations used to express direction [closed]

We express direction relative to a reference point and call a certain direction positive and it's direct opposite direction negative, by convention. But, what notation should we use to express a ...
0
votes
2answers
55 views

Locally remove a gravitational field

Let $K$ be an inertial frame of reference on $\mathbb{R}^3$ and $g=g(t,x)$ a nonuniform and nonstatic gravitational field. How I can choose a system of reference $\bar K$ such that mechanical effects ...
2
votes
0answers
48 views

Noether charge in light-cone coordinates (1+1D)

I have read in this article http://arxiv.org/abs/1107.2917 that the noether charge (in 1+1 D) $$ Q= \int dx \; q_t$$ could be written in terms of lightcone coordinates $x^\pm = t\pm x$ as $$Q=\int dx^...
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4answers
110 views

Does the $t = 0$ hypersurface of simultaneity constitutes the universal present?

Does the hypersurface of simultaneity in the diagram below represent the universal present moment? Source: Einstein for Everyone - Spacetime
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2answers
53 views

Difference between space of reference and system of coordinates

In the book "The meaning of the relativity" by A. Einstein, it is referring to two different concepts: space of reference and system of coordinates. What it is the difference? It says: "we ...
0
votes
1answer
21 views

Heliocentric to barycentric coordinates

I have a system with a central body and "particles" orbiting around it. The system is described in heliocentric coordinates. I am trying to obtain the velocity of the central body in barycentric ...
0
votes
2answers
86 views

Components of Velocity in polar co-ordinates

Consider a point moving along a curve in a plane. The position of a point P on a coordinate system can be specified by a single vector $\vec{r}$=$r\hat{r}$. A rough sketch describing the situation is ...
0
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0answers
43 views

A question from “The meaning of the relativity, by A.Einstein” - Lorentz transformations [duplicate]

Let $K$ and $\bar K$ be two cartesian co-orditate systems in $\mathbb{R}^3$. The element: $$s^2=(\Delta x^1)^2+(\Delta x^2)^2+(\Delta x^3)^2$$ is an invariant in all co-ordinate system. I want prove ...
0
votes
1answer
46 views

How to measure time in presence of a strong gravitational field? [duplicate]

I need an operative definition of "measuring time in general relativity" that takes in consideration also the presence of strong gravitational fields between me and clock, able to deviate the light ...
1
vote
1answer
72 views

“Measure of time in general relativity” [duplicate]

Suppose to be in an arbitrary gravitational field and you are moving in it arbitrarily with a clock in your hand. In this general situation I ask: if I read the positions of the hands of the clock, ...
1
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1answer
38 views

dot product of two vectors in spherical polar coordinates, do I have to convert to cartesian coordinates?

For two vectors $p1=(r1,\theta_1, \phi_1)$ and $p2=(r2, \theta_2, \phi_2)$ I want the dot product p1.p2. However, the solutions I have seen, involve finding the components in Cartesian coordinates and ...
0
votes
1answer
40 views

Why are dimensions regarded as square/perpendicular?

Starting from the second dimension, the dimensions are basically represented by a square, cube, tesseract, and so on. I don't know if this is a stupid question or not, but is there an obvious or less-...
-6
votes
1answer
51 views

Can velocity be negative? [closed]

In free fall I use the formula V=g*t and g is negative(-9.8m/s^2). It gives me negative.
2
votes
2answers
85 views

What is the difference between time and space in general relativity?

I know that similar questions have been asked before, I will try to be specific. In special relativity time is the coordinate with minus sign in metric tensor. In general relativity the components of ...
0
votes
0answers
30 views

Are general coordinate transformations and diffeomorphisms the same? [duplicate]

Infinitesimal diffeomorphisms $x{}^\mu \rightarrow x{}^\mu + \xi{}^\mu$ (with $\xi{}^\mu \ll 1$) change geometric objects by means of the Lie derivative, that is, $X \rightarrow X + \mathcal{L}_\xi \, ...
1
vote
0answers
30 views

Coordinate time difference between emiting and detecting a photon in bent spacetime

Consider an arbitrary non-trivial metric $g_{ij}$ - like the Schwarzschild metric. Now, consider two observers $A$ and $B$, staying at fixed radii $R_A$ and $R_B$, respectively, with $R_A > R_B$. ...
1
vote
0answers
17 views

Lapse Function and Shift Vector in Minkowski and de Sitter

I'd like to find the lapse function and shift vector in 1+1 Minkowski as well as 1+1 de Sitter (flat foliation) for a region foliated this way: The $y$-axis represents time while the x-axis ...
4
votes
2answers
109 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
vote
1answer
42 views

Time variable in Lorentz transformations

When an object goes with a speed near from the light celerity, it inflates in the direction of its speed. The inflation rate is given by Lorentz transformations as follows: $x'= γ(x-vt)$ where $v$ ...
-2
votes
2answers
33 views

Projectile motion dependency [closed]

I have read that in projectile motion both vertical and horizontal components are independent of each other but i don't get it that how it is possible i think that they are dependent. If they are not ...
9
votes
4answers
184 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 Carroll's GR book, it is claimed that $M$ is the mass of the ...
1
vote
4answers
4k views

When an object moves downward, is its height negative?

The question is: A ball is thrown directly downward with an initial speed of 8.00m/s from a height of 30.0m. After what time interval does it strike the ground. So I went through the problem ...
20
votes
6answers
2k views

Proving that interval preserving transformations are linear

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 ...
7
votes
1answer
186 views

What coordinate system is used to describe planets positions in the universe?

How are planets positions described in the space and in respect to what? For example is Sun the origo and right now at this moment Earth has [coord_X, coord_Y, coord_Z]? or maybe [lng, lat]? Edit1:...
3
votes
1answer
88 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'...
0
votes
0answers
30 views

What are the components of r-hat in spherical coordinates?

so I've found a lot of identities that relate the spherical unit vectors to cartesian unit vectors. What is the expression for the spherical unit vectors IN spherical coordinates? I'm tying my brain ...
0
votes
0answers
31 views

Inertial coordinate systems [duplicate]

In Newtonian mechanics, by the following two assumptions: (i) The time is absolute. (ii) The length is absolute. it is easy find the relations betweem two coordinate systems with uniform motion ...