A set of numbers used to quantify location in space.

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30 views

Relativity Coordinate transformation of Vector [closed]

I'm taking a first course in General Relativity but I've been struggling with coordinate system transformation. For example, if I have a Vector defined in Cartesian (x,y) coordinates as $V_x=x^2+3y$ ...
11
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4answers
936 views

Coordinates vs. Geometries: How can we know two coordinate systems describe the same geometry?

Specifically, I'm asking this because I'm taking a class on General Relativity, and in Hartle's book Gravity, in Ch. 12, after having spent some time using Schwarzschild coordinates, we are introduced ...
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3answers
99 views

Thinking about the properties of 'nothing' [closed]

If a certain identifiable part of space that has no type of measurable energy fields manifesting 'in it' for a given duration ; is such a totally empty space the same as 'nothing'? Anything with any ...
5
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2answers
125 views

Kerr Metric from rotated Schwarzschild?

Say we have got a system in GR that is described by the Schwazschild metric. Then we perform a coordinate transform that gives the metric in a rotating system. Why is the transformed metric not the ...
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1answer
63 views

Is it correct to think about a point in time as the set of positions of all “things”?

Is it correct to think about a point in time as the set of positions of all "things" (photons, electrons, etc) that exist in the universe at that moment, despite the fact that simultaneity is ...
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3answers
123 views

Is the local Lorentz transformation a general coordinate transformation?

There is a saying in Nakahara's Geometry, Topology and Physics P371 about principal bundles and associated vector bundles: In general relativity, the right action corresponds to the local Lorentz ...
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0answers
31 views

Divergence theorem for cylindrical coordinates [closed]

I have a Vector field in a cylinder where x^2+y^2=4 and goes from z=0 to z=3 and a vector field A=(4x)i-(2y^2)j+(z^2)k and I'm trying to verify the divergence theorem for the vector field i set set ...
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5answers
2k views

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, ...
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1answer
66 views

Difference between local inertial frame and coordinate chart

In the most cases the local inertial frame is definied "physically" but I'm searching for a mathematically meaningful definition of the local inertial frame to solve my problem: Is the local ...
3
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1answer
61 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 ...
5
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5answers
834 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 ...
0
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2answers
101 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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1answer
45 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
545 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
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5answers
753 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
66 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
176 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 ...
2
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0answers
46 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
93 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 ...
0
votes
1answer
34 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
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3answers
583 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 ...
3
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3answers
129 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
179 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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2answers
107 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: ...
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1answer
50 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
247 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 ...
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1answer
111 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 ...
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1answer
71 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 ...
6
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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
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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
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1answer
63 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
84 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 ...
4
votes
0answers
121 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
78 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 ...
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1answer
202 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
33 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
70 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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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. ...
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1answer
54 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
31 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
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1answer
28 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
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0answers
218 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 ...
0
votes
1answer
106 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 ...
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1answer
64 views

$\frac{d}{dr}=0$ and $\frac{d}{dz}=0$ (cylindrical coordinates) for a 1D ring

In http://ritchie.chem.ox.ac.uk/Grant%20Teaching/2010/Lecture%204%202010.pdf slide 21 of 26, he says "Radius of ring is fixed and so derivatives in $r$ are 0." Presumably this goes for ...
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votes
1answer
93 views

Coordinate velocity and free-falling past an event horizon

Ws can re-arrange the standard Schwarzschild metric as $$\left(\frac{d\tau}{dt}\right)^2=\frac{r-1}{r}\left[1-\left(\frac{r}{r-1}\frac{dr}{dt}\right)^2\right]$$ Where $\frac{r-1}{r}$ is the local ...
0
votes
1answer
56 views

In central-force mechanics, how do we substitute $ξ=\frac{1}{r}$?

I have taken a look at central-force mechanics in the past, but I still cannot understand how $ξ=\frac{1}{r}$ is substituted to find $\frac{d^2r}{dt^2}$ in terms of ξ. So I know from $F=ma$ that: ...
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3answers
81 views

Prove that the spacetime interval is not invariant under Galilean transformations [closed]

The spacetime interval $(\Delta s)^2 = (\Delta x)^2 + (\Delta y)^2 + (\Delta z)^2 - c^2(\Delta t)^2$ is invariant under the Lorentz transformation and this isn't the case for the Galilean ...
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0answers
293 views

Hamiltonian for Electron in Magnetic Field with Symmetric Gauge in Polar Coordinates

I am new on the board and have a question about how to write the Hamiltonian for an electron in a magnetic field rotating at a fixed radius. I would like to write the hamiltonian using the symmetric ...
4
votes
2answers
1k views

Light-cone coordinates

The light-cone coordinates are defined as $$x^{\pm} ~=~\frac{x^0 \pm x^3}{\sqrt{2}}.$$ Then in the light cone coordinates the position 4-vector becomes: $(x^+, x^-, x^1, x^2)$ . Zwiebach, in his A ...