A set of numbers used to quantify location in space.

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15
votes
6answers
908 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 ...
12
votes
3answers
810 views

Is “now” or “the present moment” properly defined in GR?

My question is about the extent to which "now" is defined in GR. In Minkowski spacetime, it's possible to define a "now" for an inertial observer by finding a spacelike 3-plane such that, in the ...
11
votes
1answer
222 views

Polar Decomposition of a Complex Scalar Field

People often write a complex scalar field via polar decomposition. What does this parametrization precisely mean? To be more explicit consider the following Lagrangian of a complex scalar field with ...
9
votes
5answers
932 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 $$ ds^{2} = -(cdx^{0})^{2} +(dx^{1})^{2} + (dx^{2})^{2} + (dx^{3})^{2} $$ where $ ...
7
votes
2answers
540 views

Is there any situation in Physics where the Right Hand Rule is not arbitrary?

We use Right Hand Rule in calculating Torque not because that's the direction torque is pointing in the real, physical world, but because it's a convenient way to indicate the "sign" of the rotation ...
7
votes
1answer
1k views

How to calculate roll, yaw and pitch angles from 3D co-ordinates (Euler Angles)

I have digitized a video of a flying fly in a 3-dimensional space. At all instants I know the x, y, and z co-oridinates of the following points on the fly's body --- The points are my choice, and ...
7
votes
2answers
352 views

Tensor equations in General Relativity

In the context of general relativity it is often stated that one of the main purposes of tensors is that of making equations frame-independent. Question: why is this true? I'm looking for a ...
7
votes
0answers
223 views

Understanding and deriving ellipsoidal coordinates geometrically

If one were to read old texts on mathematical physics, like Maxwell, Morse & Feshbach, Hilbert and Courant, Jacobi, etc... they'd find ellipsoidal coordinates popping up, but the authors derive ...
6
votes
3answers
198 views

From Manifold to Manifold?

Tensor equations are supposed to stay invariant in form wrt coordinate transformations where the metric is preserved. It is important to take note of the fact that invariance in form of the tensor ...
6
votes
2answers
497 views

Why isn't invariant notation common?

In principle, one can write quantities in a manifestly invariant - rather than covariant - fashion in e.g. special relativity. For example, rather than writing just $x^\mu$, we could write the basis ...
6
votes
1answer
88 views

Curvilinear Coordinates and basis vectors

In these notes, $\frac{\partial \vec{r}} {\partial q_i}$ is stated to form a basis set for the vector space. How does this happen? Also, how does one justify this equation from Goldstein's ...
5
votes
3answers
249 views

A thought experiment on vision and curved spacetime

What follows is a long self-made example to deal with my conceptual issues of visualizing curved spacetime. Imagine an observer floating somewhere in space. He feels no strain on his body, ...
5
votes
3answers
139 views

Integral in different coordinate systems

In Griffiths' electrodynamics book, he uses the equation, $$\nabla^2\mathbf{A}=-\mu_0 \mathbf{J},$$ to state that $$\mathbf{A}(\mathbf{r}) = ...
5
votes
2answers
374 views

Lorentz transformation in light cone coordinates in string theory

What is the explicit form of the Lorentz transformation changing the light cone coordinates in the light cone gauge in string theory? The extended nature of the strings complicate matters, especially ...
5
votes
1answer
2k 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 ...
5
votes
1answer
449 views

What do up-left orthogonality has in common with up-down and what is their relationship?

I am familiar with the true (or general) notion of orthogonality, given in the Linear Algebra and Pythagoras theorem derived from the $\vec x \cdot \vec y = 0$. I have also recently got to know that ...
5
votes
1answer
311 views

Is General Relativity applicable for all coordinate systems?

My understanding was that relativistic physics can be expressed in any inertial coordinate system, but not arbitrary systems. That is, no experiment can determine if we are "still" or "moving" at a ...
5
votes
3answers
307 views

Is the equivalence principle strictly fulfilled by general relativity?

The equivalence principle states The outcome of any local experiment in a freely falling laboratory is independent of the velocity of the laboratory and its location in spacetime. Any real local ...
5
votes
1answer
346 views

6 independent Einstein field equations?

I can't understand the comment on page 409, Gravitation, by Misner, Thorne, Wheeler It follows that the ten components $G_{\alpha\beta} =8\pi T_{\alpha\beta}$ of the field equation must not ...
5
votes
0answers
103 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]? ...
4
votes
3answers
330 views

First Postulate of Special Relativity: What does it mean?

Wikipedia has this quote: Special principle of relativity: If a system of coordinates K is chosen so that, in relation to it, physical laws hold good in their simplest form, the same laws hold ...
4
votes
3answers
253 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
votes
2answers
299 views

Metric tensor in special and general relativity

I'm having trouble understanding the metric tensor in general relativity. What I've understood so far has come from my course lecture notes used in conjunction with "The Road to Reality" by Roger ...
4
votes
4answers
190 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 ...
4
votes
1answer
130 views

Integrating Radial Vector Fields

Given a integral $$\int_vd^3{r} \;\vec{r}\;\rho(r)$$ and How do you convert it to spherical coordinate system, noting that $\rho(r)$ is indeed as it is without vector, i.e. it is spherically symmetric ...
4
votes
3answers
356 views

Does the Relativity Principle of Special Relativity imply homogeneity and isotropy of all the reference frames?

In Rindler's book: Relativity, Special, General and Cosmological, is stated on page 40 that the Relativity Principle (RP), when applied to just one Inertial Frame (IF), guarantees the homogeneity and ...
4
votes
1answer
121 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: $$ ...
4
votes
1answer
173 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 ...
4
votes
1answer
126 views

How to assign coordinates to the elements of a flat metric space

Consider the metric space $(M, d \,)$ where set $M$ contains sufficiently many (at least five) distinct elements, and consider the assignment $c_f$ of coordinates to (the elements of) set $M$, $c_f ...
4
votes
1answer
422 views

failing to see the conundrum in the Einstein hole argument

I've been reading about the Einstein hole argument, and i fail to understand what makes active diffeomorphisms "special" compared to passive diffeomorphismsm also known as good old coordinate ...
4
votes
1answer
964 views

Convert ECI coordinates to latitude/longitude?

I have been given output in (what I believe to be) ECI format (from OrbitTools): ...
3
votes
4answers
497 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
votes
4answers
121 views

What is the motivation for the definition of a manifold?

In Wald's General Relativity, an $n$-dimensional $C^{\infty}$ manifold $\mathit{M}$ is defined as a set, with subsets $\lbrace{O}_{\alpha}\rbrace$, which satisfies 3 properties. In particular, the ...
3
votes
1answer
190 views

Is the apparent lack of (Ricci) curvature in the Schwarzschild metric due to a choice of coordinates?

I've been lightly studying GR lately. Something that has been bothering me has been the lack of (Ricci) curvature produced from the Schwarzschild metric in the few lectures I've watched, as well as ...
3
votes
2answers
287 views

The wave equation in general relativity, special relativity, and Cartesian coordinates

The relativistic wave equation is $$\square\varphi=\rho$$ where $\varphi$ is the field, $\rho$ is the source, and $\square$ is the D'Alembert operator, defined by ...
3
votes
2answers
343 views

Why don't global coordinates always exist for a manifold?

Let $M$ be a manifold and $(\phi,U)$ a patch. Then $\phi(P)=\bar{x}=\begin{bmatrix} x^1\\ x^2\\ \vdots\\ x^n \end{bmatrix}$ for each $P$ in $U$. But each $P$ in $M$ is in some patch, so this ...
3
votes
2answers
614 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 ...
3
votes
1answer
182 views

Why does the Kruskal diagram extend to all 4 quadrants?

Why is it that the Kruskal diagram is always seen extended to all 4 quadrants when the definitions of the $U,V$ coordinates don't seem to suggest that the coordinates are not defined in, say, the 3rd ...
3
votes
2answers
130 views

Cartesian Coordinates to Polar Coordinates

I apologize if this question is trivial, but I am new to physics and am struggling with some of the basic concepts. Working in $\mathbb{R}^2$ with standard coordinates $(x,y)$, suppose we have a ...
3
votes
2answers
304 views

Explanation for Negative $\rho$ (radial distance) in Cylindrical Coordinates

My question : What does it mean when we arrive at negative values for distance variables like $\rho$ in cylindrical coordinates? (after some discussion here,I revised the question, at the end of the ...
3
votes
2answers
635 views

Coordinate Transformation of Scalar Fields in QFT

By definition scalar fields are independent of coordinate system, thus I would expect a scalar field $\psi [x]$ would not change under the transformation $x^\mu \to x^\mu + \epsilon^\mu $. Correct? ...
3
votes
7answers
461 views

Relation between coordinates and frames of reference

I always get a little uneasy that all the theories I can think of (at least since Newton) are constructed in a way such that they would be true in heaven and on earth ... but we can never go ...
3
votes
2answers
113 views

Ball Bearing Inside a Hollow, Spinning Rod: where is the logical flaw?

As described in the title, suppose we have a frictionless, hollow rod that is rotating in the $xy$-plane with some fixed angular velocity $\omega$. The rod is pivoting around its midpoint. Suppose we ...
3
votes
1answer
300 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 ...
3
votes
1answer
69 views

Quantum mechanics with non-cartesian coordinates

Let say we have the classical hamiltonian of a harmonic oscillator: $$H=\frac{p_x^2+p_y^2+p_z^2}{2m}+\frac{k_1x^2+k_2y^2+k_3z^2}{2}$$ and we want to find the hamiltonian operator in quantum mechanics, ...
3
votes
1answer
104 views

Is there a technical term for “meaningfulness” of mathematical operations?

Is there a technical term for "meaningfulness" of mathematical operations? For example, adding vectors that represent forces has a meaning regardless of the coordinate frame, but an elementwise ...
3
votes
1answer
38 views

Are the cylindrical and spherical form of Jeans' equations equivalent?

The question kind of says it all, what I really want to know is are the differences in their forms only due to the co-ordinate transform? And as such should a suitable spherical system satisfy ...
3
votes
2answers
171 views

Can a curvature in time (and not space) cause acceleration?

I realize that the curvature of space-time causes acceleration (gravity). Is it possible to have a curvature only of space, or a curvature only of time? If so, would a curvature only of space, or a ...
3
votes
1answer
161 views

QM: How to compute position/momentum relation in polar coordinates

So if we are working in one dimensional space, we have the formula: $$\langle x|p\rangle = \frac{1}{\sqrt{2\pi\hbar}} e^{ipx/\hbar}$$ Suppose instead we are confined to a circle of radius $R$ so that ...
3
votes
1answer
80 views

Primary direction in planet-centered equatorial reference frame

I am given the classical orbital elements of the orbit of a spacecraft around a planet which is not the Earth, say Venus. I assume those are referred to a reference frame whose fundamental plane is ...