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16
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2answers
1k views

Why is there no Gravitational Magnetic Field?

We think that the electric field and gravitational field operate similarly with their corresponding charges/masses. With just a difference that the electric field is sometimes attractive and sometimes ...
10
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4answers
350 views

Electromagnetic field and continuous and differentiable vector fields

We have notions of derivative for a continuous and differentiable vector fields. The operations like curl,divergence etc. have well defined precise notions for these fields. We know electrostatic and ...
9
votes
3answers
328 views

Field created by varying Gravitational field

Changing Electric Field causes Magnetic filed and changing Magnetic Field causes Electric Field. Is there anything similar in relation to Gravitational Field? What sort of field is created by varying ...
8
votes
5answers
521 views

Is there a fourth component to the electric field and magnetic field?

The Question If the three vector electric and magnetic fields come from the four component four-potential, then is there a fourth component to the electric and magnetic field? Related Question I ...
8
votes
2answers
181 views

What do the names “E mode” and “B mode” mean? Where do they come from?

This has been bugging me a bit since the BICEP announcement, but if there are any resources that answer my question in a simple way, they've been buried in a slew of over-technical or over-popularized ...
6
votes
2answers
910 views

Decomposition of a vectorial field in free-curl and free-divergence fields

Is it always possible to do that decomposition? I'm asking it because Helmholtz theorem says a field on $\mathbb{R}^3$ that vanishes at infinity ($r\to \infty$) can be decomposed univocally into a ...
6
votes
2answers
290 views

Must every isometry have an associated Killing vector?

I understand that the flows of Killing vector fields are isometries, and that one-parameter groups of isometries have an associated Killing vector which generates them, but are your Killing vectors ...
6
votes
1answer
2k views

Physical significance of Killing vector field along geodesic

Let us denote by $X^i=(1,\vec 0)$ the Killing vector field and by $u^i(s)$ a tangent vector field of a geodesic, where $s$ is some affine parameter. What physical significance do the scalar quantity ...
6
votes
2answers
379 views

How to show that every Killing vector field is a matter collineation?

Various texts make this claim, but no proof is given. Explicitly, let $L$ denote the Lie derivative. Suppose $L_X g_{ab} = 0$ for some vector field $X$, called a Killing vector field. Suppose that ...
6
votes
3answers
731 views

How can we describe the polarization (of light) coming from an arbitrary angle?

In an optics lab, where all optical beams pretty much reside in a plane, it is fairly simple to describe (linear) polarizations as vertical or horizontal (or s and p). When we start talking about ...
5
votes
3answers
2k views

What is a Killing vector field?

I recently read a post in physics.stackexchange that used the term "Killing vector". What is a Killing vector/Killing vector field?
5
votes
3answers
181 views

Wind's Sources and Drains (see live map!!)

I was pointed out by a friend to this website that shows live map of wind in US. It sometimes show interesting places where all the wind seems to converge and vanish. What's the origin of such "wind ...
5
votes
1answer
162 views

Are conformal, Killing and homothetic vector fields the same in pseudo-riemannian manifolds?

I work in the Lorentzian manifolds, more generally in pseudo Riemannian manifolds and applications to general relativity. I know the definitions of conformal, Killing and homothetic vector fields in ...
5
votes
1answer
80 views

Irrotational fluid

Often, when threating some problem of fluid dynamics I have read that people make the approximation of irrotational fluid, i.e. the velocity field is assumed irrotational: $$ \nabla \times \vec{v}=0 ...
4
votes
3answers
644 views

Difference between spinor and vector field [duplicate]

How do we distinguish spinors and vector fields? I want to know it in terms of physics with mathematical argument.
4
votes
1answer
2k views

Uniqueness of Helmholtz decomposition?

Helmholtz theorem states that given a smooth vector field $\pmb{H}$, there are a scalar field $\phi$ and a vector field $\pmb{G}$ such that $$\pmb{H}=\pmb{\nabla} \phi +\pmb{\nabla} \times \pmb{G},$$ ...
4
votes
1answer
166 views

Conservative Vector Fields

I was always told that to find whether or not a vector field is conservative, see if the curl is zero. I have now been told that just because the curl is zero does not necessarily mean it is ...
4
votes
4answers
1k views

Are the field lines the same as the trajectories of a particle with initial velocity zero

Is it true that the field lines of an electric field are identical to the trajectories of a charged particle with initial velocity zero? If so, how can one prove it? The claim is from a german ...
4
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2answers
2k views

Killing vector fields

I am facing some problems in understanding what is the importance of a Killing vector field? I will be grateful if anybody provides an answer, or, refer me to some review or books.
4
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6answers
2k views

Concrete example of divergence of a vector field

I'm studying vector analysis and it is hard for me to understand what divergence of a vector field really is. I know that $divF=\nabla\cdot F$ but I don't understand what kind of quantity it gives and ...
4
votes
2answers
187 views

A wonky gravitational potential and its critical points

I have tough problem I am not sure how to solve: For this question, we are confined to a plane. Consider a gravitational field that is proportional to $\frac{1}{r^3}$ instead of $\frac{1}{r^2}$, and ...
4
votes
1answer
76 views

field solutions for covariant derivative of vector field constrained to zero

Question: What do the solutions of $\nabla_\mu A^\nu = 0 $ look like? And is it possible for spacetime curvature to somehow restrict the solution to $A^\nu = 0$? Here is my current ...
4
votes
1answer
126 views

Where does energy in a field come from?

Let us consider for example Earth's gravitational field. If we put a ball somewhere in this field, the ball starts to accelerate due to the gravitational force exerted on it. I understand the ...
4
votes
1answer
241 views

What are the generators of spherical symmetry?

The title says it all. I think this should be a pretty simple question but I just couldn't find the answer. Ok -- I'll give a bit more context to my question. I'm encountering this in the context of ...
4
votes
2answers
163 views

Hilbert action's invariance under general coordinate changes

In an article, when considering invariance of the Hilbert action under a general coordinate change this formula appears for how the metric changes ...
4
votes
1answer
65 views

Conformal Killing fields on Schwarzschild

I am trying to understand which are the conformal Killing Fields on the Schwarzschild spacetime. I say that $X$ is a conformal Killing field on $S$ ($S$ is Schwarzschild) if there exists a function ...
4
votes
2answers
142 views

Geometric meaning of parallel transport

The definition of parallel transport of a vector $v^b$ along a curve $C$ with tangent field $\it{t}^a$ is given by Wald's GR as $$t^a \nabla_a v^b = 0$$ Is it correct to think of $\nabla_a v^b$ as ...
4
votes
1answer
116 views

Bondi-Metzner-Sachs (BMS) symmetry of asymptotically flat space-times

I started studying the BMS symmetry in connection with the paper: http://arxiv.org/abs/1312.2229 and there are a few strange things I noticed. First of all, from reading the original papers by Bondi, ...
3
votes
3answers
168 views

Relation between component and algebraic definition of covariant vectors

I studied contravariance and covariance concepts in following way: For any vector if we get its components by parallelogram way we achieve contravariant components, and if we want to get its ...
3
votes
4answers
3k views

Divergence of $\frac{\hat{r}}{r^2}$

In David J. Griffiths's Introduction to Electrodynamics, the author gave the following problem in an exercise. Sketch the vector function $$ \vec{v} ~=~ \frac{\hat{r}}{r^2}, $$ and compute ...
3
votes
3answers
73 views

Is it possible to have a non conservative vector field, such that the closed loop integral is $0$ for only some specific path(s)?

I was wondering whether there exists some non conservative field in which the closed loop integral over some specific path(s) is $0$, even if it's not $0$ for all the closed loop integrals. Or to put ...
3
votes
3answers
233 views

Are Field Lines an accurate depiction of reality?

Field lines are used for explaining a wide variety of phenomenon. But is it really an accurate depiction of reality? Is it more accurate to imagine a field in a different manner. For instance, using ...
3
votes
3answers
3k views

Why the heat flux vector at a point must be perpendicular to the temperature isothermal surface? Is it a definition or a deduction?

Before the question: I am working on numerical calculation of three dimension parabolic equation that based on Fourier's Law of which I am a little confused. Here comes the law in modern mathematics ...
3
votes
2answers
634 views

Metric coefficients in rotating coordinates

Let $(t,x,y,z)$ be the standard coordinates on $\mathbb{R}^4$ and consider the Minkowski metric $$ds^2 = -dt^2+dx^2+dy^2+dz^2.$$ I am trying to compute the metric coefficients under the change of ...
3
votes
1answer
724 views

'Easy way' of finding out the Killing vector fields?

Is there a way for calculating the Killing vector fields of a given metric in a quick way? Sure I can guess looking at the metric at the symmetries, and then guess some of them, but, for instance, in ...
3
votes
1answer
87 views

Why can a killing vector field be determined globally by its value and first derivative at one point?

It is said in Weinberg's Book, Gravitation and Cosmology, page 377, that a killing vector field (which we a priori assume exists globally) can be uniquely determined by its value and first derivative ...
3
votes
1answer
89 views

Christoffel symbol

For two nearby points in General Theory of Relativity. The change in the vector components when parallel transported is given by Now, since the parallel transport change must depend on the path ...
3
votes
1answer
162 views

What is the math/physics behind maximizing tinker's damage?

Tinker is a hero in the popular game Dota 2. He has a spell called 'March of the Machines' that creates a stampede of little robots in a rectangular area. The robots do damage on impact of an enemy ...
3
votes
2answers
80 views

Line integral definition of work clarification

So I am kind of confused about the role of force when calculating work. Specifically, when defining work using a line integral. There is a paragraph in my calculus book that is really throwing me off ...
3
votes
2answers
187 views

Changing vector basis in AdS$_3$

I have AdS${}_3$ given as a surface embedded in a 4 dimensional pseudo-Riemannian space $$x^2+y^2-u^2-y^2=-l^2$$ With metric: $$ds^2=dx^2+dy^2-du^2-dv^2$$ I have Killing vectors of that space ...
3
votes
0answers
519 views

Killing vectors for 2-sphere as generators of $SO(3)$ symmetry

How to get Killing vectors in a form of generators of $SO(3)$ group symmetry? By using Killing equations for metric $ds^{2} = d\theta^{2} + \sin^{2}(\theta^{2}) d\varphi^{2}$ I got $$ ...
3
votes
0answers
210 views

Pseudo scalar mass and Pure scalar mass

Since the only difference between pseudo scalar and a scalar term is just a change of sign under a parity inversion, is it possible that both of them be present in the same field and interact? For ...
2
votes
4answers
258 views

Gradient is covariant or contravariant?

I read somewhere people write gradient in covariant form because of their proposes. I think gradient expanded in covariant basis $i$, $j$, $k$, so by invariance nature of vectors, component of ...
2
votes
3answers
91 views

Generators of the Diffeomorphism Group

So what are the generators of a Diffeomorphism Group? For simplicity, let's consider $ Diff(R^2) $ (diffeomorphisms of the euclidean plane.) Diffeomorphisms are differentiable, invertible ...
2
votes
4answers
215 views

Why does $E=\nabla\phi$ follow from $\nabla\times E=0$?

I understand that using one of Maxwell's equations, $$\vec{\nabla} \times \vec{E}(\vec{x})=0,$$ it can be said that $$\vec{E}(\vec{x})=-\vec \nabla \phi(\vec{x}).$$ However, I can't find or ...
2
votes
3answers
304 views

Gravitational field has no curl? What about gas discs around stars, black holes, etc.?

So everybody says the gravitational field has no curl, and is not comparable to a liquid swirling around a drain. Observationally, of course, there are many examples of vector fields (which I think ...
2
votes
2answers
253 views

Time-like Killing vector in FRW metric?

The spatially flat FRW metric in cartesian co-ordinates is given by: $$ds^2 = -dt^2 + a^2(t)(dx^2 + dy^2 + dz^2)$$ As I understand it there are Killing vectors in the $x$, $y$, $z$ directions implying ...
2
votes
1answer
75 views

Killing vector contractions along isometric curves

Imagine $\xi_{\nu}$ is a Killing vector field on a manifold. Does $\xi_{\nu}\xi^{\nu}$ remain constant along any isometric curve defined by the Killing vector field? My guess is that yes since as ...
2
votes
3answers
567 views

How to get an integral formula for the flux time derivative

$$\frac{d}{dt}\int \limits_{A} \mathbf B d \mathbf A = \int \limits_{A} \left( \frac{\partial \mathbf B}{\partial t} + \mathbf v (\nabla \cdot \mathbf B ) + [\nabla \times [\mathbf v \times \mathbf B ...
2
votes
2answers
1k views

Electric field at a point from a square surface

I'm trying to determine the electric field at a point P (located on the +Z axis) due to a square of side length [L] and centered at the x-y plane origin. The square has a constant surface density [s]. ...