Vector-fields are vector valued functions which define a vector at each point in space. Examples of the vector field include the electric field and the velocity of a fluid.

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1answer
85 views

Killing vector and one-form [closed]

p. 21 in this paper (http://arxiv.org/abs/0704.0247) $V$ is Killing vector, where $V^2 = −4b\bar{b}$, which means it is timelike Killing vector. The authors say: From $V^2 = −4|b|^2$ and $V = ∂...
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1answer
72 views

How do we know if a Killing Spinor is Time-like or Null?

How to know whether a Killing spinor orbit is time-like or null? This is present in a paper like this 29/39 here. I'm not asking for a technical answer, just a logical cliche answer chit-chat answer. ...
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1answer
68 views

Variation of a tensor

Let a change of coordinates be given by $x^{\mu}\to x^{\mu '}=x^{\mu}+\varepsilon \xi^{\mu}(x)$ with epsilon a small quantity. Given a tensor $T$ we define $\delta T:=T'(x)-T(x)$. I guess this means $...
2
votes
2answers
111 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: $$X_{\mu,\nu}+X_{\nu,\mu}=...
3
votes
1answer
119 views

Hyperbolic flow / vector field - irrotational and divergence-free?

My text book on meteorology claims that a hyperbolic flow pattern is both divergence-free and irrotational: (d) Hyperbolic flow that exhibits both diffluence and stretching, but is ...
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3answers
137 views

Is there any physical interpretation for $\nabla\cdot(\nabla \times F)=0$?

It is well known that the divergence of the curl is always 0. Mathematically I understand why this happens ($d^2=0$ where $d$ is the exterior derivative) but today I was wondering what is the physical ...
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1answer
84 views

How can I prove that for a Killing vector $\nabla^a \nabla_a \xi^\mu = -R^b_a \xi^a$? [closed]

I'm taking a course on General Relativity and I'm trying to prove that for a Killing vector field $\xi^\mu$ the following equation holds: $$\nabla^a \nabla_a \xi^\mu = -R^\mu_a \xi^a$$ Where $R_ab$ ...
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6answers
3k views

Why are Killing fields relevant in physics?

I'm taking a course on General Relativity and the notes that I'm following define a Killing vector field $X$ as those verifying: $$\mathcal{L}_Xg~=~ 0.$$ They seem to be very important in physics ...
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1answer
47 views

Why does a system whose equations of movement are $\lambda^2U^{\alpha} + \partial_{\mu}F^{\mu \alpha} = 0$ have three degrees of freedom?

I'm trying to understand the solution of a problem where I have to study a field ($U^\mu$) which Lagrangian is: $$\mathscr{L} = - \frac{1}{4} F_{\mu \nu} F^{\mu \nu} + \frac{1}{2} \lambda^2 U_{\mu} U^...
2
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1answer
49 views

Local Coordinate Expressions for Lie Derivatives

I'm currently working through the math chapters of Norbert Straumann's book on General Relativity. I have trouble understanding the coordinate expression of the Lie derivative of a basis vector. The ...
0
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0answers
51 views

Electric field of a dipole in cartesian

I have the function that describes the electric field of a dipole in cartesian coordinates: $$E_z= \frac{p}{4\pi\epsilon_0} \cdot \left(\frac{3z^2}{r^5}- \frac{1}{r^3}\right) $$ and $$E_x=\frac{p}{4\...
2
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1answer
122 views

Are all maximally symmetric spacetimes constant curvature spacetimes?

A $d$ dimensional maximally symmetric spacetime is a spacetime with the maximum allowed number of Killing vectors. This number is $\frac{d(d+1)}{2}$. Constant curvature spacetimes are spacetimes ...
2
votes
3answers
113 views

Why can the divergence of vector potential be anything?

Purcell in his book was deriving the vector potential $\bf A$ using $\text{curl}\;(\text{curl}\; \mathbf A)= \mu_0 \mathbf J\; .$ After some algebra, he came to this: $$-\frac{\partial^2 A_x}{\...
5
votes
1answer
183 views

Finding diffeomorphism given vector fields [closed]

Given a vector field how do you find the associated diffeomorphisms? Say I am given a vector field in Minkowski space $$\xi = x \frac{\partial}{\partial t} + t \frac{\partial}{\partial x}.$$ How do ...
2
votes
0answers
74 views

Integral curves in null hypersurfaces [closed]

Let be $(M^{n+1},g)$ a spacetime (Lorentz manifold, connexe and time-oriented), $n\ge 2$, and $S\subset M$ a null hypersurface (codim $S=1$ and the restriction of $g$ to each tangent space $T_p S$ is ...
0
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2answers
73 views

What is the criterion for a vector field (dim > 1) to be conservative?

We know that if the curl vanishes: $$ \nabla \times \vec{v} = 0$$ then the field is irrotational and is conservative, but what about in higher dimensions than 1? The cross product is not defined there,...
0
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1answer
61 views

Wightman function for massless vector fields in Coulomb gauge

I've been looking for quite some time an expression for the Wightman functions for a massless vector field in the Coulomb gauge $\nabla\cdot\mathbf{A}=0$ (I think it is equivalent to the Feynman gauge ...
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2answers
207 views

Conservation of energy and Killing-field

In general relativity we have no general conservation of energy and momentum. But if there exists a Killing-field we can show that this leads to a symmetry in spacetime and so to a conserved quantity. ...
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2answers
91 views

Divergence-free vector field on a non-simply connected domain

We know that divergence-free vector fields are themselves curls of vector fields on simply connected domains. I want to construct a counterexample in the case the domain is not simply connected. So ...
0
votes
1answer
79 views

Finding the electric and magnetic fields from the vector potential $\vec A=\vec{E_0} e^{i(\vec k\cdot\vec r-\omega t)}$

I am trying to find the electric and magnetic fields from the vector potential $$\vec A=\vec{E_0} e^{i(\vec k\cdot\vec r-\omega t)},$$ I know $$\vec B=\vec \nabla\times \vec A$$ and $$\vec E=-\vec\...
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2answers
267 views

Direction of H and B inside and outside a bar magnet

I seem to have encountered a contradiction when thinking about the directions of $\textbf{H}$ and $\textbf{B}$ inside and outside a bar magnet. Suppose that a bar magnet has a roughly constant ...
-1
votes
1answer
46 views

A leaf floating on the surface of water (in $x$-$y$ plane) curl along positive $z$ axis [closed]

A leaf floating on the surface of water (in x-y plane) show that for a very small circular leaf ($\nabla \times \overrightarrow v $) is equal to twice the angular velocity of rotation of the leaf, ...
2
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1answer
44 views

Details on the magnetic field of a linearly polarized electric wave

Suppose we are in vacuum and we have an electric field $\vec{E}$ which we assume is simple harmonic wave that propagates through $z$ and is linearly polarized in the $x$-$y$ plane along $x$ i.e. $\vec{...
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0answers
126 views

Constructing Killing tensors from Killing vectors

Background: After reading about Carter constant and symmetries in GR, I became interested in Killing tensors. I tried reading this paper by Alan Barnes, Brian Edgar and Raffaele Rani, discussing ...
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1answer
220 views

Derivation of one-form/vector equation in Carroll confusion

I don't understand the derivation of Equation 2.14$$\mathrm{d}f\left(\frac{d}{d\lambda}\right)=\frac{df}{d\lambda} \tag{2.14}$$ in Carroll's Lecture Notes on General Relativity (http://ned.ipac....
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votes
2answers
648 views

Killing Vectors in Schwarzschild Metric

Given the Schwarzschild metric with $(-,+,+,+)$ signature, $$\text ds^2=-\left(1-\frac{2M}{r}\right)dt^2+\left(1-\frac{2M}{r}\right)^{-1}dr^2+r^2(d\theta^2+\sin^2\theta\,d\phi^2)$$ the lack of ...
2
votes
1answer
89 views

Killing vectors in General Relativity?

I'm looking to derive the surface area of the event horizon of a Schwarzschild black hole. I was just wondering if it were possible for someone to explain to me this: $$ \sqrt{g_{\theta\theta}g_{\phi\...
1
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0answers
75 views

How do I derive Feynman rules for vectors involving derivatives?

Suppose I have a term in the Lagrangian: $$\cal{L} \equiv (\partial_\mu B^+_\nu) B^{-\mu} A^\nu $$, where $B^\pm$ are charged massive vector particles and $A$ is photon. Now, how can we derive the ...
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1answer
250 views

Gradient, divergence and curl with covariant derivatives

I am trying to do exercise 3.2 of Sean Carroll's Spacetime and geometry. I have to calculate the formulas for the gradient, the divergence and the curl of a vector field using covariant derivatives. ...
-1
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1answer
90 views

To prove, $\nabla.(\nabla\phi \times \nabla\psi)$ =0 [closed]

Please Help me solving the problem using levi-cevita symbol : Prove That, $\nabla.(\nabla\phi \times \nabla\psi)$ =0 where $\phi =\phi(x,y,z)$ & $\psi=\psi(x,y,z)$
0
votes
2answers
51 views

Faraday's law and potential along a closed curve

$$\int(\nabla \times \vec{E}) \cdot d\vec{S} = \oint \vec{E}\cdot d\vec{l} = -\frac{\mathrm{d} \phi_b}{\mathrm{d} t}$$ The second expression is the potential difference along a closed curve. Isn't ...
2
votes
3answers
120 views

Why exactly are fields so crucial to modern physics?

I have read that fields were the mathematical tool that allowed people like Laplace to develop a working model of the Solar System where Newton could not. But my understanding is that fields are ...
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4answers
178 views

What is the physical significance of curl $\nabla\times\boldsymbol{V}$?

What is the physical significance of curl $$\nabla\times\boldsymbol{V}~?$$ I mean I read 'curl V represents the rotation of the vector $V$. My question what is it about the term $\nabla\times\...
0
votes
3answers
134 views

Work done by a constant vector field is 0?

We know that $$\oint \boldsymbol{F}\cdot d\boldsymbol{r}= \iint (\nabla \times \boldsymbol{F})\cdot d\boldsymbol{s}.$$ Now if $\boldsymbol{F}$ is a constant vector, then $\nabla \times \boldsymbol{F}=...
0
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1answer
75 views

Clarification about some steps in the derivation of the Lie derivative (mechanics)

First of all, this question may seem to be undefined, because I'm not sure how to connect this (to me) newly introduced concept with the abstract notion of the Lie derivative. I'm not even sure if I ...
1
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1answer
42 views

Norm of Dilatation operator [closed]

The dilatation operator is given by $$D=x^{a}\frac{\partial}{\partial x^{a}}+z\frac{\partial}{\partial z}$$ How the norm can be $$D^{2}=\frac{L^{2}}{z^{2}}(\eta_{\mu\nu}x^{\mu}x^{\nu}+z^{2})$$ where ...
4
votes
1answer
169 views

Gradient one-form [duplicate]

I am trying to understand what gradient one-form means actually. In the book that I'm following (A first course on General Relativity by Schutz) it's told that gradient is a one-form and it's ...
0
votes
1answer
121 views

Proving the invariance of the inner product

If we define the inner product as ${\textbf{u}\cdot\textbf{v}=g_{ij}u^{i}v^{j}}$, where ${g_{ij}}$ is the metric tensor, ${S}$ and ${T}$ are transformation matrices, ${S}$-for covariant indices and ${...
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votes
1answer
56 views

Vector calculus trouble in Hamilton's equations / particle in E.M.field [closed]

As part of applying Hamilton's equations to a particle in an electromagnetic field, one step is to take $\dot{\mathbf{p}} = - \dfrac{\partial H}{\partial \mathbf{r}} = -\nabla H = - \nabla [\dfrac{1}{...
0
votes
0answers
55 views

Little problem with surface integral

I've read a problem and I think there's a mistake in it. The problem is as follows. Note: The surface is the surface of a sphere. \begin{equation} \oint\frac{d\vec{a}}{d} \end{equation} Where $d=|\...
3
votes
0answers
78 views

Tangent Vectors as Infinitesimal Displacements

I'm reading Wald's General Relativity, and I'm stuck on something that is stated very early on in the book. For an abstract manifold $M$, he goes through the usual definition of a tangent vector at $p\...
4
votes
1answer
256 views

How can I mathematically describe the parallel transport in the Roman soldiers example?

I've been trying to understand parallel transport. Many of the descriptions present a mathematical version: $\nabla_V X = 0$. And/or they present an example involving soldiers (usually Roman) ...
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0answers
45 views

Helical killing vector

A killing vector X is defined as a vector field that satisfies the relation $$\mathcal{L}_X g_{\mu\nu}=0.$$ which basically means that if one were to transport the metric along this vector, there ...
2
votes
1answer
205 views

Differentiation of a vector with respect to a vector

Does differentiation of a vector with respect to a vector make any sense? Even if it makes sense, how does it make any physical meaning? I mean what is the physical interpretation?
0
votes
3answers
244 views

Curl of a vector field [closed]

What is the physical interpretation of curl of a vector field? Just as divergence implies flux through a surface. I mean if $\vec A$ is a vector field, what does $\left(\nabla \times \vec A \right)$ ...
0
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0answers
11 views

Calculating Trajectory in non-Uniform electric field? [duplicate]

Above is the equation of the electric field. Where k is the electric constant, q is the charge on the particle and m is the mass of particle. The particle q initial position is (0,10) and starts off ...
-1
votes
1answer
53 views

Proving that $\vec F$ is conservative field [closed]

I need to prove that $\vec F$ is coservative field: $$\vec F=\underbrace{\bigg(yz+\frac{1}{yz} \bigg)}_{Q} \hat i+\underbrace{\bigg(xz-\frac{x}{y^2z} \bigg)}_{P}\hat j+\underbrace{\bigg(xy-\frac{x}{...
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4answers
650 views

Why is $F=-\nabla V$?

I came across this equation $$F=-\nabla V$$ where $V$ is potential energy. I do understand that $$F(r)=-\frac{dV}{dr}.$$ Hence does this mean the nabla operator in this case means derivative? Because ...
0
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1answer
179 views

Deriving gravitational potential energy using vectors

Here is my attempt at derivation: First you must find a vector function for the gravitational force. By the inverse square law, the magnitude of gravitational force between two bodies of mass $m$ ...
1
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1answer
117 views

Trajectories piecewise smooth?

In my studies of calculus and real analysis I have found the proofs of several theorems, commonly used in physics, such as those concerning the conservativity of fields, for example like If $\...