All Questions
463 questions with no upvoted or accepted answers
8
votes
0
answers
222
views
Has the Helmholtz decomposition of the $\mathbf{E}$ field from the Liénard–Wiechert potentials been worked out?
If you look at Maxwell's equations for $\mathbf{E}(\mathbf{x},t)$ they split neatly into two categories. They are:
\begin{align}
\nabla\cdot\mathbf{E}(\mathbf{x},t)&=\frac{\rho(\mathbf{x},t)}{\...
- 22.8k
8
votes
0
answers
1k
views
Do divergence and curl of Lorentz force have some physical meaning?
Time ago I started thinking about this: if we take the well known Lorentz Force expression, namely
$$\mathbf{F} = q\left(\mathbf{E} + \mathbf{v}\times\mathbf{B}\right)$$
and we operate $\nabla\cdot \...
- 3,735
6
votes
1
answer
514
views
Covariant derivative of the vielbein determinant
The vielbein postulate says that
$$\nabla_\mu e_v^{\,a}=\partial_{\mu}e_\nu^{\,a}+\omega_{\mu\,\, b}^{\,\,a}\,e^b_\nu-\Gamma^\sigma_{\mu\nu}\,e^{\,a}_\sigma=0.$$
$\nabla$ is the coordinate covariant ...
- 99
6
votes
0
answers
242
views
Defining the covariant derivative on bitensors
Bitensors (tensors defined on two different points) are an extension of tensors found in some applications of general relativity, where objects such as the world function, parallel transport operator, ...
- 16.7k
5
votes
0
answers
1k
views
Bianchi identity of gauge theory
How to prove Bianchi identity?
\begin{align*}
\varepsilon^{\mu\nu\rho\sigma}D_{\nu}F_{\rho\sigma}=0
\end{align*}
using Jacobi identity;
\begin{align*}
\epsilon^{\mu\nu\rho\sigma}[D_{\mu},[D_{\rho},D_{\...
- 111
4
votes
1
answer
473
views
Conformal Casimir as Differential Operator
my question regards equation (165) of [1], namely, how to write the conformal Casimir as a differential operator in the "usual" $z,\bar{z}$ coordinates. If one inspects the definition of the ...
4
votes
0
answers
277
views
Conformal Killing Group volume in string amplitude calculations
In string amplitude calculations, the volume conformal Killing group (CKG), if finite, can enter into the calculations when the number of external vertex operator insertions is less than the (complex) ...
- 823
4
votes
0
answers
384
views
Schwarzian derivative from conformal factor
Suppose I have a 2D Lorentzian conformally flat metric
$$ ds^2 = -\Omega(u, v) du dv.$$
I consider a conformal field theory whose stress-energy tensor $T_{ab}$ is known on the flat metric
$$ds^2 = -...
- 4,321
4
votes
0
answers
743
views
Tangent Vectors as Infinitesimal Displacements
I'm reading Wald's General Relativity, and I'm stuck on something that is stated very early on in the book. For an abstract manifold $M$, he goes through the usual definition of a tangent vector at $p\...
- 729
4
votes
0
answers
1k
views
How is Infinitesimal coordinate transformation related to Lie derivatives?
I am reading the book "Gravitaion and Cosmology" by S. Weinberg. In section 10.9, while discussing Lie derivatives of tensors of different ranks, he makes a general comment:
The effect of an ...
- 437
4
votes
0
answers
3k
views
Killing vectors for 2-sphere as generators of $SO(3)$ symmetry
How to get Killing vectors in a form of generators of $SO(3)$ group symmetry?
By using Killing equations for metric $ds^{2} = d\theta^{2} + \sin^{2}(\theta^{2}) d\varphi^{2}$ I got
$$
\varepsilon_{\...
- 353
4
votes
0
answers
668
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 ...
- 511
3
votes
1
answer
67
views
"Deriving" the covariant derivative
Suppose we are working in scalar QED with Lagrangian
$$\mathscr{L} = (D_\mu \phi)(D^\mu \phi)^* - \frac{1}{4}F_{\mu\nu}F^{\mu\nu}.$$
I now want to find the form of the covariant derivative $D_\mu$ ...
- 283
3
votes
0
answers
78
views
Explicitly Computing a Lie Transported Vector or Covector
I'm having difficulty understanding how to physically compute how a covector changes under Lie transport.
Suppose on $\mathbb{R}^2$ I have a vector field $V=x\frac{\partial}{\partial x}+y\frac{\...
- 31
3
votes
0
answers
118
views
Reduce multiplicative noise to additive noise with singular matrices
I have a stochastic differential equation as
\begin{equation}
\dot{\textbf{X}}=\textbf{A}\textbf{X}+\alpha(t)\textbf{B}\textbf{X}-\alpha^*(t)\textbf{B}^T\textbf{X}
\end{equation}
where $T$ ...
- 95
3
votes
0
answers
153
views
d'Alembertian operator in presence of torsion
Consider a Riemann-Cartan 4-dimensional spacetime with torsion. In such a spacetime, I have been asked to compute the d'Alembertian operator acting on a scalar field. Here's what I tried:
$$ g^{\mu\nu}...
- 702
3
votes
0
answers
358
views
Integration by parts of covariant derivative
There already exists posts to discuss this question, but I don't think it's totally done!
We can write the covariant derivative as
$$D_i=\partial_i-igA_i^aT^a \tag{1}$$
There are two kinds of opinions ...
- 1,461
3
votes
4
answers
638
views
Derivation of covariant derivative
I'm currently doing Introductory QFT and was confused about the origin of the additional terms in the covariant derivate. My understanding is as follows:
If we begin with the Dirac Lagrangian ...
- 100
3
votes
0
answers
66
views
How does being spherically symmetric imply this particular form for the metric?
If $(M, g)$ is a 4-dimensional Lorentzian manifold whose isometry group contains a subgroup isomorphic to
$\text{SO}(3)$, then we can put $g$ in the form:
$$
g = -A(r, t) \mathrm{d} t ^2 + B(r, t) \...
- 181
3
votes
2
answers
160
views
Acceleration in terms of displacement
I am having problems understanding the derivation of acceleration in terms of displacement. The first step is fine:
$$a(x) = \frac{\mathrm dv(x)}{\mathrm dt}
= \frac{\mathrm dv(x)}{\mathrm dx} \frac{\...
- 131
3
votes
0
answers
92
views
Vector representation of the Virasoro Algebra
The Witt algebra is defined by the following Lie Bracket :
$$
[l_n,l_m]=(n-m)l_{(n+m)}.
$$
One representation can be obtained by an algebra of conformal killing vectors of Minkowski space:
$$
ds^2 = -...
- 2,302
3
votes
0
answers
89
views
About the equation $\frac {d^2} {dt^2}\vec x(t) = \nabla \times \vec F(x(t))$. Motion in a curl vector field
I was wondering if there is a physical interpretation of ODEs of the form
$$\frac d{dt}\vec x(t)=\vec y(t)$$
$$ \frac d{dt} \vec y(t) = \nabla \times \vec F(x(t))$$
(or equivalently $\frac {d^2} {dt^2}...
- 208
3
votes
0
answers
636
views
Do divergence and curl uniquely determine a 3D vector field?
I feel this should be true, but I'm not sure how. If it's true, can someone tell me how? If it's false, just say so in a comment.
Why I feel it's feel true? I have a clumsy reason. Divergence is like ...
- 959
3
votes
0
answers
68
views
Lattice differentiation and Locality
Assume we define the locality of a theory in the following way:
Assume we have a theory of real scalars, so this theory is non local if the action has terms like
$$\int d^dx\,\phi(x)V(x-y)\phi(y).$$
...
- 1,429
3
votes
0
answers
549
views
longitudinal and transverse components in higher dimensions
I am familiar with the Helmholz decomposition of a vector field in three dimensions:
$$\vec{V}=\vec{\nabla}\wedge\vec{A}+\vec{\nabla}\phi$$
But I am interested to show that something similar can be ...
- 1,914
3
votes
0
answers
184
views
Time derivative in rotating frame
In Goldstein (2ed) sec 4.9 - Rate of change of a vector, why does he say that the instantaneous angular velocity $\omega$ is not a derivative of any vector?
$$ (d\textbf{G})_{space} = (d\textbf{G}...
- 158
3
votes
0
answers
378
views
Uncertainty calculation - when to use absolute value bars?
I'm asking this because I saw (at least) two questions here on this Stack that seemed very similar and caused the same confusion to me in reading the answers to both.
Suppose we have a formula:
$A = ...
- 4,574
3
votes
0
answers
476
views
Curl operator in Schwarzschild metric
I'm trying to write down the curl operator explicitly for a Schwarzschild metric in cylindrical coordinates. I am trying to use the general expression of the curl operator in orthogonal curvilinear ...
- 31
3
votes
0
answers
438
views
How to diagonalize Lagrangian with two gauge vector fields?
I have recently stumbled upon a Lagrangian i had never seen before. Given two real vector fields $A_\mu$ with force tensor $F_{\mu\nu}$ and $V_\mu$ with $G_{\mu\nu}$.
The theory that describes their ...
- 2,223
3
votes
0
answers
371
views
Why is surface gravity named this way?
I've come across two definitions of surface gravity.
We look at the killing horizon of a Killing vector field $\xi$ and we find from the definition that $\xi_\nu \nabla^\nu \xi^\mu = \kappa \xi^\mu$....
- 2,889
3
votes
0
answers
1k
views
How to calculate topological charge?
For a complex vector field in two dimensions with one or more phase singularity - a point where the field amplitude is zero and the phase is undefined - how do you explicitly calculate the total ...
- 31
3
votes
0
answers
396
views
Laplacian and Dirac Delta function
Although I find it mathematically dubious, we said that $$\Delta \frac{1}{r} ~=~ -4\pi \delta^3({\bf r}).$$ Now, I was wondering is there a similar relation to the delta function if we look at $$\...
- 1,928
3
votes
0
answers
222
views
Questions about deduction the dual form of Frobenius's Theorem
I am reading Page 435, General Relativity by Wald.
Let $T^*\subset V^*$ be a subspace of the dual tangent space of a manifold, $W\subset V$ be the subspace of the tangent space annihilated by $T^*$, ...
- 787
2
votes
0
answers
58
views
Why general relativity agrees with Newtonian theory in the limit?
I'm currently reading "The Large Scale Structure of Spacetime" by Hawking and Ellis. While the mathematical computations are clear to me, I find myself puzzled by the physics, particularly ...
- 21
2
votes
0
answers
46
views
Is a spin connection with torsion possible whereas the affine connection is only Levi-Civita (torsion-free) in Supergravity?
In the paper "Simple Supergravity" from G. Dall'Agata & M. Zagermann (arXiv:2212.10044v2 15 Feb. 2023) on page 8 when it comes to the antisymmetric part of the covariant derivative of ...
- 10.1k
2
votes
0
answers
47
views
Finding condition for Adiabaticity
I have a differential equation describing a resonator that looks like this:
$$ \frac{d\alpha(t)}{dt} = [j a - b]\alpha(t) + \sqrt b e^{jct}$$ where I can solve it putting: $$\alpha(t) = \alpha e^{jct}$...
- 21
2
votes
0
answers
49
views
Applications of time derivative of unit vector
A math methods textbook I'm currently reading went into great detail deriving the following expression for the time derivative of a generic unit vector $\hat{r}$.
$$
\frac{d\hat{r}}{dt} = \frac{1}{r^2}...
- 21
2
votes
0
answers
404
views
Static spherically symmetric spacetimes
I would like to better understand a hypothesis that Wald uses to derive the general local formula of a static spherically symmetric spacetime.
A spacetime is said to be spherically symmetric if its ...
- 31
2
votes
1
answer
152
views
Confirming an action is invariant under a supersymmetric transformation
I am studying chapter 9 of the book Mirror Symmetry, available here. My question is relating to page 156/157 where Supersymmetry is being introduced for the first time in QFT in 0-dimensions.
We are ...
- 213
2
votes
0
answers
57
views
How does the divergence change under a change of frame (with geometric algebra)?
I'm trying to prove equations (85) and (86) from Hestenes' paper Gauge Theory Gravity with Geometric Calculus (ResearchGate version).
$$
\dot{\nabla}^\prime \cdot \dot{\underline{f}}(A) = J_f^{-1}[ (\...
- 163
2
votes
0
answers
109
views
Lie derivative of surface gravity on killing horizon
Carroll's 'spacetime and geometry' says surface gravity $\kappa$ is defined by $K^\mu \nabla_\mu K=-\kappa K$ where $\nabla$ and $K$ each denote covariant derivative and Killing vector field.
This is ...
- 21
2
votes
0
answers
117
views
Tetrad formalism: Cartesian-like tetrad?
I'm confused about tetrad formalism.
In the article the have the Kerr metric in Boyler-Lindquist coordinates. They then define the tetrad at a point $r,t,\theta,\varphi$ as the one-form basis
$$
e^{(t)...
- 268
2
votes
0
answers
61
views
Ostrogradsky instability and fractional derivatives
Are fractional derivatives (or even more generally differentegrals) also under the scope of the Ostrogradsky instability theorem?
- 6,727
2
votes
0
answers
91
views
Derivation of a partial derivative equation by Albert Einstein in Special Theory of Relativity
I was reading Albert Einstein's "On the Electrodynamics of Moving Bodies". In section "Kinematical Part", on $3 (Theory of the transformation of coordinates and times from a ...
- 21
2
votes
0
answers
225
views
Thermodynamics Chain Rule And Independent Variables
I was reading my textbook and I came up across the entropy $S(T,V,N)$ where temperature $T$, volume $V$, and number of particles $N$ are the independent variables. According to the chain rule the ...
- 73
2
votes
1
answer
336
views
Vectors in AdS/CFT: scaling dimension and near boundary behaviour
I'm trying to understand how the near boundary expansion of a field in AdS$_{d+1}$ is related to the conformal dimension of the corresponding operator in the dual CFT$_d$.
I use coordinates in which ...
2
votes
0
answers
86
views
Doubt on $SU(2)_{L} \times U(1)_{Y}$ covariant derivative and its action on a fermion
I) Introduction
I.1)
The mathematical structure is quite clear: given a spacetime $M$, and a Lie group $G$ (the gauge group), we can construct the Principal bundle $P^{G}_{M}$. The connection $1$-form ...
- 3,159
2
votes
1
answer
64
views
Implications of Galilei-Invariance on a time-independent potential
I'm trying to compute a result shown in my classical mechanics lecture on my own. Namely, consider that a system composed of $n$ particles follows a law of force
$m_k\ddot{\vec{x_k}} = \vec{F_k}(\vec{...
- 324
2
votes
0
answers
391
views
Mayard's mechanical differentiator
I've recently been reading an article on a mechanical diferentiator (Nouvelles solutions de calcul grapho-mécanique. Dérivographe et planimètre).
The author describes a mechanical device, which, given ...
- 21
2
votes
1
answer
1k
views
Proof for covariant vector transformation law
(I'm asking this on the physics exchange not on the math one because i don't need an extremely rigorous explanation)
I understand the derivation for the contravariant vector transformation law is ...
- 21