All Questions
58 questions with no upvoted or accepted answers
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 ...
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, ...
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 ...
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:
$$ g^{\mu\nu}...
3
votes
0
answers
477
views
Curl operator in Schwarzschild metric
I'm trying to write down the curl operator explicitly for a Schwarzschild metric in cylindrical coordinates. I am trying to use the general expression of the curl operator in orthogonal curvilinear ...
2
votes
0
answers
46
views
Is a spin connection with torsion possible whereas the affine connection is only Levi-Civita (torsion-free) in Supergravity?
In the paper "Simple Supergravity" from G. Dall'Agata & M. Zagermann (arXiv:2212.10044v2 15 Feb. 2023) on page 8 when it comes to the antisymmetric part of the covariant derivative of ...
2
votes
0
answers
57
views
How does the divergence change under a change of frame (with geometric algebra)?
I'm trying to prove equations (85) and (86) from Hestenes' paper Gauge Theory Gravity with Geometric Calculus (ResearchGate version).
$$
\dot{\nabla}^\prime \cdot \dot{\underline{f}}(A) = J_f^{-1}[ (\...
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 ...
2
votes
0
answers
83
views
Is my geometric interpretation of $T \left(\frac{\partial S}{\partial T}\right)_P = \left(\frac{\delta Q}{dT}\right)_P$ correct?
I originally started writing this as just a question, but in the process of writing it I may have solved it myself. Still, I would very much appreciate if someone more knowledgeable than myself took a ...
2
votes
0
answers
504
views
How can I derive the covariant derivative of the Ricci tensor using the Ricci scalar?
\begin{align*} R&=g^{\mu\nu}R_{\mu\nu}\\ \Rightarrow \nabla^{\gamma} R&=\nabla^{\gamma}(g^{\mu\nu}R_{\mu\nu})\\ &=\nabla^{\gamma}(g^{\mu\nu})R_{\mu\nu}+g^{\mu\nu}\nabla ^{\gamma}(R_{\mu\nu}...
2
votes
0
answers
240
views
Torsion form and exterior covariant derivative
The torsion form can be defined as the exterior covariant derivative of a solder form, $\Theta=d_\omega\theta$. This derivative is always in the fundamental representation of the algebra $\mathfrak g$ ...
2
votes
0
answers
433
views
Thermodynamics and differential forms
In Potter's Thermodynamics: Demystified (page 68), the author wrote:
Since only two independent variables are necessary to establish the state of a simple
system, the specific internal energy can ...
2
votes
0
answers
3k
views
Are Laplace Operator and mean curvature exactly the same thing for 2D function?
Let's assume we study 2D function/surface f(x,y).
Then Laplace Operator is defined as: $$\nabla^2 f=\frac{\partial^2f}{\partial x^2}+\frac{\partial^2f}{\partial y^2}$$
And the mean curvature: let $\...
1
vote
0
answers
62
views
Adjoint of the covariant derivative of a field?
Let's call $D$ the covariant derivative, $T$ the transposition of a field and $*$ its complex conjugate, so $T*$ is the "adjoint".
Is: $$(D_{\mu}\Phi)^{T*} (D_{\mu}\Phi)=D^{\mu}\Phi^*D_{\mu}\...
1
vote
1
answer
72
views
Covariant derivative for spin-2 field
I have mostly seen the concept of covariant derivative with regard to spin-1 fields. Is it possible to define the covariant derivative for spin-2 fields as well?
1
vote
0
answers
113
views
Derivation of covariant derivative by means of parallel transport
I've studied covariant derivative in many courses so far, but I got stuck on the definition given by the teacher notes of the exam of Topological QFT.
I think that he improperly used the name "...
1
vote
0
answers
118
views
Lie derivative of a one-form
I am going through Nakahara's textbook on geometry and topology in physics. Intuitively, I understand the definition of a lie derivative of a vector field
$$\mathcal{L}_xY = \lim_{\epsilon_\to 0}\frac{...
1
vote
0
answers
47
views
Scalar curvature in ADM Formalism (coordinate to coordinate-free transition)
I am attempting to express the scalar curvature in a coordinate-independent manner. Following the works of Bojowald, Thiemann, we have:
$$ {}^{(4)}R= {}^{(3)}R+K_{a b}K^{a b}- (K_a^a)^2 - 2\nabla_a v^...
1
vote
0
answers
67
views
Confusions about partial and covariant derivatives
Let, we are in 1d cartesian space with metric $g_{xx} = x^2$. Let we have a vector $v = 1/x e_x$. Since the vector is designed to shrink its components as the basis grows - its total length will ...
1
vote
0
answers
63
views
Wald: 2-dim Covariant Derivative for Null Hypersurfaces
On pp. 221-222, Wald introduces the 2-dim "hatted" manifold of null geodesics. He moves from 9.2.30 to 9.2.31 and he is allowed to do so because the tensors have the special properties that ...
1
vote
0
answers
345
views
On the definition of the Van Vleck-Morette determinant
Let $M$ be a Riemannian manifold and $\sigma$ the world function. The Van-Vleck-Morette determinant $D$ is defined by
$$D(x,x')=\det(-\sigma_{;\mu\nu{}'})$$
Regarding the semi-colon: In chapter $4.1$ ...
1
vote
0
answers
64
views
How to transform a partial derivative to a directional derivative with respect to some affine parameter?
Suppose an affine parameter $\lambda$ is defined along a null geodesic with $dx^\mu/d\lambda=k^\mu$. How could I write the partial derivative $\partial f/\partial x^\mu$ by using $df/d\lambda$? If $k^\...
1
vote
0
answers
186
views
Derivation of covariant derivative of Spinor
I am following this derivation for the covariant derivative of spinors. I have some questions about this derivation:
On page 3 they use the fact, that
\begin{align*}
V^a(x) = \bar{\Psi}(x)\gamma^a\...
1
vote
0
answers
82
views
Scalar curvature from Riemannian metric
I want to compute the scalar curvature for points on an empirical manifold (sampled data).
I have already an algorithm that learns the Riemannian metric and computes geodesics, so from the metric I ...
1
vote
0
answers
170
views
What is the meaning of $\nabla _{\mu}\nabla _{\nu}\phi(r)$ in general relativity?
I know the covariant derivative of a tensor is
$$\nabla_{\mu} V_{\nu}=\partial_\mu V_\nu-\Gamma_{\mu\nu}^{\lambda}V_{\lambda}$$
Now I want to obtain $\nabla_{\mu}\nabla_{\nu}\Phi(x)$ where $\Phi(x)$...
1
vote
0
answers
135
views
Are covariant derivative on associated bundles exterior covariant derivatives?
The gauge covariant derivative we encounter in gauge theory $D\psi = d\psi + A\wedge \psi$ is a covariant derivative on the associated vector bundle, right? Here $\psi$ is the matter field, $A$ the ...
1
vote
1
answer
286
views
Covariant derivative with an upper index in terms of Christoffel symbols
I have encountered expression
$$\frac{1}{2}\left(2 \dot{g}_{\mu}{}^{\lambda ; \mu}-\dot{g}_{\mu}{}^{\mu ; \lambda}\right)$$
in a GR paper.
Here we assume to be working with the de Sitter metric $g$ ...
1
vote
0
answers
477
views
Covariant derivative in spherical coordinates
Let's say I have a 4-vector $A^{\nu}$ and I take its covariant derivative (I'm using cartesian coordinates), so:
$\nabla_{\mu} A^{\nu}=\partial_{\mu}A^{\nu}+\Gamma^{\nu}_{\mu \alpha} A^{\alpha}$
But ...
1
vote
0
answers
32
views
Looking for differential operators satisfying a specific commutation relation with the Laplace operator
Consider the Laplace operator on some manifold, $\Delta=(\det g)^{-1/2} \frac{\partial}{\partial x^j} \left( (\det g)^{1/2}g^{jk} \frac{\partial}{\partial x^k}\right) $.
I am looking for differential ...
1
vote
0
answers
91
views
Affine Connection
On page 74 of Weinberg's General Relativity textbook he writes the following:
Equation 3.2.4:
$$\Gamma_{\mu \nu}^{\lambda} \equiv \frac{\partial x^\lambda}{\partial \xi^\alpha}\frac{\partial^2 \xi^\...
1
vote
0
answers
216
views
Lie derivative of the non-coordinate metric being 0
I'm trying to answer a question about a the Lie derivative of a metric in a non-coordinate basis.
Here, the $C^{a}_{bc}$ are from the Lie derivative of one basis vector with respect to another, or ...
1
vote
0
answers
63
views
Is my thinking correct for partial derivatives and tensors?
So I was transforming the affine connection and I ended up with a term like this:
$$
\frac{\partial^2 x'^a}{\partial x'^b \partial x^p}
$$
where $x$ and $x'$ are two different coordinate systems
...
1
vote
0
answers
796
views
Covariant derivative with respect to commutator
I have some confusion with the notion of $\nabla_{[A, B]}\bf{v}$, that expression, with a commutator of vector fields as the subindex of the connection appears for instance in the definition of the ...
1
vote
0
answers
103
views
Question about Covariant Derivatives in General Relativity
I'm following the differential approach of Schutz' book where vectores are geometrical objects written as
\begin{equation}
\vec{V}=V^a\ \vec{E}_a
\end{equation}
Where $V^a$ are the components of the ...
1
vote
0
answers
278
views
A Lie derivative $\mathcal{L}_{\alpha^A}$ with respect to a spinor $\alpha^A$?
Suppose we work with Minkowski flat space $M$ (just to make things easy). If $\textbf X$ is a Killing vector field it is possible to define the Lie derivative of an spinor $\alpha^A$ with respect to $\...
0
votes
0
answers
59
views
Covariant derivative with torsion
The covariant derivative is defined (on contravariant vectors) as:
$$\nabla_\mu V^\nu = \partial_\mu V^\nu + \Gamma^\nu_{\mu \rho} V^\rho \tag{1}$$
The purpose of the covariant derivative is to ...
0
votes
0
answers
159
views
What's the difference? $\nabla_\mu e_\nu=\Gamma_{\mu \nu}^\rho e_\rho~\text{ and }~\partial_\mu e_\nu=\Gamma_{\mu \nu}^\rho e_\rho~?$
What's the difference? $$\nabla_\mu e_\nu=\Gamma_{\mu \nu}^\rho e_\rho~\text{ and }~\partial_\mu e_\nu=\Gamma_{\mu \nu}^\rho e_\rho~?$$
In John Dirk Walecka's book 'Introduction to General Relativity',...
0
votes
0
answers
58
views
Partial derivatives of Christoffel symbols to Covariant derivatives
I wanted to express this thing: $g^{ab}\partial_c\Gamma^c_{ab} - g^{ab}\partial_a\Gamma^c_{cb}$, in terms of a covariant derivative. I figured out that if you swap $a$ and $c$ in the $\partial \Gamma$ ...
0
votes
0
answers
46
views
Application of Fermi-Walker derivative to specific problem
I am now reading about the tetrad formalism in GR and I am starting (how not) with the Wikipedia Article:
Frame fields in general relativity.
In this article, as an example, they show how tetrads can ...
0
votes
1
answer
115
views
Double covariant derivative of a mixed tensor
Let's say, we have a mixed tensor of type (2,1) denoted by $T^{mn}{}_p$ and the goal is to find the expression of $[\nabla_a, \nabla_b] T^{mn}{}_p$ in terms of fundamental tensors.
Firstly, I am ...
0
votes
0
answers
103
views
Conceptual confusion about the formula for parallel transport
I am examining the covariant derivative of a vector according to the formula $$\nabla_{\mu}V^{\nu} = \partial_{\mu}V^{\nu} + \Gamma^{\nu}_{\mu \lambda}V^{\lambda}$$ and also operating under the ...
0
votes
1
answer
118
views
Covariant and partial derivative of a vector field (not component)
Is the covariant derivative of a vector field (not the components of a vector) same as the partial derivative?
I am adding a screenshot from page 69 from General Relativity: An introduction for ...
0
votes
0
answers
201
views
Commuting material time derivative and material space derivative
Let's note $x$ the coordinates in the current configuration and $\nabla$ the associated gradient; similarly, let's note $X$ and $\nabla_0$ for the reference configuration. I will also note the ...
0
votes
1
answer
72
views
What is the physical meaning of non-commuting tetrads?
I'm reading about the tetrad formalism in GR and one main difference between the coordinate and the tetrad frame is that coordinate derivatives commute $\partial_\mu \partial_\nu = \partial_\nu \...
0
votes
0
answers
45
views
Covariant Derivative of New Timelike coordinate
In another forum, someone was explaining to me how to define a new time coordinate in a metric tensor to diagonalize it. They said that if I want spacelike hypersurfaces, the covariant derivative of $...
0
votes
1
answer
102
views
Higher dimension derivatives
In the case of higher dimensions (e.g. 4+1 dimensions) how would the 5 derivative ($\partial_5$) change?
For example if $\\X=(x^{\mu},z)$, would the 5 derivative change as $$\partial_5\partial^5X=\...
0
votes
0
answers
104
views
Derivatives of the metric in the local Lorentz frame
In the local Lorentz frame (local flatness at point P) we have:
$$
g_{\alpha \beta} (P) = \eta_{\alpha \beta}, \quad \Gamma^{\rho}_{\alpha \beta}(P) = 0.
$$
In the reference that I am following (The ...
0
votes
2
answers
139
views
Question on varying the Ricci tensor
When varying the Ricci tensor, there’s an in-between step that allows for example
$$\delta\left(\partial_{\alpha}\Gamma^{\alpha}_{\phantom{\alpha}\beta\gamma}\right)= \partial_{\alpha}\left(\delta\...
0
votes
0
answers
83
views
Doubt of gauge covariant derivatives: how can I derive it?
In the context of general relativity (GR) it is necessary to introduce the notion of covariant derivatives. From the point of view of a basic introduction, we always start to deal with GR in a highly ...
0
votes
0
answers
59
views
Covariant derivative in $f(Q,T)$ field equations
I am trying to evaluate the field equations in $f(Q,T)$-gravity and I'm not sure how to evaluate this term:
$$-\dfrac{-2}{\sqrt{-g}}\nabla_{\alpha}(f_{Q} \sqrt{-g}\enspace P^{\alpha}_{\enspace \mu \nu}...