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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 ...
gammadragon's user avatar
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, ...
Slereah's user avatar
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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 ...
Sucheta's user avatar
  • 437
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}...
Faber Bosch's user avatar
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 ...
Iraklolo's user avatar
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 ...
Frederic Thomas's user avatar
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}[ (\...
foghorn's user avatar
  • 163
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 ...
M.N.Raia's user avatar
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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 ...
ummg's user avatar
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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}...
Bruce Wayne's user avatar
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 $\mathfrak g$ ...
Bellem's user avatar
  • 258
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 ...
M.N.Raia's user avatar
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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 $\...
physixfan's user avatar
  • 371
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}\...
Mathieu Krisztian's user avatar
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?
physics_2015's user avatar
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 "...
polology's user avatar
  • 177
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{...
Souroy's user avatar
  • 343
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^...
Powder's user avatar
  • 403
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 ...
Nayeem1's user avatar
  • 1,248
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 ...
mster8390's user avatar
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 $4.1$ ...
Filippo's user avatar
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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^\...
Haorong Wu's user avatar
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\...
Aralian's user avatar
  • 527
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 ...
can't stop me now's user avatar
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 $\Phi(x)$...
Alice's user avatar
  • 67
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 ...
Alex's user avatar
  • 143
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 $g$ ...
Student's user avatar
  • 1,122
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 ...
Mathew's user avatar
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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 ...
WLV's user avatar
  • 63
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^\...
Jbag1212's user avatar
  • 2,740
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 ...
baker_man's user avatar
  • 420
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 ...
Toby Peterken's user avatar
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 ...
David's user avatar
  • 275
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 ...
P. C. Spaniel's user avatar
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 $\...
raul's user avatar
  • 428
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 ...
Frederic Thomas's user avatar
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',...
Jianbingshao's user avatar
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 $\partial \Gamma$ ...
Stargazer's user avatar
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 ...
T. ssP's user avatar
  • 533
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 ...
raf's user avatar
  • 151
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 ...
Relativisticcucumber's user avatar
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 ...
Nayeem1's user avatar
  • 1,248
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 ...
mekano's user avatar
  • 1
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 \...
Pau Bañón Pérez's user avatar
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 $...
user345249's user avatar
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=\...
DespStudent's user avatar
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 ...
Edison Santos's user avatar
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\...
user avatar
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 ...
M.N.Raia's user avatar
  • 3,159
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}...
Mark Pace's user avatar
  • 164