All Questions
Tagged with differentiation differential-geometry
397 questions
3
votes
1
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118
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Covariant derivative acting on Dirac delta function
Pardon my naive computational question. In my calculations, I encounter the following expression:
\begin{equation} \label{eq1}
\frac{\delta}{\delta g^{\gamma \epsilon}(z)} \left( g_{\mu \alpha}(x) \...
- 85
3
votes
2
answers
340
views
Understanding the definition of the covariant derivative
I'm currently working my way through the book "Mathematical Methods for Physics - An Introduction to Group Theory, Topology and Geometry" and I think I have a very fundamental ...
- 61
0
votes
1
answer
80
views
The definition of the Lie Derivative
I am aware that an answer to an almost identical question already exist, however, I found the already existing answer not helpful (at least to my current question).
Carroll defines, in his book, the ...
-1
votes
0
answers
17
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How to prove that a Lie algebra-valued differential form is exact for the covariant derivative [migrated]
Given a differential $p$-form $\omega^A$ over a smooth manifold with values on some Lie algebra, I wanted to know how could one prove that it can be written as an exact form for the exterior covariant ...
- 381
3
votes
1
answer
114
views
Relationship between covariant derivative and metric tensor
In general relativity, the covariant derivative of the coordinate vector is a tensor, equal to $$x^{\mu}_{:\rho} = x^{\mu}_{,\rho} + \Gamma^{\mu}_{\rho\nu}x^{\nu},$$ is it meaningful to equate this ...
- 562
1
vote
0
answers
62
views
A trick for derivatives of thermodynamic quantities [closed]
Starting from
$$dU=TdS-PdV$$
We can write, for instance $U(T,V)$ and $S(T,V)$ to obtain:
$$\left(\frac{\partial U}{\partial T}\right)_VdT+\left(\frac{\partial U}{\partial V}\right)_T dV=T\left(\frac{\...
- 225
3
votes
1
answer
94
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What happens to $g^{\alpha\beta}_{,\sigma}=-g^{\alpha\mu}g^{\beta\nu}g_{\mu\nu,\sigma}$ when $g_{\mu\nu}\rightarrow \eta_{\mu\nu}$ (weak field limit)?
The equation
$$g^{\alpha\mu}_{\,\,\,\, ,\sigma}\,g_{\mu\nu} + g^{\alpha\mu}\,g_{\mu\nu,\sigma} = (g^{\alpha\mu}g_{\mu\nu})_{,\sigma} = \delta^\alpha_{\nu,\sigma} = 0 $$
gives the useful relation
$$g^{\...
- 467
0
votes
0
answers
38
views
Four-divergence of a vector [duplicate]
The covariant derivatives of a four-vector is
$$
\nabla_{\nu}U_{\mu} = \partial_{\nu}U_{\mu} - \Gamma^{\lambda}_{\mu\nu}U_{\lambda}
$$
It has the following identity:
$$
\nabla_{\mu}U^{\mu} = \frac{\...
- 49
0
votes
1
answer
69
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Is 4-velocity a Vector in the Sense of Covariant Derivative along Worldline
The definition of 4-velocity $U^{\mu} \equiv dx^{\mu}(\tau)/d\tau$, however, we've learnt that the covariant derivative for a vector along a curve parametrized by proper time is,
$$\frac{DA^{\mu}}{D\...
- 938
1
vote
1
answer
98
views
Proving a Superfunction Identity
I am trying to figure out the proof of the identity given between equations (1.11.7) and (1.11.8) in ref. [1], i.e.
\begin{align}
\Phi'(e^{-K}\,z\,e^K)=e^{-K}\Phi'(z) \tag{1}
\end{align}
where $z=(...
- 49
5
votes
1
answer
330
views
Divergence of vector field term-wise
In a spacetime $(M, g)$ the following identity for the divergence of a vector field $X$ holds
$$ \nabla_{\mu} X^{\mu} = \frac{1}{\sqrt{-\det g}} \,
\partial_{\mu} \big( \sqrt{- \det g} \ X^{\mu} \big)...
- 695
1
vote
0
answers
62
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Adjoint of the covariant derivative of a field?
Let's call $D$ the covariant derivative, $T$ the transposition of a field and $*$ its complex conjugate, so $T*$ is the "adjoint".
Is: $$(D_{\mu}\Phi)^{T*} (D_{\mu}\Phi)=D^{\mu}\Phi^*D_{\mu}\...
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 ...
- 10.1k
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
0
votes
0
answers
159
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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',...
- 51
1
vote
3
answers
98
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Does (covariant) divergence-freeness of the stress-energy tensor ${T^{\mu\nu}}_{;\nu}=0$ follow from the Bianchi identity?
I'm working through Chap. $30$ of Dirac's "GTR" where he develops the "comprehensive action principle". He makes a very slick and mathematically elegant argument to show that the ...
- 467
3
votes
1
answer
133
views
A theorem on page 72 in The Large Scale Structure of Space-Time [closed]
In chapter 3 of the book, page 72, a static observer is defined as $V^{a}\equiv f^{-1}K^{a}$, where $K^{a}$ is a timelike Killing vector field and $f^{2}=-K^{a}K_{a}$. Then, Hawking & Ellis claim ...
- 33
0
votes
1
answer
81
views
Covariant Directional Derivative
How is the covariant directional derivative $\frac{D}{d\lambda}=\frac{dx^{\mu}}{d\lambda}\nabla_{\mu}$ in GR related to acceleration? I am motivated to ask this question because I’ve seen it stated ...
- 79
3
votes
1
answer
163
views
Laplace-Beltrami operator for a vector field
For a scalar field $\varphi$, the "wave" operator is defined as follows:
$$\Box \varphi \equiv g^{ab}\nabla_a\nabla_b~\varphi = \frac{1}{\sqrt{|g|}}\partial_a\left\{\sqrt{|g|}~g^{ab}~\...
- 603
0
votes
0
answers
58
views
Partial derivatives of Christoffel symbols to Covariant derivatives
I wanted to express this thing: $g^{ab}\partial_c\Gamma^c_{ab} - g^{ab}\partial_a\Gamma^c_{cb}$, in terms of a covariant derivative. I figured out that if you swap $a$ and $c$ in the $\partial \Gamma$ ...
- 23
1
vote
0
answers
50
views
A covariant derivative computation in General Relativity [duplicate]
I am trying to compute $\nabla^\mu\nabla^\nu R_{\mu\nu}$.
I proceed as follows:
\begin{align}
\nabla^\mu\nabla^\nu R_{\mu\nu}&=g^{\mu\rho}g^{\nu\lambda}\nabla_\rho\nabla_\lambda R_{\mu\nu} \\
&...
- 372
1
vote
1
answer
72
views
Covariant derivative for spin-2 field
I have mostly seen the concept of covariant derivative with regard to spin-1 fields. Is it possible to define the covariant derivative for spin-2 fields as well?
- 387
1
vote
1
answer
77
views
Does the divergence theorem require the covariant derivative to be metric compatible?
I know this is more of a mathematical question, but it arises in the context of general relativity and uses its language so I thought it would be best to ask it here. I understand that the divergence ...
- 4,737
1
vote
1
answer
114
views
How is this deduced? (Differentiation of tensors)
In Schutz's An Introduction to General Relativity, he talked about how to differentiate tensors. Here is a step that I cannot understand.
$$\frac{d\mathbf{T}}{d\tau} = \left( T^{\alpha}_{\beta, \gamma}...
- 63
0
votes
1
answer
83
views
What is the relation between gauge field and Levi-Civita connection?
In field theory, covariant derivative is something like
$$D_{\mu}\phi=(\partial_{\mu}-igA_{\mu})\phi$$
while in differential geometry, covariant derivative is something like
$$D_{\mu}V^{\nu}=\partial_{...
- 101
0
votes
1
answer
75
views
Lie derivative: moving boat on a flowing river
Lie derivatives signifies how much a vector (Tensor) changes if flown in the direction of some other vector. I am thinking the typical moving boat on a flowing river problem where the river is flowing ...
- 85
0
votes
1
answer
346
views
Isomorphism of the tangent space and the space of directional derivatives [closed]
I have already constructed the tangent space to a manifold, denoted $T_pM$, and I have a good basis for it $\{\hat e_{(\mu)}\}$. (I followed the method of equivalence classes of curves tangent at $p$....
- 713
0
votes
0
answers
46
views
Application of Fermi-Walker derivative to specific problem
I am now reading about the tetrad formalism in GR and I am starting (how not) with the Wikipedia Article:
Frame fields in general relativity.
In this article, as an example, they show how tetrads can ...
- 533
2
votes
1
answer
190
views
Covariant derivative to the metric determinant?
I am reading the paper Alternatives to dark matter and dark energy, but cannot obtain one specific equation no matter how I tried. So I wrote an email to the author, the following is what he replies ...
- 51
2
votes
1
answer
74
views
Closed interval in variation of a field
Let's suppose that $\psi$ is a field so that $$\psi\in\Gamma^\infty(\pi):=\{\psi\in C^\infty(M,E)\ |\ \pi\circ\phi=I_M\}$$ where $E$ is a spacetime bundle over spacetime $M$ and $\pi : E\rightarrow M$ ...
user361492
0
votes
3
answers
240
views
What is the correct term for $\nabla\phi$? Co-vector or 1-form or both?
In the olden days, $\nabla\phi$ was used to be called a covariant vector (Weinberg used this language in his book Gravitation & Cosmology). But this terminology is considered bad for several ...
- 12.3k
0
votes
1
answer
115
views
Double covariant derivative of a mixed tensor
Let's say, we have a mixed tensor of type (2,1) denoted by $T^{mn}{}_p$ and the goal is to find the expression of $[\nabla_a, \nabla_b] T^{mn}{}_p$ in terms of fundamental tensors.
Firstly, I am ...
- 151
1
vote
4
answers
445
views
How to find the double covariant derivative of a general vector?
I have been reading through Carroll's GR textbook and there is a line in the derivation of the Riemann tensor that I do not understand.
$$\nabla_\mu \nabla_\nu V^\rho=\partial_\mu(\nabla_\nu V^\rho) - ...
- 71
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 "...
- 177
1
vote
0
answers
94
views
Del operator confusion [closed]
The very first thing my textbook says is that the Del operator is defined as:
$$\vec{\nabla}=\vec{a}^i\nabla_i$$
Where $\nabla_i$ is the covariant derivative and " $\vec{a}^i$ is the curvilinear ...
- 371
0
votes
1
answer
113
views
What is the intuition or the derivation of covariant derivative?
I asked this question in mathematics but the answer I got was a bit too abstract for me so I hope that my fellow physicists can give me more of an intuition or an easier explaination of my question. ...
- 371
4
votes
2
answers
231
views
Looking for the geometric meaning of the curl of Killing vector fields
From Killing equation
$$\nabla_\nu \xi_\mu + \nabla_\mu \xi_\nu = 0$$
it can be shown that $\nabla_\nu \xi_\mu$ is antisymmetric.
From it we can construct an antisymmetric tensor $\mathcal{A}_{\mu\nu}$...
- 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
$$\mathcal{L}_xY = \lim_{\epsilon_\to 0}\frac{...
- 343
2
votes
2
answers
152
views
How to calculate the rotation at a singularity?
An electrodynamics lecture asks me to prove that
$$
\nabla \times \left( \frac{\vec{M} \times \vec{x}}{ |\vec{x}|^3} \right) = \frac{8 \pi}{3} \vec{M} \delta^3(\vec{x})- \frac{\vec{M}}{|\vec{x}|^3}+ \...
- 151
8
votes
2
answers
915
views
How does the covariant derivative satisfy the Leibniz rule?
In Carroll's "Spacetime and Geometry", he states on page 95 (section 3.2) that the covariant derivative, $\nabla$, is a map from $\left(k, l\right)$ tensor fields to $\left(k, l+1\right)$ ...
- 103
1
vote
1
answer
94
views
Is the Lie derivative in a coordinate direction covariant?
Considering a partial derivative of a vector field $w^a$ in x-direction (also called here 1-direction) I can write it as $$\frac{ \partial w^a}{\partial x^1 } = \partial_1 w^a - \Gamma^a_{1c} w^c + \...
- 10.1k
2
votes
0
answers
57
views
How does the divergence change under a change of frame (with geometric algebra)?
I'm trying to prove equations (85) and (86) from Hestenes' paper Gauge Theory Gravity with Geometric Calculus (ResearchGate version).
$$
\dot{\nabla}^\prime \cdot \dot{\underline{f}}(A) = J_f^{-1}[ (\...
- 163
1
vote
1
answer
102
views
Variation of Torsion-Free Spin Connection
In the book 'Supergravity' by Freedman and van Proeyen, in exercise (7.27) it is written
To calculate [the variation $\delta\omega_{\mu ab}$ of the torsion-free spin connection], consider the ...
- 372
3
votes
2
answers
201
views
What is difference between an infinitesimal displacement $dx$ and a basis one-form given by the gradient of a coordinate function?
In general relativity, we introduce the line element as $$ds^2=g_{\mu \nu}dx^{\mu}dx^{\nu}\tag{1}$$ which is used to get the length of a path and $dx$ is an infinitesimal displacement But for a ...
- 644
2
votes
1
answer
109
views
Spin connection for a vector field
Using Birrel & Davies convention, the covariant derivative for a field of arbitrary spin in curved spacetime is given by
$$\nabla_\mu=\partial_\mu+\Omega_\mu,\tag{1}$$
with
$$\Omega_\mu=\frac{1}{2}...
- 2,349
0
votes
2
answers
329
views
Transformation of Lie derivative of one-form
In the textbook Supergravity ( by Freedman and Proeyen, 2012), they have defined the Lie derivative of a covariant vector with respect to a vector field V on page 139:
$$ \mathcal{L}_V \omega_\mu = V^\...
- 542
2
votes
1
answer
194
views
Derivation of Leibniz Rule for Exterior Derivative
I was reading Sean Carrol's GR book, when on page 85 he introduces the Leibniz rule analogue for exterior derivatives:
$$\text d(\omega\wedge\eta) = (\text d\omega)\wedge\eta + (-1)^p\omega\wedge(\...
- 129
1
vote
0
answers
47
views
Scalar curvature in ADM Formalism (coordinate to coordinate-free transition)
I am attempting to express the scalar curvature in a coordinate-independent manner. Following the works of Bojowald, Thiemann, we have:
$$ {}^{(4)}R= {}^{(3)}R+K_{a b}K^{a b}- (K_a^a)^2 - 2\nabla_a v^...
- 403
1
vote
1
answer
289
views
What is the difference between $\partial_{\mu}$ and $\partial^{\mu}$? [closed]
I've seen in many books both expressions $\partial_{\mu}$ and $\partial^{\mu}$, which are the covariant and contravariant partial derivatives, respectively, and in one of Susskind's books he defined ...
- 751
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 ...
- 1,397