All Questions
397 questions
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Intuitive Definition of Curl and Stokes' Theorem
I am asking this question on SE Physics because it's about an intuitive derivation, not a rigorously made proof of a theorem.
Reading Mathematical Methods for Physics and Engineering by Riley, et al. (...
- 485
3
votes
2
answers
184
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Does covariant derivative include magnitude change of a vector as well as direction change of the same vector?
Does covariant derivative include magnitude change of a vector as well as direction change of the vector? In some explanations I followed I have not noticed mentioning of magnitude change along with ...
- 1,835
0
votes
1
answer
121
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Hodge Laplacian and scalar
I'm reading Nakahara GEOMETRY, TOPOLOGY AND PHYSICS now.Then, Hodge Laplacian is given by
\begin{align}
\Delta=(d+d^{\dagger})^2=dd^{\dagger}+d^{\dagger}d
\end{align}
For example, we consider 0-form $...
- 135
2
votes
1
answer
262
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Defining the exterior derivative with torsion [duplicate]
As introduced in most standard GR textbooks, the exterior derivative of a p-form $X$ is defined in a coordinate basis as $(dX)_{\mu_1....\mu_{{p+1}}}:= (p+1) \partial_{[\mu_1}X_{\mu_2....\mu_{{p+1}]}}$...
- 786
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1
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421
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Tensor contraction and covariant derivative
I have naive question about GR and Covariant derivative.
You know
\begin{align}
\nabla_{\gamma} g_{\alpha \beta}=\nabla_{\gamma} g^{\alpha \beta}=0
\end{align}
And I would like to compute covariant ...
- 135
4
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1
answer
305
views
How to derive $\nabla_{\mu} T^{\mu\nu}=\frac{1}{\sqrt {-g}}\partial_{\mu}(\sqrt{-g}\,T^{\mu\nu})$?
This $$\nabla_{\mu} T^{\mu\nu}=\frac{1}{\sqrt {-g}}\partial_{\mu}(\sqrt{-g}\,T^{\mu\nu})\tag{3.39}$$ is from a textbook on general relativity on black hole Vaidya metric, where only non-zero term of ...
- 175
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2
answers
1k
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Covariant derivative of the spin connection
I wish to compute $$[\nabla_{\mu}, \nabla_{\nu}]e^{\lambda}_{~~a}. $$
To do so, I make use of $\nabla_{\nu}e^{\lambda}_{~~a} = \omega_{a~~~\nu}^{~~b}e^{\lambda}_{~~b}$, so that I may write
$$\nabla_{\...
- 1,415
1
vote
1
answer
484
views
Second derivative of a function in a manifold [closed]
Suppose we have a function curve $\gamma(t)$ on a manifold $M$ . Define the function $$f(t)=\frac{1}{2}g_{ab}\dot\gamma(t)^a\dot\gamma(t)^b$$
Introducing coordinates $x^i$ the first derivative of the ...
- 485
1
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2
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892
views
Dirac equation and Dirac operator
A Dirac operator is a differential operator acting on a vector bundle $V$ over a Riemannian manifold $M$:
$$
\tag{1}
D^{2}=\Delta
$$
Where $\Delta$ in the Laplacian (in the Euclidean space).
An ...
- 394
0
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1
answer
64
views
The contravariant derivative of a substitution for the de Sitter metric
Consider the de Sitter metric:
$$ds^2 = (1-\frac{r^2}{a^2})dt^2 -(1-\frac{r^2}{a^2})^{-1}dr^2-r^2d\Omega^2$$
I know that we rewrite the metric as $(u,r,\theta \phi)$ using the substitution $$u = t-...
- 988
1
vote
3
answers
698
views
Covariant gradient - What am I missing?
I know that the components of the gradient should transform covariantly, and have used this many times in special relativity, etc. However, I also know that covariant components transform like the ...
- 1,245
0
votes
1
answer
399
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Killing equation in coordinates
In proving that it is possible to write the killing equation in coordinates as $$L_X g=0\iff X_{\alpha;\beta}+X_{\beta;\alpha}=0$$
I have read that the key observation, to write the equation in ...
- 187
4
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3
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1k
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Why is the covariant derivative of the metric tensor with UPPER indices equal to zero? [closed]
I've shown that $\nabla_{\lambda} g_{\mu\nu} = 0 $ rigorously by the following method:
$ \nabla_{\lambda} g_{\mu\nu} = \partial_{\lambda}g_{\mu\nu} - \Gamma^{\rho}_{\lambda\mu} g_{\rho\nu} - \Gamma^{\...
2
votes
2
answers
339
views
Different definitions of exterior derivative
In Sean Carroll's GR book, pg 84, the exterior derivative $d$ is defined as
$$(dA)_{\mu_1\mu_2...\mu_{p+1}} = (p+1) \partial_{[\mu_1}A_{\mu_2...\mu_{p+1}]}\tag{2.76}$$
where $A$ is a $p$-form and the ...
- 4,276
2
votes
1
answer
576
views
Using Christoffel symbols to derive formulas for div, grad, curl
In Sean Carroll's GR book, pg. 1oo, it was said that in flat space, the Christoffel symbols vanish in Cartesian coordinates. However, in curvilinear coordinates, they do not vanish. For example, for ...
- 4,276
1
vote
1
answer
1k
views
Chain rule for covariant derivative?
Does a chain rule for the covariant derivative exist so that we can evaluate an expression like
$$\nabla_c\sqrt{t_{ab}}?$$
where $t_{ab}$ are tensor components?
More generally, how do we take the ...
- 4,276
2
votes
3
answers
266
views
Is the equation $[\nabla_{\mu},\nabla_{\nu}]=F_{\mu\nu}$ correct? If yes, how does it have to be interpreted?
It seems like simply using the equation
\begin{equation}
\nabla_{\mu}=\partial_{\mu}+A_{\mu}
\end{equation}
isn't enough: One obtains
\begin{equation}
[\nabla_{\mu},\nabla_{\nu}]=\underbrace{[\...
- 1,911
2
votes
1
answer
3k
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Commutator of covariant derivative for rank 2 tensor
I am a newbie at tensor notation and I have been told to prove the identity
$$ (\nabla_a\nabla_b - \nabla_b\nabla_a)X^a_{\ \ \ b}=- R^e_{\ \ \ bcd}X^a_{\ \ \ e}+ R^a_{\ \ \ ecd}X^e_{\ \ \ b} $$
I am ...
- 181
2
votes
1
answer
154
views
How obtain the last expression of the Killing equation?
In order to write down the Killing equation, if by definition a vector field $X$ is said to be Killing $\iff$ $L_X g=0$, then I can rewrite this condition as:
$$L_X g=X g(U, V)-g(L_XU, V)-g(U,L_XV)=g(\...
- 133
1
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1
answer
526
views
Vanishing covariant derivative of a vector field
I'm asked to prove the following statement in my physics book:
A vector field with covariant components $v^b$, in order to have a vanishing covariant derivative everywhere in a manifold, must satisfy:...
- 577
3
votes
1
answer
340
views
Definition of exterior derivative
In Sean Carroll's GR book, pg 84, the exterior derivative $d$ is defined as
$$(dA)_{\mu_1\mu_2...\mu_{p+1}} = (p+1) \partial_{[\mu_1}A_{\mu_2...\mu_{p+1}]}\tag{2.76}$$
where $A$ is a $p$-form and the ...
- 4,276
1
vote
1
answer
404
views
Difference between covariant derivatives in general relativity and electromagnetism
There are many similarities between the gauging of a $U(1)$ symmetry to obtain the physics of electromagnetism and the gauging of diffeomorphisms to obtain the physics of general relativity. In ...
1
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1
answer
145
views
If $g_{ij}$ is a tensor of type $(0,2)$, what is kind of tensor is $\partial_{i}g_{jk}$?
Suppose $g_{ij}$ is a tensor of type $(0,2)$, then what type of object is $\partial_{i}g_{jk}$? Is it even a tensor, and if so, of what type? Is the $\partial_{i}$ still a differential with respect to ...
- 123
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votes
2
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284
views
Derivative Operators on a manifold
I am having some trouble coming to terms with the notion of a derivative operator on a manifold. In Robert M. Wald's General Relativity, the definition in the textbook is given in terms of 5 ...
- 2,740
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2
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268
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Gradient, one-form and Sean Carroll
"A tensor (k,l) is a multilinear map from k dual vectors and l vectors to R (...) The gradient, ..., is an honest (0,1) tensor."
These citations are retired from Sean Carrol Spacetime and ...
- 990
1
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1
answer
270
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Metric independent affine connections
Normally, the affine connections are objects that define parallel transport. In general Relativity they are the Christoffel symbols of the second kind. Consequently, they depend on the metric tensor. ...
1
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2
answers
1k
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Determining the partial derivative of a metric tensor
Im new to the Tensor Calculus and General Theory of Relativity, and I have one question. I want to determine the Christoffel symbols in FRW metric.
This is the general equation of Christoffel symbols:
...
- 25
7
votes
2
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765
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What is meant when we say that a differential takes on a certain value?
As far as i understand it, total differentials are linear maps that map vectors to numbers. In thermodynamics we encounter statements that a we have reached equilibrium when a total differential of a ...
- 1,616
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0
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91
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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^\...
- 2,740
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0
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504
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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}...
- 35
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3
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4k
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Why is the covariant derivative of a one-form $\nabla_{i}v_j=\frac{\partial v_j}{\partial u^{i}}-\Gamma^k_{~ij}v_k$?
I understood that the covariant derivative of a vector field is
$$
\nabla_{i}v^j=\frac{\partial v^j}{\partial u^{i}}+\Gamma^j_{~ik}v^k
$$
Then why is the covariant derivative of a covector field
$$
\...
- 157
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votes
0
answers
83
views
Doubt of gauge covariant derivatives: how can I derive it?
In the context of general relativity (GR) it is necessary to introduce the notion of covariant derivatives. From the point of view of a basic introduction, we always start to deal with GR in a highly ...
- 3,159
2
votes
1
answer
262
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Exterior Derivative on Curved Manifold (SpaceTime)
Considering the 2-Form Gauge Potential $B_{\mu \nu}$, we can write $dB=H$.
In a flat manifold we have that $H_{\mu \nu \rho}=\partial_{\mu}B_{\nu \rho}+\partial_{\rho}B_{\mu \nu}+\partial_{\nu}B_{\rho ...
- 782
6
votes
1
answer
934
views
What is the analogy between the gauge covariant derivative and the covariant derivative in General Relativity?
A particle in the Dirac field can be described with the following equation
$$i\gamma^\mu\partial_\mu\psi-m\psi=0$$
This is if the particle is non-interacting. However, if we impose a local $U(1)$ ...
- 805
1
vote
1
answer
115
views
Covariant derivative of contracted tensor: why is it not obvious
In Wald's GR book (1984), he writes on page 221,
In the timelike case, we restricted consideration to deviation vectors $\eta^a$ orthogonal to $\xi^a$ [$\xi^a$ is the normalized vector field of ...
- 131
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votes
3
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1k
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Christoffel symbol and covariant derivative
I came across the Christoffel symbols via the geodesic equation, and I understand the extrinsic form and the intrinsic form and can prove that they are identical:
extrinsic form:
$$\Gamma^{j}_{~ik}=\...
- 157
2
votes
3
answers
456
views
An identity between the d'Alembertian and the covariant derivative
Suppose $f$ is function which depends on $\phi$, $f = f(\phi)$; and $\phi$ is a scalar field. We define
$$\square \equiv g^{\mu\nu} \nabla_\mu \nabla_\nu$$ and $$\nabla_\mu f(\phi) \equiv f_{;\nu}$$
...
- 25
3
votes
1
answer
1k
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Covariant derivatives in a rank 2 tensor
I was trying to prove that for any second order tensor:
$$A^{\mu\nu}_{;\mu\nu}=A^{\mu\nu}_{;\nu\mu}$$
considering the torsion free property and locally flat coordinates. Considering the point where ...
- 1,154
1
vote
3
answers
219
views
Why isn't $g^{\mu \nu}\partial_{\nu}$ equal to $\partial^{\mu}$ in General Relativity?
The question is pretty short. I have been told that unlike for flat spacetime, in curved spacetime we cannot write
$g^{\mu \nu}\partial_{\nu}f=\partial^{\mu}f$
With $f$ an scalar function, but I don't ...
- 108
4
votes
1
answer
1k
views
Explicit expression of gradient, laplacian, divergence and curl using covariant derivatives
For my course in General Relativity I am given the problem to find the expressions for the gradient, laplacian, divergence and curl in spherical coordinates using covariant derivatives.
I have tried ...
- 53
2
votes
2
answers
548
views
Definition of parallel transport
The definition of parallel transport is $t^iD_i u^j=0$, where $\vec{t}$ is the tangent vector to the curve and $\vec{u}$ is the vector being parallel transported along the curve. In flat space, using ...
- 503
1
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1
answer
79
views
Alternative formula for the affine connection in a new coordinate basis
In Hobsons's General Relativity: An Introduction for Physicists, pg. 64,
he gave two different expressions for the affine connection $\Gamma'^a_{bc}$ in a transformed coordinate basis $x'^a$ (the ...
- 4,276
0
votes
2
answers
55
views
Isn't the following addition wrong on manifold as done in Frankel book?
In ch-$4$ when calculating expression of Lie derivative using Hadamard's Lemma before $(4.4)$ Frankel's do following manipulation:
$$\lim_{t\rightarrow0}\frac{\textbf{Y}_{\phi_tx}(f)-\textbf{Y}_x(f)}{...
- 3,073
4
votes
1
answer
167
views
What motivates defining vectors as first order differential operators?
I have read some introductions to geometrical ideas and tensors and physics and what some of them do (see, for example, Frankel's Geometry of Physics) is define a vector as a first order differential ...
- 73
0
votes
0
answers
59
views
Covariant derivative in $f(Q,T)$ field equations
I am trying to evaluate the field equations in $f(Q,T)$-gravity and I'm not sure how to evaluate this term:
$$-\dfrac{-2}{\sqrt{-g}}\nabla_{\alpha}(f_{Q} \sqrt{-g}\enspace P^{\alpha}_{\enspace \mu \nu}...
- 164
1
vote
1
answer
654
views
How is d'Alembertian operator is defined in differential geometry?
Which general formula for the box operator is correct, $\Box=g^{ij}\partial_i\partial_j$ or $\Box=\frac{1}{\sqrt{g}}\partial_i(\sqrt{g}g^{ij}\partial_j)$? I have seen both the definition being used ...
- 175
1
vote
2
answers
654
views
Invariance of differential operators
How do we prove that del operator is invariant under any kind of change of coordinates, specifically under galilean transformations? I am getting an extra term containing the relative velocity of two ...
1
vote
1
answer
303
views
How to ("geometrically") differentiate unit vectors of spherical coordinates?
I have been trying to derive the expressions of partial derivatives of unit vectors with respect to each other in the spherical coordinate system. I was able to get all of them except $\frac{\partial \...
- 39
2
votes
1
answer
338
views
Rewriting the Laplacian on a curved manifold
I guess there is a sense in which the following is true:
"The Laplacian written on a Riemannian manifold $(M,g)$ can be seen as adding a coordinate dependent mass field to the Laplacian on ...
- 131
3
votes
1
answer
66
views
Covariant and controvariant bases derivative
How to show that
$\overrightarrow{\textbf{e}}_\sigma\cdot\partial_\mu \overrightarrow{\textbf{e}}_\nu = \overrightarrow{\textbf{e}}_\sigma\cdot\partial^\mu \overrightarrow{\textbf{e}^\nu}$
where $\...
- 133