All Questions
21 questions
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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
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1
answer
289
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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
1
answer
153
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Differentiating the index notation
I am always confused with the algebra of differentiating the index notation, and have browsed many other posts but still confused. There must be details I have been missing. It would be really ...
- 41
0
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1
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61
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What is $A'$ in the Reissner-Nordstrom metric?
So I was reading this paper on the Reissner-Nordstrom metric and on it they define $A$ as:
But they don't define $A'$. Yet $A'$ still ends up in other equations like defining the Ricci tensors:
So ...
- 43
6
votes
1
answer
161
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What does $\delta/\delta t$-derivative represent in tensor calculus?
Some texts, such as Pavel Grinfeld's, talk about a $\delta/\delta t$-derivative whose role (in trajectory analysis of particles using tensor calculus) is pretty obscure to me. For example, the ...
user89243
2
votes
1
answer
89
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Does the expression "$𝑑𝑠^2$..." mean the same thing as "$\Delta 𝑠^2$... "?
I reviewed this question but sometimes I'm unsure about delta ($\Delta$) versus differential ($d$) notation.
Does the expression "$ds^2=-c^2dt^2+a^2(t)[dr^2 + S_k^2(r)d\Omega^2 ]$" mean the ...
- 181
1
vote
1
answer
286
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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 $g$ ...
- 1,122
1
vote
1
answer
394
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How to compute divergence of a metric tensor?
I am reading a paper where the author defines the divergence to be
$$\left(\delta_{g} \dot{g}\right)_{\mu}:=-\dot{g}_{\mu \kappa;}{}^{\kappa}$$
where $g$ looks like the De Sitter metric,
$$g=(3 / \...
- 1,122
0
votes
1
answer
209
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Tensor notation of covariant derivative
I'm trying to apply Wald's General Relativity equation $3.1.14$:
$$\nabla_a{T^{b_1\dots b_k}}_{c_1\dots c_{\ell}}=\overline{\nabla}_a{T^{b_1\dots b_k}}_{c_1\dots c_{\ell}}+\sum_i{C^{b_i}}_{ad}{T^{b_1\...
1
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4
answers
2k
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Deriving the Covariant Derivative of the Metric Tensor
First off, I did look through some other questions:
Covariant Derivative of Metric Tensor
Why is the covariant derivative of the metric tensor zero?
https://math.stackexchange.com/q/2174588/
But they ...
- 81
1
vote
1
answer
314
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What are Connection Forms in General Relativity?
I'm trying to follow an article by H. Ellis (1973), where he developed the first ever metric of a traversable Wormhole (more info here).
In pages 105-106 (the end of the 3rd page in the linked file ...
- 167
0
votes
1
answer
197
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4-velocity lowering index question
The 4-velocity in contravariant form is given by
$$V^\mu=\frac{dx^\mu}{d\tau}$$
for some general co-ordinates $x^\mu$ and proper time $\tau$.
Is the 4-velocity in covariant form given by
$$V_\nu=V^\...
- 5,991
0
votes
2
answers
152
views
Covector basis derivation
On page 65 of Schutz's A first course in General Relativity, he introduces the notation $\phi_{,\alpha}=\partial\phi/\partial x^\alpha$. He then says that $x^\alpha_{\ \ ,\beta}=\delta^\alpha _{\ \ \ \...
- 183
2
votes
1
answer
2k
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How is semicolon derivative notation defined for multiple derivatives?
I have a covector $\eta_\mu$. Then I have some notation which says $$\eta_{\alpha;\beta\gamma}$$ What does this mean? I understand that given a vector $A^\alpha$, that $$A^\alpha_{;\beta}=\nabla_\beta ...
- 491
2
votes
2
answers
241
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Misconception about index notation
I'm going to give an example in General Relativity but this is a question about index notation and coordinate transformations in general. In "Spacetime and Geometry" by Sean Caroll, there is this ...
- 4,737
0
votes
1
answer
905
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On covariant derivative
Let us denote a 1 form on manifold M with $\eta$ which in a chart looks like $\eta=\eta_{\mu}dx^{\mu}$ where $\eta_{\mu}$ are smooth functions on M. Now given the coordinate vector fields $\frac{\...
- 25
8
votes
3
answers
3k
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Are indices conventionally raised inside or outside of partial derivatives in general relativity?
If $A_\mu$ is a one-form, then is there a widely accepted convention among physicists about whether the notation $$\partial_\mu A^\mu \tag{1}$$ means "the partial-derivative four-divergence of the ...
- 49.4k
-1
votes
1
answer
124
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$x'^i_j x^j_k = n\delta^i_k$ rather than $1\delta^i_j$?
These are my calculations
$$x'^i_j x^j_k = \sum_{j=1}^n \frac{\partial x'^i}{\partial x^j}\frac{\partial x^j}{\partial x'^k} = \sum_{j=1}^n \frac{\partial x'^i}{\partial x'^k} =n \delta^i_k\ne \delta^...
- 107
1
vote
2
answers
185
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Tensor index question
I am looking at the solution in the book "Problem book in Relativity and Gravitation" for problem 10.6. I don't think I need to go into the details of the problem (I will do so if need be) because I ...
- 1,001
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1
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186
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What is the difference between $\frac{DA^\mu}{D\lambda}$ and $\frac{DA^\mu}{d\lambda}$?
I earlier asked this question How can you have $\frac{DA^\mu}{d\tau}$? I am now wondering:
What is the difference between $\frac{DA^\mu}{D\lambda}$ and $\frac{DA^\mu}{d\lambda}$?
In the linked ...
0
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1
answer
119
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How can you have $\frac{DA^\mu}{d\tau}$?
If a covariant derivative is given by:
$$D_\nu A^\mu=\partial_\nu A^\mu +\Gamma^\mu_{\nu \lambda} A^{\lambda}$$
Then how does $\frac{DA^\mu}{d\tau}$ make any sense? Since there are no 'differentials' ...