All Questions
Tagged with differentiation metric-tensor
125
questions
7
votes
5
answers
634
views
What does the metric condition $\nabla_\rho g_{\mu\nu}=0$ in General Relativity intuitively mean for an observer measuring distances?
In General Relativity, the following condition hold: $\nabla_\rho g_{\mu\nu}=0$, where $g_{\mu\nu}$ is the metric of spacetime which has to do with measuring distances and angles and $\nabla$ is the ...
- 7,720
4
votes
1
answer
948
views
Why is the absolute gradient of the metric tensor $\nabla_{\alpha} g_{\mu \nu} = 0$ in every coordinate system? [duplicate]
Is there any intuitive explanation for why the absolute gradient of the metric tensor $\nabla_{\alpha} g_{\mu \nu} = 0$ in every coordinate system?
- 41
0
votes
1
answer
103
views
Calculate Tensor and its transformations, transformation of derivatives of dependent variables
I had been learning tensor notation for a while and here's what's I have read:
1 Tensor had rank, denote two types covarient or contravariant.
2 $T(\__\alpha,\__\beta,\_\gamma)$ in place naming ...
- 1,048
0
votes
1
answer
210
views
Is the variation of a metric with respect to a metric with a different signature, zero?
I have a problem that involves calculating the variation of a metric $ \bar{g}_{\alpha\beta} $ with +3 signature with respect to a metric $ g_{\alpha\beta} $ with a signature of +1. Both metrics have ...
- 1
0
votes
1
answer
111
views
Computer software with symbolic representation of differential operator
I wrote a matlab code for curl, divergence, gradient, and laplacian as shown below. By changing the value of $h1,h2,h3$, I was able to obtain the Symbolic Representation of result in the different ...
- 1,048
3
votes
1
answer
327
views
How come $\frac{\partial(\partial_{\beta}A_{\gamma})}{\partial(\partial_{\mu}A_{\nu})} = g_{\beta\mu}g_{\gamma\nu}$?
For context, this equation is used in the following (from Schwartz's QFT 3.44)
$$\partial_{\mu} \frac{\partial(\partial_{\alpha}A_{\alpha})^2}{\partial(\partial_{\mu}A_{\nu})} = \partial_{\mu}\left[2(\...
- 1,982
0
votes
1
answer
1k
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Is the vanishing of the covariant derivative of the metric necessary? [duplicate]
Does the covariant derivative of the metric metric always vanish?
I.e. $$\nabla_a g_{bc}=0$$
Are there situations where this can be assumed to not hold? For instance in case of an asymmetric metric?
- 1,488
0
votes
2
answers
9k
views
What is the derivative of an angle? [closed]
What is the derivative of an angle? I don't understand
1
vote
1
answer
1k
views
Lagrange density for massless scalar field [duplicate]
I am reading a book on QFT which is stating the following.
For a massless scalar field $\phi$ the simplest possible Lagrangian is given by
$$
\mathcal{L}(x) = \frac{1}{2} \partial^\mu\phi \partial_\...
- 1,259
0
votes
1
answer
377
views
Scalar Field Theories
The Lagrangian density for a single real scalar field theory is \begin{equation}\mathcal{L}=\frac{1}{2}(\partial_{\mu}\phi)^{2}-V(\phi)\end{equation} I have often seen this written \begin{equation}\...
- 1,209
-1
votes
1
answer
98
views
What is $\frac{\delta (\partial_\kappa \sqrt{g})}{\delta g^{\mu\nu}}$?
Title says it all, is there a closed expression for
$$\frac{\delta (\partial_\kappa \sqrt{g})}{\delta g^{\mu\nu}}$$
where $g = \det g_{\mu\nu}$?
- 1,323
0
votes
1
answer
554
views
Difference between covariant and contravariant differential operator
What is the difference between $\partial_\mu$ and $\partial^\mu$ ? Is there a need for the distinction between these guys in special relativity at all?.
As far as I know $\partial_\mu=\frac{\partial}{\...
- 562
4
votes
2
answers
2k
views
Difference tensor between two connections
I am using these supergravity lecture notes by Gary W. Gibbons. On page 18, the author claims that geodesics and autoparallels coincide for a theory with totally antisymmetric torsion, and proves it ...
- 330
1
vote
1
answer
117
views
The dimensional analysis of the GR geodesic equation
The geodesic equation parametrized by the proper time contains two terms:
$$
{d^{2}x^{\mu } \over ds^{2}}=-\Gamma ^{\mu }{}_{{\alpha \beta }}{dx^{\alpha } \over ds}{dx^{\beta } \over ds}\
$$
The ...
3
votes
1
answer
1k
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Affine connection in general relativity
In The GR lecture my teacher deduced the relation between affine connection and the metric tensor according to the following way:
He firstly wrote the relationship of two tensors like this (I ...
- 1,061