All Questions
16
questions
1
vote
1
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54
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Tensor equation
What is a valid tensor equation. In the book by Bernard Schutz, it is often argued that a valid tensor equation will be frame invariant. So the conclusions reached by relatively easy calculation done ...
- 137
3
votes
2
answers
714
views
Partial derivatives vs Covariant derivatives in polar coordinates
Covariant derivatives take into account for both component and basis changes, thereby applicable for curved spaces - where partial derivatives only take component changes into account - is this ...
- 1,141
3
votes
2
answers
345
views
Difference and meaning of index the derivative operator
I'm a beginner in this type of math, we are just starting to study it, but I need some clarifications about the meaning and the difference of when we write
$$\partial_i \qquad \text{and}\qquad \...
- 361
1
vote
0
answers
244
views
On the definition of the Van Vleck-Morette determinant
Let $M$ be a Riemannian manifold and $\sigma$ the world function. The Van-Vleck-Morette determinant $D$ is defined by
$$D(x,x')=\det(-\sigma_{;\mu\nu{}'})$$
Regarding the semi-colon: In chapter $4.1$ ...
- 1,801
2
votes
2
answers
281
views
Is the contracted Christoffel symbol a tensor?
The coordinate transformation law (from coordinates x to coordinates y) for the Christoffel symbol is:
$$\Gamma^i_{kl}(y)=\frac{\partial y^i}{\partial x^m} \frac{\partial x^n}{\partial y^k} \frac{\...
- 633
2
votes
2
answers
219
views
Confusion about Transforming Christoffel Symbols
I'm trying to understand how transforming Christoffel symbols works. Specifically I'm thinking about the transformation between Schwarzschild and Eddington-Finkelstein coordinates,
$$\Gamma^v_{\;vv}=\...
- 323
0
votes
1
answer
113
views
Partial differentiation of tensors
We know that $∂x^ρ /∂x^μ = δ^ρ_μ$
Τhen, $∂x_ρ /∂x^μ = η_{ρμ}$
Should be correct, right?
Similarly,
$\frac{∂x_ρ} {∂x_μ} = δ^μ_ρ$
Also, if
$x'^μ = e^α x^μ $, then
$∂'_μ$ should be $e^α ∂_μ$
I am new to ...
- 348
1
vote
2
answers
333
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Are there differences in notation for the d'Alembert operator?
On Wikipedia the d'Alembert operator is defined as
$$\square = \partial ^\alpha \partial_\alpha = \frac{1}{c^2} \frac{\partial^2}{\partial t^2}-\nabla^2 $$
However, my professor uses the notation:
$$ \...
0
votes
3
answers
990
views
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:
$$\Gamma^{j}_{~ik}=\...
- 157
4
votes
1
answer
868
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Explicit expression of gradient, laplacian, divergence and curl using covariant derivatives
For my course in General Relativity I am given the problem to find the expressions for the gradient, laplacian, divergence and curl in spherical coordinates using covariant derivatives.
I have tried ...
- 53
1
vote
2
answers
519
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Which derivative to use in the change of metric tensor due to a gauge transformation?
I'm used to calculating the change in the metric due to a gauge transformation in the following way:
The gauge transformation up to linear order is
\begin{equation}
x^\mu \rightarrow x' ^\mu =x^\mu ...
- 4,827
1
vote
2
answers
559
views
Calculate the expression of divergence in spherical coordinates $r, \theta, \varphi$
Hi this is my first question in [Physics.SE] I saw a lot of posts and I liked them. I hope that my question will be answered too.
While I'm solving a problem in vector calculus. I recognized that I ...
user262095
106
votes
4
answers
10k
views
Why does nature favour the Laplacian?
The three-dimensional Laplacian can be defined as $$\nabla^2=\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}+\frac{\partial^2}{\partial z^2}.$$ Expressed in spherical coordinates, it ...
- 1,327
4
votes
1
answer
946
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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
2
answers
9k
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What is the derivative of an angle? [closed]
What is the derivative of an angle? I don't understand
