All Questions
17 questions
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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
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201
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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 ...
- 634
2
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1
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258
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Infinitesimal coordinate transformation and Lie derivative
I need to prove that under an infinitesimal coordinate transformation $x^{'\mu}=x^\mu-\xi^\mu(x)$, the variation of a vector $U^\mu(x)$ is $$\delta U^\mu(x)=U^{'\mu}(x)-U^\mu(x)=\mathcal{L}_\xi U^\mu$$...
- 372
1
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0
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64
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How to transform a partial derivative to a directional derivative with respect to some affine parameter?
Suppose an affine parameter $\lambda$ is defined along a null geodesic with $dx^\mu/d\lambda=k^\mu$. How could I write the partial derivative $\partial f/\partial x^\mu$ by using $df/d\lambda$? If $k^\...
- 105
9
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1
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598
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Inverse of the covariant derivative
Given the covariant derivative of some tensor, for the sake of this example a covariant vector:
$$\nabla_\mu A_\nu$$
Is there a well-defined inverse operation on the covariant derivative such that it ...
- 613
4
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2
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510
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Converting differential to gradient
Landau & Lifschitz's fluid mechanics book proposes the following statement for an isentropic proccess:
$$dH=vdp \Rightarrow \nabla H=v\nabla p$$
What's the rigorous way to get this result (...
- 43
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1
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87
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How do I reconcile these two definitions of acceleration?
How do I reconcile these two definitions of acceleration?
$$a=\frac{d\bar{v}}{dt}=(\frac{dv^k}{dt}+v^i v^j \Gamma^k_{ij})\bar{e}_k \tag{1}$$
and
$$a^k=v^{\small\beta} \nabla_{\small\beta} v^k.\tag{2}$$...
- 971
0
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1
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81
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Volume of a two-dimensional sphere in a fixed three sphere geometry
I'm just starting to read Hartle's Gravity and he gives the following equation for the volume of a 2D sphere of radius $r$ if space was a fixed 3-sphere geometry on page 20.
$$V = 4\pi a^3\left(\frac{...
- 5
0
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2
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186
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What's the physical meaning of $\vec A \times \vec \nabla$? [closed]
What's the physical meaning of $\vec A \times \vec \nabla$?
Gradient represents a field. What if we try we write "opposite" of that? I know that if direction of $\vec \nabla \times \vec A$ ...
- 91
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2
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204
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Physical meaning of the exterior derivative of the first law of thermodynamics
We know, $$ dU = d \overline{q} - d \overline{W}.$$ suppose we took the exterior derivative on both sides, then:
$$ 0= d( d \overline{q}) - d( d \overline{W})$$
This means, $$ d^2 \overline{q} = d^2 \...
- 8,040
1
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1
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79
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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
-1
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2
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203
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Divergence of vector fields intuitive understanding
I am troubled by the idea of positive and negative divergence of a vector field. I understand that the idea of e.g. positive div is that a gas expands everywhere (for the velocity field of a gas ...
- 111
1
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1
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50
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Path Coordinates: direction problem (doubt) in derivative of tangential vector
Why is the direction of derivative of tangential vector perpendicular to the direction of the tangential vector?
- 29
4
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2
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5k
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How is dot or cross product possible using the del operator?
Yesterday in class my teacher told me that the del operator has a direction but no value of its own (as its an operator). So it can't be called exactly a vector. But in vector calculus we see that div ...
- 1,430
5
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1
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1k
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Is the inverse of the deformation gradient simply the deformation gradient of the inverse transformation?
If we have a continuum where the initial positions are denoted $X$ and the positions after some deformation are denoted $x$, the deformation gradient is defined:
$$ F = \frac{\partial x}{\partial X} $...
- 103
3
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2
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969
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Differential operators in curvilinear coordinates
In the appendix A of Griffith's Electrodynamics text, he cites Spivak's Calculus on Manifolds as a reference more a more complete treatment of taking the gradient, curl, divergence, and Laplacian in ...
15
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3
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44k
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Derive vector gradient in spherical coordinates from first principles
Trying to understand where the $\frac{1}{r sin(\theta)}$ and $1/r$ bits come in the definition of gradient.
I've derived the spherical unit vectors but now I don't understand how to transform ...
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