All Questions
Tagged with differentiation tensor-calculus
190
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Confusion about contraction and covariant derivatives [closed]
Understanding Contraction and Second Covariant Derivatives in Tensors
I am confused about contraction in tensors and the second covariant derivative in tensors. Consider a tensor $T_{\mu\nu}$ and the ...
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90
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Component notation and matrix notation for gradient of vector
I'm trying to understand vector and tensor notation, but I'm coming across some difficulties. Say I have vector $\vec{u}$ and I compute its gradient $\nabla \vec{u}$. Then I get a tensor $\frac{\...
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1
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68
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Covariant derivative for spin-2 field
I have mostly seen the concept of covariant derivative with regard to spin-1 fields. Is it possible to define the covariant derivative for spin-2 fields as well?
- 337
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105
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How is this deduced? (Differentiation of tensors)
In Schutz's An Introduction to General Relativity, he talked about how to differentiate tensors. Here is a step that I cannot understand.
$$\frac{d\mathbf{T}}{d\tau} = \left( T^{\alpha}_{\beta, \gamma}...
- 63
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68
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Lie derivative: moving boat on a flowing river
Lie derivatives signifies how much a vector (Tensor) changes if flown in the direction of some other vector. I am thinking the typical moving boat on a flowing river problem where the river is flowing ...
- 65
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53
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Tensor equation
What is a valid tensor equation. In the book by Bernard Schutz, it is often argued that a valid tensor equation will be frame invariant. So the conclusions reached by relatively easy calculation done ...
- 137
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119
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Covariant derivative to the metric determinant?
I am reading the paper Alternatives to dark matter and dark energy, but cannot obtain one specific equation no matter how I tried. So I wrote an email to the author, the following is what he replies ...
- 41
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71
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Double covariant derivative of a mixed tensor
Let's say, we have a mixed tensor of type (2,1) denoted by $T^{mn}{}_p$ and the goal is to find the expression of $[\nabla_a, \nabla_b] T^{mn}{}_p$ in terms of fundamental tensors.
Firstly, I am ...
- 151
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85
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Del operator confusion [closed]
The very first thing my textbook says is that the Del operator is defined as:
$$\vec{\nabla}=\vec{a}^i\nabla_i$$
Where $\nabla_i$ is the covariant derivative and " $\vec{a}^i$ is the curvilinear ...
- 267
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99
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What is the intuition or the derivation of covariant derivative?
I asked this question in mathematics but the answer I got was a bit too abstract for me so I hope that my fellow physicists can give me more of an intuition or an easier explaination of my question. ...
- 267
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91
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Is the Lie derivative in a coordinate direction covariant?
Considering a partial derivative of a vector field $w^a$ in x-direction (also called here 1-direction) I can write it as $$\frac{ \partial w^a}{\partial x^1 } = \partial_1 w^a - \Gamma^a_{1c} w^c + \...
- 9,778
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308
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Covariant derivative of the Ricci tensor using pure algebra
I want to differentiate the Ricci tensor covariantly, namely without using Bianchi identities and with pure algebra, I want to prove:
$$
D _{\mu} R^{\mu\nu} = {{1}\over{2}} g^{\mu\nu} \partial_{\mu}R
$...
- 101
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2
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242
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Transformation of Lie derivative of one-form
In the textbook Supergravity ( by Freedman and Proeyen, 2012), they have defined the Lie derivative of a covariant vector with respect to a vector field V on page 139:
$$ \mathcal{L}_V \omega_\mu = V^\...
- 513
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43
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Scalar curvature in ADM Formalism (coordinate to coordinate-free transition)
I am attempting to express the scalar curvature in a coordinate-independent manner. Following the works of Bojowald, Thiemann, we have:
$$ {}^{(4)}R= {}^{(3)}R+K_{a b}K^{a b}- (K_a^a)^2 - 2\nabla_a v^...
- 403
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1
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86
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Bianchi identity in EMT [closed]
$ ∇_a∇_b F_{ab} = 0 $ ($F_{ab}$ Faraday tensor in EMT.)
proof is given by
"To see this, assume a Minkowski spacetime for simplicity and adopt
Cartesian coordinates, so that the covariant ...
- 81