All Questions
Tagged with tensor-calculus differential-geometry
789
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Notation for vector density in Lagrangian density
Consider a manifold $M$ and a Lagrangian density $\mathcal{L} \equiv \mathcal{L}(\phi, \nabla \phi)$. By varying the action, one obtains the equation
$$\int_M \, dV \; \Big( \frac{\partial \mathcal{L}}...
- 743
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Derivation of Einstein-Cartan (EC) action for parametrized connection $A$ & introduction of torsion
I have some trouble with one missing step when I want to get the teleparallel action from general EC theory, which I am not fully understanding.
The starting form of action (3-Dimensional) is:
$$
S_{...
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1
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90
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How to use the definition of a rank-$2$ tensor for this kind of examples?
Suppose that, a rank-$2$ tensor transforms as
\begin{align}
T'^{ij}=\frac{\partial x'^i}{\partial x^k}\frac{\partial x'^k}{\partial x^l}T^{kl}.
\end{align}
How to use this criterion to investigate if ...
- 195
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1
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What is the Lie derivative of Ashtekar connection and its conjugate momentum in LQG?
I am using the reference Black hole entropy from an SU(2)-invariant formulation of Type I isolated horizons for this question. I am trying to understand the two equations (30) that give the variation ...
- 43
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1
answer
76
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How to derive $\partial^{\nu}F^{\mu\alpha} + \partial^{\alpha}F^{\nu\mu} + \partial^{\mu}F^{\alpha\nu}=0$ for the Electromagnetic field tensor? [closed]
The problem says to show that
$$\partial_{[\mu}F_{\alpha\nu]}=F^{\mu\alpha, \nu} + F^{\nu\mu,\alpha} + F^{\alpha\nu,\mu}=0$$ stems from Maxwell equations.
I haven't been able to find this anywhere on ...
- 21
4
votes
1
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292
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Torsion and Compatibility with the Metric
Compatibility with a metric, also referred to as metricity, means, I believe, that the covariant derivative of the metric is zero:
$$g_{ij;k}=g_{ij,k}-\Gamma^m_{ik}g_{mj}-\Gamma^m_{jk}g_{im}=0$$
This ...
- 133
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68
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Selecting Indices for the Riemann Tensor
How do I know when computing the Riemann Tensor (in two dimensional) which indices to select? Consider the Riemann Tensor $R^a_{bcd}$ how do I know what values to take for $a$?
As an example, consider ...
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Timelike normal vector becomes null
I have a metric given by
\begin{equation}
ds^2 = \frac{e^{2 A(z)}}{z^2} \left(-g(z) dt^2 + \frac{dz^2}{g(z)} + dx^2 + dx^2_1 + dx^2_2 \right)
\end{equation}
where $A(z) = -a \ln(b z^2 + 1)$ and $g(z)$ ...
- 404
1
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1
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Books that approach General Relativity via differential forms, without coordinates [duplicate]
Does someone know about some books about differential geometry applied to General Relativity that are written using the language of differential forms, fiber bundles, & spin connection, and not ...
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Bitensors at three or more space-time points
Bitensors, i.e. tensors at two points that have indices belonging to either of them, have been used in the literature quite a bit and there are many calculations involving them. They are the go-to ...
1
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1
answer
69
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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?
- 357
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2
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135
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Why is the 4-current a tensor rather than a tensor density?
I am trying to understand electromagnetism better in terms of tensors and differential geometry. First recall that (in the Lorenz gauge) the equation of motion for the four-potential $A^\mu$ is
$$(-\...
- 151
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1
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80
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Exponential of the metric tensor
Exponential of an arbitrary matrix can be written as
$$e^A = \displaystyle\sum_{n=0}^\infty \dfrac{A^n}{n!}$$
In Einstein notation, how this expression will look like?
In Einstein notation, what ...
- 756
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1
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106
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How is this deduced? (Differentiation of tensors)
In Schutz's An Introduction to General Relativity, he talked about how to differentiate tensors. Here is a step that I cannot understand.
$$\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 ...
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