Questions tagged [tensor-calculus]
Tensor calculus (tensor analysis) is a systematic extension of vector calculus to multivector and tensor fields in a form that is independent of the choice of coordinates on the relevant manifold, but which accounts for respective sub-spaces, their symmetries, and their connections.
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Why is the $\theta$ term of QCD violating charge and parity (CP) symmetries?
From the non-trivial nature of the QCD vacuum, the Lagrangian is augmented with a term like
\begin{equation}
\theta \frac{g^2}{32 \pi^2} G_{\mu \nu}^a \tilde{G}^{a, \mu \nu}
\end{equation}
where $ \...
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Covariant derivative of the vielbein determinant
The vielbein postulate says that
$$\nabla_\mu e_v^{\,a}=\partial_{\mu}e_\nu^{\,a}+\omega_{\mu\,\, b}^{\,\,a}\,e^b_\nu-\Gamma^\sigma_{\mu\nu}\,e^{\,a}_\sigma=0.$$
$\nabla$ is the coordinate covariant ...
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A question from cosmological perturbation theory
We consider the following scalar perturbation on the FRW metric
$$ds^2=-(1+2\Phi)dt^2+2a(\partial_iB)dx^idt+a^2[(1-2\Psi)\delta_{ij}+2\partial_{ij}E]dx^idx^j,$$
where $\Phi$, $B$, $\Psi$ and $E$ are ...
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Uses of the Angular Momentum 4-Tensor
The angular momentum 4-tensor has 6 independent components, three angular momentum components and three new guys. Some call these new guys the 'boosts', but since they are the conjugate momentum of ...
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Question about derivation of tensor in Di Francesco's CFT
This is a question for anyone who is familiar with Di Francesco's book on Conformal Field theory. In particular, on P.108 when he is deriving the general form of the $2$-point Schwinger function in ...
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Variation of the purely covariant Riemann tensor
I need to find the variation of the purely covariant Riemann tensor with respect to the metric $g^{\mu \nu}$, i.e. $\delta R_{\rho \sigma \mu \nu}$.
I know that,
$R_{\rho \sigma \mu \nu} = g_{\rho \...
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Proportionality Constant in Einstein Field Equations
The Einstein Field Equations: $$G_{ab}~=~8\pi T_{ab}.$$ I am familiar with how to obtain the $8\pi$ proportionality factor through correspondence with Newtonian gravity, but am wondering if this ...
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Why is the Lie derivative of a differential 1-form tensorial?
It says in Appendix B of Sean Carroll's "Spacetime and Geometry" that the Lie derivative of a differential 1-form, defined by
$$
\mathcal{L}_{V} \omega _{\mu} = V^{\nu} \partial _{\nu} \...
user319197
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Divergence of tensor fields
I have found numerous definitions for the divergence of a tensor which makes me confused as to trust which one to use.
In Itskov's Tensor Algebra and Tensor Analysis for Engineers, he begins with ...
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Interesting tensor constructed from Ricci tensor
Does the tensor
$$\frac{1}{2}g^{\rho\lambda}(R_{\lambda\mu;\nu}+R_{\lambda\nu;\mu}-R_{\mu\nu;\lambda})$$
have any interesting uses in the context of General Relativity? This is a quantity which has an ...
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Why do the definitions of ADM-energy, -linear momentum and -mass make sense?
In asymptotically flat spacetimes, the ADM-energy, linear momentum and mass are defined as
$$E:= \frac{1}{16\pi}\lim_{r\to\infty} \int_{S^2_r}\sum_{i,j}\partial_ig_{ij}-\partial_jg_{ii}\frac{x^j}{r}\...
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Timelike, spacelike etc. for higher-order tensors
Vectors $V^\mu$ in relativity can be classified into those which are timelike, spacelike and null. A similar classification is available for tensors: A tensor
$$T^{\mu_1\mu_2...\mu_p}_{\phantom{\mu_1\...
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Can the permittivity tensor always be diagonalized?
For an anisotropic medium, permittivity is a symmetric tensor, with elements as $\epsilon_{ij}$
How can one be sure that it is diagonalizable, if some elements, say $\epsilon_{xy}$ and $\epsilon_{yx}$...
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How is Infinitesimal coordinate transformation related to Lie derivatives?
I am reading the book "Gravitaion and Cosmology" by S. Weinberg. In section 10.9, while discussing Lie derivatives of tensors of different ranks, he makes a general comment:
The effect of an ...
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Covariant versus "ordinary" divergence theorem
Let $M$ be an oriented $m$-dimensional manifold with boundary. As stated in Harvey Reall's general relativity notes (here) or Sean Carroll's book, the "covariant" divergence theorem (i.e. with ...
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