# 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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### Tensor perturbation inflation

During inflation the metric is de-Sitter so $dt^2-d\underline{X}^2$. I know that the eqn.motion governing GW's from inflation (tensor perturbations) is $$2H\dot{h}+\ddot{h}-\nabla^{2}_{i}h~=~0,$$ ...
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### Differential Forms and Densities

I've heard that differential forms are related to densities, however I'm still a little confused about that. I thought on the case of charge density and I came to that: let $U\subset\mathbb{R}^3$ be a ...
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I just starting to study GR and I could not prove the following: if I have to tensors $T_{\mu\nu}$ and $Q_{\mu\nu}$ such that $T_{\mu\nu}=Q_{\mu\nu}$, why can I multiply both sides of the equation by $... 1answer 264 views ### Is it correct to sum over either index of the metric the same way? I don't know if the following is correct, i want to compute the following derivative $$\frac{\partial }{\partial (\partial_{\mu}A_{\nu})}\left(\partial^{\alpha}A^{\beta}\partial_{\alpha}A_{\beta} \... 0answers 159 views ### Einstein +Maxwell 's tensor Why is it true that we can deduce that Einstein's GR equations coupled with Maxwell's EM equations may be written in the form$$R_{ij}=C(F_{ik}F_j^{\,\,k}-{1\over 4}g_{ij}F_{mn}F^{mn})$$without ... 3answers 3k views ### Maxwell Stress Tensor in the absence of a magnetic field I'm having some trouble calculating the stress tensor in the case of a static electric field without a magnetic field. Following the derivation on Wikipedia, Start with Lorentz force:$$\mathbf{F} = ... 1answer 343 views ### Relationship between a formal vector derivative and time evolution of an operator I'm an undergraduate in physics, with all the lack of knowledge inherent in that. In two of my classes, my professors introduced two equations which look eerily similar. The first, from general ... 1answer 94 views ### Two General Relativity questions Hi When contracting$T^{\mu \nu}$with$ g_{\mu \nu}$does one get$T^{\mu \nu}_{\mu \nu} = T$? is the metric tensor already a sum over its component, so it is effectively a trace of a matrix with ... 4answers 701 views ### Is “entanglement” unique to quantum systems? My text shows (sections 0.2 and 0.3) that the joint "state space" of a system composed of two subsystems with$k$and$l$"bits of information", respectively, requires$kl$bits to fully describe it. ... 0answers 116 views ### Equivalence of simple formulations of qubit entanglement I'm reading some very elementary treatments of quantum computation and am unsure about the correspondence among "definitions" of qubit entanglement. One definition states that (1) the bits of a two-... 2answers 1k views ### What is the Riemann curvature tensor contracted with the metric tensor? Can the Ricci curvature tensor be obtained by a 'double contraction' of the Riemann curvature tensor? For example$R_{\mu\nu}=g^{\sigma\rho}R_{\sigma\mu\rho\nu}$. 1answer 982 views ### Do partial derivatives commute on tensors? Do partial derivatives commute on tensors? For example, is $$\partial_{\rho}\partial_{\sigma}h_{\mu\nu} - \partial_{\sigma}\partial_{\rho}h_{\mu\nu}=0$$ correct? 3answers 1k views ### Relativistic basic question - four vector, Lorentz matrix I have heard relativistics only very compressed during my student time. Now I looked up the definitions again and a question comes into my mind: A contravariant vector is transformed like this:$(a^...
I'm kind of confused. I want to calculate the electromagnetic invariant $I := F^{\mu\nu}F_{\mu\nu}$, but I'm not sure what is the easiest way to do so. So, I was trying to do it in matrix form, i.e. ...