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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Showing that $E_{\alpha}$ and $B_{\alpha}$ is spacelike

Lately, I came across the concept of treating the electric and magnetic fields as 4-vectors via: $$E_{\alpha}=F_{\alpha\beta}U^{\beta},\:B_{\alpha}=\frac{1}{2c}\epsilon_{\alpha\beta\mu\nu}F^{\beta\mu}...
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Prove that covariant differentiation obeys the product rule

So, in Hobson's general relativity, the following question is asked: Show that covariant differentiation obeys the usual product rule, e.g. $$\nabla_a(A_{bc}B^{cd})=\nabla_a(A_{bc})B^{cd}+A_{bc}\...
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Is the raised Levi-Civita symbol a tensor density of weight 1?

In Sean Carroll's GR book, pg 83, between eqs. (2.69-70), the Levi-Civita symbol with raised indices is defined as $$\tilde{\epsilon}^{\mu_1\mu_2...\mu_n}=\text{sgn}(g)\tilde{\epsilon}_{\mu_1 \mu_2...\...
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What is the sign of the metric determinant?

I read that for the Levi-Civita symbol $\tilde{\epsilon}_{ijk}$, people somtimes define another version of the symbol with upper indices: $$\tilde{\epsilon}^{ijk}=\text{sgn} (g) \tilde{\epsilon}_{ijk} ...
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Geodesics equation in a 2-space with a certain $ds^2$

This is exercise 3.20 of Hobson's general relativity. It's presented as follows: In the 2-space with line element $$ds^2=\frac{dr^2+r^2d\theta^2}{r^2-a^2}-\frac{r^2dr^2}{(r^2-a^2)^2}$$ Where r>a, ...
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Torsion tensor and affine connection symbols

I have read tons of questions about this topic but I think my particular issue is not solved. If so, please let me know. So I want to prove that the torsion tensor $\mathcal{T}$ actually transforms ...
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Vanishing covariant derivative of a vector field

I'm asked to prove the following statement in my physics book: A vector field with covariant components $v^b$, in order to have a vanishing covariant derivative everywhere in a manifold, must satisfy:...
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How to derive equation 27.4 in Dirac's “General Theory of Relativity” book?

I've been having trouble following Dirac's logic in deriving equation 27.4 in his general relativity book. If $p^\mu$ is some matter current 4-vector, satisfying $\partial_\mu p^\mu = 0$, Dirac says ...
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Swapping indices

I have a tensor $T^{\mu\nu}$ that looks like this: T^mu,nu = {{2,0,1,-1},{1,0,-3,2},{-1,1,0,0},{2,1,-1,2}} I want to find $T^{\nu\mu}$. If I swap the indices, what ...
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Hadamard gate over 2 qubits [closed]

Let H be the Hadamard gate: $$(\frac{1}{\sqrt{2}})\begin{pmatrix}\begin{array}{rrrrrrrr} 1 & 1 \\ 1 & -1 \end{array}\end{pmatrix}$$ I would like to write down the matrix associated to the ...
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Torsion tensor symmetrisation

Given that the affine connection can be written as: $$\Gamma ^a{}_{bc}= \bigg\{ {a \atop bc} \bigg\} - \frac{1}{2}(T^a{}_{bc}+T_c{}^a{}_b-T_{bc}{}^a) $$ Where $\big\{ {a \atop bc} \big\}$ denotes the ...
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If the world had four spatial dimensions, then area would be a tensor?

In three dimensions area is a vector because two dimensions have a direction relative to the third. If the world had four spatial dimensions then area would be a tensor? And what form then the laws of ...
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Wedge Product Convention

In Wald’s General Relativity textbook he defines the wedge product as: $$(w \wedge u)_{a_1 ... a_p b_1 ... b_q}= \frac{(p+q)!}{p!q!} w_{[a_1 ... a_p}u_{b_1 ... b_q]}$$ My question is relatively simple:...
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Definition of exterior derivative

In Sean Carroll's GR book, pg 84, the exterior derivative $d$ is defined as $$(dA)_{\mu_1\mu_2...\mu_{p+1}} = (p+1) \partial_{[\mu_1}A_{\mu_2...\mu_{p+1}]}\tag{2.76}$$ where $A$ is a $p$-form and the ...
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d'Alembertian of Ricci tensor

What is the d'Alembertian of the Ricci tensor $\square R_{\mu\nu}$? I know that for a scalar $\square := g^{\mu\nu} \nabla_\mu \nabla_\nu$ and I can use the fact that $\nabla_\mu \nabla_\nu F = \left( ...
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Indices Exchange

In General Relativity, we often face indices exchange; but I actually do not really understand how to change indices properly. For example: If I have $$ R_{ab}\partial_c \phi \partial^b\phi \delta g^{...
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Is there a table for hyperfine interaction tensor?

I'm doing research in biophysics and I need to find the hyperfine interaction tensors for ascorbic acid radical, FADH-, and oxigen radical (O2-). Is there a place where I can find the hyperfine ...
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Differential forms and wedge product

In Sean Carroll's GR book, the differential $p$-form is defined as a $(0,p)$ tensor that is completely antisymmetric, which I would think is something like $$\frac{1}{2!}(t_{ab}-t_{ba})\textbf {e}^a \...
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Covariant derivative of tetrad/vielbein

I am learning about the tetrad basis for manifolds from this lecture notes. On pg 52, the spin connections ${{w_\mu}^a}_b$ are defined as $${{w_\mu}^a}_b=e^a_\nu e^\lambda_b\Gamma^\nu_{\mu\lambda}-e^\...
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Does product rule of covariant derivative apply for a scalar

I'm proving a tensor identity and I wonder if it is true that $$ \nabla_\mu \left( A^\nu B_\nu \right) = \nabla_\mu\left( A^\nu \right) B_\nu + A^\nu \nabla_\mu\left( B_\nu \right) $$ where $A^\nu$ ...
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Transformation rules for quantities

When we formulate transformation laws for vectors and tensors, the transformation rule for $x^\mu$ is calculated via arguments from total derivatives considering $x^\mu=x^\mu(x^{'\nu})$ that in turn ...
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Zee Quantum Field Theory page 35

About spin 2 polarization tensors $$\varepsilon_{\mu\nu}^{(a)}, $$ it is claimed that $$\sum_{a} \varepsilon_{\mu\nu}^{(a)}(k)\varepsilon_{\lambda\sigma}^{(a)}(k) = A(G_{\mu\lambda}G_{\nu\sigma}+G_{\...
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Commutation relation involving $\gamma^5$. Spot the error

I'm trying to prove a relation that is useful when studying general properties of Dirac spinors, namely, that $\left[\gamma_5,\sigma^{\mu\nu}\right]=0$ where $\sigma^{\mu\nu}\sim i\gamma^\mu\gamma^\nu$...
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Form of General Relativity Hydrostatic Equilibrium Equation

I have been reading into general relativity and have recently gotten stuck at a manipulation of the equation for hydrostatic equilibrium of a perfect fluid. We have $$T^\mu_{\nu;\mu} = 0$$ which leads ...
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If $g_{ij}$ is a tensor of type $(0,2)$, what is kind of tensor is $\partial_{i}g_{jk}$?

Suppose $g_{ij}$ is a tensor of type $(0,2)$, then what type of object is $\partial_{i}g_{jk}$? Is it even a tensor, and if so, of what type? Is the $\partial_{i}$ still a differential with respect to ...
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Covariant derivative of a tensor

I have one question. I'm new in tensor calculus, so question may seem like stupid. Covariant derivative of a tensor $T^\alpha$: $$\nabla_\beta T^\alpha=\frac{\partial T^\alpha}{\partial x^\beta}+\...
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Einstein summation convention when a sum of terms is present

I'm reading Landau / Liftshitz vol. 6 on fluid mechanics, and I encountered the expression (page 45, top): $$\frac{\partial v_i}{\partial x_k} + \frac{\partial v_k}{\partial x_i}.$$ The expression ...
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Tensors Differentiation

I know that $\frac{\partial x^{\mu}}{\partial x^{\nu}}=\delta^{\mu}_{\nu}$ but a few days back, I read somewhere that $\frac{\partial x_{\mu}}{\partial x^{\nu}}=\eta_{\mu\nu}$. Can someone help me ...
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Calculating the coordinate basis

In Robert M. Wald's General Relativity the definition of the "coordinate basis" (of the tangent space) of a manifold is given by: Let $\psi: O \to U \subset \mathbb{R}^n$ be a chart with $p \...
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Derivative Operators on a manifold

I am having some trouble coming to terms with the notion of a derivative operator on a manifold. In Robert M. Wald's General Relativity, the definition in the textbook is given in terms of 5 ...
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How do you find coordinate transformations to LIF for the weak gravity metric?

I am working on the same problem from Schutz as this question, which discusses the weak gravitational field metric. $$ds^2=−(1+2\phi)dt^2+(1−2\phi)(dx^2+dy^2+dz^2)$$ From this metric, I was able to ...
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Gradient, one-form and Sean Carroll

"A tensor (k,l) is a multilinear map from k dual vectors and l vectors to R (...) The gradient, ..., is an honest (0,1) tensor." These citations are retired from Sean Carrol Spacetime and ...
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Contraction of Christoffel symbol and metric tensor

How can I prove this contraction of Christoffel symbol with metric tensor? $$ g^{k\ell} \Gamma^i_{\ \ k\ell} = \frac{-1}{\sqrt{|g|}}\frac{\partial\left(\sqrt{|g|}g^{ik}\right)}{\partial x^k} $$ I know ...
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What does a contravariant and a covariant tensor with the same indices result in?

For example, for two arbitrary tensors $M^\alpha N_\alpha$, can this be written in a simpler way like equivalent to a scalar? or does it even vanish entirely?
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Why is the magnetic field $B$ a pseudo-vector?

Physically speaking, "pseudo-vectors" are vectors $v\in \mathbb{R}^3$ which transform as $ v'= (\det {R})v$ if the "system were to transform as $R\in O(3)$". However, what does ...
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Help with tensor calculus identity proof (antisymmetric matrix and levi civita symbol)

I'm having some trouble with 2 identitys from tensor calculus. I need to proof these two guys: in euclidean 3-dimensional space, an antisymmetric matrix with entries $M_{ij}$ is equivalent to a ...
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Tensor Index notation vs Matrix notation Transpose

Referring to the answer in the following question: https://physics.stackexchange.com/a/349030/288587 I just cant figure out how to go from: $$ \eta_{\mu\nu} = \Lambda^\alpha_{\;\mu}\Lambda^\beta_{\;\...
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How can I derive the covariant derivative of the Ricci tensor using the Ricci scalar?

\begin{align*} R&=g^{\mu\nu}R_{\mu\nu}\\ \Rightarrow \nabla^{\gamma} R&=\nabla^{\gamma}(g^{\mu\nu}R_{\mu\nu})\\ &=\nabla^{\gamma}(g^{\mu\nu})R_{\mu\nu}+g^{\mu\nu}\nabla ^{\gamma}(R_{\mu\nu}...
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Why is the multiplication of the metric and an inverse metric the Kronecker delta?

I am having a hard time understanding \begin{align*} \delta_{\beta}^{\alpha}=g^{\alpha\nu}g_{\beta\nu}\\ \end{align*} equality. I understand the situation where the indices are the same and the ...
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Understanding tensor transformations

I am trying to learn how tensors transform under coordinate transformations. For an example, under a transformation from the coordinate system $x^\mu \longrightarrow x'^\mu$ a covariant tensor is ...
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Covariant derivative of a one-form

I understood that the covariant derivative of a vector field is $$ \nabla_{i}v^j=\frac{\partial v^j}{\partial u^{i}}+\Gamma^j_{~ik}v^k $$ Then why is the covariant derivative of a covector field $$ \...
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Are vector components “normalized” in generic vector differential operater expressions, e.g., divergence?

A typical way of writing the expression for the divergence of a vector field in general orthogonal coordinates is: $$\nabla\cdot\vec{A}=\frac{\partial_{u}\left[VWA^{u}\right]+\partial_{v}\left[WUA^{v}\...
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Under what conditions is this tensor isotropic?

The tensor I'm talking about is \begin{equation}\tag{1} A_{ij}=\int d{\Omega}\,T_{ij}(\theta,\varphi) \end{equation} where the integral is over the whole solid angle. I know that an isotropic tensor ...
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Problem proving a tensor identity

I am asked to prove $$\nabla \cdot (T \cdot v) = T : \nabla v + v\cdot (\nabla \cdot T)$$ Where $T$ is a order-2 tensor and $v$ is vector, in an orthogonal basis. Let $\delta _{ij}$ denote the ...
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Derivative of basis vector in terms of Christoffel symbols

I would like to derive the formula $$\partial_{c}\vec{e}^{\,a}=-\Gamma_{bc}^{a}\vec{e}^{\,b}$$ where $\vec{e}_{a}$ are the basis vectors on a manifold. In the lecture, we did it in the following way: $...
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Integral form of energy-momentum tensor conservation (Stokes' theorem)

Is there a way in which the conservation law of the energy momentum tensor $\nabla _\nu T^{\mu\nu}=0$ can be written in integral form using Stokes' theorem, namely as something roughly similar to: $$ ...
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Transpose of a 2x2 Tensor

This question arises after reading through several Stack Exchange posts and after a long chat with another user in a previous question I asked about this topic. The following "contradiction" ...
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Variation of Ricci scalar with respect to torsion

I'm following a paper in the derivation of field equation in a space with torsion. Now the variation of the Ricci tensor $$\delta R_{ij}=2\nabla_{[k}\delta\Gamma^k_{i]j}+2S_{ki}^{\hphantom0\hphantom0 ...
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How to show ${\epsilon^{ij}}_k {\epsilon_{ij}}^l = -2\delta^l_k$ for the Levi-Civita symbol?

I am trying to prove the following identity for the levi civita symbol $${\epsilon^{ij}}_k {\epsilon_{ij}}^l = -2\delta^l_k,$$ taken from the Ashok Das QFT book pg 153, equation 4.102. I made use of ...
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Exterior Derivative on Curved Manifold (SpaceTime)

Considering the 2-Form Gauge Potential $B_{\mu \nu}$, we can write $dB=H$. In a flat manifold we have that $H_{\mu \nu \rho}=\partial_{\mu}B_{\nu \rho}+\partial_{\rho}B_{\mu \nu}+\partial_{\nu}B_{\rho ...

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