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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Contraction of 4-dimensional epsilon tensor with four-vector components

Evaluating a feynman diagram via the trace method I encoutered the term $$ p_\alpha p'_\beta k_\mu k'_\nu\cdot \epsilon^{\alpha\beta\mu\nu} $$ where $\epsilon^{\alpha\beta\mu\nu}$ denotes the 4-...
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Derivation of residual reparameterization symmetry equation

Here is my attempt to derive the residual reparameterization equation $$\partial_{\alpha}\xi_{\beta}+\partial_{\beta}\xi_{\alpha}=\Lambda \eta_{\alpha\beta}$$ For a reparameterization $x^{\alpha}\...
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Parallel transport and Geodesic deviation

We know that when we derive the Geodesic equation, we want to actually understand the geometrical meaning of the Riemann tensor. We see from the geodesic equation that the second derivative of the ...
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Tensor equation solving

Suppose we have the following equation \begin{equation} \omega_{\rho\sigma}[A^\mu,J^{\rho\sigma}]=B^ \mu \end{equation} where $[A^\mu,J^{\rho\sigma}]=A^\mu J^{\rho\sigma}- J^{\rho\sigma}A^\mu$. I am ...
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Which types of strain tensor are positive definite?

I am taking a look at different types of strain tensor. Specifically, I am thinking about if the infinitesimal strain tensor \begin{align*} \epsilon_{ij} = \frac{1}{2} (\frac{\partial u_i}{\partial ...
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Why am I getting this tensor rotation wrong?

Let $$\rho_\theta \equiv \rho(R_\theta) = \begin{pmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{pmatrix}\tag{1}$$ be a representation of $SU(2)$, and consider the ...
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55 views

A matter of indices - Electromagnetic Field Lagrangian

I was trying to manipulate the following lagrangian : $ L = -\frac{1}{4}F^{\mu\nu}F_{\mu\nu} $ where, of course, $ F^{\mu\nu} = \partial^\mu A^\nu - \partial^\nu A^\mu $. The first steps are ...
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How to get $\vec{B}$ from $\vec{H}$ in hyperspatial ($n>3$) electromagnetism?

So I’m currently trying to formulate Maxwell’s equations in dimensions in other than 3 in order to improve my understanding of electromagnetism. In 3D, Maxwell’s equations can be described by $$\begin{...
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Clarification on electric quadrupole moment definition [duplicate]

I have encountered two (?) definitions of the electric quadrupole moment. They are: $$Q_{ij}=\frac{1}{2}\int \rho(\vec{x}')x'_i x'_j\,\mathrm{d}^3x'$$ and $$Q_{ij}=\int (3x'_i x'_j-\delta_{ij}x'^2)\...
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Which tensor should the metric raising or lowering?

For something like $g^{ij} n_i h_{kj}$, how do I know which one should the metric operate on? $n_i$ or $h_{kj}$? The results could be $n^j h_{kj}$ or $n_i h^i_{k}$, which are different. The question ...
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Proving an equation is valid in any basis [closed]

I have a question I'm having trouble with. I'm given an equation with 2nd order tensors and a few constants and we're asked to prove that it remains valid under any basis. Any ideas how I can go about ...
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Dimensions of velocity vector in differential geometry

If we have a velocity vector written in, say, cartesian coordinates: $\mathbf V$ = $\dot{x}$ i $+$ $\dot{y}$ j Note that Dim($\dot{x}$) = Dim($\dot{y}$) = $LT^{-1}$ which are the dimensions of speed ...
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1answer
55 views

Raising and lowering indices in line elements - why do we raise and lower them in line elements?

My question refers to Piattella's lecture notes on cosmology. On page 15, the Euclidean line element is defined as $$ ds^2 = \vert d\mathbf{x}\vert^2 = \delta_{ij}dx^idx^j. $$ My first question is ...
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The tensor product in the Hamiltonian of graphene

I have the Hamiltonian of pristine graphene \begin{equation} H=v_{F}.\boldsymbol{\gamma}.\boldsymbol{p} \end{equation} with $\boldsymbol{p}=(p_{x},p_{y})$ is the momentum operator, $v_{F}$ is the ...
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Small advice on doing Maxwell's covariant form

Hello fellow physicists! I really enjoyed that Carroll (Spacetime and Geometry) included how tensors can be used to rewrite Maxwell's equations. →Firstly rewriting the usual in tensor/index notation: ...
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Proving Something is a Rank-4 Tensor

Whilst going over my undergraduate notes on General Relativity, I came across the Quotient Rule for tensors: Briefly, if $\mathbf{X \, A} = \mathbf{B}$ with $\mathbf{B}$ being a non-zero tensor and $\...
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Applying gradient in spherical coordinates to vector in cartesian coordinates [closed]

I am trying to calculate the gradient of a vector field $\boldsymbol{u}$. In cartesian coordinates, I would normally do $$\left(\nabla\boldsymbol{u}\right)_{ij}=\partial_{i}u_{j}=\left(\begin{array}{...
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Contraction of antisymmetric tensor [closed]

Let $\omega^{ab}$ be antisymmetric in the indices $a$ and $b$. Why we have $$\omega^{ab}(\theta_{ab}-\theta_{ba})=2\omega^{ab}\theta_{ab}$$
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Evaluating shear tensor by gravitational wave

The shear tensor is given by: \begin{equation} \sigma_{ij} = \frac{1}{2}\nabla_\alpha U_\beta + \frac{1}{2}\nabla_\beta U_\alpha + \frac{1}{2} U_\alpha U^\mu \nabla_\mu U_\beta + \frac{1}{2} U_\beta U^...
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Identity regarding problem 2.4 of Walds General Relativity

I was going through some of the problems in Wald's General Relativity and in problem 4 chapter 2 I found something that confuses me. So, basically we are asked to show that in any coordinate chart $(...
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Geometric visualization of addition of angular momenta

Introduction Consider the Hilbert space of two $\frac{1}{2}$ spin particles (electrons, for instance), spanned by \begin{equation} {|\alpha\rangle, \, \alpha = \uparrow,\downarrow} \end{equation} \...
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Choosing an Irreducible Tensor Operator Basis where the Singular Values of Each Basis Element are the Same

Let $\mathcal{B(H)}$ be the space of all bounded linear operators on the Hilbert space $\mathcal{H}$. Let $g \rightarrow \mathcal{U}_g$, where $\mathcal{U}_g (\cdot):= U(g) (\cdot) U(g)^\dagger $, be ...
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Transformation law of vector fields on $\mathbb{R}^n$

So suppose we have a function $F$ from $\mathbb R^2$ to $\mathbb R^2$ defined by $F(x,y) = (g(x,y),h(x,y))$ where $g$ and $h$ represent temperate and pressure respectively (the point is, they are both ...
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64 views

Trace of the Riemann tensor contracting the first and the fourth indices

I know the trace of the Riemann tensor is defined by contracting over the first and the third indices (equivalent to the second and the fourth), and the trace over the first two or the last two ...
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Why scale tensor products like that?

Studying the sum of angular momentum in quantum mechanics, tensor products were introduced to us to get the general system from the individual states. There's a property that says: $(av)\otimes w = a(...
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Contraction over two indices on a symmetric tensor

I'm reading a book on GR, and it is going over tensors. They say, When contracting over a pair of upper indices that are symmetric on one tensor, only the symmetric part of the lower indices will ...
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49 views

Contraction of 2 Riemann tensors [closed]

Iz possibile to find a costant $c\neq0$ such that $R^{abcd} R_{cbad} = c R^{abcd}R_{abcd}$?
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Trying to understand electric and magnetic fields as 4-vectors

I was trying to show that the field transformation equations do hold when considering electric and magnetic fields as 4-vectors. To start off, I obtained the temporal and spatial components of $E^{\...
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24 views

Trouble with obtaining Ohm's law in proper frame

A naive generalisation of Ohm's law is as follows: $$J^{\alpha}=\frac{\sigma}{c} F^{\alpha\beta}U_{\beta}$$ It is known that the above will only work in the proper frame of the conductor. However, I ...
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Why does the torsion tensor NOT vanish?

The torsion tensor in local coordinates is $$ T^k_{ij} \enspace = \enspace \Gamma^k_{ij} - \Gamma^k_{ji} $$ However, the Christoffel symbols are symmetric in their lower indices, i.e. $$ \Gamma^k_{ij} ...
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How to express the fact that $\delta_\mu{}^\nu$ is symmetric?

In relativity, given a metric tensor $g_{\mu\nu}$ and its inverse $g^{\mu\nu}$, by definition of the inverse, we have the relation $g_{\mu\rho}g^{\rho\nu} = \delta_\mu{}^\nu$. In matrix form, $\delta_\...
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How to derive Infinitesimal Strain Tensor in Cylindrical Coordinates [closed]

How can I obtain the below formulas of infinitesimal strain in cylindrical coordinates using matrix calculation given the first formula? I find it hard to study them because I still don't know how to ...
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45 views

Curve is geodesic iff $\nabla_k g(V,V)$ vanishes

Let $V$ be a Killing vector field and let $s \longmapsto x^i(s)$ be a curve such that $$\dot{x} \enspace \equiv \enspace \frac{dx^i}{dx}(s) \enspace = \enspace V^i\big(x(s)\big)$$ Show that $x^i(s)$ ...
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Different definitions 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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Concise and concrete book for tensor calculus for physics [duplicate]

Please suggest a concise and comprehensive book for introduction and application to tensor calculus (from physics perspective). Will the book $\textbf{Tensor Calculus: A Concise Course By Barry Spain}$...
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59 views

Chain rule for covariant derivative?

Does a chain rule for the covariant derivative exist so that we can evaluate an expression like $$\nabla_c\sqrt{t_{ab}}?$$ where $t_{ab}$ are tensor components? More generally, how do we take the ...
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68 views

Contravariant tensor definition must be incorrect?

Both my textbooks (Schaums Tensor calculus and Neuenschwander's Tensor calc for physics) define a contravariant tensor of order one (which they also state is synonymous with a simple vector) as ...
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Commutator of covariant derivative for rank 2 tensor

I am a newbie at tensor notation and I have been told to prove the identity $$ (\nabla_a\nabla_b - \nabla_b\nabla_a)X^a_{\ \ \ b}=- R^e_{\ \ \ bcd}X^a_{\ \ \ e}+ R^a_{\ \ \ ecd}X^e_{\ \ \ b} $$ I am ...
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Calculating Jacobian of transformation

In Sean Carroll's GR book, pg. 89, there is this equation (2.93) involving the Jacobian of a general transformation: $$\frac{\partial x^{\mu_1}}{\partial x^{\mu_1'}}...\frac{\partial x^{\mu_n}}{\...
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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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Contraction of Levi Civita symbol

In Sean Carroll's GR book pg. 83, the contraction of two Levi-Civita symbols is defined as $$\epsilon^{\mu_1...\mu_p\alpha_1...\alpha_{n-p}}\epsilon_{\mu_1...\mu_p\beta_1..\beta_{n-p}} = (-1)^s p! (n-...
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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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63 views

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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36 views

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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31 views

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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1answer
81 views

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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54 views

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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71 views

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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