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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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Index position when varying an action with respect to the metric

I'm confused about where we should put tensor indices when we vary an action wrt the metric. For example, if I have in the Lagrangian a term such as $$ A_{\mu\nu}B^{\mu\nu}, $$ do I necessarily have ...
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Differentiating Scalar along a geodesic

I have been studying GR for sometime and doing exercises from Schutz and I have a question about differentiating along a geodesic. Here is what I know. The equation of geodesic in terms of four ...
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Could be possible to build a 4-vector in special relativity whose spatial component was the electric field E?

Hi everyone and sorry for my English. I would like to know if I can build a legitimate 4-vector as $E^\alpha=(E^0,\mathbf{E})$. I'd like you to check if my way is correct. 1- We already know that $\...
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Question about tensor integration

A (covariant)$(0,2)$-tensor can be written as: $$T = \sum_{\mu}\sum_{\nu}t_{\mu \nu} e^{\mu}\bar{\otimes}e^{\nu}\tag{1}$$ with the particular basis vectors $e^{\mu}\bar{\otimes}e^{\nu}$ that spans ...
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65 views

Line element 1-form

It was pointed out that dual vectors of a manifold, and hence differential 1-forms, are not dependent on the metric (Intuition behind dual vectors ('Bongs of a bell' does not help)). But doesn'...
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Intuition behind dual vectors ('Bongs of a bell' does not help)

Similar to the post here (How to visualize the gradient as a one-form?), I'm wondering about an intuition behind dual vectors and differential forms (and the link in that answer to Thorne's notes is ...
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Momentum in capacitor field; How can an EM field have zero momentum density but non-zero momentum flux?

Consider the case of a simple, stationary parallel plate capacitor oriented with its plates lying in the x-y plane. The E-field is simply given by: $$\vec{E} = \frac{Q}{\epsilon_0A}\hat{z} $$ with ...
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134 views

Doubts on covariant and contravariant vectors and on double tensors

I'm trying to study tensors. Given a coordinates transformation from cartesian to $u_i$ ones: $$ u_1 = u_1 (x,y,z) \qquad u_2 = u_2 (x,y,z) \qquad u_3 = u_3 (x,y,z) $$ I can write a vector $\mathbf{...
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2answers
44 views

4-Gradient vector and the Field strength tensor

Need some help evaluating the following 4-gradient, that of the gradient of the field strength tensor $$F^{\mu\nu}= \begin{bmatrix} 0 & -E_x & -E_y & -E_z\\\ E_x & 0 & -...
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1answer
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How to construct a 2-partite matrix

Let's assume we have an internal hamiltonian $H_0 = \mid 1\rangle \langle 1\mid$. Now let's assume we have two systems with identical Hamiltonians $H^1_0$,$H^2_0$ and I want to compute the joint ...
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Is there a useful way to visualize the symmetries of the relativistic Riemann curvature tensor?

I find it useful to see diagrams such as trees, colored 2D and 3D arrays, etc., which illustrate how terms combine in composite expressions. For example, the following is my visualization of the ...
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Constructing a Hamiltonian for $N$-qubits

Let us assume we have a qubit with an internal Hamiltonian $H_0 = \sum_i \varepsilon_i |i\rangle\langle i|$. Now let's assume we have 2 such qubits. How would their joint Hamiltonian look like? I ...
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MTW Box 4.1 Contraction of EM 2-form with surface element bivector to give “magnetic flux”. What does that mean?

This is my attempt to make sense of Box 4.1-4.b of MTW's Gravitation. I'm not entirely sure I have the computation correct. But, even if it is correct, I don't really understand how the final result ...
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1answer
62 views

When can we raise lower indices on “nontensors” as described in Dirac's book *General Theory of Relativity*?

This is a follow on to my previous question: Dirac's book *General Theory of Relativity*: Doesn't this show the partial derivative of the metric tensor is zero? I should not have made that ...
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Is there a good treatment of “familiar” physics using exterior calculus, AKA differential forms?

By familiar physics, I mean the physics of things I can reach out and touch. In other words, neither relativity nor analytical dynamics, etc. After re-reading chapter 4 of MTW's Gravitation yet ...
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(Lorentz etc) invariant vector fields

(Background: I know some but not much differential geometry, hopefully enough to formulate this post.) I want to ask about what physicists mean when they say scalar, vector, etc. The answer in ...
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What is the industry standard definition of $\nabla\cdot\mathfrak{T}$ (del dot a tensor)? Re: MTW

In chapter 3 of MTW's Gravitation using the example of a rank-3 tensor $\mathfrak{S}$ they define $$\text{(divergence of }\mathfrak{S}\text{ on the first slot)}\equiv{\nabla\cdot\mathfrak{S}}$$ ...
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Proof of the acceleration formula in tensor notation [closed]

From book "Introduction to Tensor Analysis and the Calculus of Moving Surfaces" by Pavel Grinfeld, Edition 2013 : The component $A^{i}(t)$ of acceleration $\mathbf{A}(t)$ of the particle from the ...
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1answer
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Dirac's book *General Theory of Relativity*: Doesn't this show the partial derivative of the metric tensor is zero?

See the bold text for my question. This question regards Dirac's General Theory of Relativity page 13. In his demonstration that the length of a vector is unchanged by parallel displacement Dirac ...
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Write down the components of metric tensor correctly [closed]

this is a FLRW metric and I want to write down the metric tensor from this FLRW metric accurately. Can anyone please help me to do this? Thanks in advance. \begin{equation}\tag{1} ds^2 = a^2 ( \tau) [...
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Derivation of Covariant Maxwell's Equations

I am trying to derive the covariant formulation of Maxwell's equations. I understand that all four of Maxwell's equations can be written compactly as $$\partial_{\mu}F^{\mu\nu} - j^{\mu} = 0 \;, \...
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Tensorality of the Lie Derivative and $i([X,Y]) = [L(X),i(Y)]$

I'm trying to understand the equation following (2.15) on p.9 of Blau's Symplectic Geometry and Geometric Quantization. For two vector fields $X,Y$ on a symplectic manifold $M$ we are told one ...
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1answer
52 views

Converting an invariant matrix to a non-invariant tensor

I'm working on the following problem: In 4-dimensional notations, given a transformation matrix Calculate the matrices $\Lambda_{\mu\nu}$, $\Lambda_\mu^\nu$ and $\Lambda^{\mu\nu}$ The ...
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1answer
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Electric quadrupole - tensor identity

In classical electrodynamics, we introduce the electric quadrupole moment $$D^{ij}\equiv\int y^i y^j \rho \mathrm{d}^3y$$ and its reduced (trace-less) version $$\mathcal{D}^{ij}\equiv D^{ij} - \frac{1}...
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3answers
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How to show vanishing entries for invariant tensors?

Given i.e., a fourth order tensor $T_{ijkl}$ with spatial indices $i,j,k,l\in\{x,y,z\}$, and rotational invariance around the $z$ axis, how do I show that $T_{ijkl}=0$ because of said rotational ...
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1answer
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GR with Torsion: Definition of contorsion

I start doing some computations in manifolds with non vanishing torsion and things are getting a bit confused, basically because of notations and definitions. I understand that in presence of non ...
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2answers
36 views

Four-velocity and the metric tensor

The metric tensor $g_{{\mu}{\nu}}$ has this property $$g_{{\mu}{\nu}}g^{{\mu}{\nu}}=4$$ and the four-velocity, $U^{\mu}=\frac{dx^{\mu}}{d\tau}$ which has this property $$U_{\mu}U^{\mu}=c^2.$$ So ...
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Tetrad/vierbein for Eddington-Finkelstein coordinates

I have trouble with obtaining vierbein (and orthonormal frame components) given by $\eta_{(a)(b)} = e_{(a)}^{\mu}e_{(b)}^{\nu}g_{\mu\nu} $ with tetrads $e_{(i)}= e_{(i)}^{\mu}\partial_\mu$ and ...
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Number of components of a symmetric contravariant tensor

I am given the following question : "What is the largest number of different components that a symmetric contravariant tensor of rank two can have when (a) N = 4, (b) N = 6? What is the number for ...
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Why aren't Christoffel symbols tensors? - asked from a fibre bundle perspective

I've been reading about connections on fibre bundles recently and it's made me think about the exact nature of the Christoffel symbols in GR. If we have a vector bundle $E$ over $M$ and put a ...
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56 views

General coordinate transformations?

Say I have a vector field expressed in Cartesian coordinates: $$\mathbf{A} = \sum_i A_i \mathbf{\hat{e}}_i$$ where the $\hat{\mathbf{e}}_i$ are the generalisation of the unit vectors $\mathbf{\hat i}, ...
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1answer
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Why is $\partial_{\mu}x^{\nu} = \delta^{\nu}_{\mu}$?

In Blundell's book on QFT, one can find the following Is this because of: $$\partial_{\mu}x^{\nu} = \partial_{\mu}x^{\nu^{'}} \partial_{\nu^{'}}x^{\mu}$$ $$\partial_{\mu}x^{\nu} = \Lambda_{\mu}^{\...
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1answer
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Basics of Tensor theory

Consider that we have an orthonormal basis $\{e_1, e_2, e_3\}$ We know that $e_2 \times e_3 = \pm e_1$, to show this in terms of tensor notation, from the Continuum Mechanics by Chadwick textbook it ...
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Proving the first Bianchi identity only from the other three Riemann curvature tensor identities [closed]

Given that $R_{abcd}=-R_{bacd}$, $R_{abcd}=-R_{abdc}$ and $R_{abcd}=R_{cdab}$ can I prove that $R_{abcd}+R_{acdb}+R_{adbc}=0$ without using the definition of the Riemann curvature tensor? Are the ...
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Effect of Co-ordinate Change on Euler-Lagrange Equations for Scalar Fields

Consider a single scalar field $\phi$ on a manifold $\mathcal{M}$. Suppose in $\{x^\mu\}$ co-ordinates, the Lagrangian density is $\mathcal{L}(\phi, \frac{\partial \phi}{\partial x^\mu})$. This means ...
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Question about derivation of four-velocity vector

In order to describe a notion of rate of change of positon, in four-dimensional spacetime, we have to introduce the concept of four-velocity. So, consider the following: For a massive particle ...
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1answer
63 views

Why the basis of vectors and one-forms can not be related through the metric as a vector and one-forms?

I know that basis vector and basis of one-forms are related through $$ \tilde{e}^\mu \cdot \vec{e}_\nu = \delta^\mu _\nu .\tag{1}$$ However, the metric has the property that allows to convert ...
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Does the quadrupole moment tensor contracted with itself yield Kronecker delta?

I have trouble understanding the Kronecker delta and how it comes up in tensor equations. I know the metric contracted with itself gives the Kronecker delta which either is 0 or 1 depending on if the ...
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Question on the invariance of Hooke's law for isotropic materials

Hooke's law for an elastic isotropic 2-d material is a rank 4 tensor with 16 elements $$ \begin{pmatrix} a_{11} & a_{12} & 0 & 0 \\ a_{12} & a_{11} & 0 & 0 \\ 0 & ...
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Help with understanding the imposition of gauge conditions

Let $s$ be a positive integer and $h_{a_1\dots a_s}$ be a traceless and totally symmetric (real) field which is defined modulo gauge transformations of the form $$\delta_{\xi}h_{a_1\dots a_s}=\...
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1answer
28 views

Double divergence of stress tensor for migration flux

I am looking to calculate migration as a function of time using equation in Image 1. SigmaP is the total particle stress tensor in the cylindrical coordinates (r, theta, z). I am only interested in ...
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1answer
74 views

A question about the expression of Riemann tensor in Landau & Lifshitz

I was reading Landau & Lifshitz "The Classical Theory of Fields" and there is a expression at the beginning of section 92-Properties of the curvature tensor I don't understand. The author ...
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Using diagonality in Einstein notation

Given a diagonal matrix $D$, with diagonal elements given by vector $\mathbf{d}$. Representing this in Einstein notation gives $$ D_{ij} = \delta_{ijk} d_k $$ where $$ \delta_{ijk} = \begin{cases}...
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Why is the scalar product of two four-vectors Lorentz-invariant?

Why is the scalar product of two four-vectors Lorentz-invariant? For instance, given two four-vector $A^\mu$ and $B^\mu$, so their scalar product is $A\cdot B=A^\mu B_\mu=A^\mu g_{\mu\nu}B^{\nu}$. ...
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1answer
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Show the density is an objective quantity

I am studying the Lagrangian and Eulerian representations of quantities within the scope of fluid dynamics. At some point I am asked to show that the density is an objective quantity, which for a ...
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1answer
34 views

Isotropy of the Minkowski metric

An isotropic tensor has the same components in all rotated coordinate systems. Scalars or tensors of rank zero are isotropic. Tensors of rank one or vectors are not isotropic. The only isotropic rank-...
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1answer
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Proof of invariance of scalar product under rotations, using index notation

So I have got the following question: Show that the scalar product of two cartesian vectors $p_i\cdot q_i$ is invariant under coordinate transformations (orthogonal transformations) Now I know ...
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2answers
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What is the way to represent a Lorentz tensor field?

For a vector field one can represent this with an array of arrows. There is a standard sort of way to represent tensors in Euclidean space as small ellipses. Is there any standard way of ...
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1answer
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Distinguishing between matrix forms when reordering indices of tensors

I'm studying general relativity and tensors. It seems that in the cooridnate independent form of the tensor, the order of indices matters even between an upper and lower index. For example, in general,...
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Maxwell Lagrangian where $F$ and its derivatives are the variables (i.e., without replacing $F={\rm d}A$) [duplicate]

The way the electromagnetic Lagrangian is usually constructed is by noticing that the EM fields are always constrained to satisfy ${\rm d}F=0$ (half of Maxwell's equations). We can immediately solve ...