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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28 views

Second order tensors as linear operators

A tensor is formally defined as an object whose components obey some transformation rules. I, however, find it more intuitive to look at (second order) tensors as a linear operator/function between ...
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What is the best way to imagine the difference between vectors and one-forms

I am studying the GR and reading the Schutz. He is defining the one-form as $\widetilde{p} = p_{\alpha}\widetilde{w}^{\alpha}$, and a vector $\vec{A} = A^{\beta}\vec{e}_{\beta}$ such that $$\...
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Alternative formulation for the gradient of divergence of a vector

I am trying to derive two identities which are needed in simplifying both solid and fluid momentum balance equations. Let $A$ be a second-order matrix and $u$ be a vector with three Cartesian ...
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What is the operator $I_\zeta$, satisfying $ d I_\zeta + I_\zeta d = 1$ that allows us to define Noether-Wald charge?

In these lectures on general relativity by Geoffrey Compère, the author, in section 1.3.3., equation 1.62, define an operator $I_\zeta$ which satisfies the identity $$ d I_\zeta + I_\zeta d = 1.\tag{...
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Calculating the metric tensor

In my lecture we just approached the metric tensor and the general form of a scalar product. So for two vectors $\vec{x}$ and $\vec{y}$ the scalar product is $\vec{x} \cdot \vec{y} \enspace = \...
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Lorentz Invariance of the Euler-Lagrange equation for fields

Given an Lorentz invariant Lagrangian density $L$ of a Lorentz invariant scalar field $\phi$, How does one show that the following term in the Euler-Lagrange equation is invariant under Lorentz ...
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What are the rules when differentiating tensor functions to the power of 2?

How do I differentiate tensor squared functions? I know that, for example, to differentiate a function like $x^\rho x^\mu$ it is as follows: $$\partial_\mu (x^\rho x^\mu) = \frac{\partial(x^\rho x^\...
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How can I rotate the piezoelectric tensor?

For a 3m crystal, the piezoelectric tensor or equivalently the nonlinear optical susceptibility is the following; $$\begin{bmatrix}0&0&0&0&d_{15}&-d_{22}\\\ -d_{22}&d_{22}&...
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Is the result of the product of metric tensors $\eta^{\mu\nu} \eta_{\mu\nu} = 1$? [duplicate]

Is the result of the product of metric tensors $\eta^{\mu\nu} \eta_{\mu\nu} = 1$? If so how would I prove this? I know that tensors are represented as matrices but I don't know how I'd prove this (...
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Effect of Levi-Civita symbols on rank-two tensors

I am trying to understand the following: $$ \epsilon^{mni} \epsilon^{pqj} (S^{mq}\delta^{np} - S^{nq}\delta^{mp} + S^{np}\delta^{mq} - S^{mp}\delta^{nq}) = -\epsilon^{mni} \epsilon^{pqj}S^{nq}\delta^{...
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Is there any meaning to the object $\partial_{\mu} \Gamma^{\rho}_{\ \nu\sigma} + \Gamma^{\rho}_{\ \mu\lambda} \Gamma^{\lambda}_{\ \nu\sigma}$?

In a calculation I am encountering the object $$ O^{\rho}_{\ \mu\nu\sigma} \ := \ \partial_{\mu} \Gamma^{\rho}_{\ \nu\sigma} + \Gamma^{\rho}_{\ \mu\lambda} \Gamma^{\lambda}_{\ \nu\sigma} \ , \tag{1}$...
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If $S_{\mu\nu\sigma} V^{\mu}V^{\nu}V^{\sigma} = T_{\mu\nu\sigma} V^{\mu}V^{\nu}V^{\sigma}$, then is it true $S_{\mu\nu\sigma} = T_{\mu\nu\sigma}$?

For any vector $V$, suppose that the following equality holds $$ S_{\mu\nu\sigma} V^{\mu}V^{\nu}V^{\sigma} = T_{\mu\nu\sigma} V^{\mu}V^{\nu}V^{\sigma} $$ for two tensors $S$ and $T$. Does it follow ...
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Every symmetric second rank covariant tensor can be transformed into diagonal form with diagonal elements 0,±1

I started learning about tensors, and this theorem was mentioned: "Every symmetric second rank covariant tensor can be transformed into diagonal form in which the diagonal elements are either 1,0 or −...
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I want to know why engineering strain is not a tensor

One of the most classical examples in the mechanics of materials is that engineering strain is not tensor. I want to know why the engineering strain doesn't meet the tensor definitation. ...
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About variation of Ricci tensor

I have been doing some calculation on variation of Ricci's tensor with respect to the metric, that, according with S. Carroll (An Introduction to General Relativity: Spacetime and Geometry, equation 4....
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How can I show the Einstein Tensor using second Identity of Bianchi? [closed]

The Einstein tensor given by: $$G_{\mu\nu}=R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}$$ Can be shown using Bianchi identity?
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How can I prove that $\Gamma_{kij}+\Gamma_{kji}=\partial_k g_{ij}$? [closed]

I want a simple proof of this identity: $$\Gamma_{kij}+\Gamma_{kji}=\partial_k g_{ij}$$ If there's no answer, give me a hint or something would help to prove it, and thanks!
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First fundamental form, induced metric and unit normal vector in general relativity

The inverse of the induced metric on a hypersurface which foliates spacetime can be expressed as $$h^{ab}=g^{ab}+n^{a}n^{b}.$$ To my knowledge in terms of the lapse and shift $$g^{00}=-\frac{1}{N^...
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How can I prove this relation? [closed]

What I want to prove : $$R_{ijrs}=\partial_r\Gamma_{ijs}-\partial_s\Gamma_{ijr}-\Gamma_{rj}^k\Gamma_{iks}+\Gamma_{sj}^k\Gamma_{ikr}$$ According to Riemann-Christoffel tensor, the covariant composant ...
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Understanding time evolution in Von Neumann's pre-measurement (Ozawa model)

I'm studying Quantum Information Theory as a non-physicist and so I'm struggling a bit with simple concepts. Let's say we have a system $\Psi$ composed of two sub-systems: $\Psi = \Gamma \cup \Xi$ ...
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1answer
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How do you define a tetrad if your metric is non-diagonal?

I have a spacetime metric with $g_{11}:=0$ and $g_{01}=g_{10}=f(r,t,\phi)$, the other two non-zero components being diagonal. I am not sure how to find a tetrad basis.
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Tensor rank of acceleration in Newtonian mechanics

Is the Wikipedia page on acceleration as of 3/16/2020 correct? https://en.wikipedia.org/wiki/Acceleration "Accelerations are vector quantities (in that they have magnitude and direction)[1][2], ...
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What is the difference between the Ricci tensor and the scalar curvature?

What is the difference of physics meaning (for beginner) between the Ricci tensor $R_{\mu\nu}$ and the scalar curvature $R$ terms ? Wikipedia gives the same explanation for the two, as we could see ...
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Tensor character of a Jacobian transformation (Tensor Calculus by J.L.Synge Ex1 #9)

I am trying to solve this question in Synge’s Tensor Calculus, Exercise 1 question #9. I don't know how to approach this question. $\frac{\partial{x^r}}{\partial{y^s}}$ are the components of the ...
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This tensor equation in Gerard t' Hooft's notes on General Relativity

This question probably has a simple answer that I'm just missing, but in Gerard 't Hooft's lectures on GR he has the following: $$X^{\mu}=Y_{\mu\alpha}Z^{\alpha\beta\beta}.$$ Why is the result of ...
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The nature of the variation of the metric tensor and the Christoffel symbols

We needed to define the covariant derivative of tensors to preserve its nature as tensors after differentiating, I mean we are so careful when we apply anything to geometric objects. Now I don't get ...
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About the decomposition of a rank 2 tensor into its irreducible components

A rank 2 tensor $T_{ij}$ of 3D rotation group $SO(3)$ is a reducible representation. It has the decomposition $9=5+3+1$ where 5 is the symmetric traceless tensor, 3 is the vector and 1 is the scalar. ...
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Spherical basis representation of a dyad

Consider a dyad $C$ formed from two real vectors $a$ and $b$ such that its Cartesian components are $$ C_{\mu\nu} = a_\mu b_\nu \qquad $$ with $a_\mu$ and $b_\nu$ being the Cartesian components of $a$ ...
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A Question On Indices Notation In General Relativity

I am trying to make sense of this simple case in my book, but I am still baffled by the notation that is used in the indices with the commas and semicolons; I also do not understand how these are ...
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3answers
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Perfect fluid stress tensor

In Thorne and Balndford's new book, they approach the subject of classical physics and tensors from the geometric viewpoint (as in relativity) that is independent from coordinates, instead from a ...
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Problem in deriving Electromagnetic tensor

I'm having troubles in understanding a mathematical step in the derivation of the electromagnetic tensor. In Landau&Lifshitz's book I found that the action of a particle in an electromagnetic ...
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Does the Lie derivative of a tensor satisfy the strong Einstein equivalence principle?

The Lie derivative of a tensor is a tensor of the same rank and type. But it is connection independent meaning it can be expressed in terms of covariant or partial derivatives. Since the Strong ...
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Constructing tensors: Matthews Walker

Consider $3$D cartesian space. Given a point $\vec{r}=(x,y,z)=(x^1,x^2,x^3)$, another point $(a,b,c)=(V_1,V_2,V_3)$ can be constructed where $a,b,c$ are the intercepts on the $x,y,z$ axes by the plane ...
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1answer
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Field equation of Palatini $f(R)$ gravity

I've been reading a paper about a Palatini formulation of $f(R, T)$ gravity theory, and when they vary the gravitational action with respect the connection $\widetilde{\Gamma}$, they obtain that \...
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Two basic questions about Christoffel symbols

I am trying to understand (rather than memorise) the derivation of the Christoffel symbols from the vanishing covariant derivative of the metric, the very first step is \begin{equation} \label{eq:...
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Is $\delta_a^b \, g_{\mu\nu}$ defined?

what is $\delta_a^b \, g_{\mu\nu}$? is the multiplication of Kronecker delta function with the metric tensor when the indices are different defined?
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Levi Civita identity

I've been trying to use the following identity (provided by wikipedia) $$\epsilon^{i_1...i_n}\epsilon_{j_1...j_n} = \delta_{j_1...j_n}^{i_1...i_n} \equiv n!\delta_{[j_1}^{i_1}...\delta_{j_n]}^{i_n}$$ ...
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Why is the metric used in physics a squared norm? [duplicate]

The length metrics used in physics are usually a squared norm, like the following: $$ds^2 = g_{\mu \nu} \, dx^{\mu} \, dx^{\nu}. \tag{1}$$ What other kinds of continuous metrics could we define? Why ...
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How can we do this tensor product $F_{\mu \nu}F^{\mu \nu}$?

Iam Studying "Quantization of the electromagnetic field using Quantum Field Theory" by Lahiri and Pal. But I don't get how they computed action in equation $8.23$. $$A=-{1\over 4} \int d^4xF_{\mu \...
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1answer
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MTW the contraction of a basis bi-vector with a basis two-form. Why am I getting this factor of 2?

My question pertains to the discussion of two-forms and bi-vectors in MTW, Chapters 3 and 4. I set out to understand how the expression for the contraction of a basis p-vector with a basis p-form ...
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Implied metric for the basis vectors when lowering indices

When we map a vector to its corresponding covector with the metric: $$g_{\mu\nu}x^\mu=x_\nu$$ is there a second (implied) metric being used to convert the basis vectors too? Written explicitly: $$...
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1answer
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Find the normalized state and function such that the equation holds for arbitrary unitary matrix [closed]

I have been recently puzzled with a problem I do not know how to solve. Here is the setting and some of my thoughts on the problem. Given: Let us denote the set of all unitary $d \times d$ matrices ...
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1answer
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Why does the covariant derivative of a $(p,q)$-tensor produce a $(p,q+1)$-tensor?

In the specific case of the covariant derivative acting on a scalar function: $$\nabla_\nu f$$ it seems strange to me that this would return a covector. Am I wrong in thinking the covariant ...
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Tensor ordering in index lowering operation

If we take two vectors and want to contract them with the metric tensor to find some frame invariant quantity: $$A^a B^b g_{ab}=\vec A\cdot \vec B$$ is there a convention on where the metric tensor ...
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Difference between the contravariant, covariant and mixed form of the electromagnetic tensor and Minkowski metric tensor?

What is the difference between the contravariant, covariant and mixed form of the electromagnetic tensor and Minkowski metric tensor? I know the difference in indices (superscript and subscript). ...
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3answers
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Why does index contraction have to be done between upper and lower indices?

If I had to give a guess based on limited understanding, I would expect it to be something to do with the resulting object no longer obeying tensor transformation properties. However, if that is the ...
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2answers
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Extending Maxwell's Equations from Flat Spacetime To Curved Spacetime

Assume we are working on a Minkowski (i.e. flat) spacetime. Let $A^{\mu} = ( \phi/c, \textbf{A})$ be the contravariant potential four-vector. Then, assuming a covariant Minkowski metric of $\eta_{\...
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Complex pseudotensor generalisation [closed]

A tensor $T$ is an object which is invariant under all coordinate transformations: $$ T\mapsto T = e^{i0}T. $$ A pseudotensor $P$ is an object which changes its sign under the inversion of a ...
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Sign of components of $\vec{L}$ seems to change under parity; book says otherwise

Book: Gravitational Waves-Vol 1 by Maggiore; pg 147, note 48 The book says that the components of angular momentum $\vec{L}$ are unchanged under parity (reversing the orientation of the axes). Now, ...

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