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-1
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0answers
10 views

Tensor product distributive property?

Is it true that for vectors a,b,c,d we have $$|a> \otimes |b> <c| \otimes <d|= |a><c| \otimes |b><d|?$$ So does this kind of distributive law hold for tensor products?
0
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0answers
18 views

Parallel transport and absolute differential between 2 vectors

$\alpha$ I try to apprehend the notion of covariant derivative. In order to undertsand better, here is a figure on which we are searching for express the difference $ d\vec{V} = \vec{V}(M') - ...
3
votes
3answers
310 views

How to define pseudovector mathematically?

The textbook describes pseudovector like this: Let $a,b$ be vectors and $c=a\times b$, $P$ be the parity operator. Then $P(a)=-a,P(b)=-b$ by definition. But $P(c)=c$ since both $a$ and $b$ reverse ...
1
vote
0answers
10 views

Writing Breit-Pauli spin-spin-coupling Hamiltonian as a sum of irreducible spin tensor operators

The spin-spin interaction part of the Breit-Pauli Hamiltonian for two electrons is essentially (omitting constants and the contact term) \begin{equation} \frac{3(\mathbf S_1 \cdot \mathbf r)(\mathbf ...
0
votes
1answer
43 views

How do I derive geodesic equation using variational principle? [duplicate]

I am trying to derive the geodesic equation using variational principle. My Lagrangian is $$ L = \sqrt{g_{jk}(x(t)) \frac{dx^j}{dt} \frac{dx^k}{dt}}$$ Using the Euler-Lagrange equation, I have got ...
2
votes
0answers
30 views

Can the permittivity tensor always be diagonalized?

For an anisotropic medium, permittivity is a symmetric tensor, with elements as $\epsilon_{ij}$ How can one be sure that it is diagonalizable, if some elements, say $\epsilon_{xy}$ and ...
3
votes
0answers
42 views

Identifying Lorentz Covariant Equations

Statement: $\phi , A^{\mu}, T^{\mu \nu}$ are a Lorentz scalar, vector, and tensor. Which of the following equations are Lorentz covariant. a. $\phi = A_{0}$ b. $\phi = A^{\mu}A_{\mu}$ c. $\phi = ...
1
vote
1answer
24 views

Transformation of self-dual and anti-self-dual tensors and irreducibility of representations

I am working out exercise 2.5 of Maggiore's book. Part of the exercise is the following: Verify that self-dual and anti-self-dual tensors are irreducible representations of (real) dimension three of ...
4
votes
1answer
218 views

Finding the components of the tensor for potential and kinetic energy

I have a rather poor understanding of what a tensor is, but enough to apply it to the biggest part of the classical mechanics I'm studying. However, I've run into a small problem while studying "Free ...
2
votes
0answers
43 views

Work by Gravity using Tensors [closed]

Now I'm familiar with the various methods for deriving work done by gravity, but I noticed a few things about the situation, and wanted to see if I could properly apply a tensor treatment to the ...
2
votes
1answer
77 views

Components of dual vectors

(This is a close retelling of Wald, problem 2.4b. Not for homework; just curiosity and an increasingly alarming suspicion that I've never actually understood anything.) Let $Y_1 ... Y_n$ be a ...
0
votes
1answer
178 views

Jaumann deviatoric stress rate

Background about terms in this question: Hookes law and objective stress rates From my understading, the Jaumann rate of deviatoric stress is written as: $$dS/dt = \overset{\bigtriangleup}{{S}} = ...
3
votes
2answers
34 views

Possible confusion, the inertia of something yields a tensor? (trying to understand an example)

I was reading the text by Dan Fleisch titled a A Student’s Guide to Vectors and on first pages he says: An example of a tensor is the inertia that relates the angular velocity of a > rotating ...
1
vote
0answers
23 views

Calculus formulas for buoyant force?

I am launching a high-altitude balloon as a part of a physics project I am working on. I know that the amount of helium I need corresponds to about 40 newtons of lift for the launch, which is all I ...
3
votes
1answer
52 views

Tensor indices and row and column labels of corresponding representation matrices

When reading undergraduate GR literature, I often see that the authors represent tensors ${\eta^\alpha}_{\beta}$, ${\eta^\beta}_{\alpha}$, $\eta_{\alpha \beta}$, $\eta^{\alpha \beta}$ as matrices. ...
1
vote
1answer
61 views

To prove uniqueness of Rotation Tensor [closed]

How can you prove that a rotation tensor which rotates some given vector is a unique tensor? Let's say we have a vector 'a' and we take a tensor product of that vector with some tensor 'Z' such that: ...
0
votes
2answers
230 views

What exactly is $T_{\mu\nu}$?

Continuous matter is described in special relativity by the matter tensor which is the so-called stress-energy-momentum tensor. I am finding a difficulty understanding how a tensorial tool ...
2
votes
1answer
137 views

Invariant tensors in a general representation and their physical meaning

I'm trying to use tensor methods to find invariant elements of representations. Specifically I'm looking at representations of $SU(5)$. I can show that the invariant element in $5\otimes\bar{5}$ (or ...
2
votes
2answers
395 views

Derivation of the Riemann tensor confusion

I'm trying to understand the derivation of the Riemann curvature tensor as given in Foster and Nightingale's A Short Course In General Relativity, p. 102. They start by giving the covariant derivative ...
-1
votes
1answer
93 views

Weyl scalar calculation

I'm trying to compute Weyl scalars, but don't really understand the formulae for them, in the sense I don't understand how to compute them. Let's take ...
2
votes
1answer
82 views

How should Christoffel symbols be written (in LaTeX)? [closed]

I'm writing a summary of a lecture on relativity, and we've recently introduced the Christoffel symbols. It seems that the upstairs indices are the "leftmost" and the downstairs indices are somewhat ...
8
votes
1answer
290 views

Shape of the state space under different tensor products

I am currently studying generalized probabilistic theories. Let me roughly recall how such a theory looks like (you can skip this and go to "My question" if you are familiar with this). Recall: In a ...
3
votes
1answer
129 views

Repeated index in covariant derivative using abstract index notation

The same index showing up twice in the charge conservation law $\nabla_a j^a = 0$, as stated using abstract index notation, highly confuses me. If we chose a coordinate basis ...
0
votes
2answers
79 views

What is the metric tensor for?

I am wondering how to use the metric tensor, in practice? I read the book and done the exercises in A student's guide to vectors and tensors by Dan Fleisch. The concept of a tensor and their ...
1
vote
3answers
129 views

Covariant and contravariant 4-vector in special relativity

I've just learned about contra- and covariant vector in the context of special relativity (in electrodynamic) and I'm struggling with some concept. From what I found, an intuitive definition of ...
0
votes
1answer
61 views

Is an event formally a 4-vector? [duplicate]

An event is a 4D point in spacetime. At every point in spacetime there is a tangent space. 4-vectors live in the tangent space. One can contract two events using a metric tensor. Is there a process ...
12
votes
4answers
846 views

Is force a contravariant vector or a covariant vector (or either)?

I don't understand whether something physical, like velocity for example, has a single correct classification as either a contravariant vector or a covariant vector. I have seen texts indicate that ...
0
votes
1answer
64 views

Variation of a tensor

Let a change of coordinates be given by $x^{\mu}\to x^{\mu '}=x^{\mu}+\varepsilon \xi^{\mu}(x)$ with epsilon a small quantity. Given a tensor $T$ we define $\delta T:=T'(x)-T(x)$. I guess this means ...
3
votes
1answer
64 views

Cauchy stress tensor in different coordinate system

The general form of the cauchy stress tensor is given by the dyadic decomposition $$\boldsymbol \sigma = \sigma_{ij}\,\,\mathbf{e}_i \otimes \mathbf{e}_j$$ I want to know how this can be expanded in ...
1
vote
0answers
11 views

Spin coefficient transformation for null rotation with $l$ fixed [closed]

In Newmann-Penrose formalism, a Null rotation with $l$ fixed is $$l^a->l^a\\ n^a-> n^a + \bar{c}m^a + c\bar{m}^a+c\bar{c}l^a \\ m^a-> m^a+cl^a \\ \bar{m}^a-> \bar{m}^a+\bar{c}l^a $$ Using ...
1
vote
2answers
84 views

Understanding Tensor-operations, covariance, contravariance, … in the context of Special Relativity

I'm currently learning about special relativity but I'm having a really hard time grasping the Tensor-operations. Let's take the Minkowski scalar product of 2 four-vectors: $$\pmb U . \pmb V = ...
4
votes
0answers
57 views

Cauchy stress tensor for a spherically symmetric problem [closed]

Given a sperically symmetric problem, I am asked to show that its Cauchy stress tensor, in spherical coordinates will assume the form: ...
29
votes
4answers
6k views

What is the physical meaning of the connection and the curvature tensor?

Regarding general relativity: What is the physical meaning of the Christoffel symbol ($\Gamma^i_{\ jk}$)? What are the (preferably physical) differences between the Riemann curvature tensor ($R^i_{\ ...
0
votes
1answer
62 views

Why is not the D'Alembert operator a scalar?

I am taking a course on classical electrodynamics and my professor has defined the D'Alembert operator to me as: $$\square=\eta^{\mu \nu} \partial_{\mu} \partial_{\nu}$$ I have been operating using ...
0
votes
1answer
68 views

What exactly does the Kretschmann scalar implies and how does it work?

From the General Relativity class lectures I understood that this particular invariant, the Kretschmann scalar namely $$R_{\mu\nu\lambda\rho} R^{\mu\nu\lambda\rho}$$ is really important because, ...
1
vote
1answer
59 views

How can I prove that for a Killing vector $\nabla^a \nabla_a \xi^\mu = -R^b_a \xi^a$? [closed]

I'm taking a course on General Relativity and I'm trying to prove that for a Killing vector field $\xi^\mu$ the following equation holds: $$\nabla^a \nabla_a \xi^\mu = -R^\mu_a \xi^a$$ Where $R_ab$ ...
0
votes
0answers
37 views

What is the significance of 'energy ellipsoid'?

Well, today I was reading Tensors by Feynman in his lectures, where he introduced the concept of 'energy ellipsoid'. This is the following excerpt: [...] the polarization tensor can be measured ...
2
votes
1answer
266 views

Perfect fluid and Cauchy momentum equation

The stress-energy tensor of a perfect fluid is given by $$T^{\mu\nu}=\left(\rho+pc^{-2}\right)u^\mu u^\nu+pg^{\mu\nu}$$ The divergence of the stress-energy tensor is zero: $\nabla_\mu T^{\mu\nu}=0$. ...
1
vote
2answers
77 views

Gradient of vector dot tensor dot vector [closed]

Im new to tensor notation. How would one take the gradient of the expression below? $$\nabla (\vec{r}\dot{}A\dot{}\vec{r})$$ $A$ is a 3 by 3 symmetric tensor independent of $\vec{r}$.
1
vote
0answers
58 views

How do I decide when to use raised/lowered indices when calculating the amplitude of a Feynman diagram?

I am learning the Feynman rules for QCD. The book I am reading tells me that gluon propagators contribute a factor of $$\frac{-ig_{\mu\nu}\delta^{\alpha\beta}}{q^2}$$ However, in one of the ...
2
votes
1answer
57 views

Magnetic Multipole Tensor

When the electric scalar potential is expanded into spherical coordinates, one gets \begin{align} \phi (\vec r) = \frac{1}{4\pi\varepsilon_0} \sum_{l=0}^{\infty} \sum_{m=-l}^l ...
3
votes
1answer
79 views

What is the difference between scalar and vector mesons?

My understanding is that vectors and pseudooscalars change sign under parity operation and pseudovectors and scalars do not. However, I don't understand what the difference between a vector and ...
0
votes
0answers
42 views

How is the Von Mises stress used in 1D or 2D?

The Von Mises Stress is given by: $\sigma_{VM} = \sqrt{\frac{3}{2}\boldsymbol{\sigma}^{'}:\boldsymbol{\sigma}^{'}}$ My understanding is that the $\frac{3}{2}$ is included to ensure that the von mises ...
0
votes
1answer
55 views

Possible inconsistency of mixed index tensor notation

I am posting this here, because in my experience, this sort of thing exists in physics-related works only. Given a local frame $\{e_{(i)}\}$ on some $n$-dimensional manifold $M$, and given a local ...
2
votes
1answer
40 views

physical meaning of major symmetry of the stiffness tensor

What happens if a stiffness tensor does not have the "major symmetry" $C_{ijkl}=C_{klij}$? Background: In linear elasticity (generalising Hooke's law from a spring to a continuous medium), the ...
21
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4answers
4k views

Irreducible tensors concept

This might be a little naive question, but I am having difficulty grasping the concept of irreducible tensors. Particularly, why do we decompose tensors into symmetric and anti-symmetric parts? I have ...
0
votes
1answer
45 views

Notation: tetrad indices

I am trying to understand the meaning of upper and lower indices as used in the Newman-Penrose formalism. The tetrad is $\lbrace l^{a},n^{a},m^{a},\overline{m}^{a}\rbrace$, where the upper index ...
5
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3answers
11k views

What does this quote about the four dimensional divergence of an antisymmetric tensor mean?

In the beginning, God said that the four dimensional divergence of an antisymmetric second rank tensor equals zero and there was light. Can someone explain what is the meaning of this quote by ...
2
votes
1answer
210 views

Riemann tensor with 2nd and last indice the same will vanish?

I calculated that Riemann tensors are antisymmetric with respect to 2nd and last indice,as the symmetry properities of $R_{\rho\nu\sigma\mu}$ goes. $$R^{\omega}_{\ \ \ ...
1
vote
1answer
56 views

Is it right to write $\varepsilon_{ijk} \delta_{jl}=\varepsilon_{ilk}$? (indices notation)

Consider the $l$ component of vector position $\vec{r}$, $r_l$, and the $i$ component of angular momentum $\vec{L}$, $L_i$. We have that $$L_i=[r\times p]_{i}=\varepsilon_{ijk}r_jp_k$$ ...