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.

Filter by
Sorted by
Tagged with
11
votes
4answers
2k views

Is partial derivative a vector or dual vector?

The textbook(Introduction to the Classical Theory of Particles and Fields, by Boris Kosyakov) defines a hypersurface by $$F(x)~=~c,$$ where $F\in C^\infty[\mathbb M_4,\mathbb R]$. Differentiating ...
4
votes
2answers
3k views

Tensor Product vs. Direct Product for three spin-1/2 particles

Let us consider three spin-1/2 particles and only focusing on their intrinsic spin $S$. The Hilbert space has then to be $\mathcal H = ℂ^2 ⊗ ℂ^2 ⊗ ℂ^2$. The spin can be described by $V ∈ \text{SU(2)}$ ...
3
votes
1answer
1k views

Correct tetrad index notation

There seems to be some different conventions on the indexes of the tetrad. I am wondering which is the standard, which is correct, and which is an abuse of notation. In Sean Carroll's notes and in ...
3
votes
1answer
565 views

Why is not ${(\Lambda^T)^\mu}_\nu = {\Lambda_\nu}^\mu$?

I am following lecture notes on SR. The author writes that the following is equivalent: $$\Lambda^T\eta\Lambda = \eta \iff \eta_{\mu \nu} {\Lambda^\mu}_\rho{\Lambda^\nu}_\sigma = \eta_{\rho \sigma}. \...
6
votes
1answer
2k views

Irreducible decomposition of higher order tensors

I am familiar with the notion of irreps. My question refers simply to tensor representations (not tensor products of representations) and how can we decompose them into irreducible parts? For example, ...
5
votes
3answers
631 views

On the uniqueness of the Riemann-Christoffel tensor

According to Section 6.2, Gravitation and Cosmology by Weinberg, the Riemann-Christoffel tensor is the only tensor that can be constructed out of the second (or lower) order derivatives of the metric ...
5
votes
1answer
110 views

What eigenvector-like tools are there for analyzing tensors of rank three and higher?

If I have a rank-two tensor that I want to analyze ─ say, an electric quadrupole moment, or a moment of inertia ─ it can often be very easy to analyze by moving to its principal-axes frame: one ...
4
votes
3answers
723 views

Rigorous mathematical definition of vector operator?

In standard quantum mechanics textbooks, the concept of operators is often introduced as linear maps that map a Hilbert space $H$ onto itself: $$ \hat{O}: H \rightarrow H \, . $$ However, directly ...
3
votes
1answer
1k views

Electromagnetic Tensor in Cylindrical Coordinates

I understand that the Electromagnetic Tensor is given by $$F^{\mu\nu}\mapsto\begin{pmatrix}0 & -E_{x} & -E_{y} & -E_{z}\\ E_{x} & 0 & -B_{z} & B_{y}\\ E_{y} & B_{z} & ...
3
votes
2answers
2k views

What is pseudo-tensor?

What is the pseudo-tensor in relativity? How do we transform tensor and pseudo-tensor under parity?
2
votes
3answers
5k views

Tensor product of two different Pauli matrices $\sigma_2\otimes\eta_1 $

I'm solving problem 3.D in H. Georgi Lie Algebra etc for fun where one is to compute the matrix elements of the direct product $\sigma_2\otimes\eta_1$ where $[\sigma_2]_{ij}\text{ and }[\eta_1]_{xy}$ ...
1
vote
2answers
336 views

Why is the cosmological constant a scalar?

Maybe my understanding is just off, but the cosmological constant is just a scalar, right? What are it's units? Why a scalar? - was a tensor 'cosmological constant' ever considered or is it just not ...
9
votes
1answer
598 views

Contracting Indices

Does anyone know how to get from (1) to (2) in the system $$ \begin{align} \mathrm{g}^{\mu\nu}_{,\rho}+ \mathrm{g}^{\sigma\nu}{{\Gamma}}^{\mu}_{\sigma\rho}+ \mathrm{g}^{\mu\sigma}{{\...
8
votes
1answer
2k views

Levi Civita covariance and contravariance

I read some older posts about this question, but I don't know if I'm getting it. I'm working with a Lagrangian involving some Levi Civita symbols, and when I calculate a term containing $\epsilon^{ijk}...
5
votes
2answers
864 views

Difference between slanted indices on a tensor

In my class, there is no distinction made between, $$ C_{ab}{}^{b} $$ and $$ C^{b}{}_{ab}. $$ All I know, and read about so far, is the distinction of covariant and contravariant, form/vector, etc. ...
3
votes
1answer
412 views

Type/Valence of the stress tensor

In classical continuum mechanics, the stress tensor is said to be of type/valence (1,1) and I do not see why. If I am correct, its maps a vector $n$ defined in $\mathbb{R}^3$ (which is the normal to ...
2
votes
1answer
3k views

What is the difference between a tensor, vector, and a matrix? [duplicate]

I'm currently going through notes on a physics course and I'm having trouble understanding the difference between a tensor, a vector, and a matrix. I know that a vector is a kind of tensor and that a ...
2
votes
2answers
1k views

Basic understanding of stress tensors in a fluid

So, after having spent the last 9 hours attempting to understand the basic tenets of stress tensors in fluids, I can honestly say that I think I know less now than when I began. My questions are ...
3
votes
0answers
211 views

Non-local gravitational energy tensor

The well-known derivation of the Landau-Lifshitz gravitational energy pseudotensor, relies on several requirements: 1) that it be constructed entirely from the metric tensor 2) that it be index ...
2
votes
0answers
302 views

Stiffness tensor

Let's have a stiffness tensor: $$ a^{ijkl}: a^{ijkl} = a^{jikl} = a^{klij} = a^{ijlk}. $$ It has a 21 independent components for an anisotropic body. How does body symmetry (cubic, hexagonal etc.)...
2
votes
1answer
692 views

Are fully raised/lowered versions of Kronecker delta tensors?

I am confused. I have two textbooks contradicting each other, at least, it seems to me so. The first one – "Field theory" by Landau & Lifshitz says that by lowering or raising one index of ...
1
vote
1answer
316 views

Physical interpretation of $Q^i = \partial _\nu T^{i \nu}$

I'm trouble with exercise 1.8 of Carroll's Space-Time and Geometry: If $\partial_\nu T^{\mu \nu} = Q^\mu$, what physically does the spatial vector $Q^i$ represent? Use the dust energy momentum ...
1
vote
2answers
510 views

Covariant and contravariant permutation tensor

I have been reading up on the permutation tensor, and have come across the following expression (in 'Generalized Calculus with Applications to Matter and Forces' by L.M.B.C Campos page 709): $$e_{i_1,\...
4
votes
2answers
525 views

Tensor Derivatives (Derivation of the electromagnetic energy stress tensor)

I have the task to derive the electromagnetic energy stress tensor. I'm pretty new to the tensor index notation and I have a problem with an occuring derivative. I have to verify that $\frac{\partial ...
2
votes
1answer
158 views

Proof that 4-potential exists from Gauss-Faraday field equation

This is a problem concerning covariant formulation of electromagnetism. Given $$\partial_{[\alpha} F_{\beta\gamma]}~=~ 0 $$ how does one prove that $F$ can be obtained from a 4-potential $A$ such ...
2
votes
1answer
214 views

Fermi-Propagated Jacobi equation in the book The Large scale structure of space-time

On page 81, equation (4.6), the author use the Fermi derivative to write the Jacobi equation \begin{equation} \tag{4.6} \frac{{D^2}_\text{F}}{\partial s^2} {}_{\bot}Z^a = -{R^a}_{bcd}{}_{\bot}Z^cV^bV^...
1
vote
1answer
487 views

Defining four-vectors in General Relativity?

Nearly every general relativity textbook I look in has a section on four-vectors. However, this section is usually titled something like 'Four vectors in Special Relativity', 'Special Relativity (...
1
vote
1answer
196 views

Why is 3D stress tensor acting only on three surfaces?

I'm trying to learn about the stress tensor (in 3D) The tensors are said to have directions (the first subindex $i$ in $\sigma_{ij}$) and specify the surface upon which they act (the second subindex $...
0
votes
1answer
648 views

Matrices as second order tensors proof?

I am trying to proof that all matrices are tensors. I have got to a stage where I need to proof that: $$\gamma_{li} \gamma_{kj}= \frac{\partial q_j}{\partial q_k'} \frac{\partial q'_l}{\partial q_i}...
8
votes
2answers
5k views

Coordinate Transformation of Scalar Fields in QFT

By definition scalar fields are independent of coordinate system, thus I would expect a scalar field $\psi [x]$ would not change under the transformation $x^\mu \to x^\mu + \epsilon^\mu $. Correct? ...
22
votes
4answers
2k views

Why do Maxwell's equations contain each of a scalar, vector, pseudovector and pseudoscalar equation?

Maxwell's equations, in differential form, are $$\left\{\begin{align} \vec\nabla\cdot\vec{E}&=~\rho/\epsilon_0,\\ \vec\nabla\times\vec B~&=~\mu_0\vec J+\epsilon_0\mu_0\frac{\partial\vec E}{\...
11
votes
1answer
2k views

Diffeomorphisms, Isometries And General Relativity

Apologies if this question is too naive, but it strikes at the heart of something that's been bothering me for a while. Under a diffeomorphism $\phi$ we can push forward an arbitrary tensor field $F$ ...
13
votes
4answers
3k views

Are covariant vectors representable as row vectors and contravariant as column vectors

I would like to know what are the range of validity of the following statement: Covariant vectors are representable as row vectors. Contravariant vectors are representable as column vectors. For ...
18
votes
2answers
504 views

In relativity, can/should every measurement be reduced to measuring a scalar?

Different authors seem to attach different levels of importance to keeping track of the exact tensor valences of various physical quantities. In the strict-Catholic-school-nun camp, we have Burke 1980,...
9
votes
6answers
8k views

Gradient is covariant or contravariant?

I read somewhere people write gradient in covariant form because of their proposes. I think gradient expanded in covariant basis $i$, $j$, $k$, so by invariance nature of vectors, component of ...
7
votes
2answers
2k views

Do Dirac Gamma Matrices act like Tensors?

Do Dirac Gamma Matrices act like Tensors? Is it true that $$ \gamma_\mu = \eta_{\mu\nu}\gamma^\nu~? $$ Also what about $\sigma_{\mu\nu}$, where $\sigma_{\mu\nu}$ is defined to be: \begin{align*} \...
3
votes
5answers
4k views

Gradient, divergence and curl with covariant derivatives

I am trying to do exercise 3.2 of Sean Carroll's Spacetime and geometry. I have to calculate the formulas for the gradient, the divergence and the curl of a vector field using covariant derivatives. ...
17
votes
5answers
6k views

In what sense can a complex number be a scalar?

A definition of a scalar like A scalar is a one-component quantity that is invariant under rotations of the coordinate system (see http://mathworld.wolfram.com/Scalar.html) seems to exclude ...
10
votes
2answers
6k views

Lowering/raising metric indexes

So, I was chatting with a friend and we noticed something that might be very, very, very stupid, but I found it at least intriguing. Consider Minkowski spacetime. The trace of a matrix $A$ can be ...
8
votes
4answers
2k views

Is the covariance or contravariance of vectors/tensors something that can be “visualized”?

I'm taking an undergrad GR course, and our text (Lambourne) mentions covariant and contravariant vectors and tensors ad-nauseum, but never really gives a formal definition for what they are, and how ...
5
votes
2answers
816 views

spinor vs vector indices of Dirac gamma matrices

I am struggling to understand the nature of the components of the Dirac matrices. If we view the four components of a Dirac spinor as $\psi^a$ with $a$ being a 'spinor' index, then if a gamma matrix ...
17
votes
3answers
2k views

How to tell that the electromagnetic field tensor transforms as a tensor?

Is any matrix a tensor in special relativity? My question is inspired by the definition of the electromagnetic field tensor in Carroll's Spacetime and Geometry book. In equation (1.69), he defines a ...
11
votes
4answers
1k views

Equivalence of two formulation of Maxwell equations on manifolds

I have read about a generalization of Maxwell equation on manifolds that employs differential forms and Hodge duality that goes as follow: $$dF = 0\qquad \text{and}\qquad d \star F = J.\tag{1}$$ As I ...
9
votes
0answers
2k views

Why is the $\theta$ term of QCD violating charge and parity (CP) symmetries?

From the non-trivial nature of the QCD vacuum, the Lagrangian is augmented with a term like \begin{equation} \theta \frac{g^2}{32 \pi^2} G_{\mu \nu}^a \tilde{G}^{a, \mu \nu} \end{equation} where $ \...
9
votes
4answers
1k views

Inconsistency with partial derivatives as basis vectors?

I have been trying to convince myself that it is consistent to replace basis vectors $\hat{e}_\mu$ with partial derivatives $\partial_\mu$. After some thought, I came to the conclusion that the basis ...
5
votes
1answer
390 views

What would be a rigorous definition of position vectors and what is their role in differential geometry?

This question is actually a couple of interrelated questions all related to the "position vector" or "displacement vector" and its different roles in differential geometry and classical mechanics, but ...
3
votes
1answer
554 views

CFT OPE: Why do only symmetric traceless operators appear in the scalar-scalar OPE?

Why do only symmetric traceless operators appear in the scalar scalar OPE? The lecture notes: http://arxiv.org/abs/1602.07982, rephrase this question as: Why does $\langle\mathcal{O}^a| \phi(x) |\phi\...
12
votes
7answers
578 views

How can a set of components fail to make up a vector?

Many books in Physics insist to define vectors are objects with components with the property that the components transform in a proper way under a change of coordinates. Now, in mathematics, on the ...
9
votes
3answers
23k 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 Michio ...
7
votes
1answer
281 views

Is the distinction between covariant and contravariant objects purely for the convenience of mathematical manipulation?

Two kinds of indices, covariant and contravariant, are introduced in special relativity. This, as far as I understand, is solely for mathematical luxury, i.e. write expressions in a concise, self-...