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0
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1answer
115 views

Confusion with anti-symmetric tensors!

I want to prove the following fact for two antisymmetric tensors: $$ G^{+\rho (\mu} H_{\,\,\, \rho}^{+\nu)} = -\frac{1}{4}\eta^{\mu \nu} G^{+\rho \sigma}H_{\rho \sigma}^{+}. $$ When I try to do it I ...
3
votes
1answer
84 views

Confusion about the Kronecker delta symbol

I am not sure I understand what the short-hand anti-symmetrization means. I.e. I know that $$\delta_{cd}^{[ab]} ~=~ \frac{1}{2}(\delta_{c}^{a}\delta_{d}^{b} - \delta_{c}^{b}\delta_{d}^{a})$$ but how ...
15
votes
2answers
881 views

Why is stress a tensor quantity?

Why is stress a tensor quantity? Why is pressure not a tensor? According to what I know pressure is an internal force whereas stress is external so how are both quantities not tensors? I am ...
0
votes
2answers
41 views

Finding $M_{ij}$ from $J_{i} = -\frac{1}{2}\epsilon_{ijk}M_{jk}$ of the Lorentz group

I cannot understand why if $$J_{i} = -\frac{1}{2}\epsilon_{ijk}M_{jk}$$ then $$M_{ij}= -\epsilon_{ijk}J_{k}.$$ Here $M_{ij}$ is the generator of the 4 dimensional Lorentz algebra (although I have ...
8
votes
1answer
85 views

How to Visualize 1-forms and $p$-forms?

I am having trouble understanding the common way of visualizing one-forms. Example of the visualization: On Wikipedia and in several math and physics texts books, I have come across visualizations ...
0
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0answers
49 views

Coordinate Symbol confusion in general relativity

In a previous post (Finding the metric tensor from the Einstein field equation?), the equation used lambda, rho mu and nu (not sure of the names of the letters!) for the Ricci tensor and swapped to a, ...
1
vote
2answers
72 views

Is four-current a vector or a vector density?

According to MTW, $$F^{\alpha\beta}{}_{;\beta} = 4\pi J^\alpha$$ and we can infer that the four-current must be an ordinary vector field because the left side is tensorial. But Wikipedia says that ...
2
votes
3answers
67 views

What is “a vector of $SO(n)$”?

I'm watching (or trying to watch) this lecture from NPTEL on classical field theory. I've understood everything in the series up till this point, including the first half of the lecture on elementary ...
2
votes
1answer
70 views

Why do derivatives act on vector fields on a worldsheet?

The covariant derivative of a vector $A^{\mu}$ at a point $x$ is defined as $$D_z A^{\mu}=\partial_zA^{\mu}+\Gamma^{\mu}_{\rho\sigma}(x)\partial_{z}x^{\rho}A^{\sigma}$$ where Greek symbols are ...
0
votes
0answers
57 views

Covariant Derivative Chain rule?

I want to prove that a covariant derivative of a vector $A^{\mu}(x(z))$ at the point $x(z)$ in general would be defined as $$D_z ...
2
votes
0answers
36 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 ...
0
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0answers
32 views

Needed small explanation of the notation in this paper [duplicate]

There is this paper: http://digital.csic.es/bitstream/10261/43192/1/p6374_1.pdf In equation (36) of this, the second line ($\omega^{+ij} =\dots$), there is a term ...
1
vote
2answers
90 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 ...
2
votes
1answer
28 views

Ambiguity in ordering of isospin states for Clebsch-Gordan coefficients

In studying isospin for nuclear physics, I am confused a bit by an ambiguity I found. If a process that goes from $K^- + p \rightarrow \Sigma^0+ \pi^0$, I can write the isospin for the left hand side ...
2
votes
0answers
58 views

How to get result $3 \otimes 3 = 6 \oplus \bar{3}$ for $SU(3)$ irreducible representations?

Let's have $SU(3)$ irreducible representations $3, \bar{3}$. How to get result that $$ 3\otimes 3 =6 \oplus \bar{3}~? $$ I'm interested in $\bar{3}$ part. It's clear that for $3 \otimes 3$ we can use ...
1
vote
1answer
68 views

Traceless multipole moments vs non-traceless moments

There are two different possibilities to define the electric quadrupole tensor: On the one hand, one can define \begin{align}Q_{kl} = \int \rho(\mathbf r') \cdot r'_k \, r'_l d^3r',\end{align} while ...
2
votes
4answers
120 views

How does the Lorentz transformation $\Lambda^{\mu}{}_{\nu}$ transform?

For example the Four-velocity transforms as $$U^{a'}=\Lambda^{a'}{}_{\nu}U^{\nu},$$ the Faradaytensor as $$F^{a'b'}=\Lambda_{\,\,\mu}^{a'}\Lambda_{\,\,\nu}^{b'}F^{\mu\nu}$$ or in Matrixnotation: ...
2
votes
0answers
33 views

Definition of Irreducible Tensor Parts in an Exercise

I am addressing exercise 23.9 on http://www.pma.caltech.edu/Courses/ph136/yr2011/1023.1.K.pdf. The exercise says that a fluid flowing through spacetime $\vec u(\mathcal P)$ can have its gradient ...
1
vote
1answer
64 views

Stress Force - Understanding Cauchy Stress Tensor

I've been trying to understand the derivation for the Cauchy Momentum Equation for so long now, and there is one part that every derivation glides over very quickly with practically no explanation ...
3
votes
2answers
114 views

The definition of transpose of Lorentz transformation (as a mixed tensor)

In the appendix of the textbook of Group Theory in Physics by Wu-Ki Tung, the transpose of a matrix is defined as the following, Eq.(I.3-1) $${{A^T}_i}^j~=~{A^j}_i.$$ This is extremely confusing for ...
11
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4answers
376 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 ...
1
vote
0answers
47 views

Learning how to use Levi-Civita symbol

I've recently started my second course in Quantum Theory and am now often required to prove more complex commutation relations. I'm aware that the Levi-Civita symbol often makes this sort of thing a ...
2
votes
2answers
92 views

Physical interpretation of order of tensor indices

Using positional index notation with tensors is common. For example, the following simple equation from Carroll's Spacetime and Geometry text (eq. 3.146): $$ R = R^\mu_{\,\,\mu} = ...
1
vote
1answer
97 views

Proof that terms in decomposition of a tensor are symmetric and antisymmetric

Any tensor of rank 2 can be rewritten as: $$A_{bc} = \frac{1}{2}(A_{bc} + A_{cb}) + \frac{1}{2}(A_{bc}-A_{cb})$$ I can understand how that works. My question is: Prove that (independently): ...
3
votes
1answer
109 views

Vector fields and tensors in E&M

I'm confused by a very basic property of electric fields. The electric field is a vector field. Vectors are tensors. Wikipedia has the following statement in the article about the electromagnetic ...
4
votes
2answers
72 views

Tensors of rotations about an arbitrary vector in C^2

I'm trying to solve the following equation: $$e^{-i\theta/2 \sigma_{\vec{i}}^A} \otimes e^{-i\theta/2 \sigma_{\vec{i}}^B} |\Psi\rangle_{AB} = e^{i\phi} |\Psi\rangle_{AB} $$ where $e^{i\phi}$ should ...
13
votes
3answers
494 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 ...
3
votes
1answer
72 views

Tensor notation

I'm trying to understand the Maxwell Stress tensor notation. I'm given that each element in the tensor is given by ...
2
votes
1answer
54 views

Why eigenvector points to principal stress plane?

I can represent a tensor by a matrix. Suppose we are talking about a 2nd order tensor, and the matrix is therefore 3x3. If I find one eigenvector of that matrix; that vector represents normal vector ...
1
vote
1answer
76 views

Riemann Curvature Tensor Symmetries Proof

I am trying to expand $$\varepsilon^{{abcd}} R_{{abcd}}$$ by using four identities of the Riemann curvature tensor: Symmetry $$R_{{abcd}} = R_{{cdab}}$$ Antisymmetry first pair of indicies ...
0
votes
0answers
45 views

Relation involving the Lorentz transformation and the inverse of its transpose

The relation I was referring to in the title is $${\Lambda_a}^b= \eta_{ac} {L^c}_d \eta^{db}$$ where ${\Lambda_a}^b$ is the inverse transpose of $L$, the Lorentz transformation. I was wondering ...
1
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4answers
198 views

Nature of Fields in QFT

I'm not exactly an expert in quantum physics, but this seems to be a simple question, and I can't find an answer anywhere! There are specific types of fields used in physics: scalar fields (i.e. as ...
1
vote
1answer
58 views

Moment of inertia as a tensor

A professor at my university briefly stated that moment of inertia is a tensor and can be represented by a $3×3$ matrix. I don't have a good idea of what a tensor is, so I would be grateful if someone ...
2
votes
2answers
77 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 ...
1
vote
0answers
13 views

Basic understanding of stress tensors in a fluid [duplicate]

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 ...
2
votes
1answer
144 views

Tensor Product of a Bra and a Ket

What does one get if the take the tensor product of a bra and a ket, for instance, $\langle\uparrow \rvert \otimes \lvert \downarrow\rangle$? What I mean it, what is this object? What does it act on? ...
1
vote
1answer
82 views

Riemann curvature tensor notation in Wald

This question is entirely on tensorial notation in Wald's General Relativity. When specifying the properties of the Riemann tensor on pg39, he states: $R_{[abc]}^{\quad \ \ \ d} = 0$ and For the ...
3
votes
1answer
70 views

How to define tensor contraction without referring to summation?

The textbook defines a tensor to be an element in $(T^*)^k×T^l→R$. It then expresses tensors as arrays of components with respect to a certain basis, and defines tensor contraction using summation ...
1
vote
1answer
99 views

Why does the second Weyl scalar describe electromagnetic radiation?

I've been reading about the null tetrad, the Weyl tensor, and the Newman-Penrose identities, and so I found out about the Weyl scalars. While the zeroth, first, third, and fourth scalars describe ...
0
votes
1answer
59 views

Clarification of Tensor, Jacobian

Is this correct? Tensors are linear mappings between two coordinate systems on a manifold. The elements of that mapping (which include the different changes of bases at each point of the manifold) ...
1
vote
0answers
96 views

Concepts in Gabriel Kron's later papers

Gabriel Kron was an important research electrical engineer known for applying differential geometry and algebraic topology to the study of electrical system. Towards the end of his career he published ...
0
votes
0answers
62 views

Jacobian for Kronecker delta

I was revising on a bit of tensor calculus, when I stumbled upon this: $$\delta^i_j = \frac{\partial y^i}{\partial x^\alpha} \frac{\partial x^\alpha}{\partial y^j}$$ And the next statement reads, ...
11
votes
3answers
323 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 ...
2
votes
1answer
62 views

Considering the theory of special relativity: Is torque still a vector?

Considering the theory of special relativity: Is torque still a vector? In classical mechanics it is easy: You have 3 axes and thus 3 planes. Every plane has its own torque so torque has 3 ...
0
votes
2answers
269 views

What is pseudo tensor?

What is the pseudo tensor in relativity? How do we transform tensor and pseudo tensor under parity?
-1
votes
1answer
116 views

Why can't we do some basic algebra in tensor calculus?

I have a very, very stupid question on the basics of tensor calculus. Consider $R_{ij} = 0$. 1)If I expand the ricci tensor $R_{ij}= g^{lm}R_{iljm}=0$. Now, my question is that, why can't we divide ...
1
vote
0answers
74 views

Double dot product in Cylindrical polar coordinates - Strain Energy

I'm working with a problem in linear elasticity, and I have to calculate the strain energy function as follows: $$2W=σ_{ij}ε_{ij}$$ Where σ and ε are symmetric rank 2 tensors. For cartesian ...
1
vote
1answer
83 views

Interpretations of (r,s) tensors [duplicate]

A tensor of type (r,s) on a vector space V is a C-valued function T on V×V×...×V×W×W×...×W (there are r V's and s W's in which W is dual space of V) which is linear in each argument. We take (0, 0) ...
3
votes
1answer
113 views

Killing Equation, trouble with tensor algebra

I'm attempting to follow a proof that the commutator of two Killing vectors is itself a Killing vector. The source that I've posted is from my course notes. I've highlighted the part I'm stuck on. ...
4
votes
3answers
253 views

How to understand the definition of vector and tensor?

Physics texts like to define vector as something that transform like a vector and tensor as something that transform like a tensor, which is different from the definition in math books. I am having ...