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0
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0answers
25 views

Needed small explanation of the notation in this paper

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 ...
0
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0answers
21 views

Constructable components on metric and its derivatives [on hold]

This is exercise 17.2 in MTW Show that there exists no tensor with components constructable from the ten metric coefficient $g_{\alpha\beta}$ and their 40 first derivatives, ...
1
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2answers
73 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
18 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
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0answers
53 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
51 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
110 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
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0answers
31 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 ...
0
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0answers
41 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
104 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 ...
0
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0answers
27 views

Gradient of a function with base vectors [migrated]

\begin{align} \nabla\left(x_1x_3\hat{e}_1+x_2x_3\hat{e}_2\right)&=x_3\left(\hat{e}_1\otimes\hat{e}_1\right)+x_3\left(\hat{e}_2\otimes\hat{e}_2\right)\\&+\mathbf{x_1\left(\hat ...
11
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4answers
327 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
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0answers
46 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
89 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
89 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
101 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
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3answers
465 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
45 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
63 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
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0answers
43 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
183 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
53 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
72 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
12 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
129 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
76 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
65 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
94 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
54 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
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0answers
73 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
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0answers
59 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, ...
10
votes
4answers
300 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
60 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
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2answers
226 views

What is pseudo tensor?

What is the pseudo tensor in relativity? How do we transform tensor and pseudo tensor under parity?
-1
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1answer
106 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
65 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
79 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
109 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. ...
3
votes
3answers
218 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 ...
1
vote
0answers
38 views

Just a contraction of indices

I came across a contraction which is not giving the desired result. This is a toy problem in how to get a supergravity theory in low energy limit of a superstring theory using the vanishing of beta ...
1
vote
1answer
163 views

Finding the metric tensor from the Einstein field equation?

I have have set my self a challenge to learn all the maths behind the Einstein field equation (EFE), and from reading it seems that the Metric tensor is the thing we are trying to find (from the 10 ...
1
vote
1answer
72 views

Ricci curvature tensor, definition of symbols

So I know that $$R_{μν}:=R^λ_{μλν}$$ is the Ricci curvature tensor (where $R^λ_{μλν}$ is the Riemann Tensor). This is in Einstein's field equations: $$R_{μν}-\frac{1}{2}g_{μν}R=\frac{8πG}{c^4}Τ_{μν}$$ ...
3
votes
3answers
156 views

Relation between component and algebraic definition of covariant vectors

I studied contravariance and covariance concepts in following way: For any vector if we get its components by parallelogram way we achieve contravariant components, and if we want to get its ...
2
votes
4answers
221 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 ...
1
vote
1answer
62 views

What kind of physical quantity is angular displacement?

Angular Displacement is neither a vector nor a scalar. What type of physical quantity it is? Are there any other examples of that physical quantity?
1
vote
2answers
85 views

How can a gas support tensile stresses?

In working through a rigorous derivation of the compressible Navier-Stokes equations, I find that the momentum flux in the X-direction should be driven not only by the normal pressure gradient ...
1
vote
0answers
71 views

Variation of the purely covariant Riemann tensor

I need to find the variation of the purely covariant Riemann tensor with respect to the metric $g^{\mu \nu}$, i.e. $\delta R_{\rho \sigma \mu \nu}$. I know that, $R_{\rho \sigma \mu \nu} = g_{\rho ...
2
votes
1answer
98 views

Hookes Law and Objective Stress Rates

Often, in papers presenting updated Lagrangian simulation methods for solid dynamics, the following procedure for updating the (Cauchy) stress tensor is presented: First, the Cauchy stress tensor is ...