The tag has no usage guidance.

learn more… | top users | synonyms (1)

6
votes
6answers
719 views

Is there a physical interpretation of a tensor as a vector with additional qualities?

What is a tensor? has been asked before, with the most highly up-voted answer defining a tensor of rank $k$ as a vector of a tensor of rank $k-1$. But if a scalar is defined as a physical quantity ...
1
vote
0answers
46 views

How can a vector have both contra-variant and co-variant components? [duplicate]

I have read that contra-variant and co-variat vectors have different transformation properties , which distinguish them, yet at the same time I have read that a vector can have contra-variant and ...
1
vote
3answers
152 views

Convention of tensor indices

Let $g_{ij}$ be the diagonal Minkowski metric tensor diag$(g) = (1,-1,-1,-1)$, then $g^{ij}$ is defined to be $(g^{-1})^{ij}$, hence $$g_{ik}g^{kj} = g_i^{\ \ j} = \text{diag}(1,1,1,1)=\delta_i^{\ \ ...
4
votes
2answers
479 views

Sign of the totally anti-symmetric Levi-Civita tensor $\varepsilon^{\mu_1 \ldots}$ when raising indices

I am confused with the sign we get when we want to raise or lower all indices of the totally anti-symmetric tensor of any rank. Take the metric to be mostly plus ($-+\ldots+$). Then is it ...
2
votes
0answers
141 views

Energy-momentum tensor

I need to show that: \begin{align} \mathcal h_i^a \, T_{ab} \, h_i^b=(\nabla_i \phi)^2-\frac{h_{ii}}{2}[\dot{\phi}^2-(\nabla \phi)^2-m^2 \phi^2] \end{align} where i) $T_{ab}=\nabla_a \phi ...
1
vote
2answers
204 views

Raising and Lowering indices of tensor

Why we use metric tensors $g$ to raise or lower indices of tensors, why not using other (invertible) order-2 tensors to do the job?
0
votes
1answer
133 views

Einstein notation in Electrodynamics

I have a question about the notation used in Electrodynamics, for the Eq.6.5.10. Can anyone tell me why is the sign of the covariant derivative swapped when it's transformed to a contravariant ...
1
vote
0answers
80 views

What is the relationship between the formal definition of a tensor and the frequently discussed notion of a “higher order matrix”?

I've been doing some self study on the principles of tensors & manifolds in preparation for a first course in general relativity. I tend to learn better when presented with the full mathematical ...
2
votes
2answers
786 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 $V ∈ \text{SU(2)}$ ...
1
vote
1answer
128 views

Physical visualisation of curvature

I was wondering-how do you visualise curvature in the context of general relativity. The gravity well and trampoline analogies are quite wrong, so I want a more realistic approach to it (say, the way ...
4
votes
1answer
75 views

How can we see that the Riemann curvature tensor is covariant?

The Riemann curvature tensor, using the conventions of wikipedia, is written in terms of Christoffel symbols as: $$ \tag{1} R^\lambda_{\,\,\mu \nu \rho} = \partial_\nu \Gamma^\lambda_{\,\,\rho \mu} - ...
0
votes
1answer
199 views

Prove $G^{+\rho(\mu}H^{+\nu)}{}_{\rho} = -\frac{1}{4}\eta^{\mu \nu}G^{+\rho \sigma}H^+_{\rho \sigma}$ [closed]

I want to prove the following fact for two antisymmetric tensors: $$ G^{+\rho (\mu} H^{+\nu)}{}_{ \rho} = -\frac{1}{4}\eta^{\mu \nu} G^{+\rho \sigma}H_{\rho \sigma}^{+}. \tag{4.39}$$ (See e.g. ...
4
votes
1answer
111 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 ...
1
vote
1answer
642 views

Raising and lowering indices of the Levi-Civita epsilon symbol in two dimensions

In two dimensions, what is the relation between $\epsilon^a{}_b$ and $\epsilon_{ab}$ where $a, b$ take the values $\{1,2\}$? By that I mean, how does the sign change in that case? In four dimensions ...
20
votes
2answers
7k 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
46 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 ...
1
vote
2answers
257 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 ...
1
vote
1answer
84 views

Electric Magnetic duality

In this paper http://arxiv.org/abs/hep-th/9705122 Section 2 We have $$S_A = \frac{1}{4g^2} \int{d^4x F_{\mu\nu}(A)F^{\mu\nu}(A)}$$ where $F_{\mu\nu}(A) = \partial_{[\mu A\nu]}$. Its Bianchi Identity ...
2
votes
3answers
92 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
95 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 ...
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 ...
0
votes
0answers
34 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 ...
2
votes
2answers
389 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
79 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 ...
5
votes
1answer
637 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 ...
2
votes
0answers
61 views

Is it possible to build a tensor with the following properties? [closed]

I am searching for a tensor in 4-dimensional space-time with two indices that satisfy: \begin{eqnarray} M_{;\mu }^{\mu \nu } &=&0 \\ M^{\mu \nu } &=&-M^{\mu \nu } \nonumber \\ ...
1
vote
1answer
247 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
327 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
74 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 ...
3
votes
2answers
294 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 ...
5
votes
2answers
615 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
votes
4answers
781 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
123 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
172 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} = ...
3
votes
2answers
398 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 ...
0
votes
0answers
67 views

Why doesn't this proof change indices?

In this pdf, in the second line of the proof, $\sigma$ was plugged in where it appears as $$\frac{\partial x^\sigma}{\partial y^{\rho'}}$$ Meanwhile in converting the coordinates of $g^{\mu'\rho'}$, ...
1
vote
1answer
342 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): ...
5
votes
3answers
339 views

Geometric meaning of parallel transport

The definition of parallel transport of a vector $v^b$ along a curve $C$ with tangent field $\it{t}^a$ is given by Wald's GR as $$t^a \nabla_a v^b = 0$$ Is it correct to think of $\nabla_a v^b$ as ...
1
vote
1answer
39 views

Tensors applied to vector and dual vector fields in GR

This is a specific question about tensor manipulation in Wald's GR. I'm asking for clarification of a trivial step, because I'm working through the text outside the context of a class, without prior ...
3
votes
1answer
242 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
77 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 ...
16
votes
3answers
779 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
103 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
156 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 ...
2
votes
2answers
298 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
2answers
260 views

Differentiation in general relativity

If we have: $$ \frac{d\phi^a}{d\tau}= \frac{\partial \phi^a}{\partial x^\mu} \frac{dx^\mu}{d\tau} \tag{1}$$ Differentiating it, we get: $$ \frac{\partial \phi^a}{\partial ...
2
votes
1answer
182 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 ...
2
votes
1answer
112 views

Why is the full eigenfunction a product of eigenfunctions and not a sum?

For example suppose there is a two electron system. Why is the full eigenfunction a product of the spatial eigenfunction and spin-wave-function for the two electron system?
2
votes
4answers
255 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 ...
3
votes
0answers
137 views

Question about derivation of tensor in Di Francesco's CFT

This is a question for anyone who is familiar with Di Francesco's book on Conformal Field theory. In particular, on P.108 when he is deriving the general form of the 2-point Schwinger function in two ...