Tensor Definition and 1000 Threads

0. Homework Statement Hi guys, I must show that the trace of the stress energy tensor is zero. The definition of it is ##T^{\mu \nu }=\frac{1}{4\pi} \left ( F^{\mu \sigma } F^{\nu \rho} \eta _{\sigma \rho}-\frac{1}{4} \eta ^{\mu \nu } F^{\sigma \rho} F_{\sigma \rho} \right )##. 1. The...

Homework Statement We will see (in Chap. 5) that the magnetic field can be derived from a vector potential function as follows: B = ∇×A Show that, in the special case of a uniform magnetic field B_{0} , one possible vector potential function is A = \frac{1}{2}B_{0}×r MUST USE TENSOR NOTATIONm...

Another trivial question from me. Homework Statement Which (if any) of the following are valid tensor expressions: (a)A^\alpha+B_\alpha (b)R^\alpha{}_\beta A^\beta+B^\alpha=0 (c)R_{\alpha\beta}=T_\gamma (d)A_{\alpha\beta}=B_{\beta\alpha} Homework Equations Nothing relevant -...

"Seeing" Tensor Products Is there a way to "visualize" the tensor product of two (or ##n##) vectors/tensors/algebras/etc.? I'm having a lot of trouble making the tensor product feel intuitive. I know its properties, and I can usually apply it without too much of a problem, but it does not...

Dirac has equation 3.4 as: x^{\lambda}_{,\mu}x^{\mu}_{,\nu}=g^{\lambda}_{\nu} Shouldn't that have a 4 on the right side? x^{\lambda}_{,\mu}x^{\mu}_{,\nu}=(4?)g^{\lambda}_{\nu}

I am new to tensor notation, but have known how to work with vector calculus for a while now. I understand for the most part how the Levi-Civita and Kronecker Delta symbol work with Einstein summation convention. However there are a few things I'm iffy about. For example, I have a problem where...

I am back again, with more tensor questions. I am getting better at this, but it is still a tough challenge of pattern recognition. Problem Statement Prove the following identity is true, using indicial notation: \nabla\times(\nabla \vec{v})^T = \nabla(\nabla\times\vec{v}) Attempt at...

Hi guys, I would like to know if the answer given to this thread is correct https://www.physicsforums.com/showthread.php?t=457405 I got the same doubt, is the expression for the tensor given in cartesian coordinates or is it general to any orthogonal coordinate system? Thanks in advance

Hi, I am having problems in constructing a stress-energy tensor representing a constant magnetic field Bz in the \hat{z} direction. The coordinate system is a cylindric {t,r,z,\varphi}. The metric signature is (+,-,-,-). I ended with the following mixed stress-energy tensor: Is this...

Some define the extrinsic curvature tensor as $$K_{\mu \nu} = h^{\ \ \ \sigma}_\nu h^{\ \ \ \lambda}_\nu \nabla_\sigma n_\lambda.$$ From the expression it seems like the index of the covariant derivative in can be any spacetime index. However, does it makes sense to ask what the...

In QM the tensor product of two independent electron's spin state vectors represents the product state which represents the possible unentangled states of the pair. I don't understand why the tensor product produces that result. |A⟩=|a⟩⊗|b⟩

Hi, Is there any good books which explain/calculate Wigner rotation, tensor operator, two-particle helicity state and related stuff in detail? Thanks.

Hi everyone! I would like to ask you a very basic question on the decomposition 3\otimes\bar 3=1\oplus 8 of su(3) representation. Suppose I have a tensor that transforms under the 8 representation (the adjoint rep), of the form O^{y}_{x} where upper index belongs to the $\bar 3$ rep and the...

Can anyone give me a geometrical interpretation of the weyl curvature tensor?

All but one of the tensor operations can be defined without reference to either coordinates or a basis. This can be done for instance by defining a ##(^m_n)## tensor over vector space ##V## as a multi-linear function from ##V^m(V^*)^n## to the background field ##F##. This allows us to define...

Hi guys! I need help on a problem from one of my GR texts. Suppose that ##\xi^a## is a killing vector field and consider its twist ##\omega_a = \epsilon_{abcd}\xi^b \nabla^c \xi^d##. I must show that ##\omega_a = \nabla_a \omega## for some scalar field ##\omega##, which is equivalent to showing...

I have spent much effort trying to prove that det|\Lambda\mu\upsilon| = +1 or -1 (following a successful effort to prove (3.50g) on p87 of MTW) From the result of \LambdaT\eta\Lambda = \eta I've produced four equations like: \Lambda00\Lambda00 - \Lambda01\Lambda01 - \Lambda02\Lambda02 -...

Hello.The Einstein has following definition (in my course): Gμε = Rμε - 1/2Rgμε. But why don't we just: gμεGμε = gμεRμε - 1/2Rgμεgμε. <=> G = R- 1/2 . R . 4 = R- 2R = -R? Is this wrong or.? Also, what is the meaning of the ricci scalair and tensor?

Homework Statement In order to show that if ##(\vec E (\vec x, t), \vec H (\vec x, t))## is a solution to Maxwell's equation then ##(\vec E (\vec x -\vec L, t), \vec H (\vec x-\vec L, t))## is also a solution, my professor used a proof and a step I do not understand. Let ##\vec x' =\vec...

Hello, I'm studying The Theory of Polymer Dynamics by Doi and Edwards and on page 98 there is a tensor, defined as a composition of two identical unit vectors pointing from the monomer n to the monomer m: \hat{\textbf{r}}_{nm}\hat{\textbf{r}}_{nm} As far as I understood, the unit vectors...

Hello, It may be trivial to many of you, but I am struggling with the following integral involving two spheres i and j separated by a distance mod |rij| ∫ ui (ρ).[Tj (ρ+rij) . nj] d2ρ The integration is over sphere j. ui is a vector (actually velocity of the fluid around i th...

I was working on the derivation of the riemann tensor and got this (1) ##\Gamma^{\lambda}_{\ \alpha\mu} \partial_\beta A_\lambda## and this (2) ##\Gamma^{\lambda}_{\ \beta\mu} \partial_\alpha A_\lambda## How do I see that they cancel (1 - 2)? ##\Gamma^{\lambda}_{\ \alpha\mu}...

I have a short question about the derivative of a given EM-Tensor. ##\rho## = mass density ##U^\mu## = 4 velocity ##T^{\mu\nu} = \rho U^\mu U^\nu## Now I do ##\partial_\mu## Should I get a) ##\partial_\mu T^{\mu\nu} = (\partial_\mu \rho) (U^\mu U^\nu) + \rho (\partial_\mu U^\mu)...

hi In a local inertial frame with g_{ij}=\eta_{ij} and \Gamma^i_{jk}=0. why in such a frame, curvature tensor isn't zero? curvature tensor is made of metric,first and second derivative of metric.

Let M be a module over the commutative ring K with unit 1. I want to prove that M \cong M \otimes K. Define \phi:M \rightarrow M \otimes K by \phi(m)=m \otimes 1. This is a morphism because the tensor product is K-linear in the first slot. It is also easy to show that the map is surjective...

I want to find the ricci tensor and ricci scalar for the space-time curvature at the Earth surface. Ignoring the moon and the sun. I have used the scwharzschilds metric, but then the ricci tensor and the scalar where equal to zero.

In order to understand how related are the theories of General Relativity and Electromagnetism, I am looking at the electric and magnetic parts of the Weyl tensor (in the ADM formalism) and compare them with the ones from Maxwell's theory. I have tried to look at the Poisson bracket, but the...

Homework Statement Show that all components of Riemann curvarue tensor are equal to zero for flat and Minkowski space. Homework Equations The Attempt at a Solution ## (ds)^2=(dx^1)^2+(dx^2)^2+...+(dx^n)^2 \\ R_{MNB}^A=\partial _{N}\Gamma ^{A}_{MB}-\partial _{M}\Gamma ^{A}_{NB}+\Gamma...

I've got a problem regarding tensors. Premise: we are considering a fluid particle with a velocity \mathbf{u} and a position vector \mathbf{x}; S_{ij} is the strain rate tensor, defined in this way: \displaystyle{S_{ij}=\frac{1}{2}\left(\frac{\partial u_i}{\partial x_j} +\frac{\partial...

I want to proof that [M\mu\nu,W\sigma]=i(g\nu\sigmaW\mu-g\mu\sigmaW\nu) I can reduce this expression but I can't find the correctly answer. Thanks!

Hi all! In a paper they say that a certain quantity is a rank-2 tensor because it transforms like a spin-2 object under rotations, that is: if the basis vectors undergo a rotation of angle \phi, then this quantity, say A, transforms like A\mapsto Ae^{i2\phi} As far as I knew, a rank-2...

Would anybody have some good recommendations for a workbook on tensor analysis? I'm looking for the kind of book that would ask a ton of basic questions like: "Convert the vector field \vec{A}(x,y,z) \ = \ x^2\hat{i} \ + \ (2xz \ + \ y^3 \ + \ (xz)^4)\hat{j} \ + \ \sin(z)\hat{k} to...

hello, The Weyl tensor is: http://ars.els-cdn.com/content/image/1-s2.0-S0550321305002828-si53.gif In 2 dimensions , the Riemann tensor is (see MTW ex 14.2): Rabcd = K( gacgbd - gadgbc ) [R] Now the Weyl tensor must vanish in 2 dimensions. However, working with the g g = [-1 0 0...

Homework Statement Show that a static, spherically symmetric Maxwell tensor has a vanishing magnetic field. Homework Equations Consider a static, spherically-symmetric metric g_{ab}. There are four Killing vector fields: a timelike \xi^{a} satisfying \xi_{[a}\nabla_{b}\xi_{c]} = 0 and...

Homework Statement We know that c[ij] = a[i]b[j] is a way to make a rank-2 tensor from two rank-1 tensors. We also know that C[abcxyz]=A[abc]B[xyz] is a way to make a rank-6 tensor from two rank-3 tensors. However, is there a matrix representation of this? I know the idea of a 6-dimensional...

If I have a 4x4 Covarient Metric Tensor g_{ik}. I can find the determinant: G = det(g_{ik}) How do I find the 4x4 Cofactor of g_ik? G^{ik} then g^{ik}=G^{ik}/G

Hi, I have been learning about tensor products from Dummit and Foote's Abstract Algebra and I'm a little confused. I understand the construction of going to the larger free group and "modding out" by the relations that will eventually end up giving us module structure. But just in the...

hi Riemann tensor has a definition that independent of coordinate and dimension of manifold where you work with it. see for example Geometry,Topology and physics By Nakahara Ch.7 In that book you can see a relation for Riemann tensor and that is usual relation according to Christoffel...

I'm studying Shankar's book (2nd edition), and I came across his equation (15.3.11) about spherical tensor operators: [J_\pm, T_k^q]=\pm \hbar\sqrt{(k\mp q)(k\pm q+1)}T_k^{q\pm 1} I tried to derive this using his hint from Ex 15.3.2, but the result I got doesn't have the overall \pm sign on the...

I have read the following three simplifications in various places, but together they give a contradiction, so at least one of them must be an oversimplification. Which one? (a) Interaction between two systems A and B is described by A\otimesB (b) An entangled state C is a pure state, and...

Hello, I would very much like someone to please clarify the following points concerning tensor summation to me. Suppose the components of a tensor Ai j are A1 2 = A2 1 = A (or, in general, Axy = Ayx = A), whereas all the other components are 0. Is this a symmetrical tensor then? How may Ai j be...

Hi, Homework Statement The components of the tensor Ai j are A1 2 = A2 1 = A, whereas all the other components are zero. I am asked to write A(BAR)i j, following a transformation to a new coordinate system, given that ∂q(BAR)k/∂qn = Rnk. I am expected to write my answer in terms of R...

I am just wondering, is there a difference in meaning/definition between the indices of a tensor being right on top of each other A_{\mu }^{\nu } and being "spaced" as in A{^{\nu }}_{\mu } I seem to remember that I once read that there is indeed a difference, but I can't remember what it...

http://im32.gulfup.com/u6n3f.jpg

Definitions like this one are common in books: For all ##k,l\in\mathbb N##, a multilinear map $$T:\underbrace{V^*\times\cdots\times V^*}_{k\text{ factors}}\times \underbrace{V\times\cdots\times V}_{l\text{ factors}}\to\mathbb R$$ is said to be a tensor of type ##(k,l)## on ##V##. Lee calls this...

Hi. I have a huge problem and without solving it I can't move forward. I will appreciate any help. Having the stress tensor S: 163.666557052527 -63.0272557558942 0.000000000000000E+000 -63.0272557558942 70.3802282767392 0.000000000000000E+000...

I was bored, so I tried to do something to occupy myself. I started going through withdrawal, so I finally just gave in and tried to do some math. Three months of no school is going to be painful. I think I have problems. MATH problems. :-p Atrocious comedy aside, Spivak provides a parametric...

I've PMd some of you with this question, but I got some conflicting replies or no replies at all lol, so I'm posting it here. I also did a Google search and found this which I'm almost sure answers my question, but I just want to confirm with you guys: ''In general, scalar fields are referred...

Homework Statement I'm confused about writing down the equation: \Lambda \eta \Lambda^{-1} = \eta in the Einstein convention. Homework Equations The answer is: \eta_{\mu\nu}\Lambda^{\mu}{}_{\rho}\Lambda^{\nu}{}_{\sigma} = \eta_{\rho\sigma} However it's strange because there seems...

While reading this article I got stuck with Eq.(54). I've been trying to derive it but I can't get their result. I believe my problem is in understanding their hints. They say that they get the result from the Gauss embedding equation and the Ricci identities for the 4-velocity, u^a. Is the...