Tensor Definition and 1000 Threads

Hi guys, Can anyone please help me to grasp a minor detail in the derivation of the Belinfante-Rosenfeld version of the Stress-Energy Tensor (SET) ? To save type, I refer to the wiki webpage http://en.wikipedia.org/wiki/Belinfante%E2%80%93Rosenfeld_stress%E2%80%93energy_tensor Using...

Homework Statement I have the following rank-2 tensor T = \nabla \cdot \sum_{i}{c_ic_ic_i} I would like to write this using index notation. According to my book it becomes T_{ab} = \partial_y \sum_{i}{c_{ia}c_{ib}c_{iy}} Question: The change \nabla \rightarrow \partial_y and c_i...

Hello, I have difficulty interpreting the following fact (I'm reading Cotinuum Mechanics by Spencer). The relative velocity between two nearby points P and Q in the current configuarion is given by: dv_i=D_{ik}dx_k + W_{ik}dx_k where D_{ik}=\frac{d}{dt}e_{ik} is the rate of deformation tensor...

I'm working on a problem that requires me to take the cartesian metric in 2D [1 0;0 1] and convert (using the transformation equations b/w polar and cartesian coords) it to the polar metric. I have done this without issue using the partial derivatives of the transformation equations and have...

Homework Statement Hi When I want to take the divergence of a rank-2 tensor (matrix), then I have to apply the divergence operator to each column. In other words, I get \nabla \cdot M = (d_x M_{xx} + d_y M_{yx} + d_zM_{zx}\,\, ,\,\, d_x M_{xy} + d_y M_{yy} + d_zM_{zy}\,\,,\,\, d_x M_{xz} +...

I am computing the \hat{I} - moment of inertia tensor - of a cylinder with height 2h and radius R, about its axis of symmetry at the point of its centre of mass. I am working in cartesian coordinaes and am not sure where I am going wrong. (I can see the cylindirical coordiates would be the...

Hi, Consider a 2D laminar only rotating about the z axis, with the axis origin at the bottom left hand corner and adjacent sides coinciding with the z and x axes. so ω = (0,0,ωz) y = 0 I don't understand how the IXZ component is 0 to just leave the IZZ component?

Hey, I have been doing a few proofs and stumbled across this little problem. Trying to show the symmetry of the Ricci tensor by using the Riemann tensor definition ##R^m_{\ ikp} = \partial_k \Gamma^m_{\ ip} - \partial_p \Gamma^m_{\ ki} + \Gamma^a_{\ ip} \Gamma^m_{\ ak} - \Gamma^a_{\ ik}...

I'm trying to clarify for myself the relation between the stress-energy tensor and the mass scalar term in metric solutions to Einstein's equations. Maybe I should also say I'm trying to understand the energy tensor better, or how it relates to boundary conditions on the solutions. My...

Q) I am using index notation to show that ε^{0123}=-1 given that ε_{0123}=1. The soluton is: ε^{0123}=g^{00}g^{11}g^{22}g^{33}ε_{0123}=-ε_{0123} where g_{\alpha\beta} is the metric tensor. I am struggling to understand the last equality. Many thanks for any assistance.

Q) I am using index notation to show that ε^{0123}=-1 given that ε_{0123}=1. The soluton is: ε^{0123}=g^{00}g^{11}g^{22}g^{33}ε_{0123}=-ε_{0123} where g_{\alpha\beta} is the metric tensor. I am struggling to understand the last equality. Many thanks for any assistance.

Ok, so the system consists of two massive spheres, m1 and m2, of radii a and b respectively, connected by a massless rod of length R, as seen in the diagram attached. The question is to calculate the moment of inertia tensor. Sol: Set the origin at the centre of mass . So that we are in...

This thread is supposed to be a continuation of the discussion of this thread: (1) https://www.physicsforums.com/showthread.php?t=88570. The previous thread was closed but there was a lot of things I did not understand. This is also somewhat related to a recent thread I created: (2)...

First of all, I'm not sure if this thread belongs here or at the "Special & General Relativity" sub-forum, if I posted at the wrong place please move it. Homework Statement I encountered this problem working in my master's degree. I need to find the stress-energy tensor of the following...

Hi, Does somebody know a link where the Einstein tensor is fully written out, i.e. only containing the metric and its derivatives? I'm just wondering how much is actually hidden in the notation.

Hi folks. Hope that you can help me. I have an equation, that has been rewritten, and i don't see how: has been rewritten to: Can someone explain me how? Or can someone just tell me if this is correct in tensor notation: σij,jζui = (σijζui),j really hope, that...

Homework Statement Hi, I have a problem calculating the variation of the action using tensor algebra because two derivative indices are causing some problem. Homework Equations Generally you have the action S = \int L(A^{\mu}, A^{\mu}_{\;,\nu}, x^{\mu})d^4x where: A ^{\mu}=...

Hi, I am currently making an effort to solve a boundary value problem of electromagnetic field. The problem is as follows: The region ##y<0## is vacuum. The region ##y \geq 0## is filled with material with ##\mu=\mu_0## and dielectric tensor ## \left( \begin{array}{ccc} \alpha & i\beta &...

I have an vector calculus identity to prove and I need to use vector notation to do it. The identity is $$\vec{\nabla}(fg)=f\vec{\nabla}{g}+g\vec{\nabla}{f}$$ I tried starting with the left side by writing $\vec{\nabla}(fg)=\nabla_j(fg)$. Now I look and that and it really looks like there is...

Hello Everyone, I came here with a question and hope you can shed some light. We know that Ricci tensor which is a contraction of Riemann tensor contains a subset of information as contained by Riemann tensor. In 3-D infact they contain the same information. I was wondering is it always...

Would there be a direct proof of the energy-stress tensor of general relativity? My lecturer only provides me with a simplified proof - 1. Guess the form of the tensor in special relativity in co-moving frame (ρ+p)uμuv+pημv Note that the pη00 term cancels the p in u0u0, to simplify the...

Homework Statement Hi I am reading about some fluid mechanics, when suddenly I read saw that someone took the derivate of a tensor. It is in this thesis, on page 26 eq. (70). It is the final equality I can't understand. So the author is taking the derivate \partial_{x_{\alpha}}...

Given a particular system, how would one construct the stress-energy tensor? I was reading Mallett's paper and the stress-energy given for an infinitely long circulating cylinder of light is of the form T_{\mu\nu}=\epsilon \eta_\mu \eta_\nu where \eta_\mu=(\eta_0,0,\eta_2,0) and ε is the energy...

Hi everyone, :) This is a question I don't understand at all. What is the annihilator in this context? Hope you can help me out with this. Problem: Find the annihilator of the tensor \(e_1\wedge e_2+e_3\wedge e_4\) in \(V=\left<e_1,\,e_2,\,e_3,\,e_4\right>\).

Hi everyone, :) Here's is a question I have trouble understanding. Hope you can help me out. :) Specifically what is meant by the structure tensor and how is it computed when given a \(2\times 2\) triangular matrix? Problem: Write the structure tensor for the algebra \(A\) of traingular...

Suppose that we have an (n+m)-dimensional tangent space ##T_p^{n+m}## which we decompose into the direct sum of two tangent spaces ##T_p^{n+m} = T_p^n \oplus T_p^m##. We have a coordinate basis in some region of the manifold ##\left\{\partial_{\mu}\right\}_{\mu=1}^{n+m}## from which we want to...

Should it be added to the Cauchy stress to calculate a "total stress", or it doesn't have such a physical interpretation as a surface force(EM field force is usually considered more of a "body force")? Certainly when the MST was first derived before aether theories were made superfluous by...

Hi everyone, :) Here's a problem that I have trouble understanding. Specifically I am not quite getting what it means by the expression \(\alpha (t)(v)\). Hope somebody can help me to improve my understanding. :) Problem: Let \(\alpha\) be the canonical isomorphism from \(V^*\otimes V\) to...

My textbook (regarding continuum mechanics) has the following identity that is supposed to be true for all tensors: a\cdotTb = b\cdotTTa But I don't get the same result for both sides when I work it out. For each side, I'm doing the dot product last. For example, I compute Tb first and...

Homework Statement Given a square and the respective distension tensor, ε, find the position on his vortices after the transformation. ε = 0.1...0.25 ...0.25...0.1 Homework Equations The Attempt at a Solution I got kind of lost in this question. I started thinking that maybe...

Hi everyone, :) Here's a problem that I recently encountered and want to get an hint on how to solve. :) Problem: Find the value \(F(v,\,f)\) of the tensor \(F=e^1\otimes e_2+e^2\otimes (e_1+3e_3)\in T_{1}^{1}(V)\), where \(v=e_1+5e_2+4e_3\), \(f=e^1+e^2+e^3\).

I am not sure if this a right place to ask what is a tensor. I already asked about vectors in Math section, but I think a tensor has more to do with physics that mathematics, so I came here. I am reading A Zee's book Einstein Gravity that students have fear of tensors. I also think that...

Hello everyone Here is the problem: Find the value $F(v,f)$ of the tensor $F=e^1\otimes e_2 +e^2\otimes(e_1+3e_3)\in T^1_1(V)$ where $v=e_1+5e_2+4e_3, f=e^1+e^2+e^3$ Does $e^1\otimes e_2=0$ in this problem?Thanks

Suppose we are given two projection operators H' and H'' such that H' + H'' = 1, i.e. that any vector can be written as V = V' + V'' = (H' + H'') V. I'm trying to prove the formula $$R(X',Y'')Z' \cdot V'' = (Z' \cdot (\nabla'_{X'}B') + \left<X'\cdot B', Z' \cdot B'\right>)(Y'', V'') + (V''...

Given a vector \vec{r}=\begin{bmatrix} x\\ y \end{bmatrix} It's possible to decompose it linearly, so: \vec{r}=x\hat{i}+y\hat{j} So, how would the linear decomposition of a tensor? Thx!

I have been trying to think about the Levi-Civita tensor in the context of Group Theory. Is there a group that it is symmetric to? I'm sorry if this is a double post but I don't think my original identical post submitted correctly. Thanks, Nate

Hey everyone, I recently learned that my certified genius weird-uncle-who-lives-at-home (IQ over 200 something, legitimate 'genius') or WULAH for short, passes his spare time by lounging around his place and doing tensor calculus. I've done some calc in 3d in college and I know that's commonly...

Are Ricci flat manifolds analogous to flat space-time ? Further for Ricci flat manifolds does the Riemann tensor vanish ?

Hello all, In Carroll's there is a brief mention of how to get an idea about the curvature tensor using two infinitesimal vectors. Exercise 7 in Chapter 3 asks to compute the components of Riemann tensor by using the series expression for the parallel propagator. Can anyone please provide a...

I have been reading an introductory book to General Relativity by H Hobson. I have been following it step by step and now I am stuck. It is stated in the book that: "It is straightforward to show that the coordinate and dual basis vectors themselves are related... "ea = gabeb ..." I have...

Homework Statement Whats up guys! I've got this question typed up in Word cos I reckon its faster: http://imageshack.com/a/img5/2286/br30.jpg Homework Equations I don't know of any The Attempt at a Solution I don't know where to start! can u guys help me out please? Thanks!

Homework Statement Hey everyone, So I've got to prove a couple of equations to do with the Levi-Civita tensor. So we've been given: \epsilon_{ijk}=-\epsilon_{jik}=-\epsilon_{ikj} We need to prove the following: (1) \epsilon_{ijk}=-\epsilon_{kji} (2)...

If the tensor tympany is loose, the ear drum is also loose. But then, the ear drum will vibrate even by weak sound waves, which will cause increased vibration of malleus, incus, and stapes. Isn't that supposed to cause hyperacusis instead?

Hello, I am not sure what the first indice in the cauchy stress tensor indicates For example, σ_xy means that the stress in the y direction, but does x mean the cross sectional area is normal to the x direction?

Our galaxy is rotating and is charged therefore the choice for the metric is the Kerr-Newman Metric. I want to solve for the Kerr-Newman Metric Tensor. There are a few questions. 1-What is the value for Q in the equation: ##r_Q^2=\frac{Q^2*G}{4*\pi*\epsilon_0*c^4}## where ##G=6.674E-20...

Generally, when we talk about moment of inertia, we talk about rotation and inherently, we talk about moment of inertia about an axis. But when we talk about inertia tensor, we calculate about a point. Is there a reason for this difference? Am I missing something? I am new to tensors.

Ok, so I'd like some advice on doing integrals that involve a laplacian and a tensor for example =\int\frac{\delta}{\delta A_{\mu}}\frac{1}{4M^{2}}(\partial_{\rho}A_{\sigma}-\partial_{\sigma}A_{\rho})\frac{\partial^{2}}{\partial x^{2}}(\partial^{\rho}A^{\sigma}-\partial^{\sigma}A^{\rho}) where...

I am looking for the Metric Tensor of the Reissner–Nordström Metric.g_{μv} I have searched the web: Wiki and Bing but I can not find the metric tensor derivations. Thanks in advance!

Homework Statement Hi all, Here's the problem: Prove, in tensor notation, that the triple scalar product of (A x B), (B x C), and (C x A), is equal to the square of the triple scalar product of A, B, and C. Homework Equations The Attempt at a Solution I started by looking at the triple...

Homework Statement Hi guys, I would like to show that if ##t^\mu## is a temporal vector then ##t^\mu t^\nu T_{\mu\nu}## is the density of energy of the EM field measured by an observer with velocity ##t^\mu##. And that it is greater or equal to 0. Density of energy is proportional to...