Tensor Definition and 1000 Threads

Is there a coordinate independent/geometric definition of a tensor density?

In my note, we have written the field strength tensor as: F^{\mu\nu} =\partial ^\mu A^\nu -\partial ^\nu A^\mu = \begin{pmatrix} 0&E_x &E_y&E_z \\ -E_x&0 &B_z &-B_y \\ -E_y&-B_z &0 &B_x \\ -E_z&B_y &-B_x&0 \end{pmatrix} But if I look into...

Hi, I'm trying to write down the stress-energy tensor for a single photon in GR, but I'm running into trouble with its transformation properties. I'll demonstrate what I do quickly and then illustrate the problem. Given a photon with wavevector p, we write {\bf T} = \int \frac{\mathrm{d}^3...

Is there a simple intuitive description of what the Ricci tensor and scalar represent? I have what seems to me a straightforward understanding of what the Riemann tensor Rabcd represents, as follows. If you parallel transport a vector b around a tiny rectangle, the sides of which are determined...

How do you find the inverse of metric tensor when there are off-diagonals? More specifivally, given the (Kerr) metric, $$ d \tau^2 = g_{tt} dt^2 + 2g_{t \phi} dt d\phi +g_{rr} dr^2 + g_{\theta \theta} d \theta^2 + g_{\phi \phi} d \phi^2 + $$ we have the metric tensor; $$ g_{\mu \nu} =...

Tensors can be of type (n, m), denoting n covariant and m contravariant indicies. Rank is a concept that comes from matrix rank and is basically the number of "simple" terms it takes to write out a tensor. Sometimes, however, I recall seeing or hearing things like "the metric tensor is a rank 2...

I noticed that sometimes exist a parallel between scalar and vector calculus, for example: ##v=at+v_0## ##s=\int v dt = \frac{1}{2}at^2 + v_0 t + s_0## in terms of vector calculus ##\vec{v}=\vec{a}t+\vec{v}_0## ##\vec{s}=\int \vec{v} dt = \frac{1}{2}\vec{a}t^2 + \vec{v}_0 t + \vec{s}_0##...

If I start with the stress-energy tensor T^{\mu\nu} of the electromagnetic field and then apply energy-momentum conservation \partial_\mu T^{\mu\nu}=0, I get a whole bunch of messy stuff, but, e.g., with \nu=x part of it looks like -E_x \nabla\cdot E, which would vanish according to Maxwell's...

According to Wikipedia, This definition doesn't sit well with me. Flux is defined as the rate that something passes through an infinitesimal surface, divided by the infinitesimal area of that surface. For example, the current flux (or current density), when dotted with a unit vector, gives...

Suppose we have some two-dimensional Riemannian manifold ##M^2## with a metric tensor ##g##. Initially it is always locally possible to transform away the off-diagonal elements of ##g##. Suppose now by choosing the appropriate equivalence relation and with a corresponding surjection we construct...

Homework Statement Problem as stated: Consider a vector A^a. Is the four-component object \left( \frac{1}{A^0},\frac{1}{A^1},\frac{1}{A^2},\frac{1}{A^3}\right) a tensor? Homework Equations Roman indices run from 0 to 3. Einstein summation convention is used. Tensors of rank 1 (vectors)...

Homework Statement Hi, I am not sure if this is the right place for my question but here goes! The stress tensor in the Si coordinate system is given below: σ’ij = {{-500, 0, 30}, {0, -400, 0}, {30, 0, 200}} MPa Calculate the stress tensor in the L coordinate system if: cos-1a33=45°, and...

In the case of swarms of particles, the stress energy tensor can be derived by considering the flow of energy and momentum "carried" by the particles along their worldlines. Is there a way to interpret the field definition of the stress energy tensor from Wald, p455 E.1.26 T_{ab} \propto...

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...

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 \(V=\left<e_1,\,e_2,\,e_3,\,e_4\right>\).

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

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!