Tensor Definition and 1000 Threads

First by antisymmetric tensor I mean the "totally antisymmetric tensor" like this: ##\epsilon^{\alpha\beta\gamma\delta} = \left\{ \begin{array}{clcl} +1 \;\; \text{when superscripts form an even permutation of 1,2,3,4} \\ -1 \;\; \text{when superscripts form an odd permutation of 1,2,3,4} \\ 0...

What do they mean by 'Contract ##\mu## with ##\alpha##'? I thought only top-bottom indices that are the same can contract? For example ##A_\mu g^{\mu v} = A^v##.

The two tensor definitions I'm (newly) familiar with, by transformation rules, and as a map from a tensor product space to the reals, don't tell me what a tensor does, and to the best of my knowledge they don't make it apparent. So, I'm looking for an operational definition, and suggesting the...

I'm looking at 'Lecture Notes on General Relativity, Sean M. Carroll, 1997' Link here:http://arxiv.org/pdf/gr-qc/9712019.pdf Page 221 (on the actual lecture notes not the pdf), where it generalizes that the energy-momentum tensor for radiation - massive particles with velocities tending to...

I am trying to prove the following. Let $V_1, \ldots, V_k$ be finite dimensional vector spaces over a field $F$. There is a natural isomorphism between $V_1^*\otimes\cdots\otimes V_k^*$ and $\mathcal L^k(V_1, \ldots, V_k;\ F)$. Define a map $A:V_1^*\times\cdots\times V_k^*\to \mathcal L^k(V_1...

(All vector spaces are over a fixed field $F$). Universal Property of Tensor Product. Given two finite dimensional vector spaces $V$ and $W$, the tensor product of $V$ and $W$ is a vector space $V\otimes W$, along with a multilinear map $\pi:V\times W\to V\otimes W$ such that whenever there is...

Homework Statement Let ##T## be a ##(1, 1)## tensor field, ##\lambda## a covector field and ##X, Y## vector fields. We may define ##\nabla_X T## by requiring the ‘inner’ Leibniz rule, $$\nabla_X[T(\lambda, Y )] = (\nabla_XT)(\lambda, Y ) + T(\nabla_X \lambda, Y ) + T(\lambda, \nabla_X Y ) . $$...

Hi, I'm looking for a program that spits out fully summed index equations. For example T_{ii} in, out comes T_{11}+T_{22}+... and so on, with Einstein summation convention.

Hi, I that <I|M|J>=M_{I}^{J} is just a way to define the elements of a matrix. But what is |I>M_{I}^{J}<J|=M ? I don't know how to calculate that because the normal multiplication for matrices don't seem to work. I'm reading a book where I think this is used to get a coordinate representation of...

What do they mean by contracting ##\mu## with ##\alpha## ?

I am making mess of the following expression.. i have following expression ## \frac{\partial{g}}{\partial{g_{\mu j}}} *g_{\nu j}=g \delta^{\mu} _{\nu} ## then I have sum over j only in the above expression. But above expression is nonzero only when ##{\mu}## is equal to ##\nu##. So we have ##...

Homework Statement (a) Show acceleration is perpendicular to velocity (b)Show the following relations (c) Show the continuity equation (d) Show if P = 0 geodesics obey: Homework EquationsThe Attempt at a SolutionPart (a) U_{\mu}A^{\mu} = U_{\mu}U^v \left[ \partial_v U^{\mu} +...

Homework Statement (a)Find Christoffel symbols (b) Show the particles are at rest, hence ##t= \tau##. Find the Ricci tensors (c) Find zeroth component of Einstein Tensor Homework EquationsThe Attempt at a Solution Part (a)[/B] Let lagrangian be: -c^2 \left( \frac{dt}{d\tau}\right)^2 +...

I'm trying to re-derive a result in a paper that I'm struggling with. Here is the problem: I wish to calculate (\nabla \otimes \nabla) h where \nabla is defined as \nabla = \frac{\partial}{\partial r} \hat{\mathbf{r}}+ \frac{1}{r} \frac{\partial}{\partial \psi} \hat{\boldsymbol{\psi}} and...

Homework Statement Consider ##T = \delta \otimes \gamma## where ##\delta## is the ##(1,1)## Kronecker delta tensor and ##\gamma \in T_p^*(M)##. Evaluate all possible contractions of ##T##. Homework Equations Tensor productThe Attempt at a Solution ##\gamma## is therefore a ##(0,1)## tensor...

I've been reading through Schutz's A First Course in General Relativity, and my solution to a particular problem has got me wondering if I'm being careful enough in my approach. The problem states: Show that, in the rest frame ##\mathcal{O}## of a star of constant luminosity ##L## (total energy...

Hello Big minds, In the book of Arfken [Math Meth for Physicists] p 134 he defined contravariant tensor...my question is about a_ij he defined them first as cosines of an angle of basis then he suddenly replaced them by differential notation...why is that? cosines are not mention in this...

Homework Statement If \sigma_{ij} = \begin{pmatrix} 3 & 3 & 3 \\ 3 & 3 & 3 \\ 3 & 3 & 3 \end{pmatrix} represents a stress tensor, on what plane(s) will the normal stress be a minimum? On what plane(s) will the shear stress be a maximum? Homework EquationsThe Attempt at a Solution The first...

I just watched susskind video on EFE but he didnt show us how to convert curvature tensor(the one with 4 indices) to that of Ricci tensor. Can anyone help me with this? Try to simplify it as I just started this.

Given an inertia tensor of a rigid body I, one can always find a rotation that diagonalizes I as I = RT I0 R (let's say none of the value of the inertia in I0 equal each other, though). R is not unique, however, as one can always rotate 180 degrees about a principal axis, or rearrange the...

Is there anywhere that teaches tensor analysis in both tensor and non tensor notation, because I'm having to pause each time i look at something in tensor notation and phrase it mentally in non tensor notation at which point it becomes staggeringly simpler. Any help apreciated

E.g - considering co variant differentiation, The issue with the normal differentiation is it varies with coordinate system change. Covariant differentiation fixes this as it is in tensor form and so is invariant under coordinate transformations.'If a tensor is zero in one coordinate system...

Hi all, I'm fairly new to GR, and I'm also somewhat new to tensors as well. I'm looking for some detailed explanation of a tensor, as I want to begin studying GR mathematically. I watched a video that was posted on PF not too long ago that was pretty good. I'm having trouble remembering who it...

I'm trying to understand what exactly it means by some tensor field to be 'well-defined' on a manifold. I'm looking at some informal definition of a manifold taken to be composed of open sets ##U_{i}##, and each patch has different coordinates. The text I'm looking at then talks about how in...

Homework Statement A system of N particles described by the vector coordinates ##\mathbf{r}_k, k = 1,2, \dots, N ## subject to 3N - f constraints can be expressed in terms of generalised coordinates ##q_i, i=1,2, \dots, f## by ##\mathbf{r}_k = \mathbf{r}_k(q_1, q_2, \dots, q_f, t)## a) Prove...

I understand that all physical laws essentially codify mathematically observed behavior. Newton codified Kepler and Brahe data, for example. Quantum Mechanics codifies observed particle behavior at relatively low speeds, etc. But Einstein had no empirical data to work from… So, I do not...

What makes it more "gravity-wavy" than the fi or psi scalar of the vector perturbations? (Im talking about metric perturbations) Thanks!

Homework Statement (a) Find faraday tensor in terms of ##\vec E## and ## \vec B ##. (b) Obtain two of maxwell equations using the field relation. Obtain the other two maxwell equations using 4-potentials. (c) Find top row of stress-energy tensor. Show how the b=0 component relates to j...

Homework Statement Suppose everything is moving slowly, How can we find the metric tensor in GR in terms of the mass contained. Homework Equations I understand in case of everything moving slowly only below equation is relevant - R00 - ½g00R = 8πGT00 = 8πGmc2 The Attempt at a Solution None.

We have ##R^{1}_{212}## as the single independent Riemann tensor component, and I'm after ##R##. From symmetry properties and contracting we can attain the other non-zero components. The solution then states that ##R_{11}=R^{1}_{111} + R^{2}_{121}=R^{2}_{121}## . I thought it would have been...

Hi guys. I am trying to understand einstein field equation and thus have started on learning tensor. For metric tensor, is it just comprised of two contra/covariant vectors tensor product among each other alone or does it requires an additional kronecker delta? I am confused about the idea...

My notes read that, the quantity ##K^{2}=K_{uv}V^{u}V^{v}## is constant along geodesics, where ##K## is a killing vector. I know my definition that the quantity on the RHS is conserved, I'm just wondering why do we call it ##K^{2}##, rather than anything else? In analogy to a killing vector, if...

I am working through an explanation in Nielson and Chuang's Quantum Computation book where they apply a CNOT gate to a state α|0>|00> + β|1>|00>. (The notation here is |0> = the column vector (1,0) and |1>=(0,1), while |00> = |0>|0>, and |a>|b>=|a>⊗|b>, ⊗ being the tensor (outer) product. I am...

In Zee's "Einstein's Gravity in a Nutshell" on page 363, while deriving the Schwarzschild solution, we have How does it work? How are the rhs and lhs equal? Where does the factor 2 come from, why just one derivative left? thanks for any replies!

Does anyone know a reference with a discussion on the experimental determination of the metric tensor of spacetime? I only know the one in "The theory of relativity" by Møller, pages 237-240. https://archive.org/details/theoryofrelativi029229mbp

Hello everyone, I'm studying Weinberg's 'Gravitation and Cosmology'. In particular, in the 'Curvature' chapter it says that the Riemann tensor cannot depend on ##g_{\mu\nu}## and its first derivatives only since: What I don't understand is how introducing the second derivatives should change...

Some subtleties of the metric tensor are just becoming clear to me now. If I take ##g_{\mu\nu}=diag(+1,-1,-1,-1)## and want to write ##\partial_\mu\phi^\mu##, it would be ##\partial_0\phi^0 -\partial_i\phi^i##, correct? ##\phi## is a 4-vector.

Hi everyone, All the books I have read until this moment only give an example of two one-forms antissymetrization, like A_{[\mu}B_{\nu]} but I want some examples like A_{[\mu\nu}B_{\sigma \rho]} or A_{[\mu}B_{\sigma \rho]} Does someone know a book or lecture notes that teach this or just...

Context: Deriving the maximally symmetric- isotropic and homogenous- spatial metric I've seen a fair few sources state that the Rienamm tensor associated with the metric should take the form: * ##R_{abcd}=K(g_{ac}g_{bd}-g_{ad}g_{bc})## The arguing being that a maximally symmetric space has...

Hi there. When I have dummy indices in a tensor equation with separate terms, I wanted to know if I can rename the dummies in the separate terms. I have, in particular: \displaystyle w_k=-\frac{1}{4}\epsilon_{kpq}\left [ \frac{\partial u_p}{\partial x_q}-\frac{\partial u_q}{\partial x_p}...

In schwarzschild metric: $$ds^2 = e^{v}dt^2 - e^{u}dr^2 - r^2(d\theta^2 +sin^2\theta d\phi^2)$$ where v and u are functions of r only when we calculate the Ricci tensor $R_{\mu\nu}$ the non vanishing ones will only be $$R_{tt}$$,$$R_{rr}$$, $$R_{\theta\theta}$$,$$R_{\phi\phi}$$ But when u and v...

Hello all, I have a quick question regarding the relation of the space-time metric and the curvature. I have determined the space-time metric, g_(alpha beta), but I am unsure as how to go from the line element ds^2 = [ 1 + (dz/dr)^2] dr^2 + r^2 dtheta^2 and the space-time metric g to the...

Hello. This is a question for the philosophers. I know just a little bit about QT and GR, but have a solid background in QM, classical physics and some particle physics. I was wondering about the stress energy tensor. I know that the graviton must have spin 2 because the source of gravity is a...

Homework Statement I have the operators ##D_{\beta}:V_{\beta}\rightarrow V_{\beta}## ##R_{\beta\alpha 1}: V_{\beta} \otimes V_{\alpha 1} \rightarrow V_{\beta}\otimes V_{\alpha 1}## ##R_{\beta\alpha 2}: V_{\beta} \otimes V_{\alpha 2} \rightarrow V_{\beta}\otimes V_{\alpha 2}## where each...

why there is a negative sign in the tensor moment of inertia??

Hi! I'm studying physics and currently taking the first mechanics course. After dealing with rotation and gyroscopes, now we're working on things like shear stress, and Hooke's law in tensor form etc. I've got Kleppner/Kolenkow but shear stress, Hooke's law in tensor form and tensors in...

I'm a mathematics major and up until now I've taken Calc 1,2,3 (so single + multivariable) a combined course in Elementary Linear Algebra + Differential Equations and PDE's. My school doesn't offer any tensor calculus classes, but I was interested in learning some of it on my own. Do I have...

Hi everybody, i have a problem that i wanted to share with you if we consider a polycrystal made of cylindrical fibers following a von mises-fisher distribution equation (17) in http://bit.do/vmisesfisher (called orientation distribution function of fibers) . i must change the probability...

The Faraday Tensor is given by: Consider the following outer product with the 4-potential: The Faraday Tensor is related to the 4-potential: F^{mn} = \Box^{m} A^n - \Box^n A^m For example, ## F^{01} = -\frac{1}{c} \frac{\partial A^x}{\partial t} - \frac{1}{c}\frac{\partial...

Hello, I am working through the MIT OCW courses 8.01 and 8.012. At my university we already learned about tensors in the first mechanics course but I don't really understand them completely. Therefore I am searching for some MIT OCW course that covers tensors. I'd be glad at any help. Apart...