Tensor Definition and 1000 Threads

I have 2 basic questions: 1. Since a type (m,n) tensor can be created by component by component multiplication of m contravariant and n covariant vectors, does this mean an (m,n) tensor can always be decomposed into m contravariant and n covariant tensors? Uniquely? 2. Since a tensor in GR , or...

So i am studying GR at the moment, and I've been trying to figure out what the derivative (not covarient) of the mixed metric tensor $$\delta^\mu_\nu$$ would be, since this tensor is just the identity matrix surely its derivative should be zero. Yet at the same time $$\delta^\mu_\nu =...

Say I want to find ##\delta \bigg( \sqrt{- \eta_{\mu \nu} \frac{dx^{\mu}}{d \tau} \frac{dx^{\nu}}{d \tau}} \bigg)##. Is the following alright: ##\delta \bigg( \sqrt{- \eta_{\mu \nu}} \bigg( \frac{dx^{\mu}}{d \tau} \bigg)^{-1/2} \bigg( \frac{dx^{\nu}}{d \tau} \bigg)^{1/2} \bigg)##?

The General Relativity text I am using gives two forms of the Electromagnetic Energy Momentum Tensor: {\rm{ }}\mu _0 S_{ij} = F_{ik} F_{jk} - \frac{1}{4}\delta _{ij} F_{kl} F_{kl} \\ {\rm{ }}\mu _0 S_j^i = F^{ik} F_{jk} - \frac{1}{4}\delta _{ij} F^{kl} F_{kl} \\ I don't see how these...

I am studying the generation of tensor perturbations during inflation, and I am trying to check every statement as carefully as possible. Starting from the metric ds^2 = dt^2 - a^2(\delta_{ij}+h_{ij})dx^idx^j I make use of Einstein's equations to find the equation of motion for the...

In my lecture we were discussing the Lagrangian construction of Electromagnetism. We built it from the vector potential ##A^\mu##. We introduced the field tensor ##F^{\mu \nu}##. We could write the Langrangian in a very short fashion as ##-\frac{1}{4}F_{\mu \nu}F^{\mu \nu}## In the end we...

Hey! :o Could you explain to me the stress tensor?? (Wondering) I haven't really understood what it is...

In the Einstein Field Equations: Rμν - 1/2gμνR + Λgμν = 8πG/c^4 × Tμν, which tensor will describe the coordinates for the curvature of spacetime? The equations above describe the curvature of spacetime as it relates to mass and energy, but if I were to want to graph the curvature of spacetime...

Homework Statement Hi all, I'm trying to learn how to manipulate tensors and in particular to differentiate expressions. I was looking at a Lagrangian density and trying to apply the Euler-Lagrange equations to it. Homework Equations Lagrangian density: \mathcal{L} = -\frac{1}{2}...

Hello buddies, Could someone please help me to understand where the second and the third equalities came from? Thanks,

I'm relatively new to differential geometry and would like to check that this is the correct definition for the tensor product of (for simplicity) two one-forms \alpha,\;\beta\;\;\in V^{\ast} : (\alpha\otimes\beta)(\mathbf{v},\mathbf{w})=\alpha (\mathbf{v})\beta (\mathbf{w}) where...

Homework Statement I am given this metric: ##ds^2 = - c^2dt^2 + a(t)^2 \left( dx^2 + dy^2 + dz^2 \right)##. The non-vanishing christoffel symbols are ##\Gamma^t_{xx} = \Gamma^t_{yy} = \Gamma^t_{zz} = \frac{a a'}{c^2}## and ##\Gamma^x_{xt} = \Gamma^x_{tx} = \Gamma^y_{yt} = \Gamma^y_{ty} =...

Why cosmological constant term ##\Lambda g_{uv}## in Einstein equation is proportional to ##g_{uv}##. Why it is even proportional to ##g_{uv}## in spacetime of MInkowski?

In textbooks, a tensor is usually defined in terms of its transformation properties. But this definition actually seems vague when it comes to checking a set of quantities to see whether they form a tensor or not. Imagine I have four functions and want to see whether they form a 2d 2nd rank...

I am reading Baez's article http://arxiv.org/pdf/gr-qc/0103044v5.pdf and I do not understand paragraph before equation (10), page 18. Equation (9) will be true if anyone component holds in all local inertial coordinate systems. This is a bit like the observation that all of Maxwell’s equations...

Homework Statement I'm just wondering if I'm doing this calculation correct? eta and f are both tensors Homework EquationsThe Attempt at a Solution \frac{\delta \left ( \gamma_{3}f{_{\lambda}}^{k}f{_{k}}^{\sigma}f{_{\sigma}}^{\lambda} \right )}{\delta f^{\mu\nu}}=\frac{\delta\left (\gamma_{3}...

as a new proposal for QGhttp://arxiv.org/abs/1502.05385 Tensor network renormalization yields the multi-scale entanglement renormalization ansatz Glen Evenbly, Guifre Vidal (Submitted on 18 Feb 2015) We show how to build a multi-scale entanglement renormalization ansatz (MERA) representation of...

Homework Statement [/B] So, I need to show Lorentz covariance of a Proca field E-L equation, conceptually I have no problems with this, I just have to make one final step that I cannot really justify. Homework Equations "Proca" (quotation marks because of the minus next to the mass part, I...

Homework Statement A sphere with dielectric constant ##\varepsilon## and radius R is placed inside a homogenous external electric field ##\vec E_0##. The sphere is divided in 2 hemispheres such that their common interface is orthogonal to the external field. Using the energy-momentum tensor...

I've been thinking about the number of degrees of freedom in a tensor with n indices in 2-dimensions which is traceless and symmetric. Initially, there are 2^{n} degrees of freedom. The hypothesis of symmetry provides n!-1 number of conditions of the form: T_{i_{1}, \ldots i_{n}}-...

When writing ##A_{a}\text{ }^{b}## one means ''The element on the a-th row and b-th column of the TRANSPOSE of A" right? Such that ##A_{a}\text{ } ^{b}= A^{b}\text{ } _{a}## ? I would just like a confirmation so I'm not learning the convention in a wrong manner.

Homework Statement Let Aijkl be a rank 4 square tensor with the following symmetries: A_{ijkl} = -A_{jikl}, \qquad A_{ijkl} = - A_{ijlk}, \qquad A_{ijkl} + A_{iklj} + A_{iljk} = 0, Prove that A_{ijkl} = A_{klij} Homework EquationsThe Attempt at a Solution From the first two properties...

Hi, my question is the title, if Ricci tensor equals zero implies flat space? Thanks for your help

Homework Statement Calculate the force of magnetic attraction between the northern and southern hemispheres of a uniformly charged spinning spherical shell, with radius R, angular velocity ω, and surface charge density σ. Use the Maxwell Stress TensorHomework Equations F=\oint \limits_S \...

Seems exist some relationship between the inverse of a matrix with the inertia tensor, looks: This relationship really exist?

Acting upon a vector say, so it is defined as: ##\frac{d}{d\lambda}V^{u}+\Gamma^{u}_{op}\frac{dx^{o}}{d\lambda}V^{p}=\frac{DV^{u}}{D\lambda}## And this can also be written in terms of the covariant derivative, ##\bigtriangledown_{k}## by ##\frac{DV^{u}}{D\lambda}=\frac{d x^{k}}{d \lambda}...

Homework Statement I was working through my text on deriving the tensor for Angular momentum of the sums of elements of a rigid body, I follow it all except for one step. Here is a great page which shows the derivation nicely - http://www.kwon3d.com/theory/moi/iten.html I follow clearly to the...

Hello, Let ##g_{jk}## be a metric tensor; is it possible for some ##i## that ##g_{ii}=0##, i.e. one or more diagonal elements are equal to zero? What would be the geometrical/ topological meaning of this?

I just have a quick question on which order around the numerator and denominator should be in the jacobian matrix that multiplies the expression. As in general Lecture Notes on General Relativity by Sean M. Carroll, 1997 he has the law as ## \xi_{\mu'_{1}\mu'_{2}...\mu'_{n}}=|\frac{\partial...

I'm looking at the informal arguements in deriving the EFE equation. The step that by the bianchi identity the divergence of the einstein tensor is automatically zero. So the bianchi identity is ##\bigtriangledown^{u}R_{pu}-\frac{1}{2}\bigtriangledown_{p}R=0##...

I have a conceptual question associated with one of the worked examples in my notes. The question is: 'Let ##\nabla## and ##\nabla^*## be connections on a manifold ##M##. Show that ##H(X,Y) = \nabla_X Y - \nabla_X^* Y## where ##X,Y## are vector fields defines a (1,2) tensor on M. To show it...

Hi, I am studying Sean Carroll's "Lecture notes on General Relativity". In the second chapter he identifies the volume element d^nx on an n-dimensional manifold with dx^0\wedge\ldots\wedge dx^{n-1}. He then claims that this wedge product should be interpreted as a coordinate dependent object...

Homework Statement I'm looking for how to calculate inverse of the 4th order tensor. That is, A:A-1=A-1:A=I(4) If I know a fourth order tensor A, then how can I calculate A-1 ? Let's just say it is inversible. Homework EquationsThe Attempt at a Solution

Do you know any good software for manipulating tensors: varying Lagrangians, checking gauge and supersymmetry transformations, etc. ? One that could deal with anti-commuting variables would be good too. One that also supplied group constants for SU(n) etc. would also be useful. I was also...

Taken from Hobson's book: How is this done? Starting from: R_{abcd} = -R_{bacd} Apply ##g^{aa}## followed by ##g^{ab}## g^{aa}g^{aa} R_{abcd} = -g^{ab}g^{aa}R_{bacd} g^{ab}R^a_{bcd} = -g^{ab}g^{aa}R_{bacd} R^{aa}_{cd} = - g^{ab}g^{aa} R_{bacd} Applying ##g_{aa}## to both sides...

The principle of least action states that the evolution of a physical system - how a system progresses from one state to another- is given by a stationary point of the action. So I think this is varying the path and keeping two points fixed- the points of the initial and final state I know...

Just one last question today if someone can help. I'm trying to derive the electromagnetic field strength tensor and having a little trouble with (i think) the use of identities, please see below: I understand the first part to get -Ei, but it's the second line of the next bit I don't...

Homework Statement Homework Equations Relabelling of indeces, 4-vector notation The Attempt at a Solution The forth line where I've circled one of the components in red, I am unsure why you can simply let ν=μ and μ=v for the second part of the line only then relate it to the first part and...

I'm trying to show that \int d^3x \,x^\mu \left(\partial_\mu \partial_0-g_{\mu 0} \partial^2 \right)\phi^2(x)=0 . This term represents an addition to a component of the energy-momentum tensor \theta_{\mu 0} of a scalar field and I want to show that this does not change the dilation operator...

I'm having difficulties understanding how I should calculate the angular velocities of a rigid body when the inertia tensor is given in body coordinates and has off diagonal elements. Let's assume I have an inertia tensor ## I = \begin{bmatrix} I_{xx} & -I_{xy} & -I_{xz} \\ -I_{yx} &...

I should calculate the variation of the Ricci scalar to the metric ##\delta R/\delta g^{\mu\nu}##. According to ##\delta R=R_{\mu\nu}\delta g^{\mu\nu}+g^{\mu\nu}\delta R_{\mu\nu}##, ##\delta R_{\mu\nu}## should be calculated. I have referred to the wiki page...

Homework Statement Given the Lagrangian density \Lambda = -\frac{1}{c}j^lA_l - \frac{1}{16 \pi} F^{lm}F_{lm} and the Euler-Lagrange equation for it \frac{\partial }{\partial x^k}\left ( \frac{\partial \Lambda}{\partial A_{i,k}} \right )- \frac{\partial \Lambda}{\partial A_{i}} =0...

I would appreciate any help with the following question: I know that for relativistic field theories, the stress tensor can be obtained from the classical action by differentiating with respect to the metric, as is explained on the wikipedia page...

Einstein's static universe obeys ##\rho = 2\lambda##. So, attractive and repelling gravity cancel each other. I'm curious about the spacetime in this universe. Because the scale factor is constant, it seems that neighboring co-moving test particles don't show relative acceleration, thus no...

Homework Statement I'm currently trying to work through some issues I'm having with tensor and vector analysis. I have an equation of the form $$\textbf{a} \bullet \textbf{b} = \textbf{c} \bullet \textbf{d}$$ where all quantities here are vectors. I want to solve for ##\textbf{b}## by finding...

Hi, I've written a little fortran code that computes the three Eigenvectors \vec{v}_1, \vec{v}_2, \vec{v}_3 of the inertia tensor of a N-Particle system. Now I observed something that I cannot explain analytically: Assume the position vector \vec{r}_i of each particle to be given with respect...

Hello all, I have a homework question that I am almost 100% sure that I solved, so I do not believe that this post should go into the "Homework Questions" section. This thread does not have to do with the answer to that homework question anyways, but rather a curiosity about whether or not this...

If you don't like indexes, look away now. I got these terms from a tensor calculus program as part of a the transformed F-P Lagrangian. \begin{align} {g}^{b a}\,{g}^{d e}\,{g}^{f c}\,{X}_{a,b c}\,{X}_{d,e f}\\ +{g}^{b a}\,{g}^{c f}\,{g}^{e d}\,{X}_{a,b c}\,{X}_{d,e f}\\ +{g}^{b a}\,{g}^{c...

In the process $$e^+e^- \rightarrow \gamma \gamma$$ for which the amplitude can be written as: $$M= \epsilon^*_{1\nu}\epsilon^*_{2\mu}(A^{\mu\nu}+\tilde{A}^{\mu\nu})$$, where $$\epsilon_i$$ is the polarization vector of a photon. How can one find the tensors $$A^{\mu\nu}$$ and...

Hello Say, the metric tensor is diagonal, ##g=\mbox{diag}(g_{11}, g_{22},...,g_{NN})##. The (null) geodesic equations are ##\frac{d}{ds}(2g_{ri} \frac{dx^{i}}{ds})-\frac{\partial g_{jk}}{\partial x^{r}}\frac{dx^{j}}{ds}\frac{dx^{k}}{ds} = 0## These are ##N## equations containing ##N## partial...