Tensor Definition and 1000 Threads

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, [FONT=Times New Roman]A:A-1=A-1:A=I(4) If I know a fourth order tensor [FONT=Times New Roman]A, then how can I calculate [FONT=Times New Roman]A-1 ? Let's just say it is inversible. Homework...

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...

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...

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...