Tensor Definition and 1000 Threads

Hi everyone! A few days ago in General Relativity class, the professor introduced the concept of Lie derivative and at the end he mentioned that the Lie derivative was a tensor itself. I've been looking everywhere, but I only find how it acts on vectors, tensors, etc. Does anyone know of any...

First of all I have this system $$\begin{pmatrix}\tau_{xx} \\ \tau_{yy} \\ \tau_{zz} \\ \tau_{xy} \\ \tau_{yz} \\ \tau_{zx} \end{pmatrix}=\begin{pmatrix}C_{11} & C_{12} & C_{12} & 0 & 0 & 0 \\ C_{12} & C_{11} & C_{12} & 0 & 0 & 0 \\ C_{12} & C_{12} & C_{11} & 0 & 0 & 0 \\ 0 & 0 & 0 & C_{44} & 0...

In the spectrum of the closed string, we encountered a graviton. Why is the symmetric 2-tensor in the open string spectrum, not a graviton?

This is exercise 1.8.3 from Foster & Nightingale: Show that if ##\sigma_{ab} = \sigma_{ba}## and ##\tau^{ab} =-\tau^{ba}## for all ##a##, ##b##, then ##\sigma_{ab}\tau^{ab}=0##. I began writing down ##\sigma_{ab}\tau^{ab}=\sigma_{ba}(-\tau^{ba})=-\sigma_{ba}\tau^{ba}##. Here I got stuck and...

In summary Tzz in maxwells stress tensor represents a force per unit area in the z direction acting on an area element that is oriented along the z direction also why it could be non zero eventhough the electric field along z is zero and I'm talking here in electrostatic

So i am confused as to what can be parallel transported , can an arbitrary tensor be transported along any curve that we wish , or do we define a curve and then solve the equation of parallel transport (which is a linear first order differential equation ) and then the solutions we get from...

Are the following two equations expressing the gradient and curl of a second-rank tensor correct? $$ \nabla R_{ij} = \frac{\partial R_{ij}}{\partial x_k} $$ $$ \nabla \times R_{ij} = \epsilon_{ijk} \frac{\partial R_{ij}}{\partial x_k} $$ If so, then the two expressions only differ by the...

In linearised theory, the polarisation tensor ##A_{\mu \nu}## (defined through ##\bar{h}_{\mu \nu} = A_{\mu \nu} e^{ik_{\rho} x^{\rho}}##) transforms under a gauge shift ##x \mapsto x + \xi## with a harmonic function ##\xi_{\mu} = X_{\mu} e^{ik_{\rho} x^{\rho}}## like: $$A'_{\mu \nu} = A_{\mu...

The abstract of my new article (Eur. Phys. J. C 84, 488 (2024)): The Dirac equation is one of the most fundamental equations of modern physics. It is a spinor equation, but some tensor equivalents of the equation were proposed previously. Those equivalents were either nonlinear or involved...

In special relativity, there's an antisymmetric rank-2 angular-momentum tensor that's "structurally" very similar to the electromagnetic field tensor. Much like you can extract from the latter (and its Hodge dual) a pair of invariants through double contractions (##\vec E \cdot \vec B## and...

Dirac (GTR, p. 37) shows simply that for a scalar function ##S## $$\int S\sqrt{-g}\,d^4 x = \int S'\sqrt{-g'}\,d^4 x'$$ and this works precisely because ##S=S'## for a scalar. But for a tensor ##T^{\mu\nu}## the same procedure gives $$\int T^{\mu\nu}\sqrt{-g} \, d^4 x = \int x^\mu_{\,\...

Is it possible to move Stress-Energy tensor to the left side of EFE? R=T => R-T=0. Relativists move cosmological constant Λ to the right side of EFE. Can we move SE tensor to make a vacuum?

Here is an action for a theory which couples gravity to a field in this way:$$S = \int d^4 x \ \sqrt{-g} e^{\Phi} (R + g^{ab} \Phi_{;a} \Phi_{;b})$$I determine\begin{align*} \frac{\partial L}{\partial \phi} &= \sqrt{-g} e^{\Phi} (R + g^{ab} \Phi_{;a} \Phi_{;b}) \\ \nabla_a \frac{\partial...

My question is about this step in the derivation: When the ##\partial_\nu \mathcal L## in 3.33 moves under the ##\partial_\mu## in 3.34 and gets contracted, I'd expect it to become ##\delta_{\mu \nu} \mathcal L##. Why is it rather ##g_{\mu \nu} \mathcal L## in the 3.34? (In this text, ##g_{\mu...

I am currently reading this book on multilinear algebra ("Álgebra Linear e Multilinear" by Rodney Biezuner, I guess it only has a portuguese edition) and the book defines an Algebra as follows: It also defines the direct sum of two vector spaces, let's say V and W, as the cartesian product V x...

Penrose demonstrates in his book "The Road to Reality" a "diagrammatic tensor notation", e.g., As I haven't seen it anywhere else, I wonder if anybody else uses it.

$$K'_{ij}A'_{jk}=B'_{ik}=a_{ip}a_{kq}B_{pq}=a_{ip}a_{kq}K_{pr}A_{rq}=a_{ip}a_{kq}K_{pr}a_{jr}a_{kq}A'_{jk}$$$$K'_{ij}=a_{ip}a_{kq}a_{kq}a_{jr}K_{pr}$$ Can someone point out my mistake? What I've found shows that K is not a tensor. It is different from my book and I cannot find my mistake...

It's a 4th-dimensional 4th-rank tensor so at first we have ##4^4=256## components. According to the book, Given that ##R_{iklm}=-R_{ikml}## 256 components reduces to 96. But I cannot see how. For one pair of i,k 16 components are dependent. We have 12 pairs of i,k(for ##i≠k## becsuse for i=k...

Hi there, have a wonderful next year! I'm here because I have a doubt. I was trying to generalize the Einstein Field Equation for Von Neumann W* Algebra, which is related with non-integer, non always positive degrees of freedom. In particular, with the sum of positive and negative fractal...

The Riemann curvature tensor contains second derivatives of metric and squares of the first derivatives. The second derivatives have to be there because they are the ones that cannot be eliminated locally by a choice of coordinates. But other than being a mathematical artifact, is there a...

Good morning friends of the Forum. For me it is difficult to geometrically imagine a tensor of order 2 and maybe that is why it is difficult for me to know, what remains invariant when making a change of coordinates of this tensor. The only thing I can think of it, is that since a tensor of...

$$ {\Lambda}^{i}_{j} $$ When indices are written on top of one another I am confused wich is the inner index and which is the lower one when we lower the upper index.

Can anyone answer me that why the description of composite system involve tensor product ? Is there any way to realize this intuitively ?

The sigma tensor composed of the commutator of gamma matrices is said to be able to represent any anti-symmetric tensor. \sigma_{\mu\nu} = i/2 [\gamma_\mu,\gamma_\nu] However, it is not clear how one can arrive at something like the electromagnetic tensor. F_{\mu\nu} = a \bar{\psi}...

I'm having trouble understanding tensor contraction. So for example, for something like AuvBvu, would this equal to some scalar?

Question: Solution: I need help with the last part. I think my numerical factors are incorrect, even if I add the last term it will get worse. What have I done wrong, or is there a better way to deal with this?

Hello, I realize this might sound dumb, but I'm having such a hard time understanding Einstein notation. For something like ∂uFv - ∂vFu, why is this not necessarily 0 for tensor Fu? Since all these indices are running through the same values 0,1,2,3?

I do private studies on my own for fun and right now I read about relativistic field theory as a preparation for later studies of quantum field theory. I simply do not understand where equation 13.78 in Goldstein's "Classical Mechanics" third edition comes from. Please explain. Please also...

I have the following question to solve:Use the metric:$$ds^2 = -dt^2 +dx^2 +2a^2(t)dxdy + dy^2 +dz^2$$ Test bodies are arranged in a circle on the metric at rest at ##t=0##. The circle define as $$x^2 +y^2 \leq R^2$$ The bodies start to move on geodesic when we have $$a(0)=0$$ a. we have to...

The Einstein tensors for the Schwarzschild Geometry equal zero. Why do they not equal something that has to do with the central mass, given that the Einstein equations are of the form: Curvature Measure = Measure of Energy/Matter Density?

In the book general relativity by Hobson the gravitational wave of a binary merger is computed in the frame of the binary merger as well as the TT-gauge. I considered what components of the Riemann tensor along the x-axis in both gauges. The equation for the metric in the source and TT-gauge are...

New to group theory. I have 3 questions: 1. Tensor decomposition into Tab = T[ab] +T(traceless){ab} + Tr(T{ab}) leads to invariant subspaces. Is this enough to imply these subreps are irreducible? 2. The Symn representations of a group are irreps. Why? 3. What is the connection between...

Consider a Lagrangian, #L#, which is a function of, as well as other fields #\psi_i#, a traceless and symmetric tensor denoted by #f^{uv}#, so that #L=L(f^{uv})#, the associated action is #\int L(f^{uv}, \psi_i)d^4x #. To vary w.r.t #f^{uv}# , I write...

Preface We know that, in Cartesian Coordinates, $$\nabla f= \frac{\partial f}{\partial x} + \frac{\partial f}{\partial y} + \frac{\partial f}{\partial z}$$ and $$\nabla^2 f= \frac{\partial^2 f}{\partial^2 x} + \frac{\partial^2 f}{\partial^2 y} + \frac{\partial^2 f}{\partial^2 z}$$ Generalizing...

My Attempted Proof ##R^{mn}_{;n} - \frac {1} {2} g^{mn} R_{;n} = 0## ##R^{mn}_{;n} = \frac {1} {2} g^{mn} R_{;n}## So, we want ##2 R^{mn}_{;n} = g^{mn} R_{;n} ## Start w/ 2nd Bianchi Identity ##R_{abmn;l} + R_{ablm;n} + R_{abnl;m} = 0## Sum w/ inverse metric tensor twice ##g^{bn} g^{am}...

Hi, I've been doing a course on Tensor calculus by Eigenchris and I've come across this problem where depending on the way I compute/expand the Lie bracket the Torsion tensor always goes to zero. If you have any suggestions please reply, I've had this problem for months and I'm desperate to...

Hi, I'm struggling with understanding the idea of tensor product and direct sum beyond the very basics. I know that direct sum of 2 vectors basically stacks one on top of another - I don't understand more than this . For tensor product I know that for a product of 2 matrices A and B the tensor...

I'm reading Carroll's GR notes and I'm having trouble deciphering a particular expression for the Riemann curvature tensor. The coordinate-free definition is (eq. 3.71 in the notes): $$R(X,Y)Z=\nabla_X\nabla_YZ-\nabla_Y\nabla_XZ-\nabla_{[X,Y]}Z$$ An index-based expression is also given in (eq...

Please, how can I get a copy of this program library?

Why do the Cauchy Stress Tensor & the Energy Momentum Tensor have the same SI units? Shouldn't adding time as a dimension changes the Energy Momentum Tensor's units? Did Einstein start with the Cauchy Tensor when he started working on the right hand side of the field equations of GR? If so, What...

So, I've been watching eigenchris's video series "Tensors for Beginners" on YouTube. I am currently on video 14. I, in the position of a complete beginner, am taking notes on it, and I just wanted to make sure I wasn't misinterpreting anything. At about 5:50, he states that "The array for Q is...

So, I've been watching eigenchris's video series "Tensors for Beginners" on YouTube. I am currently on video 14. I am a complete beginner and just want some clarification on if I'm truly understanding the material. Basically, is everything below this correct? In summary of the derivation of the...

Hello everybody! I know in classical field theory adding in the Lagrangian density a term of the form Fαβ (*F)αβ (where by * we denote the dual of the field strength tensor) does not change the EOM, since this corresponds to adding a total derivative term to the action. However when computing...

I'm reading Semi-Riemannian Geometry by Stephen Newman and came across this theorem: For context, ##\mathcal{R}_s:Mult(V^s,V)\to\mathcal{T}^1_s## is the representation map, which acts like this: $$\mathcal{R}_s(\Psi)(\eta,v_1,\ldots,v_s)=\eta(\Psi(v_1,\ldots,v_s))$$ I don't understand the...

As I understand it, parallel transport of a vector around a closed loop on a manifold can lead (in the tangent space) to 1) an angular change, given by the Riemann curvature tensor or, 2) a translational defect given by the Torsion tensor. I can see how the looping on the curvature of a 2D...

So, I have recently been trying to learn how to work with tensors. In doing this, I have come across Einstein Notation. Below is my question. $$(a_i x_i)_{e}= (\sum_{i=1}^3 a_i x_i)_r=(a_1 x_1+a_2 x_2+a_3 x_3)_r$$; note that the following expression is in three dimensions, and I use the...

Following from Wikipedia, the covariant formulation of electromagnetic field involves postulating an electromagnetic field tensor(Faraday 2-form) F such that F=dA where A is a 1-form, which makes F an exact differential form. However, is there any specific reason for expecting F to be exact...

It's possible that this may be a better fit for the Differential Geometry forum (in which case, please do let me know). However, I'm curious to know whether anyone is aware of any standard naming convention for the two principal invariants of the Weyl tensor. For the Riemann tensor, the names of...

Hi, it's about the task e) Since the density is homogeneous, I have assumed the following for ##\rho=\frac{M}{V}##. I then started the proof of ##I_{23}##, the integral looks like this: $$ I_{23}=\int_{}^{} -\frac{M}{V}r'_2r'_3 d^3r$$ Now I apply the transformation $$ I_{23}=\int_{}^{}...

Hi, unfortunately, I am not getting anywhere with the following task The inertia tensor is as follows $$\left( \begin{array}{rrr} I_{11} & I_{12} & I_{13} \\ I_{21} & I_{22} & I_{23} \\ I_{31} & I_{32} & I_{33} \\ \end{array}\right)$$ I had now thought that I could simply rotate the...