Tensor Definition and 1000 Threads

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...

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...

I have started to learn a bit about Tensor calculus and it all going above my head. May anyone give a brief outline about the topic (preferably theoretical) and the supplementary concepts attached to it.

How exactly is the EM field tensor related to the proper orthochronous Lorentz group?

Homework Statement Hi, I'm trying to show the four divergence of the stress energy tensor of the sourceless klein gordon equation is zero. I got to the part in the solution where I am left with the equations of motion which is identically zero and 3 other terms. I managed to find a solution...

Homework Statement Half disk, radius R, mass m. I need the inertia tensor about the center of mass and then find the Principal moments of inertia about another coordinate system. Struggling with the product of inertia...

Dear all, In textbooks about optics in magneto-optic materials, we often come across a Hermitian permittivity tensor with off-diagonal imaginary components. These components are relevant to the Faraday rotation of plane of polarization of light through the material. Now my question is: Is the...

How do we explains space-time through Riemann Calculus?

In SRT, the line element is ##c^2ds^2 = c^2dt^2 - dx^2 -dy^2-dz^2## and ##g_{00} = 1## (or ##-1## depending on sign conventions). In the Schwarzschild metric we have g_{00}=(c^2-\frac{2 GM}{r}) . So in the first example, ##g_{00}## is constant, in the second it depends on another coordinate...

I have come about few mathematical problems related to Riemann Tensor analysis while learning General Relativity. Should I ask these questions in this section or in the homework section. They are pretty hard!

First, I'd like to thank everyone that has helped me thus far in deriving the general relativistic tensors for the Morris-Thorne wormhole metric in an orthonormal basis. I have finally done it and grasped that concept. Now that I have done that, my new stress energy momentum tensor for this...

Homework Statement I am looking at Goldstein, Classical Mechanics. I am on page 254, and trying to reference page 190 for my confusion. I don't understand how they got from equation 6.49 to 6.50, potential energy function to tensor matrix. I really want to know how to calculate a tensor from a...

Hey guys, So in my notes I've got this statement written: If tensor with no symmetry properties, A^{\mu\nu}, contracts to a_{\mu\nu}, we can write this as A^{\mu\nu}a_{\mu\nu}=\frac{1}{2}a_{\mu\nu}(A^{\mu\nu}-A^{\nu\mu}) as a_{\mu\nu} (A^{\mu\nu}+A^{\nu\mu}) = 0. So I don't see how the...

I have recently delved into linear algebra and multi-linear algebra. I came to learn about the concepts of linear and bi-linear maps along with bases and changes of basis, linear independence, what a subspace is and more. I then decided to move on to tensor products, when I ran into a problem...

Homework Statement A tensor and vector have components Tαβγ, and vα respectively in a coordinate system xμ. There is another coordinate system x'μ. Show that Tαβγvβ = Tαβγvβ Homework Equations umm not sure... ∇αvβ = ∂vβ/∂xα - Γγαβvγ The Attempt at a Solution Tαβγvβ =...

Hi everyone! I've got a vector index notation proof that I'm struggling with. (sorry ignore the c, that's the question number) I've simplified it u * (del X del) and from there I've sort of assumed del X del = 0. Is that right and if so could somebody please explain it? Else any help on...

Hey guys, So I have the stress energy tensor written as follows in my notes for the complex Klein-Gordon field: T^{\mu\nu}=(\partial^{\mu}\phi)^{\dagger}(\partial^{\nu}\phi)+(\partial^{\mu}\phi)(\partial^{\nu}\phi^{\dagger})-\mathcal{L}g^{\mu\nu} Then I have the next statement that T^{0i} is...

In an attempt to solve the mystery of dark energy, I came across problems concerned with the General Relativity. In it, I observed that many of the problems were related with the tensor calculus. I want to know that what importance does tensor calculus hold in GR? Are there any other fields of...

\mathcal{L}_M(g_{kn}) = -\frac{1}{4\mu{0}}g_{kj} g_{nl} F^{kn} F^{jl} \\ \frac{\partial{\mathcal{L}_M}}{\partial{g_{kn}}}=-\frac{1}{4\mu_0}F^{pq}F^{jl} \frac{\partial}{\partial{g_{kn}}}(g_{pj}g_{ql})=+\frac{1}{4\mu_0} F^{pq} F^{lj} 2 \delta^k_p \delta^n_j g_{ql} I need to know how...

Suppose, I know the metric tensor of a 2D space. for example, the metric tensor of a sphere of radius R, gij = ##\begin{pmatrix} R^2 & 0 \\ 0 & R^2\cdot sin^2\theta \end{pmatrix}## ,and I just know the metric tensor, but don't know that it is of a sphere. Now I want to draw a 2D space(surface)...

Im studying Quantum Field Theory as part of my undergraduate course, and am currently looking at Noether's Theorem which has led me to the following calculation of the divergence of the Stress-Energy Tensor. I'm having difficulty in seeing how we get from line (31) to line (32). Is the 2nd term...

Some people may remember awhile back when I made a thread showing how when I derived the Einstein tensor and the stress energy momentum tensor for a certain traversable wormhole metric, that the units of the energy momentum tensor were not the same for each element and how a couple of the...

It is often stated and proved in textbooks that the momentum density is also the energy flux. The explanation is often done using the dust model. However, it is possible that in a real fluid, there is heat conduction via particle collision. There is energy flux, but since no molecules are ever...

Hello everyone! Even though I have done substantial tensor calculus, I still don't get one thing. Probably I am being naive or even stupid here, but consider $$R_{\mu\nu} = 0$$. If I expand the Ricci tensor, I get $$g^{\sigma\rho} R_{\sigma\mu\rho\nu} = 0$$. Which, in normal algebra, should...

Hello! I'd appreciate any help or pokes in the right direction. Homework Statement Show that a co-tensor of rank 2, ##T_{\mu\nu}##, is obtained from the tensor of rank 2 ##T^{\mu\nu}## by using a metric to lower the indices: $$T_{\mu\nu} = g_{\mu\alpha}g_{\nu\beta}T^{\alpha\beta}$$ Homework...

I already have the solutions emailed to me from a D H Lawden textbook. I have trouble understanding the solution as the solution is not formatted properly, and the answer seems to be a little too advanced for me. I hope that some one can help me understand the problem. 1. Homework Statement...

Recently, I used the metric for the traversable wormhole (the one in this link): http://www.spacetimetravel.org/wurmlochflug/wurmlochflug.html ds2= -c2dt2 + dl2 + (b2 + l2)(dΘ2 + sin2(Θ)dΦ2) I derived the metric tensor from this space-time interval and then from there, I derived the...

I have a few questions regarding the solution to this problem. First of all I have the Stress-Energy tensor for a scalar fields \phi^a T_{Noether}^{\mu\nu} = \displaystyle \sum_a \frac{\partial \mathcal{L}}{(\partial_{\mu}\phi_a)}\partial^{\nu}\phi^a - g^{\mu \nu}\mathcal{L} To ensure...

Can some one write for me the Symmetric part of a third order tensor (as a tensor form) Thanks .

Hi, everyone. I am having a hard time finding explicit values of non-linear susceptibility tensor values for any sort of crystals. Specifically, I'm looking for values of a BBO crystal, but I would like to know where to find others for my future research. I should say that I am looking for the...

I am a graduate student in physics. One of my biggest frustrations in my education is that I often find that my mathematical background is lacking for the work I do. Sure I can make calculations adequately, well enough to even do well in my courses, but I don't feel like I really understand...

Homework Statement Hi everyone, I need some help to know how to find the components of the inertia tensor matrix of a rigid body formed by a gruop of point masses attached to bars with no mass. I have 3 masses with cartesian coordenates: 1 (a,a,0), 2 (a,0,0) and 3 (-a,-a-0). The...

The problem statement is: Assuming that we are in vacuum, and that the only work done between mechanical systems and electricity and magnetism comes from the Lorentz force, give a full, relativistic derivation of the Maxwell stress-energy tensor.

Hi, Can someone explain the difference between, say, \Lambda_\nu^\mu, {\Lambda_\nu}^\mu and {\Lambda^\mu}_\nu (i.e. the positioning of the contravariant and covariant indices)? I have found...

I am trying to expand $$\varepsilon^{{abcd}} R_{{abcd}}$$ by using four identities of the Riemann curvature tensor: Symmetry $$R_{{abcd}} = R_{{cdab}}$$ Antisymmetry first pair of indicies $$R_{{abcd}} = - R_{{bacd}}$$ Antisymmetry last pair of indicies $$R_{{abcd}} = - R_{{abdc}}$$...