Tensor Definition and 1000 Threads

Hi, I know the generalized hookes law between stress and strain is given by the elastic tensor. This matrix has 81 constants which are reduced to 9 in the isotropic case. Can someone please help me to understand intuitively how this reduction in the elastic tensor takes place and why some of the...

Loop Quantum Gravity, Exact Holographic Mapping, and Holographic Entanglement Entropy Muxin Han, Ling-Yan Hung (Submitted on 7 Oct 2016) The relation between Loop Quantum Gravity (LQG) and tensor network is explored from the perspectives of bulk-boundary duality and holographic entanglement...

I have been trying to understand what a tensor is, still I cannot make an intuitive idea about it. I need help. Thanks in advance.

Homework Statement Not sure if this is advanced, so move it wherever. A certain rigid body may be represented by three point masses: m_1 = 1 at (1,-1,-2) m_2 = 2 at (-1,1,0) m_3 = 1 at (1,1,-2) a) find the moment of inertia tensor b) diagonalize the matrix obtaining the eigenvalues and the...

Hi, I have trouble understanding why the following relations hold true. Given the Minkowski metric \eta_{\alpha\beta}=diag(1,-1,-1,-1) and the line segment ds^2 = dx^2+dy^2+dz^2, then how can i see that this line segment is equal to ds^2 = \eta_{\alpha\beta}dx^\alpha dx^\beta . Further, we...

Hi, I am trying to figure out why a term like ## L \sim i \bar \psi_L \sigma^{\mu\nu}G_{\mu\nu}^a t^a \psi_R + h.c=## ##= i \bar \psi_L \sigma^{\mu\nu}G_{\mu\nu}^a t^a \psi_R - i \bar \psi_R \sigma^{\mu\nu}G_{\mu\nu}^a t^a \psi_L ## violates CP by looking at all the terms composing the...

Hi everyone, I want to derive the Friedmann equations from Einstein Field Equations. However, I have a problem that stems from the energy-momentum tensor. I am also trying to keep track of ## c^2 ## terms. FRW Metric: $$ ds^2= -c^2dt^2 + a^2(t) \left( {\frac{dr^2}{1-kr^2} + r^2 d\theta^2 + r^2...

Homework Statement Let ##x##, ##y##, and ##z## be the usual cartesian coordinates in ##\mathbb{R}^{3}## and let ##u^{1} = r##, ##u^{2} = \theta## (colatitude), and ##u^{3} = \phi## be spherical coordinates. Compute the metric tensor components for the spherical coordinates...

Imagine a hole drilled through the Earth from which all air has been removed thus creating a vacuum. Let a cluster of test particles in the shape of a sphere be dropped into this hole. The volume of the balls should start to decrease. However, in his article "The Meaning of Einstein's Equation"...

Suppose we are given this definition of the wedge product for two one-forms in the component notation: $$(A \wedge B)_{\mu\nu}=2A_{[\mu}B_{\nu]}=A_{\mu}B_{\nu}-A_{\nu}B_{\mu}$$ Now how can we show the switch from tensor products to wedge product below...

The antisymmetric 2-tensor ##F_{ij}## is given by ##F_{ij}\equiv \partial_{i}A_{j}-\partial_{j}A_{i}## so that ##F_{ij}={\epsilon_{ij}}^{k}B_{k}## and ##B_{i}=\frac{1}{2}{\epsilon_{i}}^{jk}F_{jk}##. I was wondering if the permutation tensor with indices upstairs is different from the...

If a large cloud of dust of constant ρ is moving with a given ##\vec v ## in some frame, then at any given time and position inside the cloud there should not be no net energy or i-momentum flow on any surface of constant ##x^i ## (i=1,2,3) because the particles coming in cancels those going out...

Homework Statement https://en.wikipedia.org/wiki/Cauchy_stress_tensor[/B] I don't understand the difference between τxy . τyx , τxz , τzx , τyz , τzy ..What did they mean ? Homework EquationsThe Attempt at a Solution taking τxy and τyx as example , what are the difference between them ? They...

Homework Statement i have a few homework question and want to be sure if I have solved them right. Q1) Write ##\vec{\triangledown}\cdot\vec{\triangledown}\times\vec{A}## and ##\vec{\triangledown}\times\vec{\triangledown}\phi## in tensor index notation in ##R^3## Q2) the spherical coordinates...

Hello everyone, There is something that has been bugging me for a long time about the meaning of Lorentz Transformations when looked at in the context of tensor analysis. I will try to be as clear as possible while at the same time remaining faithful to the train of thought that brought me...

I've seen the terms tensor calculus and tensor analysis both being used - what is the difference?

Hello I am little bit confused about calculating Ricci tensor for schwarzschild metric: So we have Ricci flow equation,∂tgμν=-2Rμν. And we have metric tensor for schwarzschild metric: Diag((1-rs/r),(1-rs]/r)-1,(r2),(sin2Θ) and ∂tgμν=0 so 0=-2Rμν and we get that Rμν=0.But Rμν should not equal to...

Hello! Why is the stress energy tensor defined as a (2 0) tensor? I understand that it needs 2 one-forms as arguments, but using the metric, can't we bring it to (1 1) or (0 2)? So is there is any physical or mathematical reason why it is defined as (2 0), or it is equally right to define it as...

From Carroll (2004) It is possible to derive the Einstein Equations (with ##c=1##) via functional variation of an action $$S=\dfrac{S_H}{16\pi G}+S_M$$ where $$S_H= \int \sqrt{-g}R_{\mu\nu}g^{\mu\nu}d^4 x$$ and ##S_M## is a corresponding action representing matter. We can decompose ##\delta...

Hello. I am confused about the notation for tensors and vectors. From what I saw, for a 4-vector the notation is with upper index. But for a second rank tensor (electromagnetic tensor for example) the notation is also upper index. I attached a screenshot of this. Initially I thought that for...

Why doesn't mass show up in the stress-energy tensor explicitly?

Hi. I'm trying to understand tensors and I've come across this problem: "Show that, in general, a (2, 0) tensor can't be written as a tensor product of two vectors". Well, prior to that sentence, I would have thought it could... Why not?

Hi, there is a book of dg of surfaces that is also about tensor calculus ? Currently i study with Do Carmo, but i am looking for a text that there is also the tensor calculus! Thank you in advance

I recently calculated the Einstein tensor for the exterior Schwarzschild solution. Here it is: G00 = 0 G11 = [-2GM/(r3c2 - 2GMr2)] - G2M2/[r2(rc2 - 2GM)2] - [-G2M2/(r4c4 - 2GMr3c2)] - (-2GM)/(r3c2) - (-2G3M3)/[r3c2(rc2 - 2GM)2] G22 = (2G2M2rc2 - 2G3M3) / (r3c6 - 2GMr2c4) G33 = G22sin2(θ)...

The Riemann-Christoffel Tensor (##R^{k}_{\cdot n i j}##) is defined as: $$ R^{k}_{\cdot n i j}= \frac{\delta \Gamma^{k}_{j n}}{\delta Z^{i}} - \frac{\delta \Gamma^{k}_{i n}}{\delta Z^{j}}+ \Gamma^{k}_{i l} \Gamma^{l}_{j n}- \Gamma^{k}_{j l} \Gamma^{l}_{i n} $$ My question is that it seems that...

Given the definition of the covariant basis (##Z_{i}##) as follows: $$Z_{i} = \frac{\delta \textbf{R}}{\delta Z^{i}}$$ Then, the derivative of the covariant basis is as follows: $$\frac{\delta Z_{i}}{\delta Z^{j}} = \frac{\delta^2 \textbf{R}}{\delta Z^{i} \delta Z^{j}}$$ Which is also equal...

I do not know if this is the proper rubric to ask this question, but I picked the one that seemed the most relevant. I have noticed some superficial resemblance between the tensor product and the ultraproduct definitions. Does this resemblance go any further? While I am on the subject of...

What 20 index combinations yield Riemann tensor components (that are not identically zero) from which the rest of the tensor components can be determined?

Homework Statement Right, so it's not really an assignment or anything, just confused of what a book says. the book is "mathematical methods for physicists." The screenshot is attached. The thing that I'm confused about is that it says "As before, aij is the cosine of the angle between x′i...

hi, when I dug up something about metric tensors, I found a equation in my attached file. Could you provide me with how the derivation of this ensured? What is the logic of that expansion in terms of metric tensor? I really need your valuable responses. I really wonder it. Thanks in advance...

Hello I've been have been done some research about Einstein Field Equations and I want to get great perspective of Ricci tensor so can somebody explain me what Ricci tensor does and what's the mathmatical value of Ricci tensor.

Hi. Why did the founding fathers of QM know that the Hilbert space of a composite system is the tensor product of the component Hilbert spaces and not a direct product, where no entanglement would emerge? I mean today we can verify entanglement experimentally, but this became technologically...

Say I have {S_{x}=\frac{1}{\sqrt{2}}\left(\begin{array}{ccc} 0 & 1 & 0\\ 1 & 0 & 1\\ 0 & 1 & 0\\ \end{array}\right)} Right now, this spin operator is in the Cartesian basis. I want to transform it into the spherical basis. Since, {\vec{S}} acts like a vector I think that I only need to...

Hi, I would like say that in this link ( ) and starting from 56.28 Suskind tries to find the energy tensor equation using \phi, afterwards he finds a equation similar to wave equation in terms of \phi. My question is: For what does \phi stand ? I could not capture the meaning of \phi. Could...

Hey everyone, I'm currently trying to understand the resistivity and conductivity tensor of a 2D sample. If a current carrying metal bar is placed inside a magnetic field the Hall Effect comes into play. I tried to search for explanations on how to obtain the resistivity tensor of the metal bar...

I was just watching a video that was reviewing some linear algebra, and it said that this was the tensor product: Let's say you have a matrix A and a matrix B (both 2 by 2 matrices). If I want to calculate the tensor A ⊗ B, then the answer is basically just a matrix of matrices. In other words...

Homework Statement We've got a line element ds^2 = f(x) du^2 + dx^2 From that we should find the geodesic equation Homework Equations Line Element: ds^2 = dq^j g_{jk} dq^k Geodesic Equation: \ddot{q}^j = -\Gamma_{km}^j \dot{q}^k \dot{q}^m Christoffel Symbol: \Gamma_{km}^j = \frac{g^{jl}}{2}...

Does there exist any 2D surface whose metric tensor is, ##\eta_{\mu\nu}= \begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix}##

I am reading Paul E. Bland's book "Rings and Their Modules ... Currently I am focused on Section 2.3 Tensor Products of Modules ... ... I need some help in order to fully understand the Remark that Bland makes on Pages 65- 66 Bland's remark reads as follows: Question 1 In the above text by...

I am reading Paul E. Bland's book "Rings and Their Modules ... Currently I am focused on Section 2.3 Tensor Products of Modules ... ... I need some help in order to fully understand the Remark that Bland makes on Pages 65- 66 Bland's remark reads as follows: Question 1 In the above text by...

hi, I tried to take the covariant derivative of riemann tensor using christoffel symbols, but it is such a long equation that I have always been mixing up something. So, Could you share the entire solution, pdf file, or links with me? ((( I know this is the long way to derive the einstein...

I understand the basics of the stress-energy tensor (I think) but I still have a couple questions about it. But first, I'd like to give a quick run down of what I do understand, and I would appreciate if one of you could correct me where I am wrong and also answer my questions afterward. So...

Homework Statement For a system of discrete point particles the energy momentum takes the form T_{\mu \nu} = \sum_a \frac{p_\mu^{(a)}p_\nu^{(a)}}{p^{0(a)}} \delta^{(3)}(\vec{x}-\vec{x}^{(a)}), where the index a labels the different particles. Show that, for a dense collection of particles...

I'm trying to show that \partial_\mu T^{\mu \nu}=0 for T^{\mu \nu}=F^{\mu \lambda}F^\nu_{\; \lambda} - \frac{1}{4} \eta^{\mu \nu} F^{\lambda \sigma}F_{\lambda \sigma}, with the help of the electromagnetic equations of motion (no currents): \partial_\mu F^{\mu \nu}=0, \partial_\mu F_{\nu...

In explanations of the importance the tensors I often see people refer to transformation properties, general covariance and the like. Now, I have also often read that in principle any physical theory, e.g. classical mechanics and special relativity, can be written in a generally covariant form...

Homework Statement Using index notation only (i.e. don't expand any sums) show that: \begin{align*} &\text{(a) } \epsilon_{ijk} \det \underline{\bf{A}} = \epsilon_{mnp} A_{mi} A_{nj} A_{pk} \\ & \text{(b) } \det \underline{\bf{A}} = \epsilon_{mnp} A_{m1} A_{n2} A_{p3} \end{align*} Homework...

Homework Statement show that \det(\underline{\bf{A}})\det(\underline{\bf{B}}) = \det(\underline{\bf{AB}}) Homework Equations \begin{align*} &\underline{\bf{A}} = A_{ij} \underline{e}_i \otimes \underline{e}_j \\ &\underline{\bf{B}} = B_{mn} \underline{e}_m \otimes \underline{e}_n \\...

Hi PF! I have a question on the dyadic product and the divergence of a tensor. I've never formally leaned this, although I'm sure it's published somewhere, but this is how I understand the operators. Can someone tell me if this is right or wrong? Let's say I have some vector ##\vec{V} = v_x i +...

Homework Statement Given two spaces described by ##ds^2 = (1+u^2)du^2 + (1+4v^2)dv^2 + 2(2v-u)dudv## ##ds^2 = (1+u^2)du^2 + (1+2v^2)dv^2 + 2(2v-u)dudv## Calculate the Riemann tensor Homework Equations Given the metric and expanding it ##~~~g_{τμ} = η_{τμ} + B_{τμ,λσ}x^λx^σ + ...## We have...

Homework Statement [ For a system of discrete point particles the energy momentum takes the form T_{\mu \nu} = \sum_a \frac{p_\mu^{(a)}p_\nu^{(a)}}{p^{0(a)}} \delta^{(3)}(\vec{x}-\vec{x}^{(a)}), where the index a labels the different particles. Show that, for a dense collection of particles...