Tensors Definition and 386 Threads

I need to prove that $$D_\mu D_\nu \xi^\alpha = - R^\alpha_{\mu\nu\beta} \xi^\beta$$ where D is covariant derivative and R is Riemann tensor. ##\xi## is a Killing vector. I have proved that $$D_\mu D_\nu \xi_\alpha = R_{\alpha\nu\mu\beta} \xi^\beta$$ I can't figure out a way to get the required...

I am reading Segei Winitzki's book: Linear Algebra via Exterior Products ... I am currently focused on Section 1.7.3 Dimension of a Tensor Product is the Product of the Dimensions ... ... I need help in order to get a clear understanding of an aspect of the proof of Lemma 3 in Section 1.7.3...

I am reading Segei Winitzki's book: Linear Algebra via Exterior Products ... I am currently focused on Section 1.7.3 Dimension of a Tensor Product is the Product of the Dimensions ... ... I need help in order to get a clear understanding of an aspect of the proof of Lemma 3 in Section 1.7.3...

Dear Friends! I am learning Tensors so my question may look simple to you. "All observers in all reference frames agree not on the basis vectors not on the components but on the combination of components and basis vectors" Q Why this happens? Please guide me where I can study it in brief and in...

What is the trace of a second rank tensor covariant in both indices? For a tensor covariant in one index and contravariant in another ##T^i_j##, the trace is ##T^k_k## but what is the trace for ##T_{ij}## because ##T_{kk}## is not even a tensor?

Hey! I'm reading Special Relativity right now and I am stuck trying to understand tensors. Can you kind people please explain to me the difference between the following 3 tensors? $$A^{\alpha \beta}$$ $$A_{\alpha \beta}$$ $$A^{\alpha}_{\beta}$$

Homework Statement I have a tensor which is given by t_{ij} = -3bx_i x_j + b \delta_{ij} x^2 + c \epsilon_{ijk} x_k And now I am asked to calculate (t^2)_{ij} : = t_{ik} t_{kj} Homework EquationsThe Attempt at a Solution At first I thought I had to calculate the square of the original...

I have learned that there is a difference between the tensors ##{T^{\mu}}_{\nu}## and ##{T_{\nu}}^{\mu}##. Does the upper index denote the rows and the lower index the columns?

I've been playing around with the Carminati-McLenaghan invariants https://en.wikipedia.org/wiki/Carminati–McLenaghan_invariants , which are a set of curvature scalars based on the Riemann tensor (not depending on its derivatives). In general, we want curvature scalars to be scalars that are...

I am still at the stage of trying to assimilate contravariant and covariant tensors, so my question probably has a simpler answer than I realize. A covariant tensor is like a gradient, as its units increase when the coordinate units do. A contravariant tensor's components decrease when the...

Homework Statement Using 26.40, show that a pseudovector p and antisymmetric second rank tensor (in three dimensions) A are related by: $$ {A}_{ij} = {\epsilon}_{ijk}{p}_{k} $$ Homework Equations 26.40: $$ {p}_{i} = \frac{1}{2}{\epsilon}_{ijk}{A}_{jk} $$ The Attempt at a Solution This...

I am reading basic cosmology but inside the books I am studying I have faced tensor so i need basic books on tensor to understand those books is it possible to suggest good books ?

'Using the following normalization in the su(3) algebra ##[\lambda_i, \lambda_j] = 2if_{ijk}\lambda_k##, we see that ##g_{ij} = 4f_{ikl}f_{jkl} = 12 \delta_{ij}## and, by expanding the anticommutator in invariant tensors, we have further that $$\left\{\lambda_i, \lambda_j\right\} =...

sorry solved

Homework Statement Is the moment of inertia matrix a tensor? Hint: the dyadic product of two vectors transforms according to the rule for second order tensors. I is the inertia matrix L is the angular momentum \omega is the angular velocity Homework Equations The transformation rule for a...

Hi, I am looking for a book that explains tensors and builds a working knowledge of tensors, like the book Linear Algebra by Friedberg Insel and Spence, which I thought explained things very well (if you haven't heard of it, its an intro. book on linear algebra). Thanks!

1. Problem statement: Assume that u is a vector and A is a 2nd-order tensor. Derive a transformation rule for a 3rd order tensor Zijk such that the relation ui = ZijkAjk remains valid after a coordinate rotation.Homework Equations : [/B] Transformation rule for 3rd order tensors: Z'ijk =...

Hello. There is one thing I can not find the answer to, so I try here. For instance, writing a general superfield on component form, one of the terms appearing is: \theta \sigma^\mu \bar{\theta} V_\mu My question is if one could have written this as \theta \bar{\theta} \sigma^\mu V_\mu ...

Homework Statement Hello everyone, can anyone help me prove this using tensors? Given three arbitrary vectors not on the same line, A, B, C, any other vector D can be expressed in terms of these as: where [A, B, C] is the scalar triple product A · (B × C) Homework Equations I know that...

Hi. I am self-studying GR and have many questions. Here are a few. If anyone can help me with any of them I would be grateful. 1 - What is the difference between Tu v and Tvu ? 2 - I have read that the order of indices matters in tensors but when transforming tensors from one coordinate...

Explain, plase to me: 2D covariant, conravariant and physical components of vector. I can not find this thematic in my official study book.

Hi. In Georgi's book page 143, eqn. (10.29) he gives an example of decomposing a tensor product into irreps: u^iv_k^j=\frac{1}{2} \left( u^iv_k^j+u^jv_k^i-\frac{1}{4}\delta_k^iu^\ell v_\ell^j-\frac{1}{4}\delta_k^ju^\ell v_\ell^i \right)\\ +\frac{1}{4} \varepsilon^{ij\ell} \left(...

I am currently an undergraduate physics and applied mathematics student, and have wanted to go ahead in my course to learn about particle physics and general relativity. However, these topics, along with Quantum field theory which I want to learn about later, are taught in tensor notation. So...

I have calculated the metric tensor, inverse metric tensor, Christoffel symbols, Ricci tensor, curvature scalar and the Einstein tensor for the Robertson Walker Metric: ds2= (cdt)2 - R2(t)[dr2/(1- kr2) + r2(dθ2 + sin2(θ)dΦ2)] Here is the metric tensor: g00 = 1 g11 = - R2(t) / (1- kr2) g22 = -...

I am well aware of an abstract definition of a general tensor as a map: \mathbf{T}:\overbrace{V\times\cdots\times V}^{n}\times\underbrace{V^{\star}\times \cdots\times V^{\star}}_{m}\longrightarrow\mathbb{R} I am happy with this definition, it makes a lot of sense to me. However, the physics...

The metric tensor has the property that it can raise and lower indices, but this is on the assumption that it (the metric) is symmetric. If we were to construct a metric tensor that was non-symmetric, would it still raise and lower indices?

Hi, I'm trying to close in on a more intuitive way of understanding tensors. For some reason, they've always held an aura of mystique for me, may be also their similarity to the word "tense" has meant that I've never really warmed to the many defintions and explanations available. So, in many...

Hey all, I'm just starting into GR and learning about tensors. The idea of fully co/contravariant tensors makes sense to me, but I don't understand how a single tensor could have both covariant AND contravariant indices/components, since each component is represented by a number in each index...

I recently derived the Einstein tensor and the stress energy momentum tensor for the Godel solution to the Einstein field equations. Now as usual I will give you the page where I got my line element from so you can have a reference: http://en.wikipedia.org/wiki/Gödel_metric Here is what I got...

First let me give the definition of tensor that my book gives: If V is a finite dimensional vector space with dim(V) = n then let V^{k} denote the k-fold product. We define a k-tensor as a map T: V^{k} \longrightarrow \mathbb{R} such that T is multilinear, i.e. linear in each variable...

On one hand, in reading Georgi's book in group theory, I comprehend the invariant tensor as a special "tensor", which is unchanged under the action of any generators. On the other hand, CG decomposition is to decompose the product of two irreps into different irreps. Now it is claimed that...

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 wasn't sure where to post this, and I hope this is the right place. I've been reading ahead of my lectures, and I've gotten a book that introduces tensors. It very quickly introduces Einstein Summation Convention, which I think I understand, \sum_{i=1}^{3} x_{i} y_{i} = x_{i} y_{i} = x \cdot...

Hello. I'm going through Ray D'Inverno's "Introducing Einstein's Relativity" and I'm stuck at a certain point and can't move forward. It deals with tensors, I'm stuck at the transformation matrix and the problem is, I can't figure out what the key equation (5.7) actually means. There is a...

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...

My text has: ##\frac{\partial x^{a}}{\partial x^{p}}V^{p}-\frac{\partial x^{a}}{\partial x^{r}}V^{r}+\frac{\partial x^{a}}{\partial x^{p}}T^{p}_{qr}V^{r}+\frac{\partial x^{a}}{\partial x^{p}}\frac{\partial }{\partial x^{q}}V^{p}=\frac{\partial x^{a}}{\partial...

They look a lot like matrices, and seem to work exactly like matrices. What is the difference between them? I have only worked with matrices, not tensors because I can't find a tutorial online but every time I have seen one they seem identical.

Not sure where to post this thread. That being said, can someone explain to me simply what covariant and contravariant tensors are and how covariant and contravariant transformation works? My understanding of it from googling these two mathematical concepts is that when you change the basis of...

It is often stated that when one tries to find a stress-energy tensor of gravitational field in GR, the resulting quantity is zero because we can always make the metric zero at a point by a coordinate transformation. So there is no local measure of energy-momentum for gravitational fields. But I...

My professor gave us a book that is still in production to use for special relativity. I am having a hard time grasping the notation and operations with Einstein upper and lower notation. Can anyone recommend a good textbook on this topic? Chris

General Relativity... Non-rotational spherically symmetric body of isotropic perfect fluid Einstein tensor metric element functions: g_{\mu \nu} = \left( \begin{array}{ccccc} \; & dt & dr & d \theta & d\phi \\ dt & g_{tt} & 0 & 0 & 0 \\ dr & 0 & g_{rr} & 0 & 0 \\ d\theta & 0 & 0 & g_{\theta...

Whats the difference between tensors and vectors?

I did some linear algebra studies and learned how to change between foreign bases and the standard basis: Change of basis matrix multiplied by the vector in coordinates with respect to the foreign basis equals the vector in coordinates with respect to the standard basis. Of course, this is...

I have been recently trying to derive the Einstein tensor and stress energy momentum tensor for a certain traversable wormhole metric. In my multiple attempts at doing so, I used a coordinate basis. My calculations were correct, but the units of some of the elements of the stress energy momentum...

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...

As you may know, the metric tensor for 3D spherical coordinates is as follows: g11= 1 g22= r2 g33= r2sin2(θ) Now, the Minkowski metric tensor for spherical coordinates is this: g00= -1 g11= 1 g22= r2 g33= r2sin2(θ) In both of these metric tensors, all other elements are 0. Now...

Draft re Ricci vs Riemann tensors This one is really just the beginning of a musing. I can't even remember if I came to any conclusion or just forgot about it. I started a thread in Jan 2014, a couple of months after this blog post, on the related issue of what the physical significance of...

What are tensors?

I have a cluster of voxels and a 2nd order stress tensor corresponding to each voxel. I was wondering as to what would be the best method to calculate an average stress tensor for the cluster as a whole? Any constructive inputs would be greatly appreciated.

Hi all, I'm trying (and failing miserably) to understand tensors, and I have a quick question: is the inner product of a rank n tensor with another rank n tensor always a scalar? And also is the inner product of a rank n tensor with a rank n-1 tensor always a rank n-1 tensor that has been...