Tensor algebra Definition and 69 Threads

Considering a vector space ##W = V\oplus V^*## equipped with quadratic form Q such that we have a clifford algebra ##Cl(W, Q)##. How can I map elements of ##V\otimes V^*## into elements of ##Cl(W, Q)##? What about elements of ##V^* \otimes V##, ##V\otimes V## and ##V^* \otimes V^*## into ##Cl(W...

Hi, I'm in trouble with the different terminologies used for tensor product of two vectors. Namely a dyadic tensor product of vectors ##u, v \in V## is written as ##u \otimes v##. It is basically a bi-linear map defined on the cartesian product ##V^* \times V^* \rightarrow \mathbb R##. From a...

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...

A brief explanation of vector and tensor concepts from A Student's Guide to Vectors and Tensors by Dan Fleisch. I found this when I was trying to better understand tensors and how they are used.

The trace of the sigma should be the same in both new and old basis. But I get a different one. Really appreciate for the help. I’ll put the screen shot in the comment part

Hello there, recently I've been trying to demonstrate that, $$\textbf{W}^2 = -m^2\textbf{S}^2$$ in a rest frame, with ##W_{\mu}## defined as $$W_{\mu} = \dfrac{1}{2}\varepsilon_{\mu\alpha\beta\gamma}M^{\alpha\beta}p^{\gamma}$$ such that ##M^{\mu\nu}## is an operator of the form $$...

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}...

In a general coordinate system ##\{x^1,..., x^n\}##, the Covariant Gradient of a scalar field ##f:\mathbb{R}^n \rightarrow \mathbb{R}## is given by (using Einstein's notation) ## \nabla f=\frac{\partial f}{\partial x^{i}} g^{i j} \mathbf{e}_{j} ## I'm trying to prove that this covariant...

As $$\hat{P_i} = \int d^3x T^0_i,$$ and $$T_i^0=\frac{\partial\mathcal{L}}{\partial(\partial_0 \phi)}\partial_i\phi-\delta_i^0\mathcal{L}=\frac{\partial\mathcal{L}}{\partial(\partial_0 \phi)}\partial_i\phi=\pi\partial_i\phi.$$ Therefore, $$\hat{P_i} = \int d^3x \pi\partial_i\phi.$$ However...

Anyone knows why musical isomorphism is called so? Why is it musical? https://en.wikipedia.org/wiki/Musical_isomorphism

Let ##\varphi## be some scalar field. In "The Classical Theory of Fields" by Landau it is claimed that $$ \frac{\partial\varphi}{\partial x_i} = g^{ik} \frac{\partial \varphi}{\partial x^k} $$ I wanted to prove this identity. Using the chain rule $$ \frac{\partial}{\partial x_{i}}=\frac{\partial...

According to my book, the equation that should meet a vector ##\mathbf{v}=v^i\mathbf{e}_i## in order to be parallel-transported in a manifold is: ##v_{, j}^{i}+v^{k} \Gamma_{k j}^{i}=0## Where ##v_{, j}^i## stands for ##\partial{v^i}{\partial y^j}##, that is, the partial derivative of the...

The covariant form for the Levi-Civita is defined as ##\varepsilon_{i,j,k}:=\sqrt{g}\epsilon_{i,j,k}##. I want to show from this definition that it's contravariant form is given by ##\varepsilon^{i,j,k}=\frac{1}{\sqrt{g}}\epsilon^{i,j,k}##.My attemptWhat I have tried is to express this tensor...

I'm trying to show that the determinant ##g \equiv \det(g_{ij})## of the metric tensor is a tensor density. Therefore, in order to do that, I need to show that the determinant of the metric tensor in the new basis, ##g'##, would be given by...

I would like to know what is the utility or purpose for which the elements below were defined in the Tensor Calculus. They are things that I think I understand how they work, but whose purpose I do not see clearly, so I would appreciate if someone could give me some clue about it. Tensors. As...

I'm trying to understand why it is possible to express vectors ##\mathbf{e}^i## of the dual basis in terms of the vectors ##\mathbf{e}_j## of the original basis through the dual metric tensor ##g^{ij}##, and vice versa, in these ways: ##\mathbf{e}^i=g^{ij}\mathbf{e}_j##...

Property (a) simply states that a second rank tensor that vanishes in one frame vanishes in all frames related by rotations. I am supposed to prove: ##T_{i_1 i_2} - T_{i_2 i_1} = 0 \implies T_{i_1 i_2}' - T_{i_2 i_1}' = 0## Here's my solution. Consider, $$T_{i_1 i_2}' - T_{i_2 i_1}' = r_{i_1...

I'm looking for literature recommendations regarding tensor networks. I never came across singular value decomposition or spectral decomposition in my linear algebra classes, so I need to brush up on the relevant mathematical background as well.

To calculate the Riemann coefficient for a metric ##g##, one can employ the second Cartan's structure equation: $$\frac{1}{2} \Omega_{ab} (\theta^a \wedge \theta^b) = -\frac{1}{4} R_{ijkl} (dx^i \wedge dx^j)(dx^k \wedge dx^l)$$ and using the tetrad formalism to compute the coefficients of the...

I need the Ricci scalar for the FRW metric with a general lapse function ##N##: $$ds^2=-N^2(t) dt^2+a^2(t)\Big[\frac{dr^2}{1-kr^2}+r^2(d\theta^2+\sin^2\theta\ d\phi^2)\Big]$$ Could someone put this into Mathematica as I don't have it?

I got stuck in this calculation, I can't collect everything in terms of ##dx^{\mu}##. ##x'^{\mu}=\frac{x^{\mu}-x^2a^{\mu}}{1-2a_{\nu}x^{\nu}+a^2x^2}## ##x'^{\mu}=\frac{x^{\mu}-g_{\alpha \beta}x^{\alpha}x^{\beta}a^{\mu}}{1-2a_{\nu}x^{\nu}+a^2g_{\alpha \beta}x^{\alpha}x^{\beta}}##...

I have an equation $$ \chi_\nu\nabla_\mu\chi_\sigma+\chi_\sigma\nabla_\nu\chi_\mu+\chi_\mu\nabla_\sigma\chi_\nu=0 $$so we also have$$ g_{\nu\rho}g_{\mu\tau}g_{\sigma\lambda}\left(\chi^\rho\nabla^\tau\chi^\lambda+\chi^\lambda\nabla^\rho\chi^\tau+\chi^\tau\nabla^\lambda\chi^\rho\right)=0 $$Does...

For instance, using the vector ##A^\alpha e_\alpha##: ##g_{\mu \nu} e^\mu \otimes e^\nu (A^\alpha e_\alpha) = g_{\mu \nu} (e^\mu, A^\alpha e_\alpha) e^\nu ## ##g_{\mu \nu} e^\mu \otimes e^\nu (A^\alpha e_\alpha) = A^\alpha g_{\mu \nu} \delta_\alpha^\mu e^\nu = A^\mu g_{\mu \nu} e^\nu = A_\nu...

I have read many GR books and many posts regarding the title of this post, but despite that, I still feel the need to clarify some things. Based on my understanding, the contravariant component of a vector transforms as, ##A'^\mu = [L]^\mu~ _\nu A^\nu## the covariant component of a vector...

Hello everyone! It seems I can't solve this exercise and I don't know where I fail. By inserting the metric on the lefthand side of I. and employing the chain rule, the equation eventually reads (confirmed by my notes from the tutorial): $$m\frac{\mathrm{d}p_\delta}{\mathrm{d}t} =...

Hello, I am calculating the krauss operators to find the new density matrix after the interaction between environment and the qubit. My question is: Is there an operational order between matrix multiplication and tensor product? Because apparently author is first applying I on |0> and X on |0>...

This was my attempt at a solution and was wondering where did I go wrong: -\frac{\partial}{\partial p_\mu}\frac{1}{\not{p}}=-\frac{\partial}{\partial p_\mu}[\gamma^\nu p_\nu]^{-1}=\gamma^\nu\frac{\partial p_\nu}{\partial p_\mu}[\gamma^\sigma...

I can see that by the tensor transformation law of the Kronecker delta that ##\frac{\partial x^a}{\partial x^b}=\delta^a_b## And thus coordinates must be independent of each other. But is there a more straightforward and fundamental reason why we don’t consider dependent coordinates? Is it...

I am reading I am reading Spacetime and Geometry : An Introduction to General Relativity -- by Sean M Carroll and have arrived at chapter 3 where he introduces the covariant derivative ##{\mathrm{\nabla }}_{\mu }##. He makes demands on this which are \begin{align} \mathrm{1.\...

Depending on the source, I'll often see EFE written as either covariantly: $$R_{\mu\nu} - \frac{1}{2}Rg_{\mu\nu} = 8 \pi GT_{\mu\nu}$$ or contravariantly $$R^{\alpha\beta} - \frac{1}{2}Rg^{\alpha\beta} = 8 \pi GT^{\alpha\beta}$$ Physically, historically, and/or pragmatically, is there a...

Homework Statement Solve this, $$\frac{\partial}{\partial x^{\nu}}\frac{3}{(q.x)^3}$$ where q is a constant vector. Homework EquationsThe Attempt at a Solution $$\frac{\partial}{\partial x^{\nu}}\frac{3}{(q.x)^3}=3\frac{\partial(q.x)^{-3}}{\partial (q.x)}*\frac{\partial (q.x)}{\partial x^{\nu}}...

What is the difference between ##{T{_{a}}^{b}}## and ##{T{^{a}}_{b}}## ? Both are (1,1) tensors that eat a vector and a dual to produce a scalar. I understand I could act on one with the metric to raise and lower indecies to arrive at the other but is there a geometric difference between the...

I need to prove that in a vacuum, the energy-momentum tensor is divergenceless, i.e. $$ \partial_{\mu} T^{\mu \nu} = 0$$ where $$ T^{\mu \nu} = \frac{1}{\mu_{0}}\Big[F^{\alpha \mu} F^{\nu}_{\alpha} - \frac{1}{4}\eta^{\mu \nu}F^{\alpha \beta}F_{\alpha \beta}\Big]$$ Here ##F_{\alpha...

Homework Statement Show that for a second order cartesian tensor A, assumed invertible and dependent on t, the following holds: ## \frac{d}{dt} det(A) = det(a) Tr(A^{-1}\frac{dA}{dt}) ## Homework Equations ## det(a) = \frac{1}{6} \epsilon_{ijk} \epsilon_{lmn} A_{il}A_{jm}A_{kn} ## The...

Hi everyone, I am trying to self study some general relativity however I met some problem in the contravarient and covarient basis. In the lecture, or you can also find it on wiki page 'curvilinear coordinates', the lecturer introduced the tangential vector ei =∂r/∂xi and the gradient vector ei...

Hi I am a person who always have had a hard time picking up new definitions. Once I do, the rest kinda falls into place. In the case of abstract algebra, Stillwell's Elements of Algebra saved me. However, in the case of Spivak's Calculus on Manifolds, I get demotivated when I get to concepts...

Consider the expression $$\left(T^{a}\partial_{\mu}\varphi^{a} + A_{\mu}^{a}\varphi^{b}[T^{a},T^{b}] + A_{\mu}^{a}\phi^{b}[T^{a},T^{b}]\right)^{2},$$ where ##T^{a}## are generators of the ##\textbf{su}(N)## Lie algebra, and ##\varphi^{a}##, ##\phi^{a}## and ##A_{\mu}^{a}## are numbers. How...

The Lorentz transformation matrix may be written in index form as Λμ ν. The transpose may be written (ΛT)μ ν=Λν μ. I want to apply this to convert the defining relation for a Lorentz transformation η=ΛTηΛ into index form. We have ηρσ=(ΛT)ρ μημνΛν σ The next step to obtain the correct...

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 am reading Bruce N. Coopersteins book: Advanced Linear Algebra (Second Edition) ... ... I am focused on Section 10.3 The Tensor Algebra ... ... I need help in order to get a basic understanding of Example 10.1 in Section 10.3 ...Example 10.1 plus some preliminary definitions reads as...

If one has two single-particle Hilbert spaces ##\mathcal{H}_{1}## and ##\mathcal{H}_{2}##, such that their tensor product ##\mathcal{H}_{1}\otimes\mathcal{H}_{2}## yields a two-particle Hilbert space in which the state vectors are defined as $$\lvert\psi ,\phi\rangle...

I am reading Bruce N. Coopersteins book: Advanced Linear Algebra (Second Edition) ... ... I am focused on Section 10.3 The Tensor Algebra ... ... I need help in order to get a basic understanding of Definition 10.5 in Section 10.3 ...Definition 10.5 plus some preliminary definitions reads as...

In the transformation of tensor components when changing the co-ordinate system, can someone explain the following: Firstly, what is the point in re-writing the indicial form (on the left) as aikTklajl? Since we're representing the components in a matrix, and the transformation matrix is also...

I'm having trouble with the mathematics of tensor products as applied to Bell states. Say I have the state \begin{align*} \left|\psi\right> &= \frac{1}{\sqrt{2}} \left(\left|0\right>_A \otimes \left|0\right>_B + \left|1\right>_A \otimes \left|1\right>_B\right) \end{align*} How would the...

Hello all! I've just started to study general relativity and I'm a bit confused about dual basis vectors. If we have a vector space \textbf{V} and a basis \{\textbf{e}_i\}, I can define a dual basis \{\omega^i\} in \textbf{V}^* such that: \omega^i(\textbf{e}_j) = \delta^i_j But in some pdf and...

I am trying to show that if (C^ab)(A_a)(B_b) is a scalar for arbitrary vectors A_a and B_b then C^ab is a tensor. I want to take the product of the two vectors then use the quotient rule to show that C^ab must then be a tensor. This lead to the question of whether or a not the product of two...

I am trying to derive the geodesic equation using variational principle. My Lagrangian is $$ L = \sqrt{g_{jk}(x(t)) \frac{dx^j}{dt} \frac{dx^k}{dt}}$$ Using the Euler-Lagrange equation, I have got this. $$ \frac{d^2 x^u}{dt^2} + \Gamma^u_{mk} \frac{dx^m}{dt} \frac{dx^k}{dt} =...

Dear All, I'm doing some tensor calculation on the divergence of gradient (of a vector) inverse. Am I allowed to first use the nabla operator on gradient and then inverse the whole product? In other words, I'm searching for the divergence of a 2nd order tensor which is itself inverse of...