Tensor Definition and 1000 Threads

I am trying to learn RGTC. It gives value of particular Riemann tensor but not all of it. What should I do?

As you may know from some other thread, I was interested through the week in finding a general way of express the energy-momentum tensor that appears in one side of the Einstein's equation. After much trials, I found that $$T^{\sigma \nu} = g^{\sigma \nu} \frac{\partial \mathcal{L}}{\partial...

first of all english is not my mother tongue sorry. I want to ask if you can help me with some of the properties of the levi-civita symbol. I am showing that $$\epsilon_{ijkl}\epsilon_{ijmn}=2!(\delta_{km}\delta_{ln}-\delta_{kn}\delta_{lm})$$ so i have this...

Homework Statement Three equal point masses, mass M, are located at (a,0,0), (0, a, 2a) and (0, 2a, a). Find the centre of mass for this system. Use symmetry to determine the principle axes of the system and hence find the inertia tensor through the centre of mass. (based on Hand and Finch...

Hi everyone, I'm currently studying Griffith's Intro to Elementary Particles and in chapter 7 about QED, there's one part of an operation on tensors I don't follow in applying Feynman's rules to electron-muon scattering : ## \gamma^\mu g_{\mu\nu} \gamma^\nu = \gamma^\mu \gamma_\mu## My...

Hello, I have a question regarding the first equation above. it says dui=ai*dr=ai*aj*duj but I wonder how. (sorry I omitted vector notation because I don't know how to put them on) if dui=ai*dr=ai*aj*duj is true, then dr=aj*duj |dr|*rhat=|aj|*duj*ajhat where lim |dr|,|duj|->0 which means...

Let we have a 2D manifold. We choose a coordinate system where we can construct all geodesics through any point. Is it enough to derive a metric from geodesic equation? Or do we need to define something else for the manifold?

Homework Statement Consider a system formed by particles (1) and (2) of same mass which do not interact among themselves and that are placed in a potential of infinite well type with width a. Let H(1) and H(2) be the individual hamiltonians and denote |\varphi_n(1)\rangle and...

I have this Hamiltonian --> (http://imgur.com/a/lpxCz) Where each G is a matrix. I want to find the eigenvalues but I'm getting hung up on the fact that there are 6 indices. Each G matrix lives in a different space so I can't just multiply the G matrices together. If I built this Hamiltonain...

Homework Statement The lecture notes states that if ##T_{ij}=T_{ji}## (symmetric tensor) in frame S, then ##T'_{ij}=T'_{ji}## in frame S'. The proof is shown as $$T'_{ij}=l_{ip}l_{jq}T_{pq}=l_{iq}l_{jp}T_{qp}=l_{jp}l_{iq}T_{pq}=T'_{ji}$$ where relabeling of p<->q was used in the second...

Suppose a second rank tensor ##T_{ij}## is given. Can we always express it as the tensor product of two vectors, i.e., ##T_{ij}=A_{i}B_{j}## ? If so, then I have a few more questions: 1. Are those two vectors ##A_i## and ##B_j## unique? 2. How to find out ##A_i## and ##B_j## 3. As ##A_i## and...

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...

Hi. I was trying to translate the divergence theorem and the Green's theorem to tensor notation that we use in Relativity. For the divergence theorem, it was easy (please tell me if I'm wrong in the below derivation). I'm using the standard electromagnetic tensor ##F_{\mu \nu}## in place of the...

Given the metric of the gravitational field of a central gravitational body: ds2 = -ev(r)dt2 + eμ(r)dr2 + r2 (dθ2 + sin2θdΦ2) And the Chritofell connection components: Find the Riemannian curvature tensor component R0110 (which is non-zero). I believe the answer uses the Ricci tensor...

Homework Statement Attached Homework EquationsThe Attempt at a Solution So the question says 'some point'. So just a single point of space-time to be isotropic is enough for this identity hold? I don't quite understand by what is meant by 'these vectors give preferred directions'. Can...

I've thought of a new way (at least I never read it anywhere) of counting the independent components of the Riemann tensor, but I am not sure whether my arguments are valid, so I would like to ask whether my argument is sound or total bonkers. The Riemann tensor gives the deviation of a vector A...

The stress-energy tensor of a perfect fluid in its rest frame is: (1) Tij= diag [ρc2, P, P, P] where ρc2 is the energy density and P the pressure of the fluid. If Tij is as stated in eq.(1), the metric tensor gij of the system composed by an indefinitely extended perfect fluid in...

Homework Statement I have several problems that ask me to prove that some quantity "transforms like a tensor" For example: "Suppose that for each choice of contravariant vector (a vector) A^nu(x), the quantities B_mu(x) are defined at teach point through a linear relationship of the form...

Hello, so my question is, if for some metric, we have found (somehow) Fμν, and we know that: Fμν=∂μAν-∂νAμ, how do we find Aν? I tried solving the differential system after imposing the Lorentz gauge ∂μAμ=0 but still, without some initial guess about which components of A are zero, the system...

Homework Statement Hello, I know about the inertia tensor about one axis, but how about a body that rotates around 3 axis x,y and z such as a spacecraft with changes in the attitude. Thanks for you help. Homework EquationsThe Attempt at a Solution

We have got a disagreement with fresh_42 in https://www.physicsforums.com/threads/the-pantheon-of-derivatives-part-ii-comments.908009/#post-5718965 So I would like to ask specialists in differential geometry for a comment 1) gradient of a function defined as follows $$\nabla...

Consider a d dimensional integral of the form, $$\int \frac{d^d \ell}{(2\pi)^d} \frac{\ell^{\sigma} \ell^{\mu}}{D}\,\,\,\text{and}\,\,\, \int \frac{d^d \ell}{(2\pi)^d} \frac{\ell^{\sigma}}{D}$$ where ##D## is a product of several propagators. One can reduce this to a sum of scalar integrals by...

δij is the Kronecker delta - is this considered a tensor or vector? I know it means the identity when i=j so I'm going to guess tensor because it's a matrix rather than just a vector but I want to make sure. A matrix is a rank 2 tensor and a vector is a rank 1 and a scalar is a rank 0? How does...

Say, we have two Hilbert spaces ##U## and ##V## and their duals ##U^*, V^*##. Then, we say, ##u\otimes v~ \epsilon~ U\otimes V##, where ##'\otimes'## is defined as the tensor product of the two spaces, ##U\times V \rightarrow U\otimes V##. In Dirac's Bra-Ket notation, this is written as...

Hi, I have a question concerning the von Kármán equations. I want to better understand the compatibility relation. The wikipedia article states that: https://en.wikipedia.org/wiki/F%C3%B6ppl%E2%80%93von_K%C3%A1rm%C3%A1n_equations "The components of the three-dimensional Lagrangian Green strain...

Consider an expression of the following form: $$I^{\mu\nu}(r) = \int d^{3}k\ \ d^{3}l\ \ \delta^{4}(r-k-l)\ (g^{\mu\nu}k\cdot{l}+k^{\nu}k^{\mu}-k^{\mu}l^{\nu})$$ ##I^{\mu\nu}## must be of the form $$I^{\mu\nu}(r) = Ar^{\mu}r^{\nu} + B\eta^{\mu\nu},$$ where ##A## and ##B## are constants. How...

I am given a hamiltonian for a two electron system $$\hat H_2 = \hat H_1 \otimes \mathbb {I} + \mathbb {I} \otimes \hat H_1$$ and I already know ##\hat H_1## which is my single electron Hamiltonian. Now I am applying this to my two electron system. I know very little about the tensor product...

Suppose we wish to use Cartesian coordinates for points on the surface of a sphere. Then all derivatives of the metric would vanish and so the Riemann curvature tensor would vanish. But it would give us a wrong result, namely that the space is not curved. So it means that if we want to get...

Are they the same object?

We've been learning about tensor products. In particular, we've been looking at index notation for the tensor products of matrices like these: ## \left( \begin{array}{cc} a_{11} & a_{12} \\ a_{21} & a_{22} \end{array} \right)## And ## \left( \begin{array}{cc} b_{11} & b_{12} \\ b_{21} & b_{22}...

By space, I mean a vector space which could be a representation of a group or even have some expanded algebraic structure. So I am not sure if this question goes here or in the Algebra subforum. Consider the tensor square r\otimes r of an irreducible group representation r with itself, and...

Hello, I have encountered the concept of tensor product between two or more different vector spaces. I would like to get a more intuitive sense of what the final product is. Say we have two vector spaces ##V_1## of dimension 2 and ##V_2## of dimension 3. Each vector space has a basis that we...

Hi I am trying to follow the derivation in some notes I have for the field strength tensor using covariant derivatives defined by Du = ∂u - iqAu . The field strength is the defined by [ Du , Dv ] = -iqFuv The given answer is Fuv = ∂uAv - ∂vAu .When I expand the commutator I get this...

Before I go any further, I do understand the ways that mechanical engineering textbooks explain why stress is a tensor. But all of those explanations seem infused with geometry (which I do NOT mean in a negative way at all); and are demonsrtrations. I am searching for a more concise/abstract...

I'm stuck on an apparently obvious statement in special relativity, so I hope you can help me. Can I define Lorenz transformations as transformations that don't change the spacetime interval in M4 and from this deduct that the metric tensor is invariant under LT? I've always read that the...

Hi, I am struggling to derive the relations on the right hand column of eq.(4) in https://arxiv.org/pdf/1008.4884.pdfEven the easy abelian one (third row) which is $$D_\rho B_{\mu\nu}=\partial_\rho B_{\mu\nu}$$ doesn't match my calculation Since $$D_\rho B_{\mu\nu}=(\partial_\rho+i g...

The Cauchy stress tensor at a material point is usually visualized using an infinitesimal cube. The stress vectors (traction vectors) on opposite sides of the cube are equal in magnitude and opposite in direction. As a result, the infinitesimal cube is in equilibrium. However, when we derive...

How to proof that εμνρσ Rμνρσ =0 ? Thanks.

Is that true in general and why: $$A^{mn}_{.~.~lm}=A^{nm}_{.~.~ml}$$

Homework Statement If f(x) is a scalar-valued function, show that ∂ƒ²/∂xi∂xj are the components of a Cartesian tensor of rank 2. Homework Equations N/A The Attempt at a Solution I don't even know where to begin. We began learning tensors in multivariable calculus (though I don't think this is...

Homework Statement Homework Equations The Attempt at a Solution I am really lost here because our professor gave us no example problems leading up to the final exam and now we are expected to understand everything about vector calculus. This is my attempt at the cross product and...

In one of our lectures we wrote Maxwell's equations as (with ##c=1##) ##\partial_\mu F^{\mu \nu} = 4\pi J^\nu## ##\partial_\mu F_{\nu \rho} + \partial_\nu F_{\rho \mu} + \partial_\rho F_{\mu \nu} = 0## where the E.M. tensor is ## F^{\mu \nu} = \begin{pmatrix} 0 & -B_3 & B_2 & E_1\\ B_3 & 0 &...

Homework Statement Suppose we have a covariant derivative of covariant derivative of a scalar field. My lecturer said that it should be equal to zero. but I seem to not get it Homework Equations Suppose we have $$X^{AB} = \nabla^A \phi \nabla^B \phi - \frac{1}{2} g^{AB} \nabla_C \phi \nabla^C...

I wonder if it is possible to write the components of the metric tensor (or any other tensor) as a summ of functions of the coordinates? Like this: g^{\mu\nu} = \sum_{\mu}^{D}\sum_{\nu}^{D} g_{_1}(x^{\mu}) g_{_2}(x^{\nu}) where g1 and g2 are functions of one variable alone and D is the...

Homework Statement Hi, I can't seem to understand the following formula in my professor's lecture notes: F_αβ = g_αγ*g_βδ*F^(γδ) Homework Equations Where g_αβ is the diagonal matrix in 4 dimensions with g_00 = 1 and g_11 = g_22 = g_33 = -1 and F^(γδ) is the electromagnetic tensor with c=1...

Homework Statement Hello, I am supposed to show that the quantity TR=JTF-t satisfies TR=∂W/∂F for some scalar function W(X, F, θ) in my continuum mechanics homework. The task is to identify this scalar function W(X, F, θ).Homework Equations This is part b) of a question. In part a), we get...

I was thinking about the metric tensor. Given a metric gμν we know that it is symmetric on its two indices. If we have gμν,α (the derivative of the metric with respect to xα), is it also valid to consider symmetry on μ and α? i.e. is the identity gμν,α = gαν,μ valid?

How would one go about setting up the stress energy tensor for a particle, say an electron subjected to electric an electric field that makes the particle oscillate with frequency \omega?

Hi, I'm looking for a modern, colourful, illustrative introductory textbook to work through on tensor calculus/continuum mechanics. I'd like one with lots of physical examples, exercises, summaries, etc. I'd like an emphasis on engineering. Something in the mould of Frank White's Fluid...

I am taking a course on GR and trying to understand Tensor calculus. I think I understand contravariant tensor (transformation of objects such as a vector from one frame to another) but I am having a hard time with covariant tensors. I looked into the Wikipedia page...