Tensor Definition and 1000 Threads

Hi, yesterday a professor of mine told me that if you have a temperature gradient along the x-axsis you could have heat flowing in the y direction. Mathematically it is pretty straightforward to find the thermal conductivity tensor required, but in real life can you name some materials that...

It is common to write e.g photon two point function in terms of manifest transverse and longitudinal form factors with lorentz structure factored out, e.g $$\Pi^{\mu \nu} = (g^{\mu \nu} - q^{\mu} q^{\nu}/q^2)T_T + q^{\mu} q^{\nu}T_L,$$ where mu and nu are polarisation indices. How do I relate...

Which properties of metric tensor are invariant of basevectors transforms? I know that metric tensor depends of basevectors, but are there properties of metric tensor, that are basevector invariant and describe space itself?

Hello, In Schawrtz, Page 116, formula 8.23, he seems to suggest that the square of the Maxwell tensor can be expanded out as follows: $$-\frac{1}{4}F_{\mu \nu}^{2}=\frac{1}{2}A_{\mu}\square A_{\mu}-\frac{1}{2}A_{\mu}\partial_{\mu}\partial_{\nu}A_{\nu}$$ where: $$F_{\mu\nu}=\partial_{\mu}...

Homework Statement The tensor force operator between 2 nucleons is defined as ##S_{12}=3\sigma_1\cdot r\sigma_2\cdot r - \sigma_1\cdot \sigma_2##. Where r is the distance between the nucleons and ##\sigma_1##and ##\sigma_2## are the Pauli matrices acting on each of the 2 nucleons. Rewrite...

Andrew Browder in his book: "Mathematical Analysis: An Introduction" ... ... defines a differential form in Section 13.1 which reads as follows: In the above text from Browder we read the following: " ... ... A differential form of degree ##r## (or briefly an ##r##-form) in ##U## is a map...

Homework Statement I have been working on defining the transpose in a tensorial view using the kroneck delta tensor.Homework Equations I will use tensor notation The Attempt at a Solution Let Tjk be a 2nd level tensor: (Tjk)TP = Tkj My first attempt is: δjkTjk δkj = Tkj However if tensor...

For a curve parametrised by ##\lambda## where ##\lambda## is along length of the curve and is 0 at one end point. At each ##\lambda## say tangent vector V and A be the two possible vectors of the tangent space. where ##V=V^\mu e_\mu## and ##A=A^\nu e_\nu##, {e} are the basis vectors. Now ##...

Hello I have been going through the cosmology chapter in Choquet Bruhats GR and Einstein equations and in definition 3.1 of chapter 5 she defines the sectional curvature with a Riemann( X, Y;X, Y) (X and Y two vectors) I don't understand this notation, regarding the use of the semi colon, is it...

If we have two points P and Q in undeformed material and after deformation they become P' and Q'. The deformation tensor is the derivative of the displacement. What is the displacement? vector PP'? or the change from PQ to P'Q'? is the second question is the strain "change in length". Why the...

Homework Statement Prove that $$\begin{align*}\mathfrak{T}_L(x) &= \frac{1}{2}\psi_L^\dagger (x)\sigma^\mu i\partial_\mu\psi_L(x) - \frac{1}{2}i\partial_\mu \psi_L^\dagger (x) \sigma^\mu\psi_L(x) \\ \mathfrak{T}_R(x) &= \frac{1}{2}\psi_R^\dagger (x)\bar{\sigma}^\mu i\partial_\mu\psi_R(x) -...

Let ##Q_ik## be a symetric tensor, so that ##Q_ik= \frac{m}{2} \dot x_i \dot x_j + \frac{k}{2} x_i x_j## (here k is also a sub, couldn't do it better with LaTeX). How do we derive such a tensor, with respect to time? And what could such a tensor mean in a physical sense? It really looks like the...

I understand, kind of, that ∇μTμυ=0 by conservation or coninuity. What would be ∂βTμυ when β=1,2,3 no time derivative.

Hello! I was talking with a friend today about electrical motors and we started talking about theoretical designs. One question came up which was could the Maxwell Stress Tensor be used to calculate the torque on a rotor of a motor where the airgap is held constant and the magnetic circuit...

How do physicists call a tensor of ## V^* \otimes V^* \otimes V##, (1,2) or (2,1)? And which part do they call contravariant and which covariant? I'm just not sure, whether the mathematical definition of funktors apply to the usances in physics. (LUP - tensor)

This is a simple and maybe stupid question. Can you take a derivative of a vector component with respect to a vector? Or even more generally,can you take the derivative of a component of a tensor with respect to the whole tensor? For instance in the cauchy tensor could you take the xx component...

If we have two sets of coordinates such that x1,x2...xn And y1,y2,...ym And if any yi=f(x1...,xn)(mutually dependent). Then dyi=(∂yi/∂xj)dxj Again dyi/dxk=(∂2yi/∂xk∂xj)dxj+∂yi/∂xk Is it the contravariant derivative of a vector?? Or in general dAi/dxk≠∂Ai/∂xk

Tensor of Riemann. Geometric interpretation.The Riemann tensor gives the variation of a vector displaced parallel in a closed loop, say a small rectangle formed by geodesic sides, (δa) and δb) first, starting from a vertex A and going to another vertex in the diagonal, B; then starting from the...

I have trouble understanding some terms in the stress-energy-tensor. For instance T^(12) stands for the flux of the x-component of momentum in the y-direction. But what does it means for the x-component of momentum to flow in the y direction? Since momentum is a vector should't the x-component...

I read in various sources (such as page 8 of these notes) that the dielectric permittivity tensor of a lossless medium is always symmetric. I am wondering how this can be the case, when: Phase accumulation in the medium could in theory depend on direction Coordinate system may be rotated to...

The energy momentum tensor of a perfect relativistic fluid is given by $$T^{\mu\nu} = (\rho + p)u^\mu u^\nu + p g^{\mu\nu}$$ I don't understand why this is a tensor, i.e. why it transforms properly under coordinate changes. ##u^\mu u^\nu## and ##g^{\mu\nu}## are tensors, so for ##T^{\mu\nu}##...

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}}...

Homework Statement This isn't exactly a problem but rather a problem in understanding the derivation of the phenomenon, or more precisely, one step in the derivation. In the following we will consider the EPR pair of two spin ##1/2## particles, where the state can be written as $$ \vert...

Consider equation (2.7.8) page 42 in the book Gravitation and Cosmology by Weinberg F' αβ = Λαγ Λβδ Fγδ Now consider the time reversal Lorenz transformation Λμν = 0 if μ ≠ ν, 1 if μ = ν = 1..3 and -1 if μ = ν = 0 then F' 00 = 0 F' 0i = -F 0i F' ij = F ij Using equation (2.7.5) of the same book...

In Chapter 11: Dynamics of Rigid Bodies, in the Classical Dynamics of Particles and Systems book by Thornton and Marion, Fifth Edition, pages 415-418, Section 11.3 - Inertia Tensor, I have three questions regarding the Inertia Tensor: 1.The authors made the following statement: "neither V nor ω...

Why are there (at least) two definitions of a tensor? For some people a tensor is a product of vectors and covectors, but for others it's a functional. While it's true that the two points of view are equivalent (there's an isomorphism) I find having to switch between them confusing, as a...

Homework Statement Prove Rijkl= k/R2 * (gik gjl-gil gjk) where gik is the 3 metric for FRW universe and K =0,+1,-1, and i,j=1,2,3, that is, spatial coordinates. . Homework Equations The Christoffel symbol definition: Γμνρ = ½gμσ(∂ρgνσ+∂νgρσ-∂σgνρ) and the Riemann tensor definition: Rμνσρ =...

Homework Statement Consider the equation for the friction force Ff = m FN. is it possible to express the friction force as a tensor? If so, what rank tensor is it, and what are the ranks of the tensor m and the normal force FN? Homework Equations Ff = mFNThe Attempt at a Solution [/B] So I...

I am still learning general relativity (GR). I know we can find the path of a test particle by solving geodesic equations. I am wondering if it is possible to derive/convert metric tensor to gravity, under weak approximation, and vice versa. Thanks!

How do we know that the stress tensor must be symmetric in the Navier-Stokes equation? Here are some papers that discuss this issue beyond the usual derivations: Behavior of a Vorticity Influenced Asymmetric Stress Tensor In Fluid Flow http://www.dtic.mil/dtic/tr/fulltext/u2/a181244.pdf...

Can I calculate a tensor of a system( lots of particles) shaped like a sphere, then get exactly the radius of the system? (I want to get lengths of three axes of ellipsoid, and I'm trying to examine the way with a sphere. )

Starting from the following definition of stress-energy tensor for a perfect fluid in special relativity : $${\displaystyle T^{\mu \nu }=\left(\rho+{\frac {p}{c^{2}}}\right)\,v^{\mu }v^{\nu }-p\,\eta ^{\mu \nu }\,}\quad(1)$$ with ##v^{\nu}=\dfrac{\text{d}x^{\nu}}{\text{d}\tau}## and...

Homework Statement Show that if the Lagrangian only depends on scalar fields ##\phi##, the energy momentum tensor is always symmetric: ##T_{\mu\nu}=T_{\nu\mu}## Homework Equations ##T_{\mu\nu}=\frac{\partial L}{\partial(\partial_\mu\phi)}\partial_\nu\phi-g_{\mu\nu}L## The Attempt at a...

Homework Statement Show that if you add a total derivative to the Lagrangian density ##L \to L + \partial_\mu X^\mu##, the energy momentum tensor changes as ##T^{\mu\nu} \to T^{\mu\nu}+\partial_\alpha B^{\alpha\mu\nu}## with ##B^{\alpha\mu\nu}=-B^{\mu\alpha\nu}##. Homework EquationsThe Attempt...

Homework Statement The Lagrangian density for the ##h=h^{00}## term of the Einstein gravity tensor can be simplified to: $$L=-\frac{1}{2}h\Box h + (M_p)^ah^2\Box h - (M_p)^b h T$$ The equations of motion following from this Lagrangian looks roughly like (I didn't calculate this, they are given...

Im trying to calculate the principals moments of inertia (Ixx Iyy Izz) for the inertia tensor by triple integration using cylindrical coordinates in MATLAB. % Symbolic variables syms r z theta R h M; % R (Radius) h(height) M(Mass) % Ixx unox = int((z^2+(r*sin(theta))^2)*r,z,r,h); % First...

<< Mentor Note -- OP has been reminded to use the Homework Help Template when posting schoolwork questions >> my think if ## \hat{r} = \sin(θ) \cos( φ) \hat{x} +\sin(θ) \sin( φ) \hat{y} +\cos(θ) \hat{z} ## ## da = R^2 \sin(θ) dθdφ \hat{r} = da_{x} \hat{x} + da_{x} \hat{y} + da_{z} \hat{z}##...

Is there a method to calculate inertia tensor form principal axes moment of inertia? Like now we have moment of inertia: (Ix,Iy,Iz)=(20,18,25), and hot to calculate the inertia tensor like (Ixx,Ixy,Ixz Iyx,Iyy,Iyz, Izx,Izy,Izz)? I have read about this page several times, but still have no idea.

I am really confused about coordinate transformations right now, specifically, from cartesian to polar coordinates. A vector in cartesian coordinates is given by ##x=x^i \partial_i## with ##\partial_x, \partial_y \in T_p \mathcal{M}## of some manifold ##\mathcal{M}## and and ##x^i## being some...

I've noticed that a very easy way to generate the Lorentz transformation is to draw Cartesian coordinate axes in a plane, label then ix and ct, rotate them clockwise some angle \theta producing axes ix' and ct', use the simple rotation transformation to produce ix' and ct', then just divide...

Given two probability distributions ##p \in R^{m}_{+}## and ##q \in R^{n}_{+}## (the "+" subscript simply indicates non-negative elements), this paper (page 4) writes down the tensor product as $$p \otimes q := \begin{pmatrix} p(1)q(1) \\ p(1)q(2) \\ \vdots \\ p(1)q(n) \\ \vdots \\...

I need to find all the non-zero components of the Riemann Tensor in a two-dimensional geometry knowing that the only two non-zero components of the Christoffel symbols are: \Gamma^x_{xx}=\frac{1}{x} and \Gamma^y_{yy}=\frac{2}{y} knowing that: R^\alpha_{\beta\gamma\delta}=\partial_\gamma...

I've been trying to understand representations of the Lorentz group. So as far as I understand, when an object is in an (m,n) representation, then it has two indices (let's say the object is ##\phi^{ij}##), where one index ##i## transforms as ##\exp(i(\theta_k-i\beta_k)A_k)## and the other index...

Greg Bernhardt submitted a new PF Insights post The 10 Commandments of Index Expressions and Tensor Calculus Continue reading the Original PF Insights Post.

Hello, I am reading Griffith's "Introduction to Electrodynamics" 4ed. I'm in the chapter on relativistic electrodynamics where he develops the electromagnetic field tensor (contravariant matrix form) and then shows how to extract Maxwell's equations by permuting the index μ. I am able to...

I am doing a problem from Schutz, Introduction to general relativity.The question asks you to find a coordinate transformation to a local inertial frame from a weak field Newtonian metric tensor ##(ds^2=-(1+2\phi)dt^2+(1-2\phi)(dx^2+dy^2+dz^2))##. I looked at the solution from a manual and it...

I understand that tensor equations are expressions that give the same answer regardless of the coordinate system when expressing the laws of nature. Does this invariance apply across different reference frames? Another words, do the tensor equations yield the same laws with respect to an...

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 am trying to calculate the Ricci tensor in terms of small perturbation hμν over arbitrary background metric gμν whit the restriction \left| \dfrac{h_{\mu\nu}}{g_{\mu\nu}} \right| << 1 Following Michele Maggiore Gravitational Waves vol 1 I correctly expressed the Chirstoffel symbol in terms...

Homework Statement From Misner, Thorne and Wheeler's text Gravitation (MTW), exercise 3.15: Show that, if F is the EM field tensor, then ##\nabla \cdot *F## is a geometric, frame-independent version of the Maxwell equation...