Tensor Definition and 1000 Threads

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...

How do the components of the stress-energy tensor act on gravity regarding a) the FRW-universe? b) a solid ball? In a FRW-universe ##\rho + 3P## determines the second derivative of the scale factor. So, there are no non-diagonal components. Just theoretically, if the perfect fluid was...

A non-degenerate Hermitian form ##(.|.)## on a vector space ##V## can be identified with a map ##L:V \to V^*## such that ##L(v)=\tilde{v}## and ##\tilde{v}(w) \equiv (v~|~w)##. Suppose we want to convert a vector ##v## to a dual vector ##\tilde{v}##. In terms of matrices, we can just construct...

Homework Statement Hi everyone! Just wondering if there's a way to prove the symmetry property of the Riemann curvature tensor $$ R_{abcd} = R_{cdab}$$ without using the anti-symmetry property $$ R_{abcd} = -R_{bacd} = -R_{abdc} $$? I'm only able to prove it with the anti-symmetry property and...

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...

I know that a vector is a tool to help with quantities that have both a magnitude a direction. At a given point in space, a vector has a particular magnitude and direction and if we take any other direction at the same point we can get a projection of this vector in that direction. Tensor is a...

Hello Everyone. I Was Wondering how excatly the Gauge invariance of the trace of the Energy-momentum tensor in Yang-Mills theory connects with the trace of an Holonomy. To be precise in what I'm asking: The Yang-Mills Tensor is defined as: $$F_{\mu \nu} (x) = \partial_{\mu} B_{\nu}(x)-...

Hi, I am studying Chapter14 in Jackson. My attached file is about field strength tensor. My question is how can I obtain the radiation and the non-radiation terms in the field strength tensor for a moving charged particle. Many thanks.

Homework Statement The energy-band dispersion for a 3D crystal is given by $$E(\mathbf k) = E_0 - Acos(k_xa) - Bcos(k_yb) - Ccos(k_zc)$$ What is the value of the effective mass tensor at ## \mathbf k = 0 ##? Homework Equations The effective mass tensor is given by $$ \left( \frac{1}{m^*}...

Many sources give explanations of the Riemann tensor that involve parallel transporting a vector around a loop and finding its deviation when it returns. They then show that this same tensor can be derived by taking the commutator of second covariant derivatives. Is there a way to understand why...

Hello, Riemann tensor ##R^i_{jkl}## 4 indexes, and it should be matrix 16x16 in spacetime if we have time coirdinate - 0 and space coordinates -1,2,3. But how should I write the components to matrix? For example ##\begin{pmatrix}R^0_{000} & R^1_{000} & R^2_{000} ... \\ R^0_{100} & R^1_{100} &...

Hello, I'm trying to calculate Christoffel symbols on 2D surface of 3D sphere, the metric tensor is gij = diag {1/(1 − k*r2), r2}, where k is the curvature. I derived it using the formula for symbols of second kind, but I think I've made mistake somewhere. Then I would like to know which of the...

Hello! Does anyone have an idea of how can I obtain information from a band diagram about the directions along which the system conducts best and worst ? Thank you in advanced! :)

Homework Statement I want to be able, for an arbitrary Lagrangian density of some field, to derive the energy-momentum tensor using Noether's theorem for translational symmetry. I want to apply this to a specific instance but I am unsure of the approach. Homework Equations for a field...

Hi all, I have some doubts regarding the strain tensors Abaqus uses for the case of Geometrical non linear analysis. In the case of geometrically linear analysis, all the strain tensors will be equal to engineering strain, So it doesn't matter which strain tensor Abaqus uses. But for...

Homework Statement Hi I am looking at part a). Homework Equations below The Attempt at a Solution I can follow the solution once I agree that ## A^u U_u =0 ##. However I don't understand this. So in terms of the notation ( ) brackets denote the symmetrized summation and the [ ] the...

Riemann tensor is defined mathematically like this: ##∇_k∇_jv_i-∇_j∇_kv_i={R^l}_{ijk}v_l## Using covariant derivative formula for covariant tensors and covariant vectors. which are ##∇_av_b=∂_av_b-{Γ^c}_{ab}v_c## ##∇_aT_{bc}=∂_av_{bc}-{Γ^d}_{ac}v_{db}-{Γ^d}_{ab}v_{dc} ##, I got these...

Suppose one has a box moving through flat space-time with a stress energy tensor ##T^{ab}## that's non-zero inside the box and zero outside the box. How does one compute the normal forces on the faces of the box associated with it's motion? I am assuming that the normal forces are measured...

Hello guys! I have to solve a problem about crystal symmetry, but I am very lost, so I wonder if anyone could guide me. The problem is the following: Using semiclassical transport theory the conductivity tensor can be defined as: σ(k)=e^2·t·v_a(k)·v_b(k) Where e is the electron charge, t...

What is g11? I am very curious, can someone briefly describe what the metric tensor is, please?

Does anyone know how can you prove that the mean value of the tensor operator S12 in all directions r is zero? S12 : http://prntscr.com/j3gn40 where s1, s2 are the spin operators of two nucleons.

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...

Homework Statement This should be pretty simple and I guess I am doing something stupid? ##T_{bv}=(p+\rho)U_bU_v-\rho g_{bv}## compute ##T^u_v##: ##T^0_0=\rho, T^i_i=-p##Homework Equations ##U^u=\delta^t_u## ##g_{uv}## is the FRW metric,in particular ##g_{tt}=1## ##g^{bu}T_{bv}=T^u_v## ##...

Homework Statement Given an electromagnetic tensor ##F^{\mu\nu}##, showing that: $$\det{F^{\mu}}_\nu=-(\vec{B}\cdot\vec{E})^2$$ Homework Equations The Attempt at a Solution I had only the (stupid) idea of writing explictly the matrix associated with the electromagnetic tensor and calculating...

I am confused about tensor invariance as it applies to velocity and energy. My understanding is a tensor is a mathematical quantity that has the same value for all coordinate systems. I also understand that a vector is a first order tensor and energy is a zero order tensor. Thus, they should...

How do astrophysicists accurately account for all of the energy and pressure within a galaxy? How is it tabulated? My understanding of general relativity predicts that space-time curvature is a consequence of mass, energy, and pressure as expressed in the Energy-Momentum tensor. The accepted...

Hi, when working with NS equations the stress tensor can be written as ##\nabla \tau = - \nabla P + \nabla \tau_{v}##, where ##\tau_{v} ## is \begin{pmatrix} \tau_{xx} & \tau_{xy} & \tau_{xz} \\ \tau_{xy} & \tau_{yy} & \tau_{yz} \\ \tau_{zx} & \tau_{zy} & \tau_{zz} \end{pmatrix} This...

Defining dS2 as gijdxidxj and given dS2 = (dx1)2 + (dx2)2 + 4(dx1)(dx2). Find gijNow here is my take on the solution: Since the cross terms are present in the line element equation, we can say that it's a non-orthogonal system. So what I did was express the metric tensor in form of a 2x2...

So the Euler-Lagrange equations give ##\partial _\mu ( \partial ^\mu A^\nu - \partial ^\nu A^\mu ) = J^\nu## with ##B=\nabla \times A##. I want to convert this to ##\nabla \times B - \frac{\partial E}{\partial t} = \vec{j}##. I reckon I am supposed to use the Minkowski metric to raise or lower...

In one General Relativity paper, the author states the following (we can assume tensor in question are tensors in a vector space ##V##, i.e., they are elements of some tensor power of ##V##) To discuss general properties of tensor symmetries, we shall use the representation theory of the...

Please forgive me if I chose the wrong thread level. I don't think this is an undergrad topic but I'm not sure. I'm looking for some info about the polarization-magnetization tensor; I can't seem to find it anywhere.

It is known that vectors change them sing under the influence of parity when ##(x,z,y)## change into ##(-x,-z,-y)## $$P: y_{i} \rightarrow -y_{i}$$ where ##i=1,2,3## But what about vectors in Minkowski space? Is it true that $$P: y_{\mu} \rightarrow -y_{\mu}$$ where ##\mu=0,1,2,3##. If yes how...

In this topic https://physics.stackexchange.com/questions/129417/what-is-pseudo-tensor one answer was the next: The action of parity on a tensor or pseudotensor depends on the number of indices it has (i.e. its tensor rank): - Tensors of odd rank (e.g. vectors) reverse sign under parity. -...

Hi everyone. Could you help me to find the way to prove some things? 1)Changing of body velocity or reference frame don't contribute to spacetime curvature 2)On the contrary the change of body mass causes the change of curvature in local spacetime I use the assumption that if we have the same...

1. The metric ##g_{\mu \nu}## of spacetime shall be constructed from tensor products of vectors (relevant are the unit vectors in the respective directions). One such vector shall be called ##A##. Homework Equations ##g_{\mu \nu} = \lambda \frac{A_\mu A_\nu}{g^{\alpha \beta} A_\alpha A_\beta}...

whyT^[ab][;b]≠T_[ab][;b] for spatially flat FLWR cosmology ((ds)^2=(c^2)* (dt)^2-a(t)^2[(dx)^2+(dy)^2+(dz)^2])? τ[ab][/;b] gives the right answer, but not τ[ab][/;b]. (T^(ab) or T_(ab)) contra-variant and co-variant energy momentum tensor of perfect fluid (;) covariant derivative, (c) spped of...

Say I wanted to set up EFE for the Earth and moon. How do I actually go about filling the stress energy tensor? I'm referencing the wikipedia page. So the time-time should be approximately E/c^2V, so for the Earth moon system ##T_{00} = \frac{3}{4\pi r_E^3}\frac{1}{c^2}(M_Ec^2 + 2/5...

Hi, I have seen the general form for the metric tensor in general relativity, but I don't understand how that math would create a Minkowski metric with the diagonal matrix {-1 +1 +1 +1}. I assume that using the kronecker delta to create the metric would produce a matrix that has all positive 1s...

I am a little bit confused about the metric tensor and would like some feedback before I proceed with my learning of GR. So I understand that metric tensor describes the geometry of the space itself. I also understand that the components of the metric tensor (any tensor for that matter) come...

I believe this thread is sufficiently different from one that was recently closed to not violate any guidelines, though there are unfortunately some similarities as the closed thread sparked the questions in my mind. If we look at the stress energy tensor of a perfect fluid in geometric units...