Tensor Definition and 1000 Threads

Homework Statement I am given c11, c12, and c44. What is poissons ratio ν and the E modulus E [100] for a single crystal for uniaxial strain in [100] (if Fe is isotropic)? ii) What is the anisotropy factor A? (iii) There is: sigma=[100 0 0; 0 100 0; 0 0 0]Mpa What is the transverse strain in...

I would like to know where one may operate with tensor quantities in quantum field theory: Minkowski tensors, spinors, effective lagrangians (for example sigma models or models with four quark interaction), gamma matrices, Grassmann algebra, Lie algebra, fermion determinants and et cetera. I...

I am trying to learn GR. In two of the books on tensors, there is an example of evaluating the inertia tensor in a primed coordinate system (for example, a rotated one) from that in an unprimed coordinate system using the eqn. ##I’ = R I R^{-1}## where R is the transformation matrix and...

Homework Statement Explain how it is possible to perform a contraction of the tensor ##T^{\beta \gamma}_{\delta \epsilon}## in order to produce a scalar T Homework EquationsThe Attempt at a Solution $$T^{\beta \gamma}_{\delta \epsilon}T_{\beta \gamma}^{\delta \epsilon}=T$$ Not sure if that is...

I am trying to figure how to get 1. from 2. and vice versa where the e's are bases for the vector space and θ's are bases for the dual vector space. 1. T = Tμνσρ(eμ ⊗ eν ⊗ θσ ⊗ θρ) 2. Tμνσρ = T(θμ,θν,eσ,eρ) My attempt is as follows: 2. into 1. gives T = T(θμ,θν,eσ,eρ)(eμ ⊗ eν ⊗ θσ ⊗ θρ)...

I have been trying to fully grasp the concept of the Cauchy stress tensor and so I thought I'd make a post where I clear up my confusion. There may be subsequent replies as I pose more questions. I am specifically confused at how the stress tensor relates to the control volume in the image...

Hello, I've found that the Faraday Tensor with both indeces down has in the first line, in MTW Gravitation book (pg 74, eq 3.7), minus the electrical field, while in Wikipedia we find that it is plus the electrical field. Which one is right? Does it depend on the signature of the metric?

I was working out the components of the Riemann curvature tensor using the Schwarzschild metric a while back just as an exercise (I’m not a student, and Mathematica is expensive, so I don’t have access to any computing programs that can do it for me, and now that I’m thinking about it, does...

Homework Statement How to proof the following property of tensor invariants? Where: ##[\mathbf{a\; b\; c}]=\mathbf{a\cdot (b\times c)} ##, ##\mathbf{T} ##is a second order tensor, ##\mathfrak{J}_{1}^{T}##is its first invariant, ##\mathbf{u, v, w}## are vectors. Homework Equations...

Hello! Kunneth fromula states that for 3 manifolds such that ##M=M_1 \times M_2## we have ##H^r(M)=\oplus_{p+q=r}[H^p(M_1)\otimes H^q(M_2)]##. Can someone explain to me how does the tensor product acts here? I am a bit confused of the fact that we work with r-forms, which are by construction...

Hi All, I am evaluating the components of the stress-energy tensor for a (Klein-Gordon) complex scalar field. The ultimate aim is to use these in evolving the scalar field using the Klein-Gordon equations, coupled to Einstein's equations for evolving the geometric part. The tensor is given by...

Hello! I was thinking about the Riemann curvature tensor(and the torsion tensor) and the way they are defined and it seems to me that they just need a connection(not Levi-Civita) to be defined. They don't need a metric. So, in reality, we can talk about the Riemann curvature tensor of smooth...

Hello! I have just started the Einstein field equations in my readings on GR and I want to make sure I understand the stress energy tensor. If we have a spherical, non-moving, non-spinning source, let's say a neutron star (I don't know much about neutron stars, so I apologize if the non-moving...

Hi, I'm getting into general relativity and am learning about tensors and coordinate transformations. My question is, how do you use the metric tensor in polar coordinates to find the distance between two points? Example I want to try is: Point A (1,1) or (sq root(2), 45) Point B (1,0) or...

(Forgive me if this is in the wrong spot) I understand how tensors transform. I can easily type a rule with the differentials of coordinates, say for strain. I also know that the moment of inertia is a tensor. But I cannot see how it transforms as does the standard rules of covariant...

The smoothed Weyl tensor can look like space that contains a non-zero Einstein tensor. To verify this, consider that gravitational waves carry mass away from (say) a rotating binary, so the apparent mass at infinity of a large sphere containing a radiating binary will be greater than the mass...

Consider the AdS metric in D+1 dimensions ds^{2}=\frac{L^{2}}{z^{2}}\left(dz^{2}+\eta_{\mu\nu}dx^{\mu}dx^{\nu}\right) I wanted to calculate the Ricci tensor for this metric for D=3. ([\eta_{\mu\nu} is the Minkowski metric in D dimensions) I have found the following Christoffel symbols...

Consider the AdS metric in D+1 dimensions ds^{2}=\frac{L^{2}}{z^{2}}\left(dz^{2}+\eta_{\mu\nu}dx^{\mu}dx^{\nu}\right) I wanted to calculate the Ricci tensor for this metric for D=3. (\eta_{\mu\nu} is the Minkowski metric in D dimensions) I have found the following Christoffel symbols...

Homework Statement I am trying to derive the following relation using inner products of vectors: Homework Equations g_{\mu\nu} g^{\mu\sigma} = \delta_{\nu}^{\hspace{2mm}\sigma} The Attempt at a Solution What I have done is take two vectors and find the inner products in different ways with...

Homework Statement In an inertial frame O calculate the components of the stress–energy tensors of the following systems: (a) A group of particles all moving with the same velocity ##v = \beta e_x##, as seen in O. Let the rest-mass density of these particles be ##\rho_0##, as measured in...

Greetings, can somebody show me how to calculate such a term? P= X E² where X is a third order tensor and E and P are 3 dimensional vectors. Since the result is supposed to be a vector, the square over E is not meant to be the scalar product. But the tensor product of E with itself yields a...

Hello! I am reading that in a perfect fluid we have no heat conduction, which implies that energy can flow out of a fluid element only if particles flow, so ##T^{0i} = 0##. I am not sure I understand why. We have ##\Delta E = \Delta Q - p \Delta V##. In our case as Q is constant, ##\Delta E = -p...

Hello! I am reading about stress energy tensor of a perfect fluid and I don't understand the ##T^{ij}## terms. They are defined to be the flux of i-th momentum through the j-th surface. Now you take a fluid element and in its momentary comoving reference frame (MCRF) you calculate these...

Hi all, I am reading Bernard Schutz's a first course in general relativity. In Chapter 4 it introduced the energy stress tensor in two ways: 1.) Dust grain 2.) Perfect fluid. The book defined the energy stress tensor for dust grain to be ## p⊗N ##, where ##p## is the 4 momentum for a single...

Hi initially I am aware that christoffel symbols are not tensor so their covariant derivatives are meaningless, but my question is why do we have to use covariant derivative only with tensors? ?? Is there a logic of this situation? ?

Say we have a matrix L that maps vector components from an unprimed basis to a rotated primed basis according to the rule x'_{i} = L_{ij} x_{j}. x'_i is the ith component in the primed basis and x_{j} the j th component in the original unprimed basis. Now x'_{i} = \overline{e}'_i. \overline{x} =...

Maxwell stress tensor ##\bar{\bar{\mathbf{T}}}## in the static case can be used to determine the total force ##\mathbf{f}## acting on a system of charges contanined in the volume bounded by ##S## $$ \int_{S} \bar{\bar{\mathbf{T}}} \cdot \mathbf n \,\,d S=\mathbf{f}= \frac{d}{dt} \mathbf...

This is an interesting question that popped through my mind. Some of us should know what is meant by „gauge transformations”, „gauge invariance/symmetry” and are used to seeing these terms whenever lectures on quantum field theory are read. But the electromagnetic field in vacuum (described in a...

Good Day, Another fundamentally simple question... if I go here; http://www-hep.physics.uiowa.edu/~vincent/courses/29273/metric.pdf I see how to calculate the metric tensor. The process is totally clear to me. My question involves LANGUAGE and the ORIGIN LANGUAGE: Does one say "one...

If F = Fxi + Fyj +Fzk is a force field, do the following derivatives have physical significance and are they related to the components of the stress tensor? I notice they have the same dimensions as stress. ∂2Fx / ∂x2 ∂2Fx / ∂y2 ∂2Fx / ∂z2 ∂2Fx / ∂z ∂y ∂2Fx / ∂y ∂z ∂2Fx / ∂z ∂x ∂2Fx / ∂x...

Homework Statement An electric field E exerts (in Gaussian cgs units) a pressure E2/8π orthogonal to itself and a tension of this same magnitude along itself. Similarly, a magnetic field B exerts a pressure B2/8π orthogonal to itself and a tension of this same magnitude along itself. Verify...

Do the field equations themselves constrain the metric tensor? or do they just translate external constraints on the stress-energy tensor into constraints on the metric tensor? another way to ask the question is, if I generated an arbitrary differentiable metric tensor field, would it translate...

I'm trying to derive the Klein Gordon equation from the Lagrangian: $$ \mathcal{L} = \frac{1}{2}(\partial_{\mu} \phi)^2 - \frac{1}{2}m^2 \phi^2$$ $$\partial_{\mu}\Bigg(\frac{\partial \mathcal{L}}{\partial (\partial_{\mu} \phi)}\Bigg) = \partial_{t}\Bigg(\frac{\partial \mathcal{L}}{\partial...

I wonder if there is a "general rule", a kind of "algorithm" for finding the components of the Stress-Energy tensor in for particular cases. For the Einstein tensor, just by knowing the metric, one can find the components of it. What about the Stress-Energy tensor? One way I thought of (and...

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fresh_42 submitted a new PF Insights post What Is a Tensor? Continue reading the Original PF Insights Post.

I've been reading Fleisch's "A Student's Guide to Vectors and Tensors" as a self-study, and watched this helpful video also by Fleisch: Suddenly co-vectors and one-forms make more sense than they did when I tried to learn the from Schutz's GR book many years ago. Especially in the video...

In coordinates given by x^\mu = (ct,x,y,z) the line element is given (ds)^2 = g_{00} (cdt)^2 + 2g_{oi}(cdt\;dx^i) + g_{ij}dx^idx^j, where the g_{\mu\nu} are the components of the metric tensor and latin indices run from 1-3. In the first post-Newtonian approximation the space time metric is...

I am trying to get a good feel for the Stress-Energy tensor, but I seem to be hung up on a few concepts and I was wondering if anyone could clear up the issues. First, when I look at the derivation of the Stress-Energy tensor for a perfect fluid (of one species, say), the 00 entry can be...

I was wondering if anyone knows how to set up a procedure in REDUZE that will decompose tensor integrals appearing in QCD loop calculations into a sum of scalar topologies with the tensor structure factored out? I've had a look at the appropriate manual but I am not entirely sure how to...

I was reading about strain rate tensors and other kinematic properties of fluids that can be obtained if we know the velocity field V = (u, v, w). It got me wondering if I can sketch streamlines if I have the strain rate tensor with me to start with. Let's say I have the strain rate tensor...

I like the spectral-flow viewpoint on chiral anomalies, as described for instance in Peskin & Schroeder, last part of Ch. 19.1 This appears to depend crucially on the concept of fermi sea level, making it specific to fermions. However, bosonic self-dual tensor fields also have an anomaly...

Hi All I would like to know if there is a way to produce simple one dimensional kinematic exercises with space-time metric tensor different from the Euclidean metric. Examples, if possible, are welcome. Best wishes, DaTario

In page 64 of David Tong's notes (http://www.damtp.cam.ac.uk/user/tong/string/four.pdf) on conformal field theory, Tong mentions that 1. the stress-energy tensor is defined as the matrix of conserved currents which arise from translational invariance, $$\delta\sigma^{a} = \epsilon^{a},$$ where...

I'm studying the component representation of tensor algebra alone. There is a exercise question but I cannot solve it, cannot deduce answer from the text. (text is concise, I think it assumes a bit of familiarity with the knowledge) (a) Convert the following expressions and equations into...

<This thread is a spin-off from another discussion. Cp. https://www.physicsforums.com/threads/wedge-product.914621/#post-5762138> Also again, be warned about this sloppy notation of indizes. You should put the prime on the symbol (or in addition to the symbol). Otherwise the equations don't...

What is a tensor quantity and how is it different from vector and scalar qantities? Also why is pressure a tensor quantity?

Hi everyone, I am currently working on a subject that involves a lot of 4th order tensors computations including double dot product and inverse of fourth order tensors. First the definitions so that we are on the same page. What I call the double dot product is : $$ (A:B)_{ijkl} =...

In the lecture notes http://top.electricalandcomputerengineering.dal.ca/PDFs/Web%20Page%20PDFs/ECED6400%20Lecture%20Notes.pdf at page 15 eq. (2.46) it says that the dielectric tensor in an isotropic media can be represented by: δi j A(k,ω) + ki kj B(k,ω) I understood that in the case of I. M...

I am doing some mathematical exercises with 3D anti-de sitter face using the metric ds2=-(1+r2)dt2+(1+r2)-1+r2dφ2 I found the three geodesics from the Christoffel symbols, and they seem to look correct to me. d2t/dλ2+2(r+1/r)*(dt/dλ)(dr/dλ)=0...