Tensor Definition and 1000 Threads

(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...

https://www.physicsforums.com/attachments/205736

Introduction Let me start with a counter-question. What is a number? Before you laugh, there is more to this question than one might think. A number can be something we use to count or more advanced an element of a field like real numbers. Students might answer that a number is a scalar. This is...

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 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

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...

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?

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...