Tensor Definition and 1000 Threads

Hey, I have not done any proper differential geometry before starting general relativity (from Sean Carroll's book: space time and geometry), so excuse me if this is a stupid question. The metric tensor can be written as $$ g = g_{\mu\nu} dx^{\mu} \otimes dx^{\nu}$$ and its also written as...

Hi, this is my first message on thi forum :D I apologize in advance for my english. I'm doing my thesis work on the theory of relativity of Einstein-Cartan. I'm following the article of Hehl of 1976; it's title is "General relativity with spin and torsion: Foundations and prospects". I can't...

Homework Statement I'm analyzing the landing gear of a plane for an extracurricular project. I know that the landing gear will undergo impact loading during touch down. Using what we've been taught, I converted the impact load to a static load, P that will act on the landing gear. Now I need to...

Hi! I am new at robotics, can you guys please help me what is the principal moment of inertia?? how to define the pose of axis about center of mass of my robotic link of a legged robot?? please guide me with some visual representation and also how to calculate Inertia Tensor?? I will really...

In the EFE, what does adding Λgμν mean and why is it not included in the Einstein tensor?

Homework Statement Is the moment of inertia matrix a tensor? Hint: the dyadic product of two vectors transforms according to the rule for second order tensors. I is the inertia matrix L is the angular momentum \omega is the angular velocity Homework Equations The transformation rule for a...

Homework Statement Demonstrate that matrix ##T## represents a 2nd order tensor ##T = \pmatrix{ x_2^2 && -x_1x_2 \\ -x_1x_2 && x_1^2}## Homework Equations To show that something is a tensor, it must transform by ##T_{ij}' = L_{il}L_{jm}T_{lm}##. I cannot find a neat general form for ##T_{ij}##...

Homework Statement In a certain system of units the electromagnetic stress tensor is given by M_{ij} = E_iE_j + B_i B_j - \frac12 \delta_{ij} ( E_kE_k + B_kB_k) where E_i and B_i are components of the 1-st order tensors representing the electric and magnetic fields \bar{E} and \bar{B}...

1. Problem statement: Assume that u is a vector and A is a 2nd-order tensor. Derive a transformation rule for a 3rd order tensor Zijk such that the relation ui = ZijkAjk remains valid after a coordinate rotation.Homework Equations : [/B] Transformation rule for 3rd order tensors: Z'ijk =...

Hello, I am working through Hughston and Tod "An introduction to General Relativity" and have gotten stuck on their exercise [7.7] which asks to prove the following non- linear wave equation for the Riemann tensor in an empty space: ∇e∇eRabcd = 2Raedf Rbecf − 2Raecf Rbedf − Rabef Rcdef I have...

Question outline: In the case of 5d Kaluza (Klein) GR with NO charge and NO gauge field we expect 5d to reduce to 4d GR exactly. So this should be a very simple useful sanity check.s the side and corner terms of Einstein, Ricci and Energy tensors are zero, then R would be the same in 4d or 5d...

Hey it might be a stupid question but I saw that the tensor product of 2 vectors with dim m and n gives another vector with dimension mn and in another context I saw that the tensor product of vector gives a metrix. For example from sean carroll's book: "If T is a (k,l) tensor and S is a (m, n)...

Hello I'm new here on this forum and on physics too. I have problem on Einstein famous equation I have a problem on the last component Tαβ I know that tensor name is Einstein stress-energy tensor and I know that Tαβ...

Hey! I'm reading a book Intermediate Physics for Medicine and Biology In it, there is a section that is describing shear forces and it says this as a side note: In general, the force F across any surface is a vector. It can be resolved into a component perpendicular to the sur- face and two...

Source: http://gmammado.mysite.syr.edu/notes/RN_Metric.pdf Section 2 Page: 2 Eq. (15) The radial component of the magnetic field is given by B_r = g_{11} ε^{01μν} F_{μν} Where does this equation come from? Section 4 Page 3 Similar to the electric charges, the Gauss's flux theorem for the...

Hi, I am working through a textbook on general relativity and have come across the statement: "A general (2 0) tensor K, in n dimensions, cannot be written as a direct product of two vectors, A and B, but can be expressed as a sum of many direct products." Can someone explain to me how this...

I am confused at why ##V_{i,j}V_{j,k}A_{km,i}## the result will end up being a vector (V is a vector and A is a tensor) What are some general rules when you are multiplying a scalar, vector and tensor?

Homework Statement M= \begin{pmatrix} 2 & -1 & 0\\ -1 & 2 & -1\\ 0 & -1 & 2\\ \end{pmatrix} Compute \frac{1}{6}\epsilon_{ijk}\epsilon_{lmn} M_{il} M_{jm} M_{kn} . The Attempt at a Solution I computed the result which is 4, by realizing that there are 36 non-zero levi-civita containing...

I'm new to working with tensors, and feel a bit uneasy about the nomenclature. I picture words like antisymmetry in terms of average random matrices where no symmetry can be found at all. However, if I understand it correctly, antisymmetry is a type of symmetry, but where signs are inverted. So...

Hi, Want to know (i) what does Riemannian metric tensor and Christoffel symbols on R2 mean? (ii) How does metric tensor and Christoffel symbols look like on R2? It would be great with an example as I am new to this Differential Geometry.

I was trying to compute the time derivative of the following expression: \mathbf{p_k} = \sum_i e_{ki}\sum_{n=0}^{\infty} \frac{(-1)^n}{(n+1)!} \mathbf{r_{ki}}(\mathbf{r_{ki}\cdot \nabla})^n \delta(\mathbf{R_k}-\mathbf{R}) I am following deGroot in his Foundations of Electrodynamics. He says...

Problem. Let $R$ be a local ring (commutative with identity) ans $M$ and $N$ be finitely generated $R$-modules. If $M\otimes_R N=0$, then $M=0$ or $N=0$. The problem clearly seems to be an application of the Nakayama lemma. If we can show that $M=\mathfrak mM$ or $N=\mathfrak mN$, where...

I'm trying to show that the lie derivative of a tensor field ##t## along a lie bracket ##[X,Y]## is given by \mathcal{L}_{[X,Y]}t=\mathcal{L}_{X}\mathcal{L}_{Y}t-\mathcal{L}_{Y}\mathcal{L}_{X}t but I'm not having much luck so far. I've tried expanding ##t## on a coordinate basis, such that...

How does one derive the general form of the Riemann tensor components when it is defined with respect to the Levi-Civita connection? I assumed it was just a "plug-in and play" situation, however I end up with extra terms that don't agree with the form I've looked up in a book. In a general...

Suppose we have a system of particles that interact via conservative forces. I wish to prove that ##K+U## is a constant of the system with tensor analysis. Here is my procedure: The Lagrangian is ##L=\frac{1}{2}m_i\dot{ r_i}^2-\Phi## Lagrange's equations ##\frac{d}{dt}(\frac{\partial...

Just a question : do we have in Dirac notation $$\langle u|A|u\rangle\langle u|B|u\rangle=\langle u|\langle u|A\otimes B|u\rangle |u\rangle$$ ?

I read in many books the metric tensor is rank (0,2), its inverse is (2,0) and has some property such as ##g^{\mu\nu}g_{\nu\sigma}=\delta^\mu_\sigma## etc. My question is: what does ##g^\mu_\nu## mean?! This tensor really confuses me! At first, I simply thought that...

OK. First of all, I'm novice at Physics so this question may be weird. Above, there are 2 expressions for strain-stress relations. Let's assume that all components in the matrix are variables, not zero, not E, nor not G in the first picture. The first one is written in 2D matrix form, whereas...

First by "this derivation" I'm referring to an online tutorial: http://farside.ph.utexas.edu/teaching/336L/Fluidhtml/node9.html It's said in the above tutorial that the ##i-th## component of the total torque acting on a fluid element is ##\tau_i = \int_V \epsilon_{ijk} \cdot x_{j} \cdot F_{k}...

Homework Statement Show that the set ##\{\mathbf{v}_1⊗\mathbf{v}_2⊗...⊗\mathbf{v}_p:\mathbf{v}_k\in V_k\}## of tensor products of vectors is strictly less than ##V_1⊗V_2⊗...⊗V_p##. Homework EquationsThe Attempt at a Solution Truly, I don't see any difference between the sets. They seem...

It seems there should be a list of tensor identities on the internet that answers the following, but I can't find one. For tensors in ##R^4##, ##S = S_\mu{}^\nu = S_{(\mu}{}^{\nu)}## is a symmetric tensor. ##A = A_{\nu\rho\sigma}= A_{[\nu\rho\sigma]}## is an antisymmetric tensor in all...

Here's a question that has bugged me for a while. The full Riemann curvature tensor R^\mu_{\nu \lambda \sigma} can be split into the Einstein tensor, G_{\mu \nu}, which vanishes in vacuum, and the Weyl tensor C^\mu_{\nu \lambda \sigma}, which does not. (I'm a little unclear on whether R^\mu_{\nu...

Homework Statement I am trying to derive the curvature tensor by finding the commutator of two covariant derivatives. I think I've got it, but my head is spinning with Nablas and indices. Would anyone be willing to check my work? Thanks Homework Equations I am trying to derive the curvature...

Is correct to say that two vectors , three vectors or n vectors as a common point of origin form a tensor ? What is the correct geometric representation of a tensor ? The doubt arises from the fact that in books on the subject , in general there is no geometric representation. Sometimes appears...

I have managed to work out parts a and b of Exercise 13.7 from MTW (attached), but can't see how part c works. I can see how it could work in (say) the example of taking a radar measurement of the distance to Venus, where we have the Euclidian distance prediction and the result of the radar...

Homework Statement A charged particle of charge q with arbitrary velocity ##\vec v_0## enters a region with a constant ##\vec B_0## field. 1)Write down the covariant equations of motion for the particle, without considering the radiation of the particle. 2)Find ##x^\mu (\tau)## 3)Find the...

I have recently derived both the purely covariant Riemann tensor as well as the purely covariant Weyl tensor for the Gödel solution to Einstein's field equations. Here is a wiki for the Gödel metric if you need it: http://en.wikipedia.org/wiki/Gödel_metric There you can see the line element I...

Say you have a scalar ##S=A^{\alpha}_{\beta}B^{\beta}_{\alpha}## . Since this just means to sum over ##{\alpha}## and ##{\beta}## , is it allowable to rewrite it as ##S=A^{\alpha}_{\alpha}B^{\beta}_{\beta}## . I don't see anything wrong with this, I simply rewrote the dummy indices, but since I...

When we define line element of Minkowski space, we also define the metric tensor of the equation. What actually is the function of the tensor with the line element.

I've started f(T) theory but I have a simple question like something that i couldn't see straightforwardly. In Teleparallel theories one has the torsion scalar: And if you take the product you should obtain But there seems to be the terms like . How does this one vanish? because we know...

Would it be possible to write the torsion tensor in terms of the metric? I know that a symmetric Christoffel Symbol can be written in terms of the partial derivatives of the metric. This definition of the christoffel symbols does not apply if they are not symmetric. Is it possible to write a...

I just want to make sure I have this right because when I go to different sites, it seems to look different every time. This is the Weyl tensor: Cabcd = Rabcd + (1/2) [- Racgbd + Radgbc + Rbcgad - Rbdgac + (1/3) (gacgbd - gadgbc)R] Is this correct?

In a multi-particle system, the total state is defined by the tensor product of the individual states. Why is it the case that operators, say position of 2 particles, is of the form X⊗I + I⊗Y and not X⊗Y where I are the identities for the respective spaces and X and Y are the position operators...

From introductory courses on EM, I was given 'sketchy' proofs that, in a EM field in vacuum, magnetic energy density is B² and electric energy density is E² (bar annoying multiplication factors; they just get under my skin, I'll skip them all in the following). Other facts of life: -FμνFμν, the...

Hello! The Einstein field equation relates the curvature of space-time to the energy tensor of mass-energy. This is fine. These field equations are derived by varying the Hilbert action. Now the Hilbert action is an integral of scalar curvature (R) over volume. So, when we vary this action, we...

Why Maxwell's stress tensor has minus sign to the corresponding components of electromagnetic momentum energy tensor ? From WP --- , where , is the Poynting vector, is the Maxwell stress tensor, and c is the speed of light. ----

<<Mentor note: This thread has been split from this thread due to going a bit off-topic.>> I would actually disagree with this. Any tensor has well defined units, but its components may not have the same units as the tensor basis may consist of basis tensors with different units. For example...

The coefficient of the stress energy tesor in the GR equation reduces to 8π/Ν, where N = {"(Kg)m/s^2.} Is it correct to conclude that all the elements of the stress energy tensor must have the dimension of N = (Kg)m/s^2 since the curvature and metric tensors on the other side of the equation are...

Homework Statement Hi I am reviewing the following document on tensor: https://www.grc.nasa.gov/www/k-12/Numbers/Math/documents/Tensors_TM2002211716.pdf Homework Equations In the middle of page 27, the author says: Now, using the covariant representation, the expression $$\vec V=\vec V^*$$...

Homework Statement So I've been trying to derive field strength tensor. What to do with the last 2 parts ? They obviously don't cancel (or do they?) Homework EquationsThe Attempt at a Solution [D_{\mu},D_{\nu}] = (\partial_{\mu} + A_{\mu})(\partial_{\nu} + A_{\nu}) - (\mu <-> \nu) =...