Tensor Definition and 1000 Threads

  1. S

    What is the metric tensor of the 4-sphere?

    After my recent studies of the curvature of the 2- sphere, I would like to move on to Minkowski space. However, I can not seem to find the metric tensor of the 4 sphere on line, nor can I seem to think of the vector of transformation properties that I would use to derive the metric tensor of the...
  2. ChrisVer

    Metric Tensor Components: Inverse & Derivatives

    I have one question, which I don't know if I should post here again, but I found it in GR... When you have a metric tensor with components: g_{\mu \nu} = \eta _{\mu \nu} + h_{\mu \nu}, ~~ |h|<<1 (weak field approximation). Then the inverse is: g^{\mu \nu} = \eta^{\mu \nu} - h^{\mu \nu}...
  3. P

    How to Compute the Inertia Tensor of a Cross-Hanger?

    I've done many exercises about inertia tensors of 3D bodies and sticks but now I have this exercise and I got stuck without any idea of how to do the integration to compute the inertia tensor. The statement is this: "Compute the inertia tensor of a cross-hanger consisting of 3 thin and linear...
  4. S

    Symmetry of Riemann Tensor: Investigating Rabmv

    We know how objects such as the metric tensor and the Cristoffel symbol have symmetry to them (which is why g12 = g21 or \Gamma112 = \Gamma121) Well I was wondering if the Riemann tensor Rabmv had any such symmetry. Are there any two or more particular indices that I could interchange and...
  5. Mr-R

    Can the Tensor Integral in GR be Bounded by a Region Outside of d^{3}x?

    Dear all, I am self studying GR and stuck on problem (23) on page 108/109. I am trying to do all of them. First I will start with (a) so you guys can breath while laughing at my attempts at (b) and (c) :blushing: (a) Attempt The tensor in the equation is bounded in the d^{3}x region. Outside...
  6. andrewkirk

    Einstein tensor in the FLRW frame

    Einstein tensor in the FLRW frame - Part 1 of 2 This note develops a formula for the ##G^{00}## component of the Einstein tensor in the FLRW coordinate system for a homogeneous and isotropic spacetime. We use the convention that tensor indices ##i, j## or ##k## are used only for spatial...
  7. kroni

    EM Tensor & GR: Does Space Curvature Impact EM Field?

    Hello, I am reading a book about General relativity, i understand that energy of the EM tensor go in the stress- energy tensor of GR equations. SO, EM field curve space. But i don't understand if space curvature impact EM field ? Is variation of space curvature can create EM field ? Clément
  8. M

    Confusion with Dot Product in Polar Coordinates with the Metric Tensor

    Alright, so I was reading up on tensors and such with non-Cartesian coordinate systems all day but now I'm a bit tired an confused so you'll have to forgive me if it's a stupid question. So to express the dot product in some coordinate system, it's: g(\vec{A}\,,\vec{B})=A^aB^bg_{ab} And, if...
  9. F

    Gauge Field Tensor from Wilson Loop

    It is possible to introduce the gauge field in a QFT purely on geometric arguments. For simplicity, consider QED, only starting with fermions, and seeing how the gauge field naturally emerges. The observation is that the derivative of the Dirac field doesn't have a well-defined transformation...
  10. WannabeNewton

    Interpretation of vorticity with non-vanishing strain tensor

    Hi guys. Let me just say at the outset that I know very little fluid mechanics but I keep coming back to the same issue over and over in a general relativity related problem so I figured I'd just ask the fluid mechanics question here. In countless places the interpretation of the vorticity...
  11. S

    Can the Ricci Tensor be Simplified Further? Suggestions Needed!

    In my studies of methods to simplify the Einstein field equations, I first decided to go about expanding the Ricci tensor in terms of the metric tensor. I have been mostly successful in doing this, but there are a couple of complications that I would like your opinions on. At the bottom of...
  12. S

    Manipulation within the Einstein Tensor in Einstein field equations

    Hello everybody. I was recently brainstorming ways to make the Einstein field equations a little easier to solve (as opposed to having to write out that monstrosity of equations that I started on some time ago) and I got an interesting idea in my mind. Here, we have the field equations...
  13. Telemachus

    Tensor algebra, divergence of cross product

    Hi there. I wanted to demonstrate this identity which I found in a book of continuum mechanics: ##curl \left ( \vec u \times \vec v \right )=div \left ( \vec u \otimes \vec v - \vec v \otimes \vec u \right ) ## I've tried by writting both sides on components, but I don't get the same, I'm...
  14. J

    Is there a convention for horizontally shifting indices in tensor notation?

    Hi, I'm teaching myself tensor analysis and am worried about a notational device I can't find any explanation of (I'm primarily using the Jeevanjee and Renteln texts). Given that the contravariant/covariant indices of a (1,1) tensor correspond to the row/column indices of its matrix...
  15. M

    Proof? Kronecker delta is the only isotropic second rank tensor

    It is pretty straight forward to prove that the Kronecker delta \delta_{ij} is an isotropic tensor, i.e. rotationally invariant. But how can I show that it is indeed the only isotropic second order tensor? I.e., such that for any isotropic second order tensor T_{ij} we can write T_{ij} =...
  16. C

    Constraints on a fourth rank tensor

    Homework Statement Consider a theory which is translation and rotation invariant. This implies the stress energy tensor arising from the symmetry is conserved and may be made symmetric. Define the (Schwinger) function by ##S_{\mu \nu \rho \sigma}(x) = \langle T_{\mu \nu}(x)T_{\rho...
  17. E

    Is the Stress-Energy Tensor for a Scalar Field Phi Isotropic in FRW Metrics?

    Hi, I am trying to show explicitly the isotropy of the stress energy tensor for a scalar field Phi. By varying the corresponding action with respect to a metric g, I obtain: T_{\mu \nu} = \frac{1}{2} g_{\mu \nu} \left( \partial_\alpha \Phi g^{\alpha \beta} \partial_\beta \Phi + m^2 \Phi^2...
  18. Greg Bernhardt

    Stress and Strain Tensor Basics: Definition, Equations & Moduli

    [SIZE="4"]Definition/Summary Stress is force per area, and is a tensor. It is measured in pascals (Pa), with dimensions of mass per length per time squared (ML^{-1}T^{-2}). By comparison, load is force per length, and strain is a dimensionless ratio, stressed length per original length...
  19. Greg Bernhardt

    Understanding the Metric Tensor: Definition, Equations, and Properties

    [SIZE="4"]Definition/Summary The metric tensor g_{\mu\nu} is a 4x4 matrix that is determined by the curvature and coordinate system of the spacetime [SIZE="4"]Equations The proper time is given by the equation d\tau^2=dx^{\mu}dx^{\nu}g_{\mu\nu} using the Einstein summation convention...
  20. Greg Bernhardt

    What is a Tensor: FAQ Definition/Summary

    [SIZE="4"]Definition/Summary A tensor of type (m,n) on a vector space V is an element of the tensor product space V\otimes\cdots\text{(m copies)}\cdots\otimes V \otimes V^*\otimes\cdots\text{(n copies)}\cdots\otimes V^*, =\ V^{\otimes m}\otimes V^{*\otimes n}, where V^* is the vector space...
  21. WannabeNewton

    Initial conditions for tensor perturbations

    Hi all. Say we have a background inflaton field ##\varphi## and that we've integrated the background equation for ##\varphi##, ##H(\eta)##, and ##a(\eta)## up to the number of e-folds of inflation corresponding to ##\epsilon = 1## in the slow-roll parameter. We then wish to solve for the ##k##...
  22. S

    1st stress-energy tensor component for an electric field

    I have recently gone over the derivation of the stress energy momentum tensor elements for the special case of dust. This case just used a swarm of particles. Now that I understand that case, I am now trying to see if I can derive the components for an electric field. I just want you guys to...
  23. I

    Exotic stress-energy tensor and causality

    Hello everybody. I would like to kindly ask your help with a hypothetical hairy question about which I think a lot recently. It is known fact, that it is not possible to construct a wormhole without exotic mass that violates the weak energy condition. It is also known that many quantum fields...
  24. N

    How to Simplify a Double Dot Product of Tensors?

    Hello, Been a long time lurker, but first time poster. I hope I can be very thorough and descriptive. So, I have been battling with a double inner product of a 2nd order tensor with a 4th order one. So my question is: How do we expand (using tensor properties) a double dot product of the...
  25. W

    Short Exact Sequences and at Tensor Product

    Hi,let: 0->A-> B -> 0 ; A,B Z-modules, be a short exact sequence. It follows A is isomorphic with B. . We have that tensor product is right-exact , so that, for a ring R: 0-> A(x)R-> B(x)R ->0 is also exact. STILL: are A(x)R , B(x)R isomorphic? I suspect no, if R has torsion. Anyone...
  26. Mr-R

    I Calculating the Riemann Tensor for a 4D Sphere

    Dear All, I am trying to calculate the Riemann tensor for a 4D sphere. In D'inverno's book, I have this equation R^{a}_{bcd}=\partial_{c}\Gamma^{a}_{bd}-\partial_{d}\Gamma^{a}_{bc}+\Gamma^{e}_{bd}\Gamma^{a}_{ec}-\Gamma^{e}_{bc}\Gamma^{a}_{ed} But the exercise asks me to calculate R_{abcd}. Do...
  27. S

    Stress Energy Tensor Components

    I have pretty much learned how to derive the left side of Einstein's field equations now (the Einstein tensor that is). Now I need to grasp that stress energy momentum tensor. Does anybody know of any good sources that will tell me how to derive the components of this tensor? I ask this...
  28. G

    What Are the Best Books for Mastering Tensor Calculus?

    Hi guys, I am interested to learn tensor calculus but I can't find a good book that provide rigorous treatment to tensor calculus if anyone could recommend me to one I would be very pleased.
  29. C

    Constructing the electromagnetic tensor from a four-potential

    *Edit: I noticed I may have posted this question on the wrong forum... if this is the case, could you please move it for me instead of deleting? thanks! :) Hello, I am having problems on building my electromagnetic tensor from a four-potential. I suspect my calculations are not right. Here are...
  30. H

    What Does the Definition of a Contravariant Vector Mean in Tensor Analysis?

    So I'm looking at Schaum's outlines for Tensors and the definition of a Contravariant vector is \bar{T}^i=T^r\frac{\partial\bar{x}^i}{\partial x^r} Where \bar{x}^i and x^r denote components of 2 different coordinates (the superscript does not mean 'to the power of') and T^i and T^r are...
  31. S

    Metric Tensor in Spherical Coordinates

    I recently derived a matrix which I believe to be the metric tensor in spherical polar coordinates in 3-D. Here were the components of the tensor that I derived. I will show my work afterwards: g11 = sin2(ø) + cos2(θ) g12 = -rsin(θ)cos(θ) g13 = rsin(ø)cos(ø) g21 = -rsin(θ)cos(θ)...
  32. A

    Is stress tensor symmetric in Navier-Stokes Equation?

    Hello, In CFD computation of the Navier-Stokes Equation, is stress tensor assumed to be symmetric? We know that in NS equation only linear momentum is considered, and the general form of NS equation does not assume that stress tensor is symmetric. Physically, if the tensor is asymmetric then...
  33. D

    Curl in 5D using levi-civita tensor

    i really lost with this. i see two possibilities: (1) something like, \epsilon_{abc}\partial_{a}A_{b}e_{c} with a,b,c between 1 and 5 or (2)like that \epsilon_{abcde}\partial_{a}A_{b} one of the options nears correct? thank's a lot
  34. electricspit

    Why Does Applying a Second Derivative to an Antisymmetric Tensor Yield Zero?

    Hello, I have two problems. I'm going through the Classical Theory of Fields by Landau/Lifshitz and in Section 32 they're deriving the energy-momentum tensor for a general field. We started with a generalized action (in 4 dimensions) and ended up with the definition of a tensor...
  35. bcrowell

    Parity of stress tensor versus stress-energy tensor

    The stress-energy tensor is an actual tensor, i.e., under a spacetime parity transformation it stays the same, which is what a tensor with two indices is supposed to do according to the tensor transformation law. This also makes sense because in the Einstein field equations, the stress-energy...
  36. G

    Can Tensor Products Define (M,N) Tensors?

    From my understanding, an arbitrary (0,N) tensor can be expressed in terms of its components and the tensor products of N basis one-forms. Similarly, an arbitrary (M,0) tensor can be expressed in terms of its components and tensor products of its M basis vectors. What about an (M,N) tensor...
  37. T

    Finding rate of change of moment of inertia tensor

    Homework Statement The Wikipedia article on spatial rigid body dynamics derives the equation of motion \boldsymbol\tau = [I]\boldsymbol\alpha + \boldsymbol\omega\times[I]\boldsymbol\omega from \sum_{i=1}^n \boldsymbol\Delta\mathbf{r}_i\times (m_i\mathbf{a}_i). But, there is another way to...
  38. T

    Tensor differentiation (element-by-element)

    Homework Statement Proof the following: \frac{\text{d}\boldsymbol\{\mathbf{I}\boldsymbol\}}{\text{d}t} \, \boldsymbol\omega = \boldsymbol\omega \times (\boldsymbol\{\mathbf{I}\boldsymbol\}\,\boldsymbol\omega) where \boldsymbol\{\mathbf{I}\boldsymbol\} is a tensor...
  39. Dale

    Expansion tensor on rotating disk

    Hi Everyone, Suppose that we have cylindrical coordinates on flat spacetime (in units where c=1): ##ds^2 = -dt^2 + dr^2 + r^2 d\theta^2 + dz^2## I would like to explicitly calculate the expansion tensor for a disk of constant radius R<1 and non-constant angular velocity ##\omega(t)<1##. I...
  40. C

    What kind of isometry? A metric tensor "respects" the foliation?

    Suppose we have a foliation of leaves (hypersurfaces) with codimension one of some Riemannian manifold ##M## with metric ##g##. For any point ##p## in ##M## we can then find some flat coordinate chart ##(U,\phi) = (U, (x^\mu, y))## such that setting ##y## to a constant locally labels each leaf...
  41. W

    Defining Functions on Tensor Products

    Hi all. This question is related to my previous one on tensor products: Is there a way of "well-defining" a function on a tensor product M(x)N (where M,N are both R-modules) ? This is the motivating example for my question : Say we want to define a map f: M(x)M-->M by f(m(x)m')=m+m' ...
  42. W

    Tensor Products and Maps Factoring Through

    Hi, I understand the tensor product of modules as a new module in which every bilinear map becomes a linear map. But now I am trying to see the Tensor product of modules from the perspective of maps factoring through, i.e., from properties that allow a commutative triangle of maps. As a...
  43. C

    Understanding tensor operators

    The definition of tensor operator that I have is the following: 'A tensor operator is an operator that transforms under an irreducible representation of a group ##G##. Let ##\rho(g)## be a representation on the vector space under consideration then ##T_{m_c}^{c}## is a tensor operator in the...
  44. C

    Ricci Tensor Proportional to Divergence of Christoffel Symbol?

    I'm reading an old article published by Kaluza "On the Unity Problem of Physics" where i encounter an expression for the Ricci tensor given by $$R_{\mu \nu} = \Gamma^\rho_{\ \mu \nu, \rho}$$ where he has used the weak field approximation ##g_{\mu \nu} = \eta_{\mu \nu} + h_{\mu \nu}## where...
  45. F

    Energy-momentum tensor, lagrangian density

    Homework Statement I try to calculate the energy tensor, but i can't do it like the article, and i don't know, i have a photo but it don't look very good, sorry for my english, i have a problem with a sign in the result Homework Equations The Attempt at a Solution In the photos...
  46. K

    Questions about tensor operator

    Hi. Before question, sorry about my bad english. It's not my mother tongue. My QM textbook(Schiff) adopt J x J = i(h bar)J. as the defining equations for the rotation group generators in the general case. My question is, then tensor J must have one index which has three component? (e.g...
  47. E

    Why shear stress components of the Stress Energy tensor not zero?

    Hi, I am having trouble understanding why Tij can be non-zero for i≠j. Tij is the flux of the i-th component of momentum across a surface of constant xj. Isn't the i-th component of momentum tangent to the surface of constant xj and therefore its flux across that surface zero? What am I...
  48. ChrisVer

    Interpreting Einstein Tensor Geometrically on a Manifold

    Do you know how could I interpret the Einstein Tensor geometrically (on a general manifold)? For example the Christoffel Symbols can show someone the divergence/convergence of geodesics and/or show how the change of metric from point to point creates an additional force/potential (through the...
  49. W

    What are the independent terms in the Magnetic Tensor

    I am trying to understand the magnetic gradient tensor which has nine components. There are three magnetic field components, but there are also three baselines. These nine gradients are organised into a 3x3 matrix. I have read that only 5 of these terms are independent. What exactly does this...
  50. TrickyDicky

    Vector fields, flows and tensor fields

    Vector fields generate flows, i.e. one-parameter groups of diffeomorphisms, which are profusely used in physics from the streamlines of velocity flows in fluid dynamics to currents as flows of charge in electromagnetism, and when the flows preserve the metric we talk about Killing vector fields...
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