What is Tensor: Definition and 1000 Discussions

In mathematics, a tensor is an algebraic object that describes a (multilinear) relationship between sets of algebraic objects related to a vector space. Objects that tensors may map between include vectors and scalars, and even other tensors. There are many types of tensors, including scalars and vectors (which are the simplest tensors), dual vectors, multilinear maps between vector spaces, and even some operations such as the dot product. Tensors are defined independent of any basis, although they are often referred to by their components in a basis related to a particular coordinate system.
Tensors have become important in physics because they provide a concise mathematical framework for formulating and solving physics problems in areas such as mechanics (stress, elasticity, fluid mechanics, moment of inertia, ...), electrodynamics (electromagnetic tensor, Maxwell tensor, permittivity, magnetic susceptibility, ...), or general relativity (stress–energy tensor, curvature tensor, ...) and others. In applications, it is common to study situations in which a different tensor can occur at each point of an object; for example the stress within an object may vary from one location to another. This leads to the concept of a tensor field. In some areas, tensor fields are so ubiquitous that they are often simply called "tensors".
Tullio Levi-Civita and Gregorio Ricci-Curbastro popularised tensors in 1900 - continuing the earlier work of Bernhard Riemann and Elwin Bruno Christoffel and others - as part of the absolute differential calculus. The concept enabled an alternative formulation of the intrinsic differential geometry of a manifold in the form of the Riemann curvature tensor.

View More On Wikipedia.org
Recent contents Top users
  1. D

    Contraction of an asymmetric tensor?

    Hey guys, So in my notes I've got this statement written: If tensor with no symmetry properties, A^{\mu\nu}, contracts to a_{\mu\nu}, we can write this as A^{\mu\nu}a_{\mu\nu}=\frac{1}{2}a_{\mu\nu}(A^{\mu\nu}-A^{\nu\mu}) as a_{\mu\nu} (A^{\mu\nu}+A^{\nu\mu}) = 0. So I don't see how the...
  2. C

    Third rank curvature tensor?

    Hi, I am curious about expressing the curvature of a manifold using a third-rank tensor. The fourth order Riemann tensor can be contracted to give the second order Ricci tensor and zeroth order Ricci scalar, but is there no way of obtaining a third- or first-order tensors, or would this simply...
  3. S

    How do you take a tensor product?

    I have recently delved into linear algebra and multi-linear algebra. I came to learn about the concepts of linear and bi-linear maps along with bases and changes of basis, linear independence, what a subspace is and more. I then decided to move on to tensor products, when I ran into a problem...
  4. sweetdreams12

    How do I correctly manipulate tensor components in different coordinate systems?

    Homework Statement A tensor and vector have components Tαβγ, and vα respectively in a coordinate system xμ. There is another coordinate system x'μ. Show that Tαβγvβ = Tαβγvβ Homework Equations umm not sure... ∇αvβ = ∂vβ/∂xα - Γγαβvγ The Attempt at a Solution Tαβγvβ =...
  5. S

    MHB Vector Index Notation Proof: Solved by Sam

    Hi everyone! I've got a vector index notation proof that I'm struggling with. (sorry ignore the c, that's the question number) I've simplified it u * (del X del) and from there I've sort of assumed del X del = 0. Is that right and if so could somebody please explain it? Else any help on...
  6. D

    Energy-Momentum tensor components for complex Klein-Gorden field

    Hey guys, So I have the stress energy tensor written as follows in my notes for the complex Klein-Gordon field: T^{\mu\nu}=(\partial^{\mu}\phi)^{\dagger}(\partial^{\nu}\phi)+(\partial^{\mu}\phi)(\partial^{\nu}\phi^{\dagger})-\mathcal{L}g^{\mu\nu} Then I have the next statement that T^{0i} is...
  7. aditya ver.2.0

    Understanding the Role of Tensor Calculus in General Relativity

    In an attempt to solve the mystery of dark energy, I came across problems concerned with the General Relativity. In it, I observed that many of the problems were related with the tensor calculus. I want to know that what importance does tensor calculus hold in GR? Are there any other fields of...
  8. N

    Partial derivative with respect to metric tensor

    \mathcal{L}_M(g_{kn}) = -\frac{1}{4\mu{0}}g_{kj} g_{nl} F^{kn} F^{jl} \\ \frac{\partial{\mathcal{L}_M}}{\partial{g_{kn}}}=-\frac{1}{4\mu_0}F^{pq}F^{jl} \frac{\partial}{\partial{g_{kn}}}(g_{pj}g_{ql})=+\frac{1}{4\mu_0} F^{pq} F^{lj} 2 \delta^k_p \delta^n_j g_{ql} I need to know how...
  9. arpon

    How to draw a 2D space in 3D Euclidean space by metric tensor

    Suppose, I know the metric tensor of a 2D space. for example, the metric tensor of a sphere of radius R, gij = ##\begin{pmatrix} R^2 & 0 \\ 0 & R^2\cdot sin^2\theta \end{pmatrix}## ,and I just know the metric tensor, but don't know that it is of a sphere. Now I want to draw a 2D space(surface)...
  10. P

    Divergence of the Stress-Energy Tensor

    Im studying Quantum Field Theory as part of my undergraduate course, and am currently looking at Noether's Theorem which has led me to the following calculation of the divergence of the Stress-Energy Tensor. I'm having difficulty in seeing how we get from line (31) to line (32). Is the 2nd term...
  11. S

    Do all elements of the stress energy tensor really have the same units?

    Some people may remember awhile back when I made a thread showing how when I derived the Einstein tensor and the stress energy momentum tensor for a certain traversable wormhole metric, that the units of the energy momentum tensor were not the same for each element and how a couple of the...
  12. T

    Energy Momentum Tensor - General Properties

    It is often stated and proved in textbooks that the momentum density is also the energy flux. The explanation is often done using the dust model. However, it is possible that in a real fluid, there is heat conduction via particle collision. There is energy flux, but since no molecules are ever...
  13. C

    Why Can't We Do Algebraic Methods with Tensors?

    Hello everyone! Even though I have done substantial tensor calculus, I still don't get one thing. Probably I am being naive or even stupid here, but consider $$R_{\mu\nu} = 0$$. If I expand the Ricci tensor, I get $$g^{\sigma\rho} R_{\sigma\mu\rho\nu} = 0$$. Which, in normal algebra, should...
  14. mef51

    [General Relativity] Prove that a tensor is a co-tensor

    Hello! I'd appreciate any help or pokes in the right direction. Homework Statement Show that a co-tensor of rank 2, ##T_{\mu\nu}##, is obtained from the tensor of rank 2 ##T^{\mu\nu}## by using a metric to lower the indices: $$T_{\mu\nu} = g_{\mu\alpha}g_{\nu\beta}T^{\alpha\beta}$$ Homework...
  15. putongren

    Introduction to Tensor Calculus, Relativity Homework

    I already have the solutions emailed to me from a D H Lawden textbook. I have trouble understanding the solution as the solution is not formatted properly, and the answer seems to be a little too advanced for me. I hope that some one can help me understand the problem. 1. Homework Statement...
  16. S

    Solving Dimensional Analysis Problems in Stress Energy Momentum Tensor

    Recently, I used the metric for the traversable wormhole (the one in this link): http://www.spacetimetravel.org/wurmlochflug/wurmlochflug.html ds2= -c2dt2 + dl2 + (b2 + l2)(dΘ2 + sin2(Θ)dΦ2) I derived the metric tensor from this space-time interval and then from there, I derived the...
  17. M

    Conservation of Stress-Energy tensor

    I have a few questions regarding the solution to this problem. First of all I have the Stress-Energy tensor for a scalar fields \phi^a T_{Noether}^{\mu\nu} = \displaystyle \sum_a \frac{\partial \mathcal{L}}{(\partial_{\mu}\phi_a)}\partial^{\nu}\phi^a - g^{\mu \nu}\mathcal{L} To ensure...
  18. M

    Writing 3rd Order Tensor Symmetric Part in Tensor Form

    Can some one write for me the Symmetric part of a third order tensor (as a tensor form) Thanks .
  19. G

    Where do I find Nonlinear Susceptibility Tensor Values?

    Hi, everyone. I am having a hard time finding explicit values of non-linear susceptibility tensor values for any sort of crystals. Specifically, I'm looking for values of a BBO crystal, but I would like to know where to find others for my future research. I should say that I am looking for the...
  20. M

    Studying Reading Bishop & Goldberg's Tensor Analysis: Prerequisites for Physicists

    I am a graduate student in physics. One of my biggest frustrations in my education is that I often find that my mathematical background is lacking for the work I do. Sure I can make calculations adequately, well enough to even do well in my courses, but I don't feel like I really understand...
  21. L

    Inertia tensor punctual masses

    Homework Statement Hi everyone, I need some help to know how to find the components of the inertia tensor matrix of a rigid body formed by a gruop of point masses attached to bars with no mass. I have 3 masses with cartesian coordenates: 1 (a,a,0), 2 (a,0,0) and 3 (-a,-a-0). The...
  22. A

    How to derive Maxwell stress-energy tensor

    The problem statement is: Assuming that we are in vacuum, and that the only work done between mechanical systems and electricity and magnetism comes from the Lorentz force, give a full, relativistic derivation of the Maxwell stress-energy tensor.
  23. B

    Difference between mixed tensor notation

    Hi, Can someone explain the difference between, say, \Lambda_\nu^\mu, {\Lambda_\nu}^\mu and {\Lambda^\mu}_\nu (i.e. the positioning of the contravariant and covariant indices)? I have found...
  24. E

    Riemann Curvature Tensor Symmetries Proof

    I am trying to expand $$\varepsilon^{{abcd}} R_{{abcd}}$$ by using four identities of the Riemann curvature tensor: Symmetry $$R_{{abcd}} = R_{{cdab}}$$ Antisymmetry first pair of indicies $$R_{{abcd}} = - R_{{bacd}}$$ Antisymmetry last pair of indicies $$R_{{abcd}} = - R_{{abdc}}$$...
  25. 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...
  26. 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}...
  27. P

    Inertia tensor of a crosshanger

    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...
  28. 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...
  29. 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...
  30. 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...
  31. 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
  32. 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...
  33. Telemachus

    Tensor calculus, gradient of skew tensor

    Hi there. I was dealing with the derivation on continuum mechanics for the conservation of angular momentum. The derivation I was studying uses an arbitrary constant skew tensor ##\Lambda##. It denotes by ##\lambda## its axial vector, so that ##\Lambda=\lambda \times## Then it defines...
  34. 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...
  35. 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...
  36. 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...
  37. 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...
  38. 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...
  39. 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...
  40. 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} =...
  41. 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...
  42. 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...
  43. Greg Bernhardt

    Stress and Strain Tensor Basics: Definition, Equations & Moduli

    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...
  44. Greg Bernhardt

    Understanding the Metric Tensor: Definition, Equations, and Properties

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

    What is a Tensor: FAQ Definition/Summary

    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 of linear...
  46. 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##...
  47. 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...
  48. 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...
  49. N

    4th order tensor double product

    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...
  50. 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...
Back
Top