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.

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  1. P

    I Irreducible Representations of EM-Tensor Under Spatial Rotations

    Can we consider the E and B fields as being irreducible representations under the rotations group SO(3) even though they are part of the same (0,2) tensor? Of course under boosts they transform into each other are not irreducible under this action. I would like to know if there is in some...
  2. K

    Calculating Angle Between E-Field and Current Vectors in Anisotropic Mat.

    In a certain anisotropic conductive material, the relationship between the current density ##\vec j## and the electric field ##\vec E## is given by: ##\vec j = \sigma_0\vec E + \sigma_1\vec n(\vec n\cdot\vec E)## where ##\vec n## is a constant unit vector. i) Calculate the angle between the...
  3. K

    Show that a (1,2)-tensor is a linear function

    I know that a tensor can be seen as a linear function. I know that the tensor product of three spaces can be seen as a multilinear map satisfying distributivity by addition and associativity in multiplication by a scalar.
  4. C

    A How are the thermal expansion of a solid and the stress tensor related?

    My idea is this: tensor stress is directly related to the internal pressure of a solid. That is to the force that the neighboring atoms exert each other in relation to a unit of surface. When I heat a solid we can have the phenomenon of thermal expansion: this is connected to the fact that a...
  5. E

    B Understanding Tensor Indices and Conservation: A Guide

    I know what Carroll refers to as 'conservation of indices' is just a trick to help you remember the pattern for transforming upper and lower components, but nonetheless I don't understand what he means in this example: E.g. on the LHS the free index ##\nu'## is a lower index, and on the RHS...
  6. M

    Covariant derivative and the Stress-enegery tensor

    My try: $$ \begin{align*} \nabla^a T_{ab} &= \nabla^a \left(\nabla_{a} \phi \nabla_{b} \phi-\frac{C}{2} g_{a b} \nabla_{c} \phi \nabla^{c} \phi\right)\\ &\overset{(1)}{=} \underbrace{(\nabla^a\nabla_{a} \phi)}_{=0} \nabla_{b} \phi + \nabla_{a} \phi (\nabla^a\nabla_{b} \phi)-\frac{C}{2}...
  7. F

    I Understanding the Stress-Energy Tensor in Special Relativity

    Hello, I try to understand how to get the last relation below ##(3)## ( from stress energy tensor in special relativity - Wikipedia ). I understand how to get equation ##(1)## but I don't grasp how to make appear the gradient operator in the dot product and the divergence operator in the...
  8. fabstr1

    Simplification of the Proca Lagrangian

    Hello, I'm trying to figure out where the term (3) came from. This is from a textbook which doesn't explain how they do it. ∂_μ(∂L/(∂(∂_μA_ν)) = ∂L/∂A_ν (1) L = -(1/16*pi) * ( ∂^(μ)A^(ν) - ∂^(ν)A^(μ))(∂_(μ)A_(ν) - ∂_(ν)A_(μ)) + 1/(8*pi) * (mc/hbar)^2* A^ν A_ν (2) Here is Eq (1) the...
  9. Diracobama2181

    Single Particle Expectation of Energy Momentum Tensor

    $$\hat{T}_{\mu v}(x)=e^{i\hat{P}x}\hat{T}_{\mu v}(0)e^{-i\hat{P}x}$$, so $$\bra{\overrightarrow{P'}}\hat{T}_{\mu v}(x)\ket{\overrightarrow{P}}=e^{iP'x}\bra{\overrightarrow{P'}}\hat{T}_{\mu v}(0)\ket{\overrightarrow{P}}e^{-i\hat{P}x}$$ Now, $$\partial^{\mu}\Phi=\int\frac{d^3 k_1}{2\omega_{k_1}...
  10. LuccaP4

    I General Form of Fourth Rank Isotropic Tensor: A Scientific Inquiry

    I have this statement: Find the most general form of the fourth rank isotropic tensor. In order to do so: - Perform rotations in ## \pi ## around any of the axes. Note that to maintain isotropy conditions some elements must necessarily be null. - Using rotations in ## \pi / 2 ## analyze the...
  11. L

    I Proving Antisymmetry of Electromagnetic Field Tensor with 4-Force

    I've already made a post about this topic here, but I realized that I didn't understand the explanation on that post. in Chapter 7 of Rindler's book on relativity, in section about electromagnetic field tensor, he states that _and introducing a factor 1/c for later convenience, we can ‘guess’...
  12. Haorong Wu

    How to calculate a trace of a tensor?

    The textbook gives some examples for ##P_1 \left ( cos \theta \right)##, ##P_2 \left ( cos \theta \right)##, and ##P_3 \left ( cos \theta \right)##. The procedure is clear. The key to the problem is to find the symmetric traceless tensor for the proper rank. I first construct a symmetric...
  13. Haorong Wu

    Classical Books about tensor analysis just good enough for physics

    Hi. I am looking for a book about tensor analysis. I am aware that there have been some post about those books, but I wish to find a thin book rather than a tome but just good enough for physics, such as group theory, relativistic quantum mechanics, and quantum field theory. I am reading...
  14. K

    A Can we always rewrite a Tensor as a differential form?

    I read in the book Gravitation by Wheeler that "Any tensor can be completely symmetrized or antisymmetrized with an appropriate linear combination of itself and it's transpose (see page 83; also this is an exercise on page 86 Exercise 3.12). And in Topology, Geometry and Physics by Michio...
  15. brotherbobby

    Independent parameters of the rotation tensor ##R_{ij}##

    I am afraid I had no credible attempt at solving the problem. My poor attempt was writing the matrix ##\mathbb R## as a ##3 \times 3## square matrix with elements ##a_{ij}## and use the matrix form of the orthogonality relation ##\mathbb R^T \mathbb R = \mathbb I##, where ##\mathbb I## is the...
  16. Q

    I Metric Tensor: Symmetry & Other Constraints

    Aside from being symmetric, are there any other mathematical constraints on the metric?
  17. phun_physics

    I Representing Quantum Gates in Tensor Product Space

    Where do I start. I want to write the matrix form of a single or two qubit gate in the tensor product vector space of a many qubit system. Ill outline a simple example: Both qubits, ##q_0## and ##q_1## start in the ground state, ##|0 \rangle =\begin{pmatrix}1 \\ 0 \end{pmatrix}##. Then we...
  18. S

    B Torsion tensor definition doubt

    Let's say we have any two covariant derivative operators ##\nabla## and ##\nabla'##. Then there exists a tensor ##C^{\alpha}_{\mu\nu}## such that for all covariant vectors ##\omega_{\nu}##,$$\nabla_{\mu}\omega_{\nu}=\nabla'_{\mu}\omega_{\nu}-C^{\alpha}_{\mu\nu}\omega_{\alpha}$$ Now I'm quoting...
  19. JD_PM

    I Computing Riemann Tensor: 18 Predicted Non-Trivial Terms

    I want to compute the Riemann Tensor of the following metric $$ds^2 = dr^2+(r^2+b^2)d \theta^2 +(r^2+b^2)\sin^2 \theta d \phi^2 -dt^2$$ Before going through it I'd like to try to predict how many non-trivial components we'd expect to get, based on the Riemann tensor basic rule: It is...
  20. Q

    A Can you numerically calculate the stress-energy tensor from the metric?

    About 10 years ago I worked on a project where I took a mater distribution and numerically solved for spatial curvature. Can this be done in the opposite direction? Can anybody point me to a resource that would allow me to calculate matter distributions when the metric is specified? What are...
  21. K

    A Differential Forms or Tensors for Theoretical Physics Today

    There are a few different textbooks out there on differential geometry geared towards physics applications and also theoretical physics books which use a geometric approach. Yet they use different approaches sometimes. For example kip thrones book “modern classical physics” uses a tensor...
  22. cianfa72

    I About Covariant Derivative as a tensor

    Hi, I've been watching lectures from XylyXylyX on YouTube. I believe they are really great ! One doubt about the introduction of Covariant Derivative. At minute 54:00 he explains why covariant derivative is a (1,1) tensor: basically he takes the limit of a fraction in which the numerator is a...
  23. P

    A Curvature Tensor for Dual Vectors

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  24. JD_PM

    A Understanding how to derive the stress-energy tensor formula

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  25. JD_PM

    Prove the well-established (in GR) stress-energy tensor formula

    Prove that the stress-energy tensor is given by the functional derivative of the action with respect to ##\delta g^{\mu \nu}## $$T_{\mu \nu} = \frac{-2}{\sqrt{-g}} \frac{\delta S}{\delta g^{\mu \nu}}$$ Tong suggests (around 25:30) we could get the desired tensor by performing a...
  26. filip97

    A Transformation law of the energy momentum tensor

    We have 4-tensor of second rank. For example energy-momentum tensor ##T_μν## , which is symmetric and traceless. Then ##T_{μν}=x_μx_ν+x_νx_μ## where ##x_μ## is 4-vector. Every 4- vector transform under Lorentz transform as (12,12). If we act on ## T_{μν}## , by representation( with...
  27. P

    MATLAB Creating the Electric Octupole Tensor of a cubic electric octupole

    I created an array, where the first three entries of each column are the x,y, and z coordinates. The last entry in each column is the charge. I called this array PCQ. l/2 l/2 -l/2 -l/2 -l/2 l/2 l/2 -l/2 -l/2 l/2 l/2 -l/2 -l/2 -l/2 l/2 l/2 l/2 l/2 l/2 l/2 -l/2 -l/2 -l/2 -l/2 q -q q -q q...
  28. V

    A Equivalence Relation to define the tensor product of Hilbert spaces

    I'm following this video on how to establish an equivalence relation to define the tensor product space of Hilbert spaces: ##\mathcal{H1} \otimes\mathcal{H2}={T}\big/{\sim}## The definition for the equivalence relation is given in the lecture vidoe as ##(\sum_{j=1}^{J}c_j\psi_j...
  29. JD_PM

    Computing an Energy-Momentum tensor given a Lagrangian

    REMARK: First of all I have to say that this Lagrangian reminds me of the Lagrangian from which we can derive Maxwell's equations, which is (reference: Tong QFT lecture notes, equation 1.18; I have attached the PDF). $$\mathcal{L} = -\frac 1 2 (\partial_{\mu} A_{\nu} )(\partial^{\mu} A^{\nu}) +...
  30. D

    I Trace of the stress energy tensor

    The stress energy tensor has many forms based on the type of matter you are describing, dust, fluid, perfect fluid... is it true that the trace of all of these matter situations is invariant?
  31. Q

    A Variation of Metric Tensor Under Coord Transf | 65 chars

    Under the coordinate transformation $\bar x=x+\varepsilon$, the variation of the metric $g^{\mu\nu}$ is: $$ \delta g^{\mu\nu}(x)=\bar g^{\mu\nu}(x)-g^{\mu\nu}(x)=-\frac{\partial{ g^{\mu\nu}}}{\partial x^{\alpha}}\varepsilon^{\alpha}+ g^{\mu\beta}\frac{\partial \varepsilon^{\nu}}{\partial...
  32. S

    Antisymmetry of the electromagnetic field tensor

    I am trying to answer exercise 5 but I am not sure I understand what the hint is implying, differentiate with respect to ##p_\alpha## and ##p_\beta##, I have done this but nothing is clicking. Also, what is the relevance of the hint "the constraint ##p^\alpha p_\alpha = m^2c^2## can be ignored...
  33. D

    I Ricci Tensor: Covariant Derivative & Its Significance

    I read recently that Einstein initially tried the Ricci tensor alone as the left hand side his field equation but the covariant derivative wasn't zero as the energy tensor was. What is the covariant derivative of the Ricci tensor if not zero?
  34. S

    I Transformation of the contravariant and covariant components of a tensor

    I have read many GR books and many posts regarding the title of this post, but despite that, I still feel the need to clarify some things. Based on my understanding, the contravariant component of a vector transforms as, ##A'^\mu = [L]^\mu~ _\nu A^\nu## the covariant component of a vector...
  35. VictorMedvil

    A Energy-Stress Tensor question

    Hello, I was wondering why the energy-stress tensor only accounts for electromagnetic Energy Density and does not include the other forces? Secondary question could this be a flaw within the mathematics of GR making it give nonsense answers for Quantum level interactions?
  36. P

    Show that the metric tensor is independent of coordinate choice

    I need to use some property of the relalation between the coordinate systems to prove that g_{hk} is independent of the choice of the underlying rectangular coordinate system. I will try to borrow an idea from basic linear algebra. I expect any transformation between the rectangular systems to...
  37. jk22

    A Should tensor sum be used in matrix mechanics?

    Suppose the Bell operator ##B=|AB(1,2)+AB(1,3)+AB(2,3)|## With ##AB\in{1,-1}## Nonlocal realism implies ##B\in{1,3}## However using usual matrix sum one eigenvalues for the result of measurement can be smaller than 1, implying nonlocal realism cannot explain the quantum result. However if...
  38. George Keeling

    I Energy Density in SR Energy-Momentum Tensor

    In Special Relativity I'm given the energy-momentum tensor for a perfect fluid:$$ T^{\mu\nu}=\left(\rho+p\right)U^\mu U^\nu+p\eta^{\mu\nu} $$where ##\rho## is the energy density, ##p## is the pressure, ##U^\mu=\partial x^\mu/\partial\tau## is the four-velocity of the fluid. In the...
  39. M

    Calculating Energy-Momentum Tensor in GR

    My attempt was to first rewrite ##S_M## slightly to make it more clear where ##g_{\mu\nu}## appears $$S_M = \int d^4x \sqrt{-g} (g^{\mu\nu} \nabla_\mu\phi\nabla_\nu\phi-\frac{1}{2}m^2\phi^2).$$ Now we can apply the variation: $$\begin{align*} \delta S_M &= \int d^4x (\delta\sqrt{-g})...
  40. T

    A Question about the irreducible representation of a rank 2 tensor under SO(3)

    When discussing how a rank two tensor transforms under SO(3), we say that the tensor can be decomposed into three irreducible parts, the anti-symmetric part, traceless-symmetric part, and a 1-dimensional trace part, which transforms as a scalar. How do we know that the symmetric and...
  41. binbagsss

    A Difference of Christoffel Symbols Transforms as Tensor

    My notes seem to imply this should be obvious : If i consider the covariant deriviative then i get something like christoffel= nabla ( cov derivative ) - partial So difference of two of them will stil have the partial derivatuves present ,assuming these are labelled by a different index ...
  42. D

    I Energy Momentum Tensor: Real Physics & Cosmology

    The energy momentum tensor and its correlation to reimannian curvature is fascinating to think about. How much are the components and which of the components are taken into consideration when doing real physics. I suppose astrophysicists and cosmologists would be the main group of scientists...
  43. Like Tony Stark

    Determine tensor of inertia of a rod

    I have to find the inertia tensor of these rods and I don't have the concept that clear... I mean, I know the formulas like: ##I_{xx}=\int y^2 + z^2 dm## ##I_{xy}=\int xy dm## But I don't know what ##x, y, z, dm## stand for. In other words, I don't know what I should replace in the formula...
  44. D

    I The Tensor & Metric: Spacetime Points & Momentum Flux

    The components of the energy tensor are defined sometimes as the flux of the ith component of the momentum vector across some component jth of constant surface. But isn't the tensor a function of points of spacetime just as the metric? How can you evaluate a surface of j when the tensor is a...
  45. W

    A Doubt about Energy Condition in Wormhole: Integral Along Null Geodesic

    I am now reading this paperhttps://arxiv.org/pdf/gr-qc/0405103.pdf, which is related to the energy condition in wormhole. Nevertheless, I got a problem in Eq.(6), which derives from so-called ANEC in Eq.(2): $$\int^{\lambda2}_{\lambda1}T_{ij}k^{i}k^{j}d\lambda$$ And I apply the worm hole space...
  46. Ibix

    A Stress-energy tensor of a rotating rod

    A rod rotates freely (edit: about an axis perpendicular to its length) in empty space. Working in an inertial coordinate system where the rod rotates around a fixed point, the rod is straight, of length ##2L## in its spinning state, and its mass distribution is symmetric along its length. The...
  47. B

    A The derivation of the volume form in Ricci tensor

    In this derivation,i am not sure why the second derivative of the vector ## S_j '' ## is equal to ## R^{u_j}{}_{xyz} s^y_j v^z y^x## could anyone explain this bit to me thank you it seems ## S_j '' ## is just the "ordinary derivative" part but it is not actually equal to ## R^{u_j}{}_{xyz}...
  48. S

    I Finding the Error in Computing Spherical Tensor of Rank 0 Using General Formula

    This should be a trivial question. I am trying to compute the spherical tensor ##T_0^{(0)} = \frac{(U_1 V_{-1} + U_{-1} V_1 - U_0 V_0)}{3}## using the general formula (Sakurai 3.11.27), but what I get is: $$ T_0^{(0)} = \sum_{q_1=-1}^1 \sum_{q_2=-1}^1 \langle 1,1;q_1,q_2|1,1;0,q\rangle...
  49. patrykh18

    Stress tensor for a parallel plate capacitor

    The question is partially taken from Griffith's book. I am confused about the physical meaning of momentum in fields. I have determined the solution and found that in part d the momentum crossing the x-y plane is some value in the positive z direction. I don't however understand the physical...
  50. S

    B What is this Tensor mapping to?

    When I'm going from a smooth manifold to R with V* X V -> R what does the R scalar stand for. Is it some length in the manifold? and Does this have to do with the way V* and V are defined, since one is a contra-variant and one is a co-variant, are they related in the way the Pythagoras formula...
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