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. 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...
  2. 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...
  3. K

    I Conservation of Energy Momentum Tensor

    Unfortunetly, I found across the web only the case where there's no source, in which case ##\partial_\alpha T^{\alpha \beta} = 0##. I'm considering Minkowski space with Minkowski coordinates here. When there's source, is it true that ##\partial_\alpha (T^{\alpha \beta}) = 0## or is it ##\int...
  4. P

    Expressing this vector integral as a tensor involving the quadrupole

    Before writing out each component I'm going to simplify ##\vec{I}## to the best of my abilities $$\vec{I} = \int \left(\hat{r}\cdot\vec{r'}\right) \vec{r'} \rho\left( \vec{r'} \right)\, d^3r'$$ $$\vec{I} = \hat{r} \cdot \int \vec{r'} \left( x' , y', z' \right) \rho\left( \vec{r'} \right)\...
  5. Vyrkk

    A Covariant derivative and connection of a covector field

    I am trying to derive the expression in components for the covariant derivative of a covector (a 1-form), i.e the Connection symbols for covectors. What people usually do is take the covariant derivative of the covector acting on a vector, the result being a scalar Invoke a product rule to...
  6. lachgar

    How to form the stress tensor component from the equilibrium equation?

    Good evening everybody. This is my suggestion for answer. The tensor is diagonal and the compression is a plane stress equilibre equation div(σ)=0 so: So, does that means that = f(y.z) = Ay+Bz and =f(x.z)= Cx+Dz A,B,C and D are constants. Is that what the question meant? Thank you in...
  7. binbagsss

    Dummy index and renaming if not a tensor

    Apologies it's been a little while since I've done this, but I believe the rule is, that if the object is not a tensor you can not rename the dummy index? For example, i have the action ##\int d^3 x \epsilon^{uvp} A_u \partial v A_p ## and I want to write this in terms of the ##i## and ##0##...
  8. A

    I Meaning of each member being a unit vector

    Summary: Meaning of each member being a unit vector, and how the products of each tensor can be averaged. Hello! I am struggling with understanding the meaning of "each member is a unit vector": I can see that N would represent the number of samples, and the pointy bracket represents an...
  9. E

    How can I use the covariant derivative to derive the Riemann curvature tensor?

    I derived this equation $$ A_{i,jk}-A_{i,kj}=R^r _{kij}A_r$$.But where do I use this $$A_{i,j}+A_{j,i}=0$$?
  10. K

    I Energy-Momentum Tensor: How Much Do University Students Learn?

    There are plentty of textbooks and online papers that talk about the energy momentum tensor, but they all look to me as if they're only covering the very introductory aspects of it. To put another way, it seems that there's much more to be learn. I would like to know if university physics...
  11. P

    A Einstein Tensor and Stress Energy Tensor of Scalar Field

    Hi All. Given that we may write And that the Stress-Energy Tensor of a Scalar Field may be written as; These two Equations seem to have a similar form. Is this what would be expected or is it just coincidence? Thanks in advance
  12. Dale

    I Invariants of the stress energy tensor

    Does anyone know of a set of invariants for the stress energy tensor? In particular, I would like to know if there is a small set of linearly independent invariants, each of which (or at least some of which) have a clear physical meaning.
  13. ohwilleke

    I How big are the non-mass parts of the stress-energy tensor?

    In Newtonian gravity, non-rest mass contributions to gravitational effects are ignored and for many purposes (e.g. low precision solar system astronomy, N-body approximations of galaxy or galaxy cluster dynamics), the other contributions that enter Einstein's field equations through the...
  14. nomadreid

    I Tensor calculation, giving|cos A|>1: how to interpret

    On pages 42-43 of the book "Tensors: Mathematics of Differential Geometry and Relativity" by Zafar Ahsan (Delhi, 2018), the calculation for the angle between Ai=(1,0,0,0) (the superscript being tensor, not exponent, notation) and Bi=(√2,0,0,(√3)/c), where c is the speed of light, in the...
  15. K

    I Energy Momentum Tensor Prerequisites: What Do I Need to Know?

    I have a feeling that topics related to the Energy Momentum tensor are the most difficult part when learning Relativity. At least to me, it seems that the textbooks I'm reading assume that readers have a previous knowledge on some other area, maybe it's classical mechanics of fluids or something...
  16. W

    Solving Metric Tensor Problems: My Attempt at g_μν for (2)

    My attempt at ##g_{\mu \nu}## for (2) was \begin{pmatrix} -(1-r^2) & 0 & 0 & 0 \\ 0 &\frac{1}{1-r^2} & 0 & 0 \\ 0 & 0 & r^2 & 0 \\ 0 & 0 & 0 & r^2 \sin^2(\theta) \end{pmatrix} and the inverse is the reciprocal of the diagonal elements. For (1) however, I can't even think of how to write the...
  17. Pencilvester

    I Three dimensional tracefree tensor?

    Hi PF, I’m working through “A Relativist’s Toolkit” by Poisson, and I’m in the section on geodesic congruances, subsection: kinematics of a deformable medium. I got through the section on the 2-dimensional example that introduced expansion, shear, and rotation just fine, but I’m having trouble...
  18. Creedence

    I Eigenvectors of the EM stress-energy tensor

    My question is that what is the physical meaning of the EM stress-energy tensor's eigenvectors? Thanks for the answers - Robert
  19. filip97

    A (A,A) representation of Lorentz group-why is it tensor?

    Why representation of Lorentz group of shape (A,A) corespond to totally symmetric traceless tensor of rank 2A? For example (5,5)=9+7+5+3+1 (where + is dirrect sum), but 1+5+3+9+7<>(5,5) implies that (5,5) isn't symmetric ? See Weinberg QFT Book Vol.1 page 231.
  20. SamRoss

    I How can the stress tensor be non-zero where there is no matter?

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  21. M

    A Tensor and vector product for Quantum

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  22. berlinspeed

    A Inquiry on Matrix Tensor Notation & Meaning in Curved Spacetime

    So if ##P_{0}## is an event, and I have ##\mathcal {g_{\mu\nu}(P_{0})}=0## and ##\mathcal {g_{\mu\nu,\alpha\beta}(P_{0})}\neq0##, does this notation mean ##\partial\alpha\partial\beta## or simply ##\partial(\alpha\beta)##? And what is the significance of it? Why can't it be zero in curved spacetime?
  23. D

    I Deriving tensor transformation laws

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  24. snoopies622

    I The vanishing of the covariant derivative of the metric tensor

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  25. hilbert2

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  26. D

    I What is the Purpose of Calculating the Christoffel Symbols in Curved Spacetime?

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  27. berlinspeed

    A Reimann Tensor Component Form: Charles&Wheeler

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  28. RicardoMP

    Derivatives on tensor components

    This was my attempt at a solution and was wondering where did I go wrong: -\frac{\partial}{\partial p_\mu}\frac{1}{\not{p}}=-\frac{\partial}{\partial p_\mu}[\gamma^\nu p_\nu]^{-1}=\gamma^\nu\frac{\partial p_\nu}{\partial p_\mu}[\gamma^\sigma...
  29. N

    I Tensor Field Notation: Einstein Gravity Explained

    Hi there, I'm just starting Zee's Einstein Gravity in a Nutshell, and I'm stuck on a seemingly very easy assumption that I can't figure out. On the Tensor Field section (p.53) he develops for vectors x' and x, and tensor R (with all indices being upper indices) : x'=Rx => x=RT x' (because R-1=RT...
  30. sergiokapone

    I Covariant derivative of the contracted energy-momentum tensor of a particle

    The energy-momentum tensor of a free particle with mass ##m## moving along its worldline ##x^\mu (\tau )## is \begin{equation} T^{\mu\nu}(y^\sigma)=m\int d \tau \frac{\delta^{(4) }(y^\sigma-x^\sigma(\tau ))}{\sqrt{-g}}\frac{dx^\mu}{d\tau}\frac{dx^\nu}{d\tau}. \end{equation} Let contract...
  31. sergiokapone

    I Derivation of Geodesics Eq from EM Tensor of Point Particle

    The energy-momentum tensor of a free particle with mass ##m## moving along its worldline ##x^\mu (\tau )## is \begin{equation} T^{\mu\nu}(y^\sigma)=m\int d \tau \frac{\delta^{(4) }(y^\sigma-x^\sigma(\tau ))}{\sqrt{-g}}\frac{dx^\mu}{d\tau}\frac{dx^\nu}{d\tau}.\tag{2} \end{equation} The covariant...
  32. G

    A How does Lorentz invariance help evaluate tensor integrals?

    We're trying to reduce the tensor integral ##\int {\frac{{{d^4}k}}{{{{\left( {2\pi } \right)}^4}}}} \frac{{{k^\mu }{k^\nu }}}{{{{\left( {{k^2} - {\Delta ^2}} \right)}^n}}}{\rm{ }}## to a scalar integral (where ##{{\Delta ^2}}## is a scalar). We're told that the tensor integral is proportional...
  33. pixel

    I Stress-Energy Tensor: Specified or Calculated?

    My understanding is that this tensor contains sources for spacetime curvature, analogous to how charge and current are sources for electric and magnetic fields. In other words, the elements of this tensor, such as mass density, are specified and used in the Einstein equation to solve for the...
  34. R

    How Does the Coriolis Force Affect Particle Motion in a Rotating System?

    m = Particles mass, Omega = Systems angular frequency, v' = particles velocity. Attempt at a Solution: $$ F_{C} = -2m \bar{\omega} \times \bar{v}^{'} = -2 \bar{\omega} \times \bar{p} = 2 \bar{p} \times \bar{\omega} $$ Let $$ \bar{\omega} = \frac {\bar{r} \times \bar{v}} {r^2}, \alpha = \frac...
  35. G

    A How to prove this property of the Dual Strength Field Tensor?

    Hi, I've found this property of Strenght Field Tensors: $$F_{\mu}^{\nu}\tilde{F}_{\nu}^{\lambda}=-\frac{1}{4}\delta_{\mu}^{\lambda}F^{\alpha\beta}\tilde{F}_{\alpha\beta}$$ Where $$\tilde{F}_{\mu\nu}=\frac{1}{2}\varepsilon_{\mu\nu\alpha\beta}F^{\alpha\beta}, \qquad \varepsilon_{0123}=1$$ I've...
  36. sergiokapone

    A Index Juggling: Angular Momentum Tensor & Inertia Tensor in 3D-Space

    Lets consider the angular momentum tensor (here ##m=1##) \begin{equation} L^{ij} = x^iv^j - x^jv^i \end{equation} and rortational velocity of particle (expressed via angular momentum tensor) \begin{equation} v^j = \omega^{jm}x_m. \end{equation} Then \begin{equation} L^{ij} =...
  37. L

    I Lorenz gauge, derivative of field tensor

    Fμν = ∂μAν- ∂νAμ ∂μFμν = ∂2μAν - ∂ν(∂μAμ) = ∂2μAνWhy ∂ν(∂μAμ) and not ∂μ∂νAμ ? And why does ∂ν(∂μAμ) drop out? thank you
  38. J

    A Electromagnetic Stress Energy Tensor Formula (-,+,+,+)

    I am trying to find the correct formula for the electromagnetic stress energy tensor with the sign convention of (-, +, +, +). Is it (from Ben Cromwell at Fullerton College): $$T^{\mu \nu} = \frac{1}{\mu_0}(F^{\mu \alpha}F^{\nu}{}_{\alpha} - \frac{1}{4}g^{\mu\nu}F_{\alpha\beta}F^{\alpha...
  39. A

    Shear and the stress tensor of a Newtonian fluid

    Similarly the paper by @buchert and @ehlers https://arxiv.org/abs/astro-ph/9510056 Here the author has defined ##v_{ij}=\frac{\partial v_i}{\partial x_j}=\frac{1}{2}(\frac{\partial v_i}{\partial x_j}+\frac{\partial v_j}{\partial x_i})+\frac{1}{2}((\frac{\partial v_i}{\partial...
  40. W

    Tensor Product of Hamiltonians

    ##U_1 \otimes U_2 = (1- i H_1 \ dt) \otimes (1- i H_2 \ dt)## We can write ## | \phi_i(t) > \ = U_i(t) | \phi_i(0)>## where i can be 1 or 2 depending on the subsystem. The ## U ##'s are unitary time evolution operators. Writing as tensor product we get ## |\phi_1 \phi_2> = (1- i H_1 \ dt) |...
  41. JD_PM

    Understanding the Maxwell Stress Tensor

    The elecromagnetic force can be expressed using the Maxwell Stress Tensor as: $$\vec F = \oint_{s} \vec T \cdot d \vec a - \epsilon \mu \frac{\partial }{\partial t} \oint_{V} \vec S d\tau $$ (How can I make the double arrow for the stress tensor ##T##?) In the static case, the second term...
  42. D

    I Help Understanding Metric Tensor

    I am trying to get an intuition of what a metric is. I understand the metric tensor has many functions and is fundamental to Relativity. I can understand the meaning of the flat space Minkowski metric ημν, but gμν isn't clear to me. The Minkowski metric has a trace -1,1,1,1 with the rest being...
  43. W

    Riemann Curvature Tensor in 2D

    Since in 2D the riemman curvature tensor has only one independent component, ## R = R_{ab} g^{ab} ## can be reversed to get the riemmann curvature tensor. Write ## R_{ab} = R g_{ab} ## Now ## R g_{ab} = R_{acbd} g^{cd}## Rewrite this as ## R_{acbd} = Rg_{ab} g_{cd} ## My issue is I'm not...
  44. A

    I No problem, happy to help! Good luck with your studies.

    Hi, I'm currently working through a tensor product example for a two qubit system. For the expression: $$ \rho_A = \sum_{J=0}^{1}\langle J | \Psi \rangle \langle \Psi | J \rangle $$ Which has been defined as from going to a global state to a local state. Here $$ |\Psi \rangle = |\Psi^+...
  45. M

    A Defining the Tensor Product of Gradients for Different Coordinate Systems

    Does anyone know where I can find the definition of ##\nabla \otimes \nabla f##? I tried googling this but nothing comes up. I know it will change depending on the coordinate system, so does anyone know the general definition OR a table for rectangular, spherical, cylindrical coordinates...
  46. K

    Investigating Belinfante Tensor: Relation to Conserved Current

    I was reading this pdf http://research.physics.illinois.edu/Publications/theses/copies/Bandyopadhyay/Chapter_3.pdf I can show myself that ##\partial_\mu T^{\mu \nu} = 0## and ##\int T^{0 \nu} = \int \Theta^{0 \nu}## if ##T^{\mu \nu} = \Theta^{\mu \nu} + \partial_\alpha B^{\alpha \mu \nu}##...
  47. M

    I Uncovering the Derivative of a Tensor: Understanding its Equations and Origins

    How/why does the first equal sign hold? Where does each derivative come from:
  48. J

    Integrals of tensor perturbations (Weinberg, p. 315)

    Hi, I stumbled upon an identity when studying tensor perturbations in cosmology. The formula states that $$ \int d^2\hat{p} f(\hat{p}.\hat{q})\hat{p}_i \hat{p}_k e_{jk}(\hat{q}) = e_{ij}(\hat{q})/2 \int d^2\hat{p} f(\hat{p}.\hat{q})(1-(\hat{p}.\hat{q})^2), $$ where ##e_{ij}(\hat{q})## is any...
  49. dRic2

    Can Anisotropic Materials Have Asymmetric Thermal Conductivity Tensors?

    Hi, yesterday a professor of mine told me that if you have a temperature gradient along the x-axsis you could have heat flowing in the y direction. Mathematically it is pretty straightforward to find the thermal conductivity tensor required, but in real life can you name some materials that...
  50. C

    I Relation between tensor decomposition and helicity amplitude

    It is common to write e.g photon two point function in terms of manifest transverse and longitudinal form factors with lorentz structure factored out, e.g $$\Pi^{\mu \nu} = (g^{\mu \nu} - q^{\mu} q^{\nu}/q^2)T_T + q^{\mu} q^{\nu}T_L,$$ where mu and nu are polarisation indices. How do I relate...
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