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.
I am confused. Why sometimes perturbation ##V'=\alpha xy## we can write as ##V'=\alpha x \otimes y##. I am confused because ##\otimes## is a tensor product and ##x## and ##y## are not matrices in coordinate representation. Can someone explain this?
During lecture today, we were given the constitutive equation for the Newtonian fluids, i.e. ##T= - \pi I + 2 \mu D## where ##D=\frac{L + L^T}{2}## is the symmetric part of the velocity gradient ##L##. Dimensionally speaking, this makes sense to me: indeed the units are the one of a pressure...
It seems the field φ(t, xi) could be integrated over all space to form a single temporal variable (which isn't a field anymore, but is just a function of time) as follows:
Φ(t) = ∫φ(t, xi)dxi
Suppose we then assume a Lagrangian from this temporal variable to be:
L1 = -1/2 Φ'(t)2 + 1/2 b2...
I'm reading "Differentiable manifolds: A Theoretical Physics Approach" by Castillo and on page 170 of the book a calculation of the Ricci tensor coefficients for a metric is illustrated. In the book the starting point for this method is the equation given by:
$$d\theta^i = \Gamma^i_{[jk]}...
Hi everyone,
studying the bending of an incompressible elastic block of Neo-Hookean material, one finds out the first Piola-Kirchoff stress tensor as at page 182 (equation 5.93)
where $e_r = cos(\theta)e_1 + \sin(\theta)e_2$ and $e_{\theta} = -sin(\theta)e_1 + \cos(\theta)e_2$
How is the...
I am working on a computational project about General Relativity. In this process, I want to code 'the stuff' that can be derivable from the metric tensor. So far, I have coded Riemann Tensor, Weyl Tensor, Einstein Tensors, Ricci Tensor, Ricci scalar. What are the other essential/needed...
Inspired by the closed thread about pressure:) Here is some of my fantasies about a definition of the stress tensor. Nothing here claims to be a correct theory but just as a matter for discussion.
The result equation doesn't fit with the familiar divergence form that are usually used in electrodynamics.
I want to know the reason why I was wrong.
My professor says about transformation of components.
But I cannot close to answer by using this hint, because I don't have any idea about "x"...
In Minkowski spacetime, calculate ##P^{\gamma}_{\alpha}U^{\beta}\partial_{\beta}U^{\alpha}##.
I had calculated previously that ##P^{\gamma}_{\alpha}=\delta^{\gamma}_{\alpha}+U_{\alpha}U^{\gamma}##
When I subsitute it back into the expression...
I am trying to understand the scalar form of the Einstein field equations. I know that you can contract the stress-energy tensor using the metric. And for a perfect fluid model, this turns out to be the energy density summed with the pressure. This also gives the Ricci scalar. However, you can...
Hi.
So I was asked the following question whose picture is attached below along with my attempt at the solution.
Now my doubt is, since the question refers to the whole system comprising of these thin rigid body 'mini systems', should the Principle moments of Inertia about the respective axes...
Hello all,
I am trying to understand the von Mises yield criterion and stumbled across two equations for the second stress invariant. Although the only difference is a difference in signs (negative and positive), it has been bothering me. Attached are the two versions. Which one is correct and...
Let us suppose we are given two vectors ##A## and ##B##, their components ##A^{\nu}## and ##B^{\mu}##. We are also given a minkowski metric ##\eta_{\alpha \beta} = \text{diag}(-1,1,1,1)##
In this case what are the
a) ##A^{\nu}B^{\mu}##
b) ##A^{\nu}B_{\mu}##
c) ##A^{\nu}B_{\nu}##
For part (a)...
I try to solve but i have 1 step in the solution that I don't understand who to solve.
Below in the attach files you can see my solution, the step that I didn't make to prove Marked with a question mark.
thanks for your helps (:
I was trying to show that the field transformation equations do hold when considering electric and magnetic fields as 4-vectors. To start off, I obtained the temporal and spatial components of ##E^{\alpha}## and ##B^{\alpha}##. The expressions are obtained from the following equations...
In the earlier years of Britgrav there were sometimes longer presentations of research done at the host university. In one such, about 2005 or so, a presentation showed that an isolated region of space could be rotated through 180 degrees by the action of extreme waves in the Weyl tensor...
I managed to write
$$F_{\alpha\beta}F^{\alpha\gamma}=F_{0\beta}F^{0\gamma}+F_{i\beta}F^{i\gamma}$$
where $$i=1,2,3$$ and $$\gamma=0,1,2,3=\beta$$.
How do I proceed?
Hello. I am doing tensor calculation with indices, such as contraction, lowering or raising indices, tensor production, etc.
I have tried the Ricci.m from https://sites.math.washington.edu/~lee/Ricci/
However, maybe because the package has not been upgraded for some time, I could not get the...
Good afternoon everyone!
I've learned that thermal conductivity has a form of second-rank tensor. As you know, diagonal components of stress tensor mean normal stress and other components mean shear stress and like that do off-diagonal components of thermal conductivity tensor have some special...
I think it is quite simple as an exercise, following the two relevant equations, but at the beginning I find myself stuck in going to identify the lagrangian for a relativistic system of non-interacting particles.
For a free relativistic particle I know that lagrangian is...
In the test of General Relativity by perihelion motion of mercury, the stress-energy tensor is set to 0 in Schwarzschild solution. Then, is the curvature caused by solar mass, or by the 0 stress-energy? Or, do we consider solar mass as the gravitating mass? Or the 0 stress-energy the gravitating...
If I have an anisotropic material with permittivity:
$$\epsilon=
\begin{pmatrix}
\epsilon_{ii} & \epsilon_{ij} & \epsilon_{ik} \\
\epsilon_{ji} & \epsilon_{jj} & \epsilon_{jk} \\
\epsilon_{ki} & \epsilon_{kj} & \epsilon_{kk} \\
\end{pmatrix}
$$
What exactly does each element represent in this...
Relevant Equations:: ##\ket{\vec{p}}=\hat{a}^{\dagger}(\vec{p})\ket{0}## for a free field with ##[\hat{a}({\vec{k})},\hat{a}^{\dagger}({\vec{k'})}]=2(2\pi)^3\omega_k\delta^3({\vec{k}-\vec{k'}})##
$$ \bra{ \vec{ p'}} T_{\mu,\nu} \ket{ \vec...
Hello.Questions: How tensor operations are done?Like addition, contraction,tensor product, lowering and raising indices. Why do we need lower and upper indices if we want and not only lower? Is a tensor a multilinear mapping?Or a generalisation of a vector and a matrix? Could a tensor be...
I am studying @Orodruin's Insight "Explore Coordinate Dependent Statements in an Expanding Universe". It looks pretty interesting. About three pages in it reads "expanding ##x^a## to second order in ##\xi^\mu## generally leads to$$
x^a=e_\mu^a\xi^\mu+c_{\mu\nu}^a\xi^\mu\xi^\nu+\mathcal{O}_3...
When I started learning about tensors the tensor rank was drilled into me. "A tensor rank ##\left(m,n\right)## has ##m## up indices and ##n## down indices." So a rank (1,1) tensor is written ##A_\nu^\mu,A_{\ \ \nu}^\mu## or is that ##A_\nu^{\ \ \ \mu}##? Tensor coefficients change when the...
I am struggling with tensor notation. For instance sometimes teacher uses
\Lambda^{\nu}_{\hspace{0.2cm}\mu}
and sometimes
\Lambda^{\hspace{0.2cm}\nu}_{\mu}.
Can you explain to me the difference? These spacings I can not understand. What is the difference between...
I have worked out (and then verified against some sources) that ##R^\theta_{\phi\theta\phi} = sin^2(\theta)##. The rest of the components are either zero or the same as ##R^\theta_{\phi\theta\phi} ## some with the sign flipped.
I was surprised at this, because it implies that the curvature...
Force lines method is used in Solid Mechanics for visualization of internal forces in a deformed body. A force line represents graphically the internal force acting within a body across imaginary internal surfaces. The force lines show the maximal internal forces and their directions.
But...
Clearly, they used the binomial expansion on this; however, I cannot figure out why [G] is sandwiched by the epsilon inverses:
$$\varepsilon^{'-1}=1/(\varepsilon+i\epsilon_{0}[G])\approx(1-i\epsilon_{0}[G]\varepsilon^{-1})\varepsilon^{-1}$$
Hello,
Throughout my undergrad I have gotten maybe too comfortable with using Dirac notation without much second thought, and I am feeling that now in grad school I am seeing some holes in my knowledge. The specific context where I am encountering this issue currently is in scattering theory...
The answer with no details is given by
First, I considered a spherical shell because I thought the velocities at different radius ##r## will be different and hence the four-momentum will be different, as well.
Then, I writed down the linear momenta by $$\epsilon^{ijk} r_i p_j = L_k$$ with...
Hello. Could we approximate a tensor of (p,q) rank with matrices of their elements? I am talking also about the general case of a tensor not only special cases. For example a (2,0) tensor with i, j indices is a matrix of ixj indices. A (3,0) tensor with i,j,k indices I think is k matrices with...
I got stuck in this calculation, I can't collect everything in terms of ##dx^{\mu}##.
##x'^{\mu}=\frac{x^{\mu}-x^2a^{\mu}}{1-2a_{\nu}x^{\nu}+a^2x^2}##
##x'^{\mu}=\frac{x^{\mu}-g_{\alpha \beta}x^{\alpha}x^{\beta}a^{\mu}}{1-2a_{\nu}x^{\nu}+a^2g_{\alpha \beta}x^{\alpha}x^{\beta}}##...
I saw briefly that the Riemann tensor can be obtained via Stoke's theorem and parallel transport along a closed curve.
If one does add winding number then it can give several results, does it imply that this tensor is multivalued ?
This proof was in my book.
Tensor product definition according to my book: $$V⊗W=\{f: V^*\times W^*\rightarrow k | \textrm {f is bilinear}\}$$ wher ##V^*## and ##W^*## are the dual spaces for V and W respectively.
I don't understand the step where they say ##(e_i⊗f_j)(φ,ψ) = φ(e_i)ψ(f_j)##...
I was reading zee's group theory in a nutshell.
I understand that we can decompose a 2 index tensor for rotation group into an antisymmetric vector(3), symmetric traceless tensor(5) and a scalar(trace of the tensor). Because "trace is invariant" it put a condition on the transformation of...
Background and Motivation
The stress energy tensor of general relativity, as conventionally defined, has sixteen components.
One of those component, conventionally component T00, also called ρ, is mass-energy density, including the E=mc2 conversion for electromagnetic fields.
The other...
Hello all,
let's suppose we have, in a flat spacetime, two observers O and O', the latter speeding away from O, with an uniform acceleration ##a##.
In the Minkowski spacetime chart of O, the world-line of O' can be drawn as a parable.
We know that the Lorentz boost at every point of the...
I know the tensor can be written as $$T^{\mu v}=\Pi^{\mu}\partial^v-g^{\mu v}\mathcal{L}$$ where $$g^{\mu v}$$ is the metric and $$\mathcal{L}$$ is the Lagrangian density, but how would I write $$T_{\mu v}$$? Would it simply be $$T_{\mu v}=g_{\mu \rho}g_{v p}T^{\rho p}$$? And if so, is there a...
I'm puzzling over Exercise 1.14 in Thorne & Blandford's Modern Classical Physics. We are given that an electric field ##\boldsymbol{E}## exerts a pressure ##
\epsilon_{0}\boldsymbol{E}^{2}/2## orthogonal to itself and a tension of the same magnitude along itself. (The magnetic field does the...
I have an equation $$
\chi_\nu\nabla_\mu\chi_\sigma+\chi_\sigma\nabla_\nu\chi_\mu+\chi_\mu\nabla_\sigma\chi_\nu=0
$$so we also have$$
g_{\nu\rho}g_{\mu\tau}g_{\sigma\lambda}\left(\chi^\rho\nabla^\tau\chi^\lambda+\chi^\lambda\nabla^\rho\chi^\tau+\chi^\tau\nabla^\lambda\chi^\rho\right)=0
$$Does...
Given the action ##S =-\sum m_q \int \sqrt{g_{\mu\nu}[x_q(\lambda)]\dot{x}^\mu_q(\lambda)\dot{x}^\nu_q(\lambda)} d\lambda## The Energy-Momentum Tensor (EMT) is defined by the variation of the metric
$$\delta S = \frac{1}{2}\int T_{\mu\nu} \delta g^{\mu\nu} \sqrt{g} d^4x$$
Then I use two...
If I have a matrix representing a 2nd order tensor (2 2) and I want to convert this matrix from M$$\textsuperscript{ab}$$ to $$M\textsubscript{b}\textsuperscript{a}$$ what do I do? I'm given the matrix elements for the 2x2 tensor. When applying the metric tensor to this matrix I understand...
A second-order tensor is comprised at least of a two-dimensional matrix, as an nth-order tensor is comprised at least of an n-dimensional matrix, but what else is in the formal definition.
A scientific definition needs to name the term being defined, and describe the meaning of that term only...