Questions tagged [tensor-calculus]

Tensor calculus (tensor analysis) is a systematic extension of vector calculus to multivector and tensor fields in a form that is independent of the choice of coordinates on the relevant manifold, but which accounts for respective sub-spaces, their symmetries, and their connections.

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Does there exist a relationship between elasticity tensor $E_{ijkl}$ and $\int_{\Gamma} F^i z^i d \Gamma$?

Does there exist a relationship between elasticity tensor $$E_{ijkl}$$ and $$\int_{\Gamma} F^i z^i d \Gamma$$ where $F^i$ is force and $z^i$ displacement. $\Gamma$ is loaded boundary. I see these two ...
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Meaning of Slot-Naming Index Notation (tensor conversion)

I'm studying the component representation of tensor algebra alone. There is a exercise question but I cannot solve it and cannot deduce answer from the text. The text is concise, I think it assumes a ...
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What are covectors in special relativity?

In special relativity the purpose of vectors makes fairly intuitive sense, they represent a point in spacetime: $$x^{\mu}=\begin{pmatrix}x^0 \\ x^1 \\ x^2 \\ x^3\end{pmatrix}$$ and we can define the ...
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Why do we write this tensor notation of space-time gradient contravariant tensor?

Why is $\partial^\mu=\frac{\partial}{\partial x_{\mu}}$ the contravariant component of space-time gradient four vector instad of $\partial^{\mu}=\frac{\partial}{\partial x^{\mu}}$?
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Is the spacetime interval a tensor?

Tensors are objects that are invarient under a change of basis representation and whose coordinates change predictably. The spacetime interval is invarient under a change of coordinate representation ...
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Definition of metric tensor and line elements

I know that the metric tensor $g_{ij}$ is defined as: $$g_{ij} = X_i \cdot X_j$$ where $x_i$ and $x_j$ are the covariant basis vectors ($X_i = \frac{\partial X}{\partial X^i}$) (the definition is ...
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Why are isotropic tensors not considered scalars?

In introductory textbooks (Griffiths, Shankar, Boas) a tensor is introduced as a mathematical objects which transform in a specific manner under changes of basis (i.e. changes of the coordinate system)...
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Tensor Method $SU(N)$

I'm working out the $SU(N)$ tensor method and reading Cheng-Li page 102, 103 (Sec. 4.3). I'm following the definition (4.94) which are $\psi^i=\psi_i^*$, $U_i^{.j}=U_{ij}$ and $U^i_{.j}=U_{ij}^*$ ...
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Is it correct to sum over either index of the metric the same way?

I don't know if the following is correct, i want to compute the following derivative $$\frac{\partial }{\partial (\partial_{\mu}A_{\nu})}\left(\partial^{\alpha}A^{\beta}\partial_{\alpha}A_{\beta} \...
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Scale transformation of $\epsilon$ and $\partial$

I got confused about scale transformation. Under \begin{align} \tilde{g}_{ab} = \Omega^2 g_{ab} \end{align} And consider scalar $\Phi$ and levi-civita symbol $\epsilon$, then how \begin{align} \...
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Which quantity is mass (tensor, vector or scalar)? [closed]

Mass and spin are fundamental characteristics of particle. Those quantities are eigenvalues of the Casimir operators of the Poincaré group. My book then writes that $$ p^\mu p_\mu = m_0^2, $$ where $...
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The spinor metric, basic spinor calculations and spinor indices

I'm currently reading the textbook "Finite Quantum Electrodynamics" by Günter Scharf, but I find myself stuck already on page 24. Background Scharf introduces the index-raising symbol (spinor metric)...
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Is this valid tensor transformation?

Is this valid or not and why? $$\Lambda^ν_i\Lambda^μ_j\frac{\partialξ}{\partial(\partial_μΨ)}=\frac{\partialξ}{\partial({\partial_j}^\primeΨ)}$$ where Λs are lorenz transformation tensors.
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Proof of contracting Christoffel Symbol Identity [duplicate]

I figured out the following: $${\displaystyle \Gamma ^{i}{}_{ki}={\frac {1}{2}}g^{im}{\frac {\partial g_{im}}{\partial x^{k}}}={\frac {1}{2g}}{\frac {\partial g}{\partial x^{k}}}={\frac {\partial \...
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Pseudo tensors and tensor densities - formal definition?

In physics, a tensor is defined as a multidimensional array with a special transformation law. Therefore, a tensor of type $(r, s)$ is an geometric object $T_{i_{1}\dots i_{s}}^{j_{1}\dots j_{r}}[\,...
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How does Schwinger transform a static $(\phi,0)$ to $(\phi ', A)$ in a moving frame?

Schwinger writes in his paper Electromagnetic mass revisited: Any spherically symmetrical charge distribution of total charge $e$, at rest, is represented by the potentials $$\phi \sim ef(r^2),...
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Biharmonic operator on manifold

The Biharmonic operator is given by $\nabla^4$, where the usual laplacian is written as the Laplace Beltrami operator on curved surfaces with a metric $g$. How to calculate the square of a laplacian? ...
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Is there a pictorial geometric interpretation of the Saint-Venant compatibility equations?

Is there a pictorial geometric interpretation of the Saint-Venant compatibility equations? Other than being a pretty system of equations possessing many symmetries, they don't mean much to me. The ...
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On the notation for the Jacobian using indices [migrated]

A contravariant vector is an object that is usually written with a superscript and it is defined by the "transformation law": $$V^{'i} = \frac{\partial x^{' i}}{\partial x^j} V^j $$ where $i,j = 0,1,...
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What exactly is the Leibnitz rule in General Relativity?

In the Differential Geometry part of a course in General Relativity (for instance in David Tong's notes here in page 99, accessed 21 Nov, 2019), one frequently comes across the Leibnitz rule when ...
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Young Tableaux and Tensors

We can represent a tensor with $(n, m)$ where $n$ are the upper indices and $m$ the lower ones. If i get the direct product of $(n,m)\otimes (n', m')$ then i will have irreducible representation. Let'...
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How to interprete this singularity? [closed]

I am calculating the Kretschmann scalar for the Schwartzchild metric. This is the graphic I get: Where $R$ is the radial coordinate and $x=\cos(\theta)$. So, there is the singularity at $R=0$ as it ...
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What is the metric tensor for?

I am wondering how to use the metric tensor, in practice? I read the book and done the exercises in A student's guide to vectors and tensors by Dan Fleisch. The concept of a tensor and their ...
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Tensor indices from Ramond [closed]

I'm working through Pierre Ramond's "Field Theory: A Modern Primer". I can't connect the steps in I.2. Eq 2.6 (p. 7) gives a property of two linear transformations in relation to the Minkowski metric,...
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What does it mean to go from a co-variant vector to a contravariant vector?

In most presentations of general-relativity I see the following statement, We can change from a covariant vector to a contravariant vector by using the metric as follows, ${ A }^{ \mu }={ g }^{ \...
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Question about Einstein summation convention

I'm dealing with the following: $$\eta^{\alpha \mu} \eta_{\alpha \nu} \phi,_{\beta \mu}$$ $$\eta^{\alpha \beta} \phi,_{\alpha \beta}$$ where $\eta$ is the Minkowski metric and $\phi$ is a function ...
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What is a three dimensional irrep ${\bf 3}$ of $SO(3)$?

What is three dimensional irreducible representation of $SO(3)$ denoted by ${\bf 3}$? Are they vectors or antisymmetric tensors of rank two each of them has three independent components. Also when ...
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How to represent a axisymmetric, stationary metric in a coordinate independent way?

A classic example of a stationary, axisymmetric metric in GR is the Kerr metric. In Boyer-Lindquist coordinates $(t,r,\theta,\phi)$ it is obvious that the metric is independent of $t,\phi$ and so is ...
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Arguments for a vanishing Riemann Tensor

Consider the following metric: $$ds^{2} = -(1+2\Phi(x,y,z))dt^{2}+dx^{2}+dy^{2}+dz^{2} \tag{1}$$ Where, in fact, $\Phi$ is the Newtonian Potential. Consider the Riemann Tensor: $$\mathrm{R}_{abcd} ...
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What is the difference between invariance and covariance? [duplicate]

In relativistic physics, paricularly in General Relativity and Quantum Field Theory, we often find the use of the two terms 'invariance' and 'covariance'. But I couldn't find any mention of the ...
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Identifying Lorentz Covariant Equations

Statement: $\phi , A^{\mu}, T^{\mu \nu}$ are a Lorentz scalar, vector, and tensor. Which of the following equations are Lorentz covariant. a. $\phi = A_{0}$ b. $\phi = A^{\mu}A_{\mu}$ c. $\phi = ...
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Properties of the electromagnetic field tensor $F^{\mu\nu}$

From the euler lagrange equations of electromagnetism in vacuum we derive the equation of motion that reads $$ \nabla_\mu F^{\mu\nu} = 0 $$ Now if we multiply both sides with a double metric we have $...
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Christoffel symbol derivation in book by Wald

In chapter 3 of Wald's General Relativity he starts by defining a covariant derivative $\nabla$ as a map on a manifold M from tensor fields $\mathscr{T}(k,l) \to \mathscr{T}(k,l+1)$ plus some required ...
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Deriving $\nabla_\mu \nabla_\sigma \mathcal{K}^\rho=R^\rho_{\sigma\mu\nu}\mathcal{K}^\nu$

I want to derive this equation from Carroll's book. $$\nabla_\mu \nabla_\sigma \mathcal{K}^\rho=R^\rho_{\sigma\mu\nu}\mathcal{K}^\nu$$ We know that $\mathcal{K}^\nu$ is a killing vector and ...
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In a coordinate change $\frac{\partial \tilde x_\lambda }{\partial x_\mu } = \frac{\partial x^\mu }{\partial \tilde x^\lambda}$?

So I was verifying wether $\partial_\mu \partial^\mu$ is a scalar. To do this I used the chain rule: $$\frac{\partial }{\partial x^\mu} = \frac{\partial \tilde x^\nu }{\partial x^\mu}\frac{\partial }{...
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Confusion about expressing an inner product using the Einstein summation convention

I think this likely comes down to the following expression, $$g’^{ab}e’_a e’_b = \delta ^a_b $$ Is this in agreement with the Einstein summation convention? Because even though the two indices are ...
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Are dual bases and the Hodge dual “entirely distinct” uses of the word “dual”, as per MTW

NB: Basis one-forms and contravariant basis vectors (which, following Menzel, I am calling reciprocal) are the same thing. See, for example, the Mathematical Appendix to Gravitation and Inertia, by ...
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Difference between $g^{\alpha\beta}$ and $g^\alpha_{\space\space\beta}$

I'm working out a problem where at some point get the following product of metric tensors and momenta: $$g^{\mu\beta}g^\nu_{\space\space\alpha}(2k+\frac{q}{2})^\alpha(\frac{q}{2}-k)_\beta$$ How can I ...
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Doubt about the “kernel of Einstein's equations”

From a coordinate-free point of view, we can rewrite the Einstein Field Equations $R^{\mu}\hspace{0.5mm}_{\nu} - \frac{1}{2}R \delta^{\mu}\hspace{0.5mm}_{\nu}=:G^{\mu}\hspace{0.5mm}_{\nu} = 8\pi T^{\...
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Where have I gone wrong in deriving Faraday's Law of Induction from its manifestly covariant form?

EDIT: ISSUE SOLVED This is simply an error in expanding tensor components on my part I'm sure but I am struggling to discover the error - where is the minus sign?? In expressing the laws of ...
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Volume form in terms of the Levi-Civita Symbol

I'm currently reading Sean Carroll's book on General Relativity, and at some point he writes: First notice that the definition of the wedge product allows us to write \begin{equation} \mathrm{d}x^...
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Is the moment of inertia calculated about an axis, or about a point? And must the point be at the center of mass?

I know that, $$L=I\omega$$ where $L$ is the angular momentum vector, $I$ is the inertial tensor, and $\omega$ is the angular velocity. Now here are my doubts :- Before I was taught the moment of ...
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Is this useful identity valid only under the integral sign?

Studying dimensional reugularization one often encounters the following identity: $$ \int d^d q\, \, q^\mu q^\nu f(q^2) = \frac{1}{d}g^{\mu\nu}\int d^d q\,\, q^2 f(q^2) $$ often justified by some ...
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Covector basis derivation

On page 65 of Schutz's A first course in General Relativity, he introduces the notation $\phi_{,\alpha}=\partial\phi/\partial x^\alpha$. He then says that $x^\alpha_{\ \ ,\beta}=\delta^\alpha _{\ \ \ \...
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How to calculate the tensor of inertia in this problem?

I have the following geometry for my problem: I'm asked to calculate the tensor of Inertia: $\stackrel{\leftrightarrow}{I}_{O}=\left[\begin{array}{ccc}{I_{11}} & {-P_{12}} & {-P_{13}} \\ {-...
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Tensors defined by transformation laws are tensors at a vector space or tensor fields?

In Physics it is common to see tensors defined by transformation properties relating components of the object in different coordinate systems. There is, however, two ways we can think of a tensor: a ...
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In electromagnetism, how do we know that either $F^{\mu\nu}$ or $A^\mu$ is a tensor?

In special relativity the partial derivative $\partial_\mu$ is a tensor. Now if some function $A^\mu$ was a tensor, then also the quantitiy $F^{\mu\nu}=\partial^\mu A^\nu - \partial^\nu A^\mu$ would ...
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Problem in odd-even decomposition of a generic metric

The metric of the unit two-sphere is given by $ \Omega_{\mu \nu} = \begin{equation} \begin{pmatrix} 1 & 0 \\ 0 & \sin^2 \theta \end{pmatrix}. \end{...
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How to raise indices on the electromagnetic tensor

How do you transform between the electromagnetic tensors $F_{\mu\nu}$ and $F^{\mu\nu}$? $$ F_{\mu \nu}= \begin{pmatrix} 0 & E_x & E_y & E_z \\ -E_x & 0 & -B_z & B_y \\ -E_y &...
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265 views

Commutation relations for inverse d'Alembertian operator

Is there a commutation relation for the inverse d'Alembertian operator in general relativity? i.e. if we define $\Box = g^{\mu\nu}\nabla_\mu\nabla_\nu$ and $\Box \Box^{-1}X_{\alpha_1,\alpha_2...}=X_{\...