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.

Filter by
Sorted by
Tagged with
4
votes
1answer
53 views

Torsion tensor symmetrisation

Given that the affine connection can be written as: $$\Gamma ^a{}_{bc}= \bigg\{ {a \atop bc} \bigg\} - \frac{1}{2}(T^a{}_{bc}+T_c{}^a{}_b-T_{bc}{}^a) $$ Where $\big\{ {a \atop bc} \big\}$ denotes the ...
4
votes
1answer
149 views

If the world had four spatial dimensions, then area would be a tensor?

In three dimensions area is a vector because two dimensions have a direction relative to the third. If the world had four spatial dimensions then area would be a tensor? And what form then the laws of ...
2
votes
1answer
76 views

Wedge Product Convention

In Wald’s General Relativity textbook he defines the wedge product as: $$(w \wedge u)_{a_1 ... a_p b_1 ... b_q}= \frac{(p+q)!}{p!q!} w_{[a_1 ... a_p}u_{b_1 ... b_q]}$$ My question is relatively simple:...
3
votes
1answer
68 views

Definition of exterior derivative

In Sean Carroll's GR book, pg 84, the exterior derivative $d$ is defined as $$(dA)_{\mu_1\mu_2...\mu_{p+1}} = (p+1) \partial_{[\mu_1}A_{\mu_2...\mu_{p+1}]}\tag{2.76}$$ where $A$ is a $p$-form and the ...
0
votes
0answers
58 views

d'Alembertian of Ricci tensor

What is the d'Alembertian of the Ricci tensor $\square R_{\mu\nu}$? I know that for a scalar $\square := g^{\mu\nu} \nabla_\mu \nabla_\nu$ and I can use the fact that $\nabla_\mu \nabla_\nu F = \left( ...
0
votes
1answer
48 views

Indices Exchange

In General Relativity, we often face indices exchange; but I actually do not really understand how to change indices properly. For example: If I have $$ R_{ab}\partial_c \phi \partial^b\phi \delta g^{...
0
votes
0answers
8 views

Is there a table for hyperfine interaction tensor?

I'm doing research in biophysics and I need to find the hyperfine interaction tensors for ascorbic acid radical, FADH-, and oxigen radical (O2-). Is there a place where I can find the hyperfine ...
6
votes
1answer
380 views

Differential forms and wedge product

In Sean Carroll's GR book, the differential $p$-form is defined as a $(0,p)$ tensor that is completely antisymmetric, which I would think is something like $$\frac{1}{2!}(t_{ab}-t_{ba})\textbf {e}^a \...
1
vote
1answer
94 views

Covariant derivative of tetrad/vielbein

I am learning about the tetrad basis for manifolds from this lecture notes. On pg 52, the spin connections ${{w_\mu}^a}_b$ are defined as $${{w_\mu}^a}_b=e^a_\nu e^\lambda_b\Gamma^\nu_{\mu\lambda}-e^\...
0
votes
1answer
41 views

Does product rule of covariant derivative apply for a scalar

I'm proving a tensor identity and I wonder if it is true that $$ \nabla_\mu \left( A^\nu B_\nu \right) = \nabla_\mu\left( A^\nu \right) B_\nu + A^\nu \nabla_\mu\left( B_\nu \right) $$ where $A^\nu$ ...
0
votes
0answers
25 views

Transformation rules for quantities

When we formulate transformation laws for vectors and tensors, the transformation rule for $x^\mu$ is calculated via arguments from total derivatives considering $x^\mu=x^\mu(x^{'\nu})$ that in turn ...
0
votes
0answers
41 views

Zee Quantum Field Theory page 35

About spin 2 polarization tensors $$\varepsilon_{\mu\nu}^{(a)}, $$ it is claimed that $$\sum_{a} \varepsilon_{\mu\nu}^{(a)}(k)\varepsilon_{\lambda\sigma}^{(a)}(k) = A(G_{\mu\lambda}G_{\nu\sigma}+G_{\...
0
votes
1answer
46 views

Commutation relation involving $\gamma^5$. Spot the error

I'm trying to prove a relation that is useful when studying general properties of Dirac spinors, namely, that $\left[\gamma_5,\sigma^{\mu\nu}\right]=0$ where $\sigma^{\mu\nu}\sim i\gamma^\mu\gamma^\nu$...
0
votes
0answers
24 views

Form of General Relativity Hydrostatic Equilibrium Equation

I have been reading into general relativity and have recently gotten stuck at a manipulation of the equation for hydrostatic equilibrium of a perfect fluid. We have $$T^\mu_{\nu;\mu} = 0$$ which leads ...
1
vote
1answer
81 views

If $g_{ij}$ is a tensor of type $(0,2)$, what is kind of tensor is $\partial_{i}g_{jk}$?

Suppose $g_{ij}$ is a tensor of type $(0,2)$, then what type of object is $\partial_{i}g_{jk}$? Is it even a tensor, and if so, of what type? Is the $\partial_{i}$ still a differential with respect to ...
0
votes
1answer
92 views

Covariant derivative of a tensor

I have one question. I'm new in tensor calculus, so question may seem like stupid. Covariant derivative of a tensor $T^\alpha$: $$\nabla_\beta T^\alpha=\frac{\partial T^\alpha}{\partial x^\beta}+\...
0
votes
1answer
83 views

Einstein summation convention when a sum of terms is present

I'm reading Landau / Liftshitz vol. 6 on fluid mechanics, and I encountered the expression (page 45, top): $$\frac{\partial v_i}{\partial x_k} + \frac{\partial v_k}{\partial x_i}.$$ The expression ...
1
vote
2answers
68 views

Tensors Differentiation

I know that $\frac{\partial x^{\mu}}{\partial x^{\nu}}=\delta^{\mu}_{\nu}$ but a few days back, I read somewhere that $\frac{\partial x_{\mu}}{\partial x^{\nu}}=\eta_{\mu\nu}$. Can someone help me ...
0
votes
2answers
132 views

Calculating the coordinate basis

In Robert M. Wald's General Relativity the definition of the "coordinate basis" (of the tangent space) of a manifold is given by: Let $\psi: O \to U \subset \mathbb{R}^n$ be a chart with $p \...
0
votes
2answers
87 views

Derivative Operators on a manifold

I am having some trouble coming to terms with the notion of a derivative operator on a manifold. In Robert M. Wald's General Relativity, the definition in the textbook is given in terms of 5 ...
0
votes
0answers
36 views

How do you find coordinate transformations to LIF for the weak gravity metric?

I am working on the same problem from Schutz as this question, which discusses the weak gravitational field metric. $$ds^2=−(1+2\phi)dt^2+(1−2\phi)(dx^2+dy^2+dz^2)$$ From this metric, I was able to ...
0
votes
2answers
82 views

Gradient, one-form and Sean Carroll

"A tensor (k,l) is a multilinear map from k dual vectors and l vectors to R (...) The gradient, ..., is an honest (0,1) tensor." These citations are retired from Sean Carrol Spacetime and ...
1
vote
1answer
97 views

Contraction of Christoffel symbol and metric tensor

How can I prove this contraction of Christoffel symbol with metric tensor? $$ g^{k\ell} \Gamma^i_{\ \ k\ell} = \frac{-1}{\sqrt{|g|}}\frac{\partial\left(\sqrt{|g|}g^{ik}\right)}{\partial x^k} $$ I know ...
0
votes
1answer
42 views

What does a contravariant and a covariant tensor with the same indices result in?

For example, for two arbitrary tensors $M^\alpha N_\alpha$, can this be written in a simpler way like equivalent to a scalar? or does it even vanish entirely?
1
vote
3answers
137 views

Why is the magnetic field $B$ a pseudo-vector?

Physically speaking, "pseudo-vectors" are vectors $v\in \mathbb{R}^3$ which transform as $ v'= (\det {R})v$ if the "system were to transform as $R\in O(3)$". However, what does ...
0
votes
1answer
75 views

Help with tensor calculus identity proof (antisymmetric matrix and levi civita symbol)

I'm having some trouble with 2 identitys from tensor calculus. I need to proof these two guys: in euclidean 3-dimensional space, an antisymmetric matrix with entries $M_{ij}$ is equivalent to a ...
0
votes
0answers
54 views

Tensor Index notation vs Matrix notation Transpose

Referring to the answer in the following question: https://physics.stackexchange.com/a/349030/288587 I just cant figure out how to go from: $$ \eta_{\mu\nu} = \Lambda^\alpha_{\;\mu}\Lambda^\beta_{\;\...
2
votes
0answers
73 views

How can I derive the covariant derivative of the Ricci tensor using the Ricci scalar?

\begin{align*} R&=g^{\mu\nu}R_{\mu\nu}\\ \Rightarrow \nabla^{\gamma} R&=\nabla^{\gamma}(g^{\mu\nu}R_{\mu\nu})\\ &=\nabla^{\gamma}(g^{\mu\nu})R_{\mu\nu}+g^{\mu\nu}\nabla ^{\gamma}(R_{\mu\nu}...
0
votes
1answer
63 views

Why is the multiplication of the metric and an inverse metric the Kronecker delta?

I am having a hard time understanding \begin{align*} \delta_{\beta}^{\alpha}=g^{\alpha\nu}g_{\beta\nu}\\ \end{align*} equality. I understand the situation where the indices are the same and the ...
2
votes
1answer
80 views

Understanding tensor transformations

I am trying to learn how tensors transform under coordinate transformations. For an example, under a transformation from the coordinate system $x^\mu \longrightarrow x'^\mu$ a covariant tensor is ...
1
vote
2answers
192 views

Covariant derivative of a one-form

I understood that the covariant derivative of a vector field is $$ \nabla_{i}v^j=\frac{\partial v^j}{\partial u^{i}}+\Gamma^j_{~ik}v^k $$ Then why is the covariant derivative of a covector field $$ \...
0
votes
0answers
43 views

Are vector components “normalized” in generic vector differential operater expressions, e.g., divergence?

A typical way of writing the expression for the divergence of a vector field in general orthogonal coordinates is: $$\nabla\cdot\vec{A}=\frac{\partial_{u}\left[VWA^{u}\right]+\partial_{v}\left[WUA^{v}\...
0
votes
2answers
63 views

Under what conditions is this tensor isotropic?

The tensor I'm talking about is \begin{equation}\tag{1} A_{ij}=\int d{\Omega}\,T_{ij}(\theta,\varphi) \end{equation} where the integral is over the whole solid angle. I know that an isotropic tensor ...
0
votes
0answers
30 views

Problem proving a tensor identity

I am asked to prove $$\nabla \cdot (T \cdot v) = T : \nabla v + v\cdot (\nabla \cdot T)$$ Where $T$ is a order-2 tensor and $v$ is vector, in an orthogonal basis. Let $\delta _{ij}$ denote the ...
2
votes
1answer
57 views

Derivative of basis vector in terms of Christoffel symbols

I would like to derive the formula $$\partial_{c}\vec{e}^{\,a}=-\Gamma_{bc}^{a}\vec{e}^{\,b}$$ where $\vec{e}_{a}$ are the basis vectors on a manifold. In the lecture, we did it in the following way: $...
1
vote
1answer
51 views

Integral form of energy-momentum tensor conservation (Stokes' theorem)

Is there a way in which the conservation law of the energy momentum tensor $\nabla _\nu T^{\mu\nu}=0$ can be written in integral form using Stokes' theorem, namely as something roughly similar to: $$ ...
1
vote
2answers
121 views

Transpose of a 2x2 Tensor

This question arises after reading through several Stack Exchange posts and after a long chat with another user in a previous question I asked about this topic. The following "contradiction" ...
0
votes
0answers
59 views

Variation of Ricci scalar with respect to torsion

I'm following a paper in the derivation of field equation in a space with torsion. Now the variation of the Ricci tensor $$\delta R_{ij}=2\nabla_{[k}\delta\Gamma^k_{i]j}+2S_{ki}^{\hphantom0\hphantom0 ...
1
vote
0answers
44 views

How to show ${\epsilon^{ij}}_k {\epsilon_{ij}}^l = -2\delta^l_k$ for the Levi-Civita symbol?

I am trying to prove the following identity for the levi civita symbol $${\epsilon^{ij}}_k {\epsilon_{ij}}^l = -2\delta^l_k,$$ taken from the Ashok Das QFT book pg 153, equation 4.102. I made use of ...
2
votes
1answer
60 views

Exterior Derivative on Curved Manifold (SpaceTime)

Considering the 2-Form Gauge Potential $B_{\mu \nu}$, we can write $dB=H$. In a flat manifold we have that $H_{\mu \nu \rho}=\partial_{\mu}B_{\nu \rho}+\partial_{\rho}B_{\mu \nu}+\partial_{\nu}B_{\rho ...
0
votes
0answers
163 views

Area element from tangent vectors

I am working on a problem in the framework of General Relativity, where light is emitted from a point source and the light bundle is described as a congruence of geodesics. Consider taking two spatial ...
1
vote
2answers
108 views

Raising and Lowering Indices with the Metric Tensor

Let $V$ be a finite-dimensional vector space and let $V^{*}$ denote its dual vector space. A tensor, $T,$ of type $(k,l)$ over $V$ is a multilinear map $$T: V^{*} \times ... V^{*} \times V \times ... ...
0
votes
0answers
49 views

Can we use Neumann's principle for microscopic properties?

Neumann's principle states if a crystal is invariant with respect to certain symmetry elements, any of its physical properties must also be invariant with respect to the same symmetry elements. We ...
0
votes
0answers
30 views

What is an example of a matrix that does not represent a tensor of rank $2$? [duplicate]

I often read that every tensor of order $2$ can be represented as a matrix (I do understand that). But not every matrix is the representation of some second order tensor. I am not sure, if I ...
0
votes
0answers
29 views

Defining the inverse of a tensor via the adjugate tensor

My professor definied the adjugate of a tensor $\mathbf{t}\in T^{1}_{1}(E)$ (E is just a vector space of dimension n) by defining its components as $adj(\mathbf{t})^{a}_{b}=\frac{1}{(n-1)!}\...
3
votes
0answers
41 views

Tetrad basis: a doubt on “Comoving” and “Static” tetrads

In the awesome paper $[1]$, Müller then gives us a plethora of spacetimes and their basic geometrical objects like the form of the line element, Christoffel symbols, Krestchmann scalars and so on. ...
1
vote
1answer
69 views

Why are vectors considered to have odd/negative/- parity while pseudovectors are even/positive/+ in parity?

Most places I read say that true/polar vectors are of odd or - parity, while axial/pseudovectors are of even + parity. But, pseudovectors gain an 'extra' sign flip after a reflection/parity ...
0
votes
1answer
22 views

Converting order parameter to director (nematics)

If I have a differential equation for the order parameter of a uniaxial nematic, like $\frac{\partial S_{\alpha \beta}}{\partial t} = f(S_{\alpha \beta})$, for $S_{\alpha \beta} = S(n_{\alpha}n_{\beta}...
0
votes
1answer
80 views

Levi-Civita and Kronecker Delta

To prove this, $$ \sum_{pq} \epsilon_{ipq} \epsilon_{jpq} = 2\delta_{ij} $$ I used Levi-Civita and delta relation $$ \sum_{q} \epsilon_{ipq}\epsilon_{jpq} = \delta_{ij}\delta_{pp}-\delta_{ip}\delta_{...
1
vote
0answers
281 views

Doubt on Cauchy Stress tensor: a partial derivative of metric tensor?

In the reference $[1]$ the author presented a definition of Stress tensor: $$ \sigma = 2 \rho \frac{\partial e}{\partial g} \tag{1}$$ In a local chart we have: $$ \sigma_{ab} = 2 \rho \frac{\...

1 2
3
4 5
39