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.

270 questions with no upvoted or accepted answers
Filter by
Sorted by
Tagged with
9
votes
0answers
2k views

Why is the $\theta$ term of QCD violating charge and parity (CP) symmetries?

From the non-trivial nature of the QCD vacuum, the Lagrangian is augmented with a term like \begin{equation} \theta \frac{g^2}{32 \pi^2} G_{\mu \nu}^a \tilde{G}^{a, \mu \nu} \end{equation} where $ \...
6
votes
1answer
229 views

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} \...
5
votes
1answer
169 views

Variation of scalar curvature upon frame deformation

Following the textbook "Ideas and Methods of Supersymmetry and Supergravity" by Ioseph Buchbinder and Sergei Kuzenko (p38 - 40): If we consider the a frame deformation in our vierbein induced by a ...
5
votes
1answer
202 views

Proportionality Constant in Einstein Field Equations

The Einstein Field Equations: $$G_{ab}~=~8\pi T_{ab}.$$ I am familiar with how to obtain the $8\pi$ proportionality factor through correspondence with Newtonian gravity, but am wondering if this ...
4
votes
1answer
107 views

How electromagnetic energy-momentum looks like for arbitrary 4-velocity vector?

I need to expresse the electromagnetic energy-momentum tensor in a vacuum $$T^\nu_{\ \ \ \mu} = F_{\mu\alpha}F^{\nu\alpha} - \frac14 F_{\alpha\beta}F^{\alpha\beta}\delta^\nu_{\ \ \mu}$$ in terms of ...
4
votes
0answers
541 views

Uses of the Angular Momentum 4-Tensor

The angular momentum 4-tensor has 6 independent components, three angular momentum components and three new guys. Some call these new guys the 'boosts', but since they are the conjugate momentum of ...
4
votes
0answers
2k views

Covariant versus “ordinary” divergence theorem

Let $M$ be an oriented $m$-dimensional manifold with boundary. As stated in Harvey Reall's general relativity notes (here) or Sean Carroll's book, the "covariant" divergence theorem (i.e. with ...
4
votes
0answers
200 views

Question about derivation of tensor in Di Francesco's CFT

This is a question for anyone who is familiar with Di Francesco's book on Conformal Field theory. In particular, on P.108 when he is deriving the general form of the $2$-point Schwinger function in ...
3
votes
2answers
128 views

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 ...
3
votes
0answers
78 views

Confused about the gauge transformation of the amplitude tensor for gravitational waves

Far away from the field sources, where the energy-momentum tensor $$T_{mn}=0 \tag{m,n=0,1,2,3}$$ The linearized EFE becomes $$\Box \bar h_{mn}=0 \tag{1}$$ where $\bar h_{mn}$ is the trace-reverse ...
3
votes
0answers
57 views

Timelike, spacelike etc. for higher-order tensors

Vectors $V^\mu$ in relativity can be classified into those which are timelike, spacelike and null. A similar classification is available for tensors: A tensor $$T^{\mu_1\mu_2...\mu_p}_{\phantom{\mu_1\...
3
votes
0answers
61 views

Variations of tensors are tensors?

Recently I posted a question about variation of metric. I thought I understood it and talked with my friend about it. After that he said he's not convinced because he can't prove variation of metric ...
3
votes
0answers
98 views

Question about Ricci Rotation Coeficients

Standard General Relativity calculations lies under, indeed, the calculations of three quantities: Christoffel Symbols of second kind, the components of Riemann tensor $R^{\mu}\hspace{1mm}_{\nu \gamma ...
3
votes
0answers
141 views

Why is a spinor not a tensor?

The title says it. why is a spinor not a tensor? I know the transformation rules for a spinor but I cant see why it is not a tensor?
3
votes
0answers
412 views

Gauge transformation of trace-reversed metric perturbation

This question is in reference to Exercise 30.4.2 in Thomas Moore's A General Relativity Workbook, which asks you to show that a gauge transformation of the trace-reversed metric perturbation $H_{\mu\...
3
votes
0answers
211 views

Non-local gravitational energy tensor

The well-known derivation of the Landau-Lifshitz gravitational energy pseudotensor, relies on several requirements: 1) that it be constructed entirely from the metric tensor 2) that it be index ...
3
votes
0answers
591 views

How is Infinitesimal coordinate transformation related to Lie derivatives?

I am reading the book "Gravitaion and Cosmology" by S. Weinberg. In section 10.9, while discussing Lie derivatives of tensors of different ranks, he makes a general comment: The effect of an ...
3
votes
0answers
188 views

Calculation of Einstein Equation

I have a 3d system with Lagrangian $$e_3^{-1} L_3 = -\frac{1}{2} R_3 + \delta_{ab} \partial_\rho q^a \partial^\rho q^b + \frac{1}{2H} V(q)$$ From this I want to calculate the Einstein equation by ...
3
votes
0answers
914 views

Variation of the purely covariant Riemann tensor

I need to find the variation of the purely covariant Riemann tensor with respect to the metric $g^{\mu \nu}$, i.e. $\delta R_{\rho \sigma \mu \nu}$. I know that, $R_{\rho \sigma \mu \nu} = g_{\rho \...
3
votes
0answers
205 views

Can the two electromagnetic field tensors be combined into a more general tensor?

Given the electromagnetic field tensor $$\begin{align} F_{\mu\nu} = \begin{pmatrix} 0 & -E_{x} & -E_{y} & -E_{z} \\ E_{x} & 0 & B_{z} & -B_{y} \\ E_{y} & -B_{z} & 0 &...
3
votes
0answers
898 views

The connection between classical and quantum spins

I have two questions, which are connected with each other. The first question. In a classical relativistic (SRT) case for one particle can be defined (in a reason of "antisymmetric" nature of ...
3
votes
0answers
210 views

Curvature and spacetime

Suppose that it is given that the Riemann curvature tensor in a special kind of spacetime of dimension $d\geq2$ can be written as $$R_{abcd}=k(x^a)(g_{ac}g_{bd}-g_{ad}g_{bc})$$ where $x^a$ is a ...
3
votes
0answers
535 views

I lost a factor of two in the electromagnetic field tensor

I apologize for this simple question, but I lost a factor of 2 and can't find it anymore, so now I'm looking on the internet, perhaps one of you has some information about its whereabouts. :-) ...
2
votes
1answer
92 views

Are there physical quantities constitute of magnitude, direction and rotation along that direction?

There are scalar quantities(magnitude) and vector quantities(magnitude and direction), but are there fundamental quantities that also depends on how it's oriented/rotated along the direction(magnitude,...
2
votes
0answers
124 views

Action & Energy-Momentum Tensor for Matter Fields

Pg 163 of "Tensors, Relativity and Cosmology" The action integral of a given matter distribution can be written in the form $$I_K=-c\int_\Omega\frac{\rho}{\sqrt{-g}}\frac{ds}{dt}\sqrt{-g} d\Omega ...
2
votes
1answer
69 views

Rewriting a lagrangian in terms of Hodge duals?

Spinors have been found to have some interesting applications in general relativity (such as Wittens positive energy proof). Recently I'd come across a series of papers 1 2 3 (there are many more ...
2
votes
0answers
97 views

Understanding Riemann Curvature Tensor in Misner, Thorne and Wheeler's Gravitation

I'm trying to understand section 11.4 of Misner, Thorne and Wheeler's Gravitation textbook, which explains how the output of the Riemann Curvature Tensor $Riemann(...,A,u,v)$ is a vector describing ...
2
votes
0answers
43 views

Doubt about the history of the tensor object in physics

The word "tensor" (or, maybe the whole physical idea of this object) could be traced date to 1898 on the work on Crystals due to Voigt [1]. It seems that the references [2] and [3], introduce the idea ...
2
votes
1answer
89 views

How to determine if a tensor is covariant or contravariant?

In special relativity, the coordenates of a event are in general written using a 4-vector: $$x^{\mu} = \binom{ct}{\textbf{x}}$$ where $\textbf{x} = (x,y,z)$ are the spacial coordenates. This is a ...
2
votes
0answers
102 views

Transformations of contravariant and covariant tensor operators

I've been able to convince myself that a set of contravariant tensor operators $\hat{O}^{x}$ for $x=1,2,...,n$ respond to a small transformation $\hat{A}$ as, \begin{equation} [\hat{A},\hat{O}^{x}]=-(\...
2
votes
0answers
92 views

Riemann curvature in orthonormal frame and Lorentz transformations

I have problem with understading how Riemann tensor in orthonormal frame transforms using Lorentz transformation of frames. I was reading Morris Thorne paper from 1988 (American Journal of Physics 56, ...
2
votes
0answers
73 views

I have been reading Mathematical methods for physicists (sixth edition) by Arfken and Weber and got stuck in section 2.9 Pseudo Tensors,Dual Tensors

In Mathematical methods for physicists (sixth edition) by Arfken and Weber the triple scalar product was defined as (in page 147-148): For $\vec A, \vec B, \vec C$ with components $A^i, B^j, C^k$ and ...
2
votes
0answers
75 views

Killing Spinor Equation in 4 Dimensions

From https://en.wikipedia.org/wiki/Killing_spinor we see that a Killing Spinor $\epsilon$ is defined as a solution of the following equation: $$\nabla_{X}\epsilon = \lambda X * \epsilon \quad , \quad ...
2
votes
0answers
158 views

Massive Real Vector Field Identity

Let's consider the classical Lagrangian density for a real vector field $V^{\mu}$ of mass $M$: $$L_{V} = -\frac{1}{4} V_{\mu\nu}V^{\mu\nu} + \frac{1}{2} M^{2} V_{\mu}V^{\mu} $$ The Eulero-Lagrange ...
2
votes
0answers
23 views

The lines of force of a point charge near Schwarzschild black hole

I am studying the article written by Hanni and ruffini. I am not so good in the field of general relativity. They have defined the components of Lorentz force in case of Schwarzschild metric as $$L_{\...
2
votes
1answer
80 views

A question from cosmological perturbation theory

We consider the following scalar perturbation on the FRW metric $$ds^2=-(1+2\Phi)dt^2+2a(\partial_iB)dx^idt+a^2[(1-2\Psi)\delta_{ij}+2\partial_{ij}E]dx^idx^j,$$ where $\Phi$, $B$, $\Psi$ and $E$ are ...
2
votes
0answers
158 views

What is the spin of an operator in QFT?

Operators in quantum field theory with $n$ Lorentz indices that are symmetrized and traceless are referred to as spin-$n$ operators. For example, a spin two operator would be \begin{equation} \bar{\...
2
votes
0answers
226 views

Rindler Coordinate Transformation (Cylindrical Coordinates)

How might one go about transforming from cylindrical flat space: $$ds^2=-dt^2+d\rho^2+dz^2+\rho^2d\phi^2,$$ to Rindler space represented by: $$ds^2=-\rho^2dt^2+d\rho^2+dz^2+d\phi^2?$$ The normal ...
2
votes
0answers
134 views

How can I convert an action in terms of differential forms to tensors?

I have an action of a gravitational theory wich is in terms of differential forms. Now, I need to transform this action (including wedge product and exterior derivative of tetrad and metric ) to an ...
2
votes
2answers
278 views

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 ...
2
votes
1answer
61 views

Shape of 3-order tensors in $O_h, O, T_d$ and $D_3$ point gruops

How does one calculate the shape of higher order $(Dimension>2)$ tensors in respect of point group symmetry? I understand that you have to use transformation matrix corresponding to a symmetry ...
2
votes
0answers
186 views

Spatial differentiation of unit vectors

I am reading a book that is related to the dynamics of meteorology and I had a question from that book. The authors have an equation that is the velocity tensor in spherical polar coordinates $$ \...
2
votes
1answer
266 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_{\...
2
votes
0answers
204 views

Vielbein formalism

Maybe this a simple problem, But I couldn't solve it: From Pierre Ramond's book (Field Theory: A Modern Primer, page 213): $$[D_{m},D_{n}]=S_{mn}^{p}D_{p}+\frac{i}{2}R_{mn}^{pq}X_{pq}$$ So in order ...
2
votes
0answers
446 views

Relation between inertia tensor and moment of inertia about an axis

As far as I understood, we define two quantities: Inertia Tensor - a $3\times3$ matrix, which describes the object "mass" of rotation in relation to a certain point, helping us calculating rotations ...
2
votes
0answers
103 views

What is the most general form of Hamiltonian to which MERA ansatz can be applied?

As far as I understand one can only use MERA(Multiscale Entanglement Renormalization Ansatz) to find ground state for Hamiltonians of following form(with possible simplifications due to additional ...
2
votes
0answers
112 views

How to prove this mathematical result about the Green's displacement tensor?

In this paper on integral equations for the scattering of elastic waves, the authors use the following result in their derivation: $$\mathbf{T}:\nabla \mathbf{G}-\nabla \mathbf{u}:\mathbf{\Sigma}=0$$...
2
votes
0answers
371 views

Notation for vectors and covectors

This is probably a very simple question, and I think I know the answer, but I cannot find a place to solidly confirm this. So if I want to write a vector $\mathbf{V}$ in terms of its contravariant (...
2
votes
0answers
57 views

Induced metric is a scalar for transformation from $x\to x'$? (Poisson E.A p.62)

I have a (simple) question about the induced metric $h_{ab}$. In Poisson E.A. (a relativist toolkit) it says in p. 62 that the induced metric $$h_{ab}=g_{{\alpha}{\beta}} \frac{\partial x^{\alpha}}{\...
2
votes
0answers
295 views

Writing Breit-Pauli spin-spin-coupling Hamiltonian as a sum of irreducible spin tensor operators

The spin-spin interaction part of the Breit-Pauli Hamiltonian for two electrons contains the term \begin{equation} 3(\mathbf S_1 \cdot \mathbf r)(\mathbf S_2 \cdot \mathbf r) - r^2 \mathbf S_1 \cdot \...