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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19
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1answer
422 views

The dual of a surface element in 4-space

In reading the classic text, "The Classical Theory of Fields", Third Edition, by Landau and Lifschitz, I found an "obvious" statement not so obvious to me. It is on p.19, the statement of the ...
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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 $ \...
7
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1answer
146 views

Total current for an arbitrary current density

Imagine a localized region $\mathcal{R}$ which contains a current density $\mathbf{j}$, which we take to be divergence-less, $\mathbf{\nabla\cdot j} = 0$. What is the total current associated with ...
5
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1answer
229 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
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613 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 ...
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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
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239 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 ...
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620 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. :-) ...
3
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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. ...
3
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68 views

Some ambiguity regarding irreducible tensors in $SU(3)$

I'm studying the tensor methods in $SU(N)$ and decided to work through Georgi book, namely chapter 10. As far as I can tell, the point is to decompose a tensor to irreducible components; symmetric and ...
3
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26 views

Why do the definitions of ADM-energy, -linear momentum and -mass make sense?

In asymptotically flat spacetimes, the ADM-energy, linear momentum and mass are defined as $$E:= \frac{1}{16\pi}\lim_{r\to\infty} \int_{S^2_r}\sum_{i,j}\partial_ig_{ij}-\partial_jg_{ii}\frac{x^j}{r}\...
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105 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 ...
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182 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 ...
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89 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\...
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111 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 ...
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167 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 ...
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131 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 ...
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635 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
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60 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}}{\...
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252 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
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732 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 ...
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206 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 ...
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1k 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
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213 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 &...
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913 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 ...
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213 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 ...
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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}...
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41 views

Closed superstrings IIA and IIB (7.22) of Clifford Johnson's book

In (7.22) the massless sectors is shown as \begin{equation} \text{IIA}:\; (\bf{8}_\text{v}\oplus\bf{8}_\text{s})\otimes (\bf{8}_\text{v}\oplus\bf{8}_\text{c});\;\; \text{IIB}:\; (\bf{8}_\text{v}...
2
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1answer
45 views

Confusion about supersymmetric Ward identities for $\mathcal{N}=4$ super Yang-Mills theory

I'm trying to understand Eq. 2.6 in this paper. I understand the idea and derivation of the SUSY Ward identity itself and I know how to apply it in the $\mathcal{N}=1$ case. What confuses me here is ...
2
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82 views

The form of QED Vertex Correction

In chapter 6 of Peskin-Schroeder's text Introduction to Quantum Field Theory it is argued that the form of vertex correction for QED can only have the following form (since we have only the constants, ...
2
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1answer
60 views

Understanding time evolution in Von Neumann's pre-measurement (Ozawa model)

I'm studying Quantum Information Theory as a non-physicist and so I'm struggling a bit with simple concepts. Let's say we have a system $\Psi$ composed of two sub-systems: $\Psi = \Gamma \cup \Xi$ ...
2
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75 views

Vanishing of covariant divergence of a superpotential term

I am working out the details in the thesis of Luke Butcher, wherein, on page 64, he claims the following: [...] of “superpotential” terms, those fields whose divergence vanishes ...
2
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446 views

Finding killing vector

I am introducing myself to the topic of killing vectors and therefore, after doing some reading, I try to solve some easy problems. For simplicity, I do my first steps in 2D. First, I chose the 2D-...
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73 views

Link between Tensor Operations and Differential Forms

On Pg 101 of MTW Gravitation, I came across the expression: $$\mathbf{F}(\mathbf{u})=\langle F,u\rangle \tag{1}$$ where $\bf{u}$ is the 4-velocity of the test charged particle, the $\bf{F}$ on the LHS ...
2
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1answer
106 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
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1answer
208 views

Tensors time derivative in moving frames

I know that the following relation exists between the time derivative of a proper vector "v" in an "absolute" frame A and the time derivative of the same vector in a "relatively moving" frame B: $$ \...
2
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1answer
93 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
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1answer
303 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 ...
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44 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 ...
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126 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
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101 views

Momentum in capacitor field; How can an EM field have zero momentum density but non-zero momentum flux?

Consider the case of a simple, stationary parallel plate capacitor oriented with its plates lying in the x-y plane. The E-field is simply given by: $$\vec{E} = \frac{Q}{\epsilon_0A}\hat{z} $$ with ...
2
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95 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
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0answers
166 views

General covariance and Maxwell equations

In General Relativity, I use the principle of general covariance such that \begin{equation} \eta_{\mu\nu}\to g_{\mu\nu}, \quad \partial_\mu\to \nabla_\mu \end{equation} so that I re-express physical ...
2
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0answers
239 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
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1answer
110 views

The foundations of geometric formulation of Newton's axioms

On Professor Frederic P. Schuller's Lecture about General relativity, where you can access it through this link: https://www.youtube.com/watch?v=IBlCu1zgD4Y, he clamed Newton's axioms can be converted ...
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31 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
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1answer
107 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 ...
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0answers
196 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
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227 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?
2
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262 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 ...

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