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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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Some formula on anti-symmetrization: general formula for anti-symmetrization $A^{i_1 \cdots i_n} X_{[a} \bar{A}_{i_1 \cdots i_n]}$ [closed]

Let $A, \bar{A}$ be the totally-anti-symmetric tensor. Here my convention for $A_{[a_1, \cdots, a_n]}= \frac{1}{n!} (A_{a_1 \cdots a_n} + \cdots)$ I want to find some general formula for \begin{...
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How to use the definition of a rank-$2$ tensor for this kind of examples?

Suppose that, a rank-$2$ tensor transforms as \begin{align} T'^{ij}=\frac{\partial x'^i}{\partial x^k}\frac{\partial x'^k}{\partial x^l}T^{kl}. \end{align} How to use this criterion to investigate if ...
Perfect Fluid's user avatar
3 votes
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Graded cyclic properties in tensor calculus formalism of supergravity

I am trying to understand the chapter 4 of https://arxiv.org/abs/hep-th/0204035. I want to obtain equation 4.19 in this article. First let me summarized some equations we need Denoting the gauge ...
phy_math's user avatar
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Is there an "obvious" reason for why the second derivative of an antisymmetric tensor with respect to coordinates over both of its indices equal to 0?

It was kind of difficult to word the title so I'll restate the question here. My professor took it almost as a given that $$\frac{\partial T^{\mu\nu}}{\partial X^{\mu}\partial X^\nu} = 0$$ If $T^{\mu\...
Copywright's user avatar
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What is the Lie derivative of Ashtekar connection and its conjugate momentum in LQG?

I am using the reference Black hole entropy from an SU(2)-invariant formulation of Type I isolated horizons for this question. I am trying to understand the two equations (30) that give the variation ...
mortimer's user avatar
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Prove that Levi-Civita tensor density is invariant [closed]

I am struggling with Exercise 7.14 ( the latter part ) of the textbook Supergravity ( by Freedman and Proeyen ). Here is the problem: We are defining the Levi-Civita form in coordiante basis as: $$ \...
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How to derive $\partial^{\nu}F^{\mu\alpha} + \partial^{\alpha}F^{\nu\mu} + \partial^{\mu}F^{\alpha\nu}=0$ for the Electromagnetic field tensor? [closed]

The problem says to show that $$\partial_{[\mu}F_{\alpha\nu]}=F^{\mu\alpha, \nu} + F^{\nu\mu,\alpha} + F^{\alpha\nu,\mu}=0$$ stems from Maxwell equations. I haven't been able to find this anywhere on ...
TiredStudent's user avatar
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Do I sum over these indices?

Question 2.11 from A First Course In General Relativity, 3rd Edition by Bernard Schutz, asks the reader to verify the following equation: $$ \Lambda^\nu_\beta(v) \Lambda^\beta_\alpha(-v) = \delta^\...
RudyJD's user avatar
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Polarization tensor of graviton in $d$ dimensions

Take the following tensor, that is the sum over the two polarizations of a gravitational wave in 3 spatial dimensions: $$E_{ijkl}(\vec{k})\equiv\sum_{\lambda = +,\times} \epsilon^\lambda_{ij}(\vec{k})\...
Flavius's user avatar
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Index notation for the generalized form of the velocity gradient with shear and rotation

Good day guys, I was studying fluid dynamics and came across the following equations (shown in the image): When we have both shear and and rotation, they each contribute to the change in velocity ...
RMS's user avatar
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Torsion and Compatibility with the Metric

Compatibility with a metric, also referred to as metricity, means, I believe, that the covariant derivative of the metric is zero: $$g_{ij;k}=g_{ij,k}-\Gamma^m_{ik}g_{mj}-\Gamma^m_{jk}g_{im}=0$$ This ...
Ric's user avatar
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Riemann-Christoffel tensor and Torsion

In Barry Spain's "Tensor Calculus", 1953, page 49, the last equation on the bottom of the page rewritten is $$A_{j;np}-A_{j;pn}=R^k{}_{jnp}A_k.$$ This relates a commutation of the 2nd ...
Ric's user avatar
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Tensor identity in L&L book 2

How does the identity $$\epsilon^{prst}A_{ip}A_{kr}A_{ls}A_{mt}=-A\epsilon_{iklm}$$ with the Levi Civita symbol $\epsilon$ and the determinant A of the matrix $A_{ik}$ follow from the equation $$\...
Takitoli's user avatar
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Variation of nonminimal derivative coupling term

all. Can I request you assistance about the following problem? How do I vary this action with respect to metric $\delta g_{ab}$ $$ \int d^4x \sqrt{-g} \Big[\kappa R+ G_{ab}\nabla^a \phi \nabla^b \phi \...
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Selecting Indices for the Riemann Tensor

How do I know when computing the Riemann Tensor (in two dimensional) which indices to select? Consider the Riemann Tensor $R^a_{bcd}$ how do I know what values to take for $a$? As an example, consider ...
missyclarke1998's user avatar
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Component notation and matrix notation for gradient of vector

I'm trying to understand vector and tensor notation, but I'm coming across some difficulties. Say I have vector $\vec{u}$ and I compute its gradient $\nabla \vec{u}$. Then I get a tensor $\frac{\...
John Vector's user avatar
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Palatini variation of Ricci tensor

I was looking at Problem 2 of chapter 4 of Sean Carroll's General Relativity book, where you were supposed to demonstrate starting from the Einstein-Hilbert action, and assuming that the connection is ...
Andreas Christophilopoulos's user avatar
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Timelike normal vector becomes null

I have a metric given by \begin{equation} ds^2 = \frac{e^{2 A(z)}}{z^2} \left(-g(z) dt^2 + \frac{dz^2}{g(z)} + dx^2 + dx^2_1 + dx^2_2 \right) \end{equation} where $A(z) = -a \ln(b z^2 + 1)$ and $g(z)$ ...
mathemania's user avatar
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1 answer
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Books that approach General Relativity via differential forms, without coordinates [duplicate]

Does someone know about some books about differential geometry applied to General Relativity that are written using the language of differential forms, fiber bundles, & spin connection, and not ...
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Equation of motion for gravity in scalar-tensor theory

I'm trying to derive equation of motion in Higgs scalar-tensor theory with the Lagrangian given by $$\mathcal{L}=[\frac{1}{16\pi}\alpha \phi^{\dagger}\phi R+ \frac{1}{2}\phi^{\dagger}_{;\mu}\phi^{;\mu}...
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Bitensors at three or more space-time points

Bitensors, i.e. tensors at two points that have indices belonging to either of them, have been used in the literature quite a bit and there are many calculations involving them. They are the go-to ...
Skybuilder's user avatar
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Derivation of transformation law for the Hilbert Stress-energy tensor [duplicate]

The Hilbert stress-energy tensor is defined as $$T_{\mu\nu}=-2 \frac{1}{\sqrt{g}}\frac{\delta S_M}{\delta g^{\mu\nu}}.$$ Given the name one expects that it transform as a tensor, but how to prove this ...
Jens Wagemaker's user avatar
2 votes
1 answer
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Confusion about tensors in $SU(3)$

I have some confusion regarding the notion of tensors in $SU(3)$ (or some other matrix Lie group, but let's keep the discussion to $SU(3)$). For concreteness, I will refer to Peskin and Schroeder's ...
Quercus Robur's user avatar
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1 answer
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Covariant derivative for spin-2 field

I have mostly seen the concept of covariant derivative with regard to spin-1 fields. Is it possible to define the covariant derivative for spin-2 fields as well?
physics_2015's user avatar
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Why is the 4-current a tensor rather than a tensor density?

I am trying to understand electromagnetism better in terms of tensors and differential geometry. First recall that (in the Lorenz gauge) the equation of motion for the four-potential $A^\mu$ is $$(-\...
Daniel Grimmer's user avatar
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Exponential of the metric tensor

Exponential of an arbitrary matrix can be written as $$e^A = \displaystyle\sum_{n=0}^\infty \dfrac{A^n}{n!}$$ In Einstein notation, how this expression will look like? In Einstein notation, what ...
SCh's user avatar
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How is this deduced? (Differentiation of tensors)

In Schutz's An Introduction to General Relativity, he talked about how to differentiate tensors. Here is a step that I cannot understand. $$\frac{d\mathbf{T}}{d\tau} = \left( T^{\alpha}_{\beta, \gamma}...
Gene's user avatar
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Tensor densities in 1 dimensional space

When we consider a 1 dimensional manifold, is a scalar density with weight (-1) the same as a covector? In particular, in a theory of gravity, if we consider $\sqrt{-g}$, with $g=\det(g_{\mu \nu})$, ...
Jens Wagemaker's user avatar
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On covariant form of Lorentz equation

The non-relativistic version of Lorentz equation has the form $$m\frac{d\vec{v}}{dt}=q(\vec{E}+\vec{v}\times\vec{B}) $$ Where $\vec{v}, \vec{E}, \vec{B}$ refers to the velocity of charged particle, ...
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Determinant of Rank-2 Tensor using Levi-Civita notation

In my Professor's notes on Special Relativity, the determinant of a rank-two tensor $[T]$ (a $4\times 4$ matrix, basically) is given using the Levi Civita Symbol as: $$T=-\epsilon_{\mu\nu\rho\lambda}T^...
V Govind's user avatar
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1 answer
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Raising and lowering indices of Kronecker delta [duplicate]

I have some confusion about how to raise the indices of the Kronecker delta. To raise and lower indices we use the metric tensor, let's suppose to use the metric (+---). I should have that $$g_{\mu\nu}...
Michael 's user avatar
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Lie derivative: moving boat on a flowing river

Lie derivatives signifies how much a vector (Tensor) changes if flown in the direction of some other vector. I am thinking the typical moving boat on a flowing river problem where the river is flowing ...
spacetime's user avatar
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Multiplying two $SO(3)$ representations

In Group Theory by Zee in Chapter IV.2, he discusses the multiplication of two $SO(3)$ representations on p. 207. Suppose you have a symmetric traceless tensor $S^{ij}$ which furnishes a $5$-...
mathemania's user avatar
2 votes
0 answers
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Measurement unit of a tensor product combinations of quantum states in Pusey-Barrett-Rudolph (PBR) theorem proof

I'm dealing with the Pusey-Barrett-Rudolph (PBR) theorem proof, in which we are using a basis built of linear combinations with tensor products. I'm trying to figure out whether measuring those linear ...
Wojciech Szyszka's user avatar
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2 answers
94 views

Einstein Summation Convention Confusion

My textbook: The second bit confuses me. I asked a question on this site yesterday (Moment of Inertia tensor confusion) which involved the moment of inertia tensor and the term $$r_{i}r_{j}$$ The ...
ED2468's user avatar
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Degrees of freedom in stress-energy tensor

The stress-energy tensor has 16 components, but this question is only about the 9 components $T^{ij}$ with $i,j=1,2,3$. According to Wikipedia, these components are defined as follows: The components ...
Riemann's user avatar
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4 votes
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Maxwell's equations with differential form formalism

I've been reading Sean Carroll's book on GR and I stumbled upon an exercise on EM using $p$-forms. I think I've solved the problem correctly but I am having problems with my answers. I'll provide the ...
user20046481's user avatar
2 votes
1 answer
182 views

How does the covariant vector transformation rule come?

As far as I understand, if a contravariant vector transforms in the form: $$\vec{x}'=A\vec{x}.$$ (Where $A$ is the transformation matrix) Then the covariant vectors shall transform as $$\tilde{w}'=(A^{...
SSsaha's user avatar
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Tensor equation

What is a valid tensor equation. In the book by Bernard Schutz, it is often argued that a valid tensor equation will be frame invariant. So the conclusions reached by relatively easy calculation done ...
Questioningmind's user avatar
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Why are the zero modes of the below operator Killing vectors? (2+1 dimensional gravity)

I'm trying to understand the eigenmodes of the following operator: $$(\Delta_{(1)}^{L L}-\frac{2}{3} R)V_\nu \equiv -\nabla^\mu \nabla_\mu V_\nu+R_{\nu \mu} V^\mu -\frac{2}{3} RV_\nu $$ Where $R_{\mu\...
faker 23's user avatar
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Is $dJ(V,V)=0$? where $J$ is a 1-form?

So is this always 0?( Where $dJ$ is the exterior derivative and $V$ a vectorial field) \begin{align} dJ(V,V)=\partial_jJ_i(dx^j\wedge dx^i)(V,V)=\\ \partial_j J_i (v^kdx^j(\partial_k)v^ldx^i(\...
Guillermo Fuentes Morales's user avatar
1 vote
1 answer
93 views

Is wedge product a tensor or a pseudo tensor?

I'm doing an exercise where $J$ is a 1-form on a manifold $M$ of dimension $N$. The exercise ask me to calculate $J∧(*J)$ with $J=dx^0+2dx^1$ in a minkowski space with metric =(-1,1,1,1) where $*J$ is ...
Guillermo Fuentes Morales's user avatar
2 votes
1 answer
131 views

In general relativity, is gauge invariance the same as coordinate invariance?

I always understood that gauge invariance of general relativity comes from the fact that the physical observables and states are the same regardless of the coordinates we choose to express them in. I ...
Níckolas Alves's user avatar
2 votes
1 answer
99 views

Reducing Tensor-rank by fixing an argument

Assume for example that you are given a (2,0) tensor $T^{\mu\nu}$ and you want to create a vector, i.e., a (1,0) tensor out of it. Is it possible to just fix an index of $T^{\mu\nu}$ while keeping the ...
Burgulence's user avatar
2 votes
0 answers
42 views

Fundamental invariants of a Lorentz tensor

As answered in this question, an antisymmetric tensor on 4D Minkowski space has two Lorentz-invariant degrees of freedom. These are the two scalar combinations of the electromagnetic tensor (as proven ...
Spinoro's user avatar
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Writing the most general form resistivity tensor

I am unsure as to write the resistivity tensor in the most general form in 3 dimensions. Using the following equation. Please can someone explain? $$\vec{E}=\frac{1}{ne}(\vec{j}\times\vec{B})+\frac{m}{...
Lauren.0's user avatar
1 vote
1 answer
91 views

Conformal Transformation of Torsion

It is well known that under a conformal transformation, we have $$\tilde{g}_{\mu \nu}=\Omega^2 g_{\mu \nu}, \; ; \tilde{w}_{\mu}=w_{\mu}-\frac{1}{\alpha} \partial_{\mu} \log(\Omega^2),$$ where $\...
ProphetX's user avatar
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2 votes
1 answer
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Independent Components of the Riemann Curvature Tensor

I am struggling to understand a general method to calculate the independent components of the Riemann Curvature Tensor (RCT). Firstly, as far as I am aware the number of independent components of the ...
Thomas's user avatar
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1 answer
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Showing that derivative of energy-momentum tensor is equal to 0

Given, \begin{equation} T^{\mu\nu} = F^{\mu\lambda} F^\nu{}_{\lambda} - \frac{1}{4} \eta^{\mu\nu} F^{\lambda\sigma} F_{\lambda\sigma}. \end{equation} Here $(T^{\mu\nu})$ is the energy-momentum ...
The Wanderer's user avatar
0 votes
2 answers
90 views

No torsion with calculating the commutator of the covariant derivatives

For simplicity, I only calculated half of the commutator. I didn't leave everything in components because I'm uncomfortable considering (I previously messed up the indices. The following is the ...
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