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Questions tagged [covariance]

How a quantity behaves under a change of basis vectors. This tag covers relativistic covariance, as well as contravariant and covariant tensors not necessarily in the context of relativity. DO NOT USE THIS TAG for statistical covariance.

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Challenging Cauchy's Stress Tensor: Objectivity and Generalization of Divergence Theorem

I'm investigating the limitations of the Cauchy stress tensor model in classical continuum mechanics, specifically focusing on its compliance with the principle of material frame indifference (MFI) ...
Foad's user avatar
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A covariant derivative computation in General Relativity [duplicate]

I am trying to compute $\nabla^\mu\nabla^\nu R_{\mu\nu}$. I proceed as follows: \begin{align} \nabla^\mu\nabla^\nu R_{\mu\nu}&=g^{\mu\rho}g^{\nu\lambda}\nabla_\rho\nabla_\lambda R_{\mu\nu} \\ &...
vyali's user avatar
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How does the bulk-to-boundary propagator transform under diffeomorphisms?

In AdS/CFT, the bulk-to-bulk propagator can be obtained as the limit of the bulk-to-bulk propagator with one point approaching the boundary. For example in the scalar case \begin{equation} K_{\Delta}(...
SouthernLion'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, ...
paul230_x's user avatar
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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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When determining the dual basis do you use the rotation matrix to rotate the basis vectors by 90 degrees?

I'm trying to teach myself general relativity, and I came across the subject of covariant and contravariant components of a vector. Suppose your basis is $e_1, e_2$, and they are separated by 45°. ...
lee pappas's user avatar
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The Role of the Kaehler Manifold in Supergravity

Actually, I already asked a similar question Coupling of supergravity to matter which has remained unanswered. So this time I will be less general. In the very interesting paper arXiv:2212.10044 [...
Frederic Thomas's user avatar
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1 answer
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Why does $S$-matrix theory end up being a covariant formalism when it is not obvious that it is?

A principle of QFT that is frequently invoked, repeated, and potentially subject to rigorous verification is that the theory in question must exhibit Lorentz covariance and be invariant under the ...
Davius's user avatar
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How to find the stability of time dependent Lyapunov equation?

After linearization of the nonlinear equations, I want to find the covariance matrix $v$ through the numerical solution of time dependent Lyapunov equation, $$dv/dt=a*v + v*a'+ d,$$ where $a$ is my ...
Ani's user avatar
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How can a vector field in $E^3$ be represented by a linear combination of only 2 basis vectors?

In Chapter I.7 of "Einstein Gravity in a Nutshell", Zee introduces the concept of covariant derivatives. I am confused by the first line in this section (see below) as it appears that we can ...
Reuven's user avatar
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2 answers
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Lorentz covariant but Lorentz inconsistent $4$-forces?

We often say that the benefit of relativistic index notation is that we can write down equations of motion that are automatically Lorentz covariant. However, I'm starting to feel that there are many ...
user196574's user avatar
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Deriving divergence in cylindrical coordinates, using covariant derivatives

Covariant derivatives are normally used to write equations covariantly in curved spaces. But in an exercise, I need to use covariant derivatives to derive Gauss' law: $\nabla \cdot \vec{E} = 4\pi\rho$ ...
Nikolaj's user avatar
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Counting independent components of Lorentz tensor

Say I have Lorentz tensors $A^{\mu\nu}$ and say this Lorentz tensor is symmetric under $\mu \Leftrightarrow \nu$ and there are only $p^\mu$ and $q^\mu$ as the physical Lorentz vectors involved. If so, ...
Quantization's user avatar
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Covariant basis vectors and scale factors in toroidal coordinates [closed]

The toroidal coordinate system $(\xi, \omega, \varphi)$ is defined in terms of the cylindrical coordinate system $(R, \phi, Z)$ as: $$ R = \frac{{R_0' \sinh \xi}}{{\cosh \xi - \cos \omega}}, \\ Z = \...
rsbb2125's user avatar
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Is the invariance of the Lagrangian under some transformation equivalent to the covariance of the motion equation? [duplicate]

Take the Lagrangian $L=\frac{1}{2}m{{\left( \frac{{\rm{d}}}{{\rm{d}}t}x \right)}^{2}}-\frac{1}{2}k{{x}^{2}}$, for example. The equation of motion of this system should be given by $m\frac{{{{\rm{d}}}^{...
aitzolander's user avatar
11 votes
7 answers
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Definition of four-velocity: why define it with proper time of the object?

The four-velocity(world-velocty) is defined by : $u^μ=\frac{dx^μ}{dτ}$ ,where $τ$ is the proper time of the object. I don't understand why it's defined with respect to the proper time but not the time ...
user381761's user avatar
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1 answer
158 views

How do we know the tensorial form of the Maxwell equations manifestly transform as tensors?

In Sean Carroll's book he derives the two tensorial Maxwell equations from the four non-tensorial equations. I noticed that one of these equations is the Bianchi identity for the electromagnetic ...
FlamePrinz's user avatar
6 votes
2 answers
175 views

Can non-inertial/fictitious forces be understood as covariant derivatives?

From classical mechanics and general relativity, we know that the natural motion of a particle in general curvilinear coordinates, assuming the Levi-Civita connection, is given by the geodesic ...
Craig's user avatar
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Derivation of covariant derivative by means of parallel transport

I've studied covariant derivative in many courses so far, but I got stuck on the definition given by the teacher notes of the exam of Topological QFT. I think that he improperly used the name "...
polology's user avatar
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Implicit assumption behind the definition of scalar, vector, and tensor fields

Let me consider a field \begin{align} A^\mu(x) \equiv dx^\mu, \end{align} which seems to be a vector field trivially. However, to check that, we calculate as \begin{align} A'^\mu(x') \equiv dx'^\mu = \...
Keyflux's user avatar
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Background field Method in non-linear $\sigma$-model: Covariantized Taylor series of geodesic between fields

In this paper, the authors try to develop an expansion of the non-linear $\sigma$-model action with covariant terms called the Background field method. We have a "fixed" background field $\...
Генивалдо's user avatar
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Del operator confusion [closed]

The very first thing my textbook says is that the Del operator is defined as: $$\vec{\nabla}=\vec{a}^i\nabla_i$$ Where $\nabla_i$ is the covariant derivative and " $\vec{a}^i$ is the curvilinear ...
Krum Kutsarov's user avatar
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1 answer
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What is the intuition or the derivation of covariant derivative?

I asked this question in mathematics but the answer I got was a bit too abstract for me so I hope that my fellow physicists can give me more of an intuition or an easier explaination of my question. ...
Krum Kutsarov's user avatar
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1 answer
192 views

Formulation of the Bianchi identity in EM

I'm trying to understand, as a self learner, the covariant formulation of Electromagnetism. In particular I've been stuck for a while on the Bianchi identity. As I've come to understand, when we ...
Luke__'s user avatar
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Tramsformation of spatial components of the 4-force

I'm trying to learn special relativity by myself. I've been reading the Griffith's chapter about relativistic dynamics and electrodynamics (chapter 12), but one thing it's not clear to me. I've been ...
Luke__'s user avatar
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0 votes
1 answer
125 views

Are the stress and strain tensor covariant or contravariant?

My question is related to this question but I don't find the answer there to be completely satisfactory. The displacement of an elastic medium is a contravariant quantity, which I think is pretty ...
Daniel Shapero's user avatar
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0 answers
68 views

How to derive the form of transformation operators in Einstein notation?

I've been reading through MWT to try and drill home some of the fundamentals a little more. I've gotten to their derivation of the form of Lorentz Transformation in Einstein notation and how they act ...
akozi's user avatar
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1 answer
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Is the Lie derivative in a coordinate direction covariant?

Considering a partial derivative of a vector field $w^a$ in x-direction (also called here 1-direction) I can write it as $$\frac{ \partial w^a}{\partial x^1 } = \partial_1 w^a - \Gamma^a_{1c} w^c + \...
Frederic Thomas's user avatar
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1 answer
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To which representation of the Lorentz group do the objects of the Duffin-Kemmer-Petiau (DKP) equation belong?

The DKP equation is, allegedly, a relativistic spin-0 or spin-1 equation. In the spin-0 case the defining algebra is a 5 dimensional representation. For spin-1 the four matrices are 10 dimensional, in ...
Craig's user avatar
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1 vote
0 answers
101 views

Under what representation of the Lorentz group do scalar $\textit{fields}$ transform?

I know that if I am sitting in a spacetime $M$ at point $p$, vectors live in the tangent space $T_pM$, and tensors in the tensor product space etc. If I want to consider general tensor fields, I ...
Craig's user avatar
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1 vote
0 answers
56 views

A curve in contravariant coordinate notation

I'm reading a general relativity text, the metric is expected to describe generally the spacetime in a gravitational field. In it there is for ex. the notation $$\sum_j g_{ij} \frac{dx^j}{ds}$$ for ...
hope12sqthou's user avatar
1 vote
2 answers
148 views

Can ideal dipoles be associated to a covariant four-current?

I am trying to check if the classical electromagnetic sources from a point electric/magnetic dipole do form a true four-current. In this SE post, it is shown that a point electric charge do transform ...
Woe's user avatar
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Contravariant or covariant tensor in electromagnetism?

I have a question about the following 2 tensors: the permittivity tensor and Maxwell's stress tensor. I was wondering if someone can explain which one is contravariant or covariant, and show why that ...
photonica's user avatar
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1 answer
91 views

Covariant and partial derivative of a vector field (not component)

Is the covariant derivative of a vector field (not the components of a vector) same as the partial derivative? I am adding a screenshot from page 69 from General Relativity: An introduction for ...
Nayeem1's user avatar
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2 votes
2 answers
135 views

Diffeomorphism invariance for derivative of scalar fields [closed]

In GR, it is well-known that the gravitational stress-energy tensor is a pseudotensor, i.e. it is not gauge-invariant. To make it gauge-invariant one needs to take it under average integral $\langle \...
gravitone123's user avatar
0 votes
1 answer
325 views

What is the difference between covariant and contravariant tensors? [duplicate]

What is the difference between covariant and contravariant tensors? I have been seeing in a lot of problems but I´m not sure what is the difference or if is only a equivalent notation.
JOSE ARTURO NORIEGA PALACIOS's user avatar
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0 answers
90 views

Divergence and Covariant/Contravariant Transformations

I am trying to understand the covariant/contravariant representation of the divergence in different coordinate systems. Normally, we would get in the holonomic basis the following divergence according ...
Kubrik's user avatar
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1 vote
1 answer
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Invariants from the covariant derivatives of a scalar field

I am reading Theoretical minimum: Special Relativity and Classical Field Theory where you construct a Lagrangian for the field by the argument that it would be invariant under the Lorentz ...
Ajaykrishnan R's user avatar
1 vote
1 answer
182 views

Valid Tensor equation

Let us have the equation as; $A_{{\mu}{\nu}}$$B_{\mu}$=$C_{{\mu}{\nu}}$ , $\mu$ and $\nu$ are free indices. Is the above equation a valid tensor equation? If not then what correction should be made to ...
Keshav shrestha's user avatar
0 votes
1 answer
116 views

Differentiating the index notation

I am always confused with the algebra of differentiating the index notation, and have browsed many other posts but still confused. There must be details I have been missing. It would be really ...
user174967's user avatar
1 vote
1 answer
150 views

Did Feynman use the equivalence principle to compute spacetime curvature correctly?

Chapter 42 of the Feynman Lectures on Physics claims that a clock "higher up" in a gravitational field will tick more slowly, and that is used to argue that space-time in a constant ...
Just Me's user avatar
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1 answer
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Quantum Angular Momentum Covariance

The angular momentum is quantified in Quantum Mechanics, it can only take multiples of $\hbar$ https://en.wikipedia.org/wiki/Angular_momentum_operator#Quantization The previous statement seems to ...
Sergio Prats's user avatar
2 votes
0 answers
22 views

Dimensional regularisation and Wick theorem [duplicate]

Consider an integral: $$I^{ij}=\int\frac{d^d\textbf{p}}{(2\pi)^d}p^i p^j f(\textbf{p}^2).$$ How can we show that this is equal to: $$I^{ij}=\frac{\delta^{ij}}{d}\int\frac{d^d\textbf{p}}{(2\pi)^d}\...
MZperX's user avatar
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1 vote
3 answers
227 views

Maxwell's equations on curved space time from principle of equivalence

I want to derive the inhomogeneous Maxwell's equations on curved space time from the principle of equivalence. I assume that for a specific space-time point there is a coordinate system $\xi^\alpha$ ...
jojo123456's user avatar
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1 answer
182 views

Principle of general covariance follows from principle of equivalence?

I am reading the book gravitation from Weinberg. On page 93 he states the principle of covariance. I attach a screenshot below. I think Weinberg is saying (below the enumeration) that the principle of ...
jojo123456's user avatar
0 votes
1 answer
87 views

Problem with understanding contravariant component transform

I am reading Susskinds book General Relativity: The theoretical minimum and I got a bit stuck on the transformation rule of contravariant components. The book defines the components of a vector $(V^{’}...
Adam Hultquist's user avatar
2 votes
1 answer
57 views

Calculations with co- and contravariant formalism in QFT

i have another question regarding calculations with the co- and contravariant formalism in QFT. It is not that i don't understand all of this, but most of the time i'm missing some "middle" ...
Marcel Moczarski's user avatar
1 vote
1 answer
157 views

Christoffel symbol and partial derivative of the metric

It may seem like a dumb question but I'm trying to solve a problem involving coordinate transformations on the Christoffel symbol and to solve it they do the product rule $$\partial_\alpha g_{\beta ' \...
Samuel Jaramillo's user avatar
9 votes
1 answer
513 views

Inverse of the covariant derivative

Given the covariant derivative of some tensor, for the sake of this example a covariant vector: $$\nabla_\mu A_\nu$$ Is there a well-defined inverse operation on the covariant derivative such that it ...
Tachyon's user avatar
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0 answers
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Can either the covariant or the contravariant version of a physical tensor be more fundamental?

This question may be too subjective, but here goes: Essentially any physically interesting quantity can be represented by a tensor on an inner product space or by a tensor field on a pseudo-Reimannian ...
tparker's user avatar
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