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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37
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8answers
4k views

Is it foolish to distinguish between covariant and contravariant vectors?

A vector space is a set whose elements satisfy certain axioms. Now there are physical entities that satisfy these properties, which may not be arrows. A co-ordinate transformation is linear map from a ...
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2answers
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Galilean invariance of the Schrodinger equation

Is the Schrodinger equation invariant under Galilean transformations? I am only asking this question so that I can write an answer myself with the content found here: http://en.wikipedia.org/wiki/...
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1answer
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Invariance of action $\Rightarrow$ covariance of field equations?

Invariance of action $\Rightarrow$ covariance of field equations? Is this statement true? I have only seen examples of this, like the invariance of Electromagnetic action under Lorentz ...
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4answers
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Definitions and usage of Covariant, Form-invariant & Invariant?

Just wondering about the definitions and usage of these three terms. To my understanding so far, "covariant" and "form-invariant" are used when referring to physical laws, and these words are ...
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3answers
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How to understand the definition of vector and tensor?

Physics texts like to define vector as something that transform like a vector and tensor as something that transform like a tensor, which is different from the definition in math books. I am having ...
3
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1answer
837 views

Transpose of (1,1) tensor

When we transpose a (1,1) tensor, shall we simply switch the two indices while keeping their upper/lower positions or switch them and also switch their upper/lower positions? In general, would the ...
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4answers
760 views

Difference between matrix representations of tensors and $\delta^{i}_{j}$ and $\delta_{ij}$?

My question basically is, is Kronecker delta $\delta_{ij}$ or $\delta^{i}_{j}$. Many tensor calculus books (including the one which I use) state it to be the latter, whereas I have also read many ...
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2answers
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Why is the “canonical momentum” for the Dirac equation not defined in terms of the “gauge covariant derivative”?

The canonical momentum is always used to add an EM field to the Schrödinger/Pauli/Dirac equations. Why does one not use the gauge covariant derivative? As far as I can see, the difference is a factor <...
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7answers
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Why should a (physical) principle be applicable to different systems in different positions in space and time?

This is a question with a philosophical, as well as physical, flavor. Why should a physical principle (or a description of one), be applicable to different systems that can be in different positions ...
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3answers
2k views

Covariant and contravariant vectors

Reading Weinberg's Gravitation and Cosmology, I came across the sentence (p.115, above equation (4.11.8)) The partial derivative operator $\partial/\partial x^\mu$ is a covariant vector, or in ...
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3answers
638 views

Why does the analogy between electromagnetism and general relativity differ if you consider them as gauge theories or fiber bundles?

Electromagnetism and general relativity can both be thought of as gauge theories, in which case there is a natural analogy between them: (Strictly speaking, the gauge symmetry of diffeomorphism ...
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2answers
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Invariance, covariance and symmetry

Though often heard, often read, often felt being overused, I wonder what are the precise definitions of invariance and covariance. Could you please give me an example from quantum field theory?
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2answers
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Understanding the difference between co- and contra-variant vectors

I am looking at the 4-vector treatment of special relativity, but I have had no formal training in Tensor algebra and thus am having difficulty understanding some of the concepts which appear. One ...
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2answers
3k views

What does it mean to transform as a scalar or vector?

I'm working through an introductory electrodynamics text (Griffiths), and I encountered a pair of questions asking me to show that: the divergence transforms as a scalar under rotations the ...
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3answers
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Is time a Scalar or a Vector?

In Wikipedia it's said that time is a scalar quantity. But its hard to understand that how? As stated that we consider only the magnitude of time then its a scalar. But on basis of time we define ...
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2answers
429 views

Under what representation do the Christoffel symbols transform?

I often read the statement, that the Christoffel symbols aren't tensors. But then, under which representation do they transform?
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2answers
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Is there an accepted axiomatic approach to general relativity?

I am reading Steven Weinberg's book Gravitation and Cosmology. He makes a big deal out of the equivalence principle and showed a bunch of deductions you can make based on it. This surprised me since ...
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2answers
274 views

Is there any way to justify or derive the form of the Lorentz force from relativity theory?

Lorentz force is in this form: $$\vec{F}=q[\vec{E}+\vec{u}\times\vec{B}]$$ As we know, it is Lorentz-invariant. Is there any way to justify or derive its form from relativity theory?
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1answer
873 views

How to check if a tensor transform a tensor?

Suppose $A^{\mu_1 \cdots \mu_n}_{\nu_{n+1}\cdots \nu_m}$ is a tensor. That means it transforms a tensor. How do I show that it transforms as a tensor? How do I see that $\cos (A^{\mu_1 \cdots \mu_n}_{\...
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4answers
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Covariant derivative for spinor fields

scalars (spin-0) derivatives is expressed as: $$\nabla_{i} \phi = \frac{\partial \phi}{ \partial x_{i}}.$$ vector (spin-1) derivatives are expressed as: $$\nabla_{i} V^{k} = \frac{\partial V^{k}}{ \...
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4answers
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Is partial derivative a vector or dual vector?

The textbook(Introduction to the Classical Theory of Particles and Fields, by Boris Kosyakov) defines a hypersurface by $$F(x)~=~c,$$ where $F\in C^\infty[\mathbb M_4,\mathbb R]$. Differentiating ...
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1answer
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Is spacetime symmetry a gauge symmetry?

In previous questions of mine here and here it was established that Special Relativity, as a special case of General Relativity, can be considered as the theory of a (smooth) Lorentz manifold $(M,g)$ ...
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1answer
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Levi Civita covariance and contravariance

I read some older posts about this question, but I don't know if I'm getting it. I'm working with a Lagrangian involving some Levi Civita symbols, and when I calculate a term containing $\epsilon^{ijk}...
3
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2answers
409 views

Momentum vector transformation

I am confused about the way momentum vector transforms in the following case: $$q_k \to q_k'= q_k + \epsilon f_k(q)$$ The Jacobian is thus $\Lambda_{ij} = \frac{\partial q'_i}{\partial q_j} \approx \...
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0answers
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composition of space expansion and movement as a gauge invariance

suppose i have a space-time where we have one point-like object* which we will call movement space probe or $\mathbf{M}_{A}$ for short, and it will be moving with constant velocity $V^A_{\mu}$ in ...
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5answers
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Why do we need coordinate-free descriptions?

I was reading a book on differential geometry in which it said that a problem early physicists such as Einstein faced was coordinates and they realized that physics does not obey man's coordinate ...
8
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2answers
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Coordinate Transformation of Scalar Fields in QFT

By definition scalar fields are independent of coordinate system, thus I would expect a scalar field $\psi [x]$ would not change under the transformation $x^\mu \to x^\mu + \epsilon^\mu $. Correct? ...
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5answers
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What problems with Electromagnetism led Einstein to the Special Theory of Relativity?

I have often heard it said that several problems in the theory of electromagnetism as described by Maxwell's equations led Einstein to his theory of Special Relativity. What exactly were these ...
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5answers
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Gradient is covariant or contravariant?

I read somewhere people write gradient in covariant form because of their proposes. I think gradient expanded in covariant basis $i$, $j$, $k$, so by invariance nature of vectors, component of ...
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5answers
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Gradient, divergence and curl with covariant derivatives

I am trying to do exercise 3.2 of Sean Carroll's Spacetime and geometry. I have to calculate the formulas for the gradient, the divergence and the curl of a vector field using covariant derivatives. ...
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4answers
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Inconsistency with partial derivatives as basis vectors?

I have been trying to convince myself that it is consistent to replace basis vectors $\hat{e}_\mu$ with partial derivatives $\partial_\mu$. After some thought, I came to the conclusion that the basis ...
7
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1answer
254 views

Is the distinction between covariant and contravariant objects purely for the convenience of mathematical manipulation?

Two kinds of indices, covariant and contravariant, are introduced in special relativity. This, as far as I understand, is solely for mathematical luxury, i.e. write expressions in a concise, self-...
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7answers
539 views

How can a set of components fail to make up a vector?

Many books in Physics insist to define vectors are objects with components with the property that the components transform in a proper way under a change of coordinates. Now, in mathematics, on the ...
7
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1answer
232 views

Canonical second quantization vs canonical quantization with multisymplectic form in AQFT

First of all, I'm a mathematician that knows less than the basics of QFT, so forgive me if this question is trivial. Please, keep in my mind that my background in physics is very poor. 1) The usual ...
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4answers
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Are there controversies surrounding the principle of general covariance in GR?

I'm a physics graduate now working with computers. I study GR in my spare time to keep the material fresh. In the Wikipedia article about the mathematics of GR, one can read the following: The term ...
7
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3answers
831 views

What is “a general covariant formulation of newtonian mechanics”?

I am a little confused: I read that there are general covariant formulations of Newtonian mechanics (e.g. here). I always thought: 1) A theory is covariant with respect to a group of transformations ...
7
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4answers
525 views

Is there a fundamental reason not to define the work vice-versa

My question arises from something which has never been really clear: in continuum mechanics, why is strain energy defined as: $$W=\int_\Omega \underline{\underline{\sigma}}:\mathrm{d}\underline{\...
5
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1answer
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Covariant derivative and Leibniz rule

I read the Wikipedia page about the covariant derivative, my main problem is in this part: http://en.wikipedia.org/wiki/Covariant_derivative#Coordinate_description Some of the formulas seem to lead ...
3
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2answers
783 views

Why are totally antisymmetric tensors more useful than totally symmetric tensors?

In an arbitrary number of dimensions, one can naturally define two tensors, Kronecker delta and Levi-Civita epsilon tensor. However, why isn't it advantageous to define some totally symmetric tensor ...
3
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2answers
279 views

Global symmetries of spacetime and general covariance

I am self learning GR. This is a rather long post but I needed to clarify few things about the effect of general coordinate transformations on the global symmetries of metric. Any comments, insights ...
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6answers
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Why define four-vectors to be quantities that transform only like the position vector transforms?

A four-vector is defined to be a four component quantity $A^\nu$ which transforms under a Lorentz transformation as $A^{\mu'} = L_\nu^{\mu'} A^\nu$, where $L_\nu^{\mu'}$ is the Lorentz transformation ...
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4answers
975 views

How did “no prior geometry” father 50 years of confusion?

I've come across this quote attributed to Misner, Thorne & Wheeler from their book, Gravitation: Mathematics was not sufficiently refined in 1917 to cleave apart the demands for "no prior ...
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1answer
384 views

Covariant derivative of a Dirac spinor and Kosmann lift

In [1] I have found a definition of the covariant derivative of a Dirac field with a general connection $\omega_{\mu a}{}^{b}$ (with torsion and non-metricity) [see eq. (29)]: $$\nabla_{\mu}\psi=\...
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3answers
243 views

Integral in different coordinate systems

In Griffiths' electrodynamics book, he uses the equation, $$\nabla^2\mathbf{A}=-\mu_0 \mathbf{J},$$ to state that $$\mathbf{A}(\mathbf{r}) = \frac{\mu_0}{4\pi}\int\frac{\mathbf{J}(\mathbf{r}')}{|\...
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0answers
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Is classical electromagnetism conformally invariant? (and a bit of general covariance)

The contest is a flat $4d$ Minkowsky space. A conformal transformation is a diffeomorphism $\tilde x(x)$ such that the metric transforms as \begin{equation*} \tilde g_{\tilde \mu \tilde \nu} = w^2(x) ...
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2answers
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Independent components in a 4-vector representing massless fields

In Ryder Page141, it is written "the electromagnetic field, like any massless field, possesses only two independent components, but is covariantly described by a 4-vector $A_{\mu}$". Why are there ...
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1answer
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Does Haag's theorem say covariant transformation of interacting field is not possible?

In https://www.physicsforums.com/threads/haags-theorem-perturbation-existence-and-qft.177865/#post-1384425 #2 post by meopemuk (Eugene) say that Haag's theorem says: $$U(\Lambda)\Phi(x) U^{-1}(\...
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1answer
985 views

What is a covariant derivative in gauge theory?

I've been studying electroweak theory and you need to keep the Lagrangian covariant by introducing covariant derivatives. What is a covariant derivative? And what does it mean to keep the Lagrangian ...
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3answers
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Does it make sense to ask how the covariant derivative act on the partial derivative $\nabla_\mu ( \partial_\sigma)$? If so, what is the answer?

I want to find out how the covariant derivative acts on terms containing a partial derivative, e.g. $ \nabla_\mu(k^\sigma\partial_\sigma l_\nu)$. But I don't know how to evaluate the terms of the form ...
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1answer
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How does a vector field transform under an infinitesimal coordinate transformation?

If I have a vector $X^{\mu}(x)$, and then I consider an infinitesimal coordinate transformation of the form $x^{\mu} \to x^{\mu} + v^{\mu}(x)$, then how does my vector $X^{\mu}(x)$ transform? From ...