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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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Transformation of $g^{\mu\nu}\partial_\mu f \partial_\nu f$

I have the expression $$g^{\mu\nu}\partial_\mu f \partial_\nu f$$ e.g. inside a Lagrange density, where $g$ is a metric tensor and I want to transform this expression to a new set of coordinates. Do ...
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Why does the factor $\sqrt{-g}$ make the volume element invariant?

My question is an extension on this and this question. The question is, how or "in what sense" does the factor $\sqrt{-g}$ make the measure invariant? Suppose, I do not add this factor to the measure....
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Divergence of a tensor

On pg.70 of Dalarsson's "Tensors, Relativity and Cosmology" For a mixed tensor of contravariant order 2 and covariant order 1 $(T^{mn}_{p,m})$, the divergence with respect to m is defined as:$$T^{...
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Covariant, Contravariant & Invariant [closed]

What are the difference between "Covariant" and "Invariant"? What does "contravariant" mean?
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Why do we need invariants to represent real life quantities?

Often it is said that one of the most useful properties of eigenvalues of a matrix is that they are invariant under change of basis. This in turn is said to be useful in physics because real, physical ...
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Indices of the Riemann Tensor of the first kind

When establishing the identity $V^i_{,kl}-V^i_{,lk}=-R^i_{tkl}V^t$ (, denotes covariant differentiation), one of the steps involves raising one of the indices of the Riemann Tensor of the first kind . ...
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Einstein notation: can a free index be upper in one term and lower in another term?

Consider a linear combination of terms written using Einstein notation. Consider one free index in the linear combination: is it necessary that the index is upper in all terms or lower in all terms, ...
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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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Why is the partial derivative a contravariant 4-vector?

The contravariant partial derivative is defined as following: $$\partial ^\mu = \frac{\partial}{\partial x_\mu}$$ where the index $\mu$ runs from 0 to 3. A contravariant vector under Lorentz ...
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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 ...
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Coordinate transformation of basis vectors

The question Let $e_a$ be the coordinate basis vectors in a manifold described by coordinate system $x^a$. The vector displacement between two nearby points is given by \begin{equation} ds=dx^ae_a=dx'...
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Dual space and Metric tensor

So I know that the dual space is the set of all linear transformations that map a vector from a vector space to the field of the space itself (the real number line, complex, quaternions). From YouTube ...
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Do Bianchi identities hold in all coordinates?

I understand by expanding out the Riemann tensor, that the Bianchi identities can be derived within a local inertial frame (LIF) by taking the partial derivatives of the Riemann tensor relations in a ...
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1answer
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Confusion Einstein notation polar coordinates

I'm having issues using Einstein notation in polar coordinates in flat space, I must be missing something basic. Consider the following example. Take the following metric on a 2+1 spacetime; $ds^2 = ...
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General covariance and a running cosmological constant

A running cosmological constant $\Lambda(t)$ can always be included in the perfect fluid source tensor. By the transformation properties of tensors Einstein's field equation is still independent of ...
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3answers
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Origin of $\sqrt{-g}$ in the integral of action $S$

I have a question that might (and probably will) be stupid: I do not understand where does the factor $\sqrt{-g}$ (i.e. $\sqrt{-\det\left(g_{\mu\nu}\right)}$) come from in the action integral S when ...
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Is the addition of a Christoffel symbol and the partial derivative of a vector a tensor?

The partial derivative of a vector $V^\lambda , _\nu$ is not a tensor. Neither is a Christoffel symbol $\Gamma^\lambda _{\mu \nu}$. Is the addition of these two objects a tensor? If they were ...
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Generally Covariant Dirac equation: The spin connection

Wikipedia, an answer on stackexchange and a few papers in the Arxiv I've found all have different definitions of the spin connection found in the Dirac equation. Can anyone please tell me what the ...
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Physical significance of one-form in a velocity field

Still tentatively feeling my way through this stuff, so please go easy. The velocity of a fluid at a point P are the components $V^{a}$ of a contravariant vector:$$v^{x},v^{y},v^{z}\equiv\frac{dx}{dt}...
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Derivation of Covariant derivative for fermionic fields

I've been reading about the Dirac equation in curved spacetime and understand the nature of the verbien, but am wondering what the relationship is between the two definitions of the Fermionic ...
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Re-Writing the Dirac Equation in True Covariant Form

This is a rather brief inquiry, but to get to the point it's always frustrated me that in non-relativistic and relativistic quantum mechanics spin matrices are written as a "vector of matrices" ...
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Covariance of applying a four-force to a stress-energy tensor with forward Euler

I've run into a great deal of confusion on what I expected to be a very simple issue of covariance. I have an equation $$T^{\mu\nu}_{~~~~;\mu} = -G^{\nu}.$$ This is manifestly covariant; so far so ...
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What are some of the reasons for raising/lowering indices of a tensor?

In Dirac's paper: Classical theory of radiating electrons, he decides to raise and lower the indices on the same object multiple times: \begin{align*} \frac{\partial{A_{\mu}}}{\partial{x_{\mu}}} &...
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Transformation of ADM parameters under diffeomorphisms

I am trying to prove the invariance of the ADM formalism under (infinitesimal) diffeomorphisms. I have checked Wald and other textbooks on the subject but have been unable to find expressions for how ...
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2answers
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Covariant Maxwell equations invariant under parity transformation

I tried to proof that the Maxwell equations are invariant under parity transformations. Therefore I used the covariant formulation of the Maxwell equations \begin{align} \partial_{\nu}F^{\nu\mu} &...
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Does string theory violate general covariance?

In a 2007 note on ArXiv, it said: String theory unifies all interaction but provides a perturbative background dependent formulation which violates general covariance. However, another 2012 paper ...
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Help with the proof of the independence of the form of Lagrange equation wrt. choice of coordinates

I am reading about Lagrange-Euler equation here. When they prove that the formula is independent of the choice of coordinate, there is this reasoning, but I could not understand (probably my calculus ...
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Four velocity, acceleration, momentum and force in general relativity

I am getting slightly confused regarding the formal definitions of different four vectors in General Relativity. Many texts on relativity begin with four vectors and dynamics in Minkowski space, and ...
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Varying a scalar field Lagrangian density

I was varying a scalar field density and I look at this term $${\cal L}~=~-\frac{1}{2}\partial _\mu\phi\partial^\mu\phi.$$ The result that I need to come is $$-\frac{1}{2}\delta(\partial _\mu\phi\...
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Doubts on covariant and contravariant vectors and on double tensors

I'm trying to study tensors. Given a coordinates transformation from cartesian to $u_i$ ones: $$ u_1 = u_1 (x,y,z) \qquad u_2 = u_2 (x,y,z) \qquad u_3 = u_3 (x,y,z) $$ I can write a vector $\mathbf{...
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(Lorentz etc) invariant vector fields

(Background: I know some but not much differential geometry, hopefully enough to formulate this post.) I want to ask about what physicists mean when they say scalar, vector, etc. The answer in ...
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“It is the gist of general relativity that it admits, on an equal footing as it were, every possible coordinatization.”

The title is a quote from Hermann Weyl in a 1955 article: Weyl, Hermann. "Why is the world four-dimensional?" In Levels of infinity: Selected writings on mathematics and philosophy. Courier ...
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Derivatives in Poincare' gauge theory

I have been reading the lectures: http://www.damtp.cam.ac.uk/research/gr/members/gibbons/gwgPartIII_Supergravity.pdf about Poincare' gauge theory. The Poincare' group is considered as semidirect ...
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Why aren't Christoffel symbols tensors? - asked from a fibre bundle perspective

I've been reading about connections on fibre bundles recently and it's made me think about the exact nature of the Christoffel symbols in GR. If we have a vector bundle $E$ over $M$ and put a ...
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General coordinate transformations?

Say I have a vector field expressed in Cartesian coordinates: $$\mathbf{A} = \sum_i A_i \mathbf{\hat{e}}_i$$ where the $\hat{\mathbf{e}}_i$ are the generalisation of the unit vectors $\mathbf{\hat i}, ...
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Effect of Co-ordinate Change on Euler-Lagrange Equations for Scalar Fields

Consider a single scalar field $\phi$ on a manifold $\mathcal{M}$. Suppose in $\{x^\mu\}$ co-ordinates, the Lagrangian density is $\mathcal{L}(\phi, \frac{\partial \phi}{\partial x^\mu})$. This means ...
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1answer
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Berry phase covariant derivative

I have been studying some simple examples of the covariant derivative for 2D surfaces and the way that it is constructed is by taking the usual derivative in the 3D Euclidean space at a point $p$ on ...
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Using diagonality in Einstein notation

Given a diagonal matrix $D$, with diagonal elements given by vector $\mathbf{d}$. Representing this in Einstein notation gives $$ D_{ij} = \delta_{ijk} d_k $$ where $$ \delta_{ijk} = \begin{cases}...
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Varying the Einstein-Hilbert action without reference to a chart

In most treatments of General Relativity, when the the Einstein-Hilbert action over some manifold $\mathcal{M}$ (plus Gibbons-Hawking-York term if $\mathcal{M}$ has a boundary), given by $$S=\frac{1}{...
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Question about the true nature of the Spinor mathematical object [closed]

My question is kind of a silly one,but,I really would like to know what truly is a Spinor. I will explain what is my concept of "truly". Throught all the question post, consider finite vector spaces ...
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Different weights for time and spatial derivative in Lagrangian Density

I'm new to QFT and trying to understand the form of the Lagrangian densitys used. As a simple model you often see a Lagrangian density of the form $${\mathcal L} = \frac{1}{2} \partial_j \phi_n \...
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Stuck on Weinberg's quick derivation of Thomas precession

In Weinberg's Gravitation and Cosmology he has a pretty concise derivation of the Thomas precession formula (Eq. 5.1.13). But I don't get the first step... A particle with intrinsic spin is under the ...
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Is $∂_\mu + i e A_\mu$ a “covariant derivative” in the differential geometry sense?

I have heard the expression "$∂_\mu + i e A_\mu$" referred to as a "covariant derivative" in the context of quantum field theory. But in differential geometry, covariant derivatives have an ostensibly ...
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1answer
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Poisson Bracket in General Relativity and tensor weight

I'm a bit confused about the tensor density weight of Poisson brackets in general relativity and their covariance. It's perhaps related to being unclear as to what happens when I integrate a scalar ...
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1answer
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Notation issue for mixed tensors

When I am asked to evaluate, $\mathbf{U^{\alpha}_{~,~~\beta}}$ for all $\alpha$ and $\beta$, what does it mean? I have not able to understand this notation. In case of $\mathbf{g(~~,~\bar{A})}$ I ...
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Why are coordinate systems used in General Relativity if it is a background independent theory?

I am studying topological manifolds as a prerequisite to studying General Relativity and although this question is premature since I have not yet begun the latter it is bothering me. From basic ...
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Gauge transformations and Covariant derivatives commute

I would like to understand the statement "Gauge transformations and Covariant derivatives commute on fields on which the algebra is closed off-shell" which was taken from section 11.2.1 (page 223)...
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What's the Lagrangian density means?

I remember from my classical mechanics class the Lagrangian function, but I don´t understand what's the meanning of the lagrangian density in General Relativity. What's the difference between one and ...
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Intuition behind covariant and contravariant vectors

sorry is there any good intuition behind the following definitions. I am having trouble understanding these. Or is it recommended to just continue reading and accept these definitions for now? Update:...
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Generalization of the Coulomb Force to the Lorentz-Force - Is it “guessing”?

it's me again, and I'm still stuck with the paper Generalization of Coulomb’s law to Maxwell’s equations using special relativity by Kobe, like in my previous question. My problem now lies in ...