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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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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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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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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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 ...
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Coordinate transformation in Tensor Calculus

I am doing a problem from Schutz, Introduction to general relativity.The question asks you to find a coordinate transformation to a local inertial frame from a weak field newtonian metric tensor $$ds^...
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Is $\partial_\alpha f^\alpha$ coordinate-independent?

At this point in Schuller's 9th lecture on GR, he claims that Poisson's equation for the Newtonian gravitational field strength is $$-\partial_\alpha f^\alpha=4\pi G \rho,$$ where $\alpha=1,2,3$. But ...
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Definition and visualization of a covector

Covector: A linear map from some set of vectors into real numbers. Also, on its own, it is an element of a vector space. Visualisation: visualize a covector as a stack of hypersurfaces of some ...
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How is Einstein's postulate about the invariance of the laws of physics justified? [duplicate]

According to one of Einstein's postulates related to special relativity, > "the laws of physics remain invariant in their form and nature in all inertial frames". But global inertial frames don't ...
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Questions about deriving Maxwell equation in the Newman-Penrose formalism

I had some questions while reading the Chandrasekhar textbook "The Mathematical Theory of Black Holes", in particular about the scalars introduced to reformulate the Maxwell equations ($g^{ik} F_{ij;k}...
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Gamma matrices in curved spacetime

How to raise and lower indices of gamma matrix in curved spacetime? Do we raise and lower the index of gamma matrix with $ g_{\mu \nu} $?
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covariant and contravariant form of a matrix

I'm following a paper to solve this equation: $y_{j}=y_{o}$ + A$\eta^{T}$ (Eq. 2) My question is about the term $\eta^{T}$. In the paper says: "With symbol $\eta$, we denoted a 1 × 6 ...
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Covariant version of the Coulomb gauge

In curved spacetime, it is possible to define the covariant version of the Lorenz gauge, going from $\partial_\mu A^\mu =0$ to $\nabla _\mu A^\mu =0$ in some curved spacetime $g_{\mu \nu}$. What is ...
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General Covariance, what does Einstein mean?

I have read the papers by Einstein and I am convinced I understand what he means completely. Given there are controversies, maybe I over understood it: It is I am convinced, can be said in two ...
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Covariant eigenvalue equation

I was working with a rank-2 tensor $G^{\mu \nu}$ and exploring the effect of covariant differentiation on it, until I found that it satisfied a "covariant eigenvalue equation" of the form: $$\...
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Testing the Lorentz invariance of tensors

Often one encounters statements like, "We know $X$ has to be $Y$ because it is the only Lorentz invariant object that exists." What is the most expeditious way to demonstrate that a tensor object is, ...
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Matrix multiplication and tensorial summation convention

I'm reading this introduction to tensors: https://arxiv.org/abs/math/0403252, specifically rules concerning summation convention (ref. page 13): Rule 1. In correctly written tensorial formulas free ...
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Difference in covariant/contravariant indexation order in Tensors

Warning: You might be wondering why this isn’t in Math Stack Exchange. In fact, it is. I asked the same question there a few days ago but got no answer, and since I think that this question is more ...
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A three rank Levi-Civita tensor in four dimensional spacetime

Is it possible to construct a Lorentz invariant, three rank Levi-Civita tensor in Minkowski Spacetime? If not, why so? I am talking about something like this $\epsilon_{\alpha\beta\gamma}$ or $\...
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The covariant derivative from the contravariant derivative

I know that the following is true: $$V^{\mu}_{~~~~~;\nu} = \frac{\partial V^{\mu}}{\partial x^{\nu}} +\Gamma^{\mu}_{~\sigma\nu} V^{\sigma}.$$ Also, by definition, we have that $V_{\rho} = g_{\rho\nu}...
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Different forms of the covariant derivative

I am working through lecture notes on General Relativity and I am at the section on covariant derivatives. I know that the following is true: $$V_{\mu;\nu} = \frac{\partial V_{\mu}}{\partial x^{\nu}}...
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Is there any example of a physical theory which isn't invariant under translations?

Isn't it trivial that all physical theories in spacetime are invariant under local translations? Is there an example of a theory which isn't invariant under translations? Please, take note that I'm ...
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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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Can cut-off regularisation cause a Poincaré anomaly?

Momentum cut-off regularisation leads to non-covariant results, i.e., it breaks the Poincaré covariance of the theory. Is there any guarantee that Poincaré covariance is always restored when we remove ...
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For a Levi-Civita connection $\nabla$, what does $\nabla^a \nabla_a$ mean? [closed]

Here $\nabla$ is the levi-civita connection of the given metric $g$. I am stuck at the last equality (g). What on earth does $$\nabla^a \nabla_a$$ mean? Isn't it just $$g^{ab}\nabla_a \nabla_b~?$$ But,...
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What is the mathematical formulation of the universality of physics in spacetime?

Consider a general spacetime manifold $\mathcal{M}$ of a given dimension (usually $D = 4$). I call two physical constraints that should be imposed on any reasonable classical theory of physics : ...
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Tensor relations on a manifold

In Hawking and Ellis, "The Large-Scale Structure of Spacetime" the following interesting remark appears: "...the only relations defined by a manifold structure are tensor relations..." Why is the ...
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Christoffel symbol derivation in book by Wald

In chapter 3 of Wald's General Relativity he starts by defining a covariant derivative $\nabla$ as a map on a manifold M from tensor fields $\mathscr{T}(k,l) \to \mathscr{T}(k,l+1)$ plus some required ...
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436 views

Difference between contravariant and covariant vector multiplication

I am having trouble in distinguishing the difference between two types of multiplication. Basically if we have $X^{\mu}=(ct,\textbf{x})^{\mu}$, what is the difference between $$\textbf{j}_{\mu}~X^{\...
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Showing that the angular momentum transforms as a vector

I define a vector as any object $(a_i,a_j, a_k)$ such that it transforms the same way as the coordinates themselves. That is if $x'_i = R_{ij}x_j$, then $a'_i = R_{ij}a_j$. Please correct me if this ...
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Showing path integral formalism is Lorentz-invariant without resorting to Hamiltonian formalism

I think people typically say that path integral formalism is manifestly Lorentz-invariant, because Lagrangian density is Lorentz-invariant. However, path formalism is typically defined with time ...
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Covariant Formulation of E&M

Can anybody explain me what does mean the "covariant formulation of electrodynamics"? What does the covariant here mean? Invariance of Maxwell equations under Lorentz Transformations? In what way? ...
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How does effective potential transform under coordinate transformation?

Let us say we have an equation of motion of the following form, $$\ddot{x}=g\tag{1}$$ For this system an effective potential can be defined as, $$\ddot{x}=-\dfrac{d}{dx}U_\text{eff}$$ $$U_\text{...
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Can an equation containing a specific tensor be Lorentz invariant?

Let $A_u$ be a vector field in spacetime. If we restrict to a $2+1$ spacetime, and define the Levi-Civita tensor $\epsilon^{uvp}$ by $\epsilon^{123}=1$, then is the following equation Lorentz ...
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A fundamental question about tensors and vectors [closed]

Studying relativity, I am deeply confused with the fundamental concept of vectors and tensors. Are they some specific "realities" that "exist" independently of coordinates? If so, given a vector $\...
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Action in curved space

I am reading Carroll's book on GR and am confused about the generalization of the action principle to curved space. Please refer to the snippet from the book below. After writing equation 4.47, we ...
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What is invariance of an equation?

I'm confused. Suppose we have a Schrodinger equation with a time-independent Hamiltonian: \begin{align} i\frac{\partial}{\partial t}\psi(x, t) = H\psi(x, t). \tag{1} \end{align} Under time reversal ...