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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Contravariance and covariance of vectors

My main source of confusion is the following. Suppose I have a scalar potential $V(x,y,z)$. The electrostatic field for this potential is $ -\vec{E} =\vec{\nabla}V = \frac{\partial{V}}{\partial{x}}\...
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On general covariance

If newton's theory could be formulated in the language of differential geometry (symplectic manifolds), what do we really mean when we say that the theory is covariant under the Galilean group when it'...
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Why does four-momentum have the same transformation matrix as spacetime coordinates?

I will outline my question in 1+1D for brevity. We can passively transform our coordinate system using a Lorentz boost; $\Lambda^{\bar{\nu}}_{\mu}x^{\mu}=x^{\bar{\nu}}$. I've seen that, by stipulating ...
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What does $C^{XY}_{\ell}$ mean when we weasure $a_{\ell m}$ in the sky?

In cosmology context, we have the general formula for the angular power spectrum $C_{\ell}$ : $$C_{\ell}=\left\langle a_{l m}^{2}\right\rangle=\frac{1}{2 \ell+1} \sum_{m=-\ell}^{\ell} a_{\ell m}^{2}=\...
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Show that the contraction of a covector and a vector is Lorentz invariant

I just got Sean Carroll's Spacetime and Geometry: An Introduction to General Relativity a couple of weeks ago, and I have resolved to go through the entire book. In the first chapter, he prompts the ...
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Components of the fully contravariant Kronecker Delta in Schwarzschild Metric

I thought the kronecker delta $\delta^{\mu\nu}$ should always be of value $1$ if both indices are equal and $0$ if they are different. However it seems that the delta has components different from $1$ ...
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Can the metric tensor be treated as a linear transformation?

In general relativity, the metric tensor $g$ is a covariant, second rank, symmetric tensor that can be written down as a 4x4 matrix. The metric tensor generalizes the notion of distance between points ...
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How is it that adding a random field to the partial derivative results in a tensorial operation?

We know that the partial derivative of a tensor is not a tensor. But how is this problem fixed by adding to the partial derivatives, a field of Christoffel symbols? Christoffel symbols are a ...
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Product of Covariant and Contravariant rank 2 Tensors

By the definition, Contravariant tensor transforms like $ (A')^{ij}=\sum_{k,l}{\frac{\partial (x')^i}{\partial x^k}\frac{\partial (x')^j}{\partial x^l}A^{kl}}$ Covariant tensor transforms like $ (A')...
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How to find covariant derivative of a contravariant tensor?

Let I have a contravariant tensor $A^\alpha$, I want to find covariant derivative of the contravariant tensor, From the transformation of the contravariant tensor ($A^\alpha=\partial_\gamma x^\alpha A^...
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How does the $\partial _{\mu} (\frac{\partial L}{\partial [\partial _{\mu} \phi]})$ term expand into a sum?

From QFT Demystified page 31: This term is from the Euler-Lagrange equation of a scalar field. How does this expand into a sum? Do we just sum over all $\mu$ from $\mu =0$ to 3, or are we supposed to ...
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Showing that dirac delta point charge densities is covariant

Consider a point charge $Q$ with a trajectory of $\textbf{s}(t)$ in frame $O$. The densities are: $$\rho(\textbf{x},t) = Q\delta^{(3)}(\textbf{x} - \textbf{s}(t))$$ $$\textbf{J}(\textbf{x},t) = Q\...
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Problems with index position of angular momentum

I want to calculate the poisson bracket of two angular momentum components: $$\{L^i,L^j\}=\frac{\partial\epsilon^{\:i\:m}_{\:\:l}q^lp_m}{\partial q^k} \frac{\partial\epsilon_{\:\:s}^{\:j\:r}q^sp_r}{\...
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Which parameter to use in relativistic Lagrangian mechanics?

According to Wikipedia the formulation of the Relativistic Lagrangian is: $$L = -mc\sqrt{g_{\alpha\beta}\dot{x}^{\alpha}\dot{x}^{\beta}}+L_I(x,\dot{x}).$$ However, I have read that using both ...
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Index manipulation in Lorentz scalars

I have been trying to show that: $ \vec{B}^{2} - \vec{E}^{2} =\frac{1}{2} f^{\mu \nu }f_{\mu \nu}$ where $\vec{B}^{2}$ and $\vec{E}^{2}$ are the square of the magnitude of the magnetic field and ...
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Proof for covariant vector transformation law

(I'm asking this on the physics exchange not on the math one because i don't need an extremely rigorous explanation) I understand the derivation for the contravariant vector transformation law is ...
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Wick's theorem on $C_\ell$ : where does factor $\dfrac{1}{2\ell+1}$ come from?

Just a question that bothers me. This concerns Wick's theorem, which my book gives as: Wick's theorem: If $G = (G_1, \dots, G_n)$ is a centered gaussian multivariate random variable ($\langle G_1 \...
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Steven Weinberg's principle of general covariance

In the book of General Relativity "Gravitation and Cosmology", in section 1 of chapter 4, Steven Weinberg states something that he calls the Principle of General Covariance which claims that:...
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Charge invariance law in GR

In SR in any inertial frame electric charge is invariant, i.e. is independent of the frame. I have seen a claim that it was confirmed experimentally with some accuracy. Is this law true in GR, i.e. in ...
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Relation between rank-2 antisymmetric tensor and other bilinear covariants

Given a spinor $\psi$, if one defines the bilinear covariants $J=\bar{\psi} \psi$, $J_{5}=i \bar{\psi} \gamma_{5} \psi$, the current $J_{\mu}=i \bar{\psi} \gamma_{\mu} \psi$, the axial current $J_{5 \...
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Fisher matrix - add an extra Fisher matrix to another but surely correlations between both matrices of observables

I have a given Fisher matrix "F" used in astrophysics to estimate the errors on cosmological parameters. This matrix is actually the combination of 2 probes (let's notice $A$ and $B$ : $B$ ...
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Given a formula in Minkowski spacetime, how can we transform it so it works in curved spacetime?

To bring a concrete example, let's say I know that the stress-energy tensor related to the electromagnetic field in flat spacetime is $$T_{\mu\nu}=F_{\mu\lambda}F_{\nu}^{\;\lambda} - \frac{1}{4}\eta_{\...
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What is the form of the conformal covariance equation in a 2D Lorentzian CFT?

The naive expectation is that a $2D$ Lorentzian correlator should obey a covariance equation of the form $$A_n(z_{i})=\Bigg(\prod_i (c z_i+d)^{h_i}(\bar{c}\bar{z}_i+\bar{d})^{\bar{h}_i}\Bigg)A_n\Big(\...
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Covariant divergence with Bianchi identity

The covariant derivative of some vector component $A^\lambda $ can be defined as \begin{equation} ∇_\mu A^\lambda = \partial_\mu {A^\lambda}+ \Gamma^\lambda_{\mu \nu} A^\nu. \end{equation} Similarly, ...
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Is there an intuitive reason for why tensors are so ubiquitous in physics?

As a beginner, I'm able to see where different individual tensors come from in physics, but I'm trying to generate some intuition for why this object - defined by a fairly specific transformation law -...
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Details of using flat metric to raise/lower indices in linearized GR. I'm getting first order discrepancies

This question is about the use of the unperturbed (Minkowski) metric $\eta_{\mu\nu}$ (and its inverse $\eta^{\mu\nu}$) to raise and lower indices in linearized gravity. There are already several ...
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Equation for Covariance of 2 Operators

In quantum mechanics, we can find a theoretical variance of operator $A$ with $\langle A^2\rangle-\langle A\rangle ^2$. Is there a similar equation for the covariance of two operators $A$ and $B$? ...
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What kind of equations can use the comma-goes-to-semicolon rule?

I am wondering what kind of equations in Minkowski spacetime can transform into a curved spacetime by using the comma-goes-to-semicolon rule. For example, consider the Schrodinger's equation for a ...
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Special Relativity: are these postulates equivalent?

At the beginning of the developing of special relativity the following principles are assumed true: Principle 1: every physical law is equal in form in every inertial frame. Principle 2: there is a ...
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About general covariance

\begin{equation} u^{\mu}=\frac{d}{d\tau}x^{\mu} \end{equation} \begin{equation} \partial_{\lambda}(u_{\nu} u^{\nu}) = (\partial_{\lambda}u_{\nu}) u^{\nu} + u_{\nu}(\partial_{\lambda}u^{\nu}) = 0 \end{...
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Demonstration of the variance on a $C_{\ell}$ : can't make appear into demonstration a term "$-1$"

Regardings the definition of $C_{\ell}$ on a survey, we measure all the $2 \ell+1$ coefficients. We are thus led to define an estimator of the observed power spectrum $$ \hat{C}_{\ell}=\frac{1}{2 \ell+...
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Some questions on proper acceleration in General Relativity

I’m trying to solve an exercise in which I have to use the definition of proper acceleration, which is: $$ a^{\mu}= u^{\nu}\nabla_{\nu}u^{\mu} $$ In the exercise, I deal with the acceleration along a ...
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Covariant and contravariant for a beginner

I saw that people were representing matrices in two ways. $$\sum_{j=1}^n a_{ij}$$ It is representing a column matrix (vector actually) if we assume $i=1$. $$\begin{bmatrix}a_{11} & a_{12} & ...
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Deriving covariant derivative identitiy of an antisymmetric tensor

If $T^{ab}$ is an antisymmetric tensor, prove: $$T^{ab}_{;b} = \frac{1}{\sqrt{g}}\partial_b(\sqrt{g}T^{ab}).$$ In this example, $g=|\text{det}g_{ab}|$. I already proved in a previous example that $\...
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2 votes
2 answers
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Lowering indices of the coordinate function $x^\mu$

In QFT, we frequently encounter expressions with upper or lower indices. I wonder how can one lower the index of the coordinate function $x^\mu$ in terms of differential geometric language. Let $M$ be ...
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Definition of acceleration and inertial frames of reference

Sorry for my math lacks, I hope you'll be patient even if this question will probably not be clear. How can we define acceleration in special relativity (and in Newtonian mechanics were we apply ...
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Are differential geometric and physics conventions for covariant derivatives consistent?

In a differential geometric setting, the covariant derivative can be defined as a map $\nabla_X:\Gamma(TM)\to\Gamma(TM)$, for any vector field $X\in\Gamma(TM)$, satisfying certain conditions. In other ...
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Why is $\frac{d^2x^{\mu}}{d\lambda^2}=0$ not a tensorial equation?

In flat space, the motion of freely falling particles given by the parametrized path $x^\mu(\lambda)$ is given by the geodesic equation $$\frac{d^2x^{\mu}}{d\lambda^2}=0.$$ Why is this not a tensorial ...
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Covariant Maxwell equations

As we know, the covariant form of Maxwell's equations (there are 2 equations in this formulation) are covariant under Lorentz transformation. Are these equations covariant under general transformation,...
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Covariant form of the electric field

I'm a bit confused about some expressions about the relativity of the electric field. The usual treatment ($c=1$) defines the four potential $A_\mu$ and $F = dA$ is the anti-symmetric 2-tensor. Then $...
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In relation to General Relativity, Invariant form of Lorentz force

I am a High school student, so am new to this topic. Using my own understanding I want to come up with the covariant(or invariant) form of Lorentz force. But I am unable to do it. MY IDEA: Let $\...
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Are there Schwarzschild solutions to EFE with the Landau-Lifschitz Pseudotensor?

I read that solving the Einstein Field Equations can sometimes lead to the problem of non-conservation of energy and that the Landau-Lifschitz Pseudotensor resolves this problem. I can't however find ...
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How does the General Theory Of Relativity achieve the goal of showing that the laws of nature have the same form in all frames of reference?

The General Theory Of Relativity is often presented as both a theory that extends relativity to all frames of reference (or, at least, to inertial frames and uniformly accelerated ones) and, at the ...
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6 votes
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How is Lorentz force frame-independent?

I have studied that the net force on a charged object moving with velocity $v$ under both electric and magnetic fields in given as $\vec{F}=q(\vec{E}+\vec{v}\times \vec{B})$. I have also been told by ...
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Transformation law for the Levi-Civita symbol under a change of basis

I'm trying to prove that the Levi-Civita symbol $\epsilon_{i_1 ... i_n}$ is a tensor density of weight $w=-1$. For this purpose, it has to be shown that the transformation law for the components of ...
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Question on the Einstein-Hilbert action

Does it make sense to write that the Einstein-Hilbert action as \begin{equation} S=\int\mathcal{L}\left(g^{\mu\nu},\partial^{\lambda}g^{\mu\nu}\right)\sqrt{-g}\,\mathrm{d}^4x=\frac{1}{2\kappa}\int R\,\...
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What physical properties are invariant under relativistic transformation?

Most of the familiar physical properties vary according to the relativistic observer's reference frame - speed, mass, energy, time, length, etc. Which properties remain invariant, so everybody will ...
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Autocorrelation and variance: can the fluctuation-dissipation theorem actually be written in terms of fluctuations?

I am considering the theorem in a statistical mechanics context, but I suppose the question could be extended to other fields where it applies as well. If we have a system with property $A$ and apply ...
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Why do the author define energy using one-form? [duplicate]

I was reading the book First course on GR from Schurz. In the latter chapters the author is going to calculate how does the motion of a photon is affected by a spherical symmetric metric. See define ...
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How do we know that solving Euler-Lagrange equation gives us the correct equation of motion in any coordinate?

As far as I understand it: If we defined a quantity called the Lagrangian as the difference between the kinetic energy and potential energy for a particle in one dimension in Cartesian coordinates, ...
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