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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Covariance/contravariance and parallel transport

I am considering following problem. Let $\varphi$ be some scalar, $\eta^{\alpha \beta}$ be the Minkowski metric tensor and $g^{\alpha \beta}$ be the metric tensor of the curved spacetime. Let’s say ...
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Gaussian path integral in QFT in curved spacetime

We have the expression for a Gaussian integral $$I(A,b)=\int d^nx~ \exp\left[-\frac{1}{2}\sum_{n,m}x_n A_{nm}x_m+\sum_{n}x_n b_n\right]=I(A,0)\exp\left[\frac{1}{2}\sum_{n,m}b_n\left(A^{-1}\right)_{nm}...
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Relativistic invariants of a classical field in 4D fashion: why the relation between the components of the current density holds?

I'm trying to understand how is justified the following relation between the first component of the current density integrated over the volume and the scalar product of the 4-vector current density ...
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A basic relation derivation in Feynman parameter method in Peskin and Schroeder's QFT book [duplicate]

I just stuck in a basic relation derivation in Feynman parameter calculation, which lie in Peskin and Schroeder's QFT book, page.191, eq.(6.45) and (6.46) $$\int \frac{d^4 \ell}{(2 \pi)^4} \frac{\ell^\...
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When can we take the action between two fixed times in a relativistic classical field theory?

Peskin and Schroeder give a brief outline of Lagrangian field theory on page fifteen in their Quantum Field Theory book, where they write: Lagrangian Field Theory The fundamental quantity of ...
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How is tensor analysis useful to Relativity? [duplicate]

How does the knowledge of tensor analysis and Differential Geometry help us understand the equations of General and Special Relativity?
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The Relationship between Coordinates and SpaceTime

I was reading a paper describing the contributions integral mathematicians and physicists have made in the advancement of physics by Michael Atiyah (https://www.jstor.org/stable/24111066), but have ...
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If velocity $v^\mu$ is a contravariant tensor of rank 1 then shouldn't force be a mixed tensor of type (1,1)?

The covariant derivative of a (1,0) tensor is a mixed tensor (1,1) but this doesn't seem to be the case. Force is always regarded as a rank 1 tensor. The derivative of velocity is acceleration. I'm ...
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If momentum is a covector, how does $p=mv$?

There are several explanations on this site [1] [2] [3] about why momentum is a covector while velocity is a vector. This distinction is important for the geometric description of classical mechanics. ...
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How to tell if the covariant derivative of something is timelike or spacelike

How can I tell if the covariant derivative of something is timelike or spacelike?
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Valid tensor expression?

The following given tensor expression is:$$A^i=B_i+C_i\tag{1}$$ which is invalid expression since the free index on L.H.S is in upper part and on R.H.S it is in lower part so to make valid tensor ...
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Fokker-Planck equation for the Wigner function to covariance matrix

I cannot understand the derivation in Louis Garbe article (https://arxiv.org/abs/1910.00604) about how to obtain the covariance matrix equation from Fokker-Planck equation for the Wigner function in ...
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Classical electromagnetism field strength with index up and down

In Classical electromagnetism, we know the Lagrangian density read $$ \mathcal{L}=-\frac{1}{4}F_{\mu \nu}F^{\mu \nu} $$ where $$ F_{\mu \nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu} $$ However, I ...
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Riemann curvature tensor

I am little bit confused on Riemann curvature tensor, Riemann curvature tensor written in component form as; $$R^d_{cab}=\partial_a\Gamma^d_{bc}-\partial_b\Gamma^d_{ac}+\Gamma^i_{bc}\Gamma^d_{ai}-\...
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Covariance matrix of Fokker-Planck equation [closed]

Rewrite the Lindblad equation above into a Fokker-Planck equation for the Wigner function is: \begin{equation} \frac{\partial W}{\partial t}(x,p)=-\omega_0p\frac{\partial W}{\partial x} - \omega_0(Xg^...
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Why does the Lagrangian have $O(4)$ symmetry after Wick rotating (previously Lorentz symmetry)?

Pertaining to the answer within link. Why is it the case, that for Lorentz invariant Lagrangian $\mathcal{L}$, after Wick rotation, the $O(4)$ invariance is established, thus manifesting itself as ...
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How to get the contravariant magnetic vector potential from the covariant vector potential?

The vector potential is a divergence-free field. When it is subjected to the operation $\widetilde{F}=\nabla\vec{A}-\left(\nabla\vec{A}\right)^T$, you get the covariant Faraday tensor. The ...
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Tensor symmetry: A symmetric mixed $(1,1)$ tensor is necessarily a multiple of the identity tensor

I found this observation in a book. "It is not possible to have an invariant definition of symmetry in one contravariant and one covariant index". That's all right, my problem is that to ...
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Is it "mathematically wrong" to ignore dual spaces, 1-forms, and covariant/contravariant indices in classical mechanics?

If everything you are working with is in Euclidean 3-space (or $n$-space) equipped with the dot product, is there any reason to bother with distinguishing between 1-forms and vectors? or between ...
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Rigorous Definition of Scalars and Vectors? [closed]

What is the rigorous definition of "Vector" (& " Scalar")? Best I got was: https://www.youtube.com/watch?v=Ncx98PmXbZc
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Covariance of Euler-Lagrange equations under arbitrary change of coordinates

I'm trying to prove that the Euler-Lagrange equation $$\frac{d}{dt}(\frac{\partial L}{\partial \dot{q}_i})-\frac{\partial L}{ \partial q_i}=0$$ is invariant under an arbitrary change of coordinates $$...
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If the scattering amplitudes are Lorentz scalars, why is S-matrix Lorentz covariant?

All observers should agree on the probabilities: $\mathcal{P}(\mathcal{R}_1 \rightarrow \mathcal{R}_2)$ in an inertial frame $\mathcal{O}$ = $\mathcal{P}(\mathcal{R}_1' \rightarrow \mathcal{R}_2')$ ...
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Mathematical characterisation of diffeomorphisms in General Relativity

Considering the diffeomorphism covariance/invariance of General Relativity, is it possible to characterise mathematically the various kinds of possible transformations $x'^{\mu} = f^{\mu}(x)$? All of ...
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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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