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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114 views

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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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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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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Williamson decomposition from Schur decomposition

How to obtain the Williamson decomposition from canonical Schur decomposition numerically, for example in Mathematica or other? I need a Mathematica code or a major packages (such as Mathematica, ...
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1answer
85 views

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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Are the covariance matrices of the displacement thermal state and of the thermal state with zero mean the same?

Here,The displacement thermal state is generated by the coupling of the coherent state and thermal noise with $N_B/(1-k)$ photons on average on the BS with a reflectivity of k, i.e $ a_{dth}=\sqrt{k} ...
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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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Co-ordinate invariance in Lagrangian form of equations

I have read that in his Mecanique Analytique [1788], Lagrange sought a “coordinate invariant expression for mass times acceleration”. The discussion regarding this is given in 'Marsden and Ratiu [15, ...
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Has the Klein-Gordon equation in curved spacetimes the same form as in flat ones? [duplicate]

The KG equation in curved geometries has the following form: $$\frac{1}{\sqrt{-g}}\partial_\mu(\sqrt{-g}~g^{\mu \nu}\partial_\nu\phi) + m^2\phi = 0,$$ where $g$ is the determinant of the metric tensor ...
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Is there a difference between base vectors and projection method of calculating coordinates? [duplicate]

I'm trying to grasp the idea of co- and contra-variance. During my study I meet something like that: Let's say we have a vector $\vec{v}$ and want to calculate its coordinates in base $\hat{b}$. I've ...
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Vectors as functions?

In my study of general relativity, I came across tensors. First, I realized that vectors (and covectors and tensors) are objects whose components transform in a certain way (such that the underlying ...
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Change of coordinate vs change of reference axes

Does basis vectors change opposite to coordinate scaling ? For example, suppose I have some oblique coordinate system, and I decide to scale up both 'axes' by a factor of $a$ and $b$ respectively. The ...
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Covariant or contravariant nature of Gradient

I've been having this confusion regarding the gradient being a covariant vector. Intuitively I seem to have understood the concept. However, mathematically, I'm unable to show this, in a single ...
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Geometrical representation of Contravariant and covariant vectors

After cruising through a lot of material online, and answers over here, my understanding of contravariant and covariant vectors are, in a finite-dimensional vector space, suppose we have a vector, ...
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Vector representation in dual space

I'm new to tensor analysis, and came across the topic of vectors and duals, and faced a massive confusion. Are vectors and duals different representations of the same object ? I had another doubt ...
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Covariance of Partial Derivative with Specific Coordinate Transformation

In a context of differentiable manifolds partial derivative are not "covariant" in the sense that if applied to a tensor the result is not a tensor anymore: $\partial^{'}_{\mu}=\frac{\...
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How are these two different definitions of covariant vector related?

Definition 1 If under a coordinate transformation $x^i\to \bar{x}^i(x^i)$ certain objects $A^i$ transform as $$A^i\to \bar{A}^{i}=\sum_{j}\frac{\partial \bar{x}^{i}}{\partial x^j}A^j,$$ those objects ...
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Diffeomorphism invariance in tetrad formalism

How do we show general coordinate invariance of Einstein-Hilbert action in the tetrad formalism or rather the Einstein-Cartan formalism where the frame field and the spin-connection are independent?
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Transform two-mode squeezed state (TMSS) to its covariance matrix

The two-mode squeezed state can be written as: ${\left| \chi \right\rangle _{AB}} = \sqrt {1 - {\chi ^2}} \sum\nolimits_n^\infty {{\chi ^n}} {\left| n \right\rangle _A}{\left| n \right\rangle _B}$ ...
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Are inner product equations invariant everywhere in spacetime?

For example, in Minkowski space, the energy of a massive particle is given by $$E=-P_{\mu}U^{\mu},$$ where the sign depends on the metric convention, $P$ is the particle 4-momentum and $U$ is the 4-...
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Proca Field Hamiltonian density

Having the most general Lagrangian of the Proca Field given by $$\mathcal{L}=C_1(\partial_\nu A_\mu)(\partial^\nu A^\mu)+C_2(\partial_\nu A_\mu)(\partial^\mu A^\nu)+C_3 A_\mu A^\mu$$ the canonical ...
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Converting covariant objects into non-covariant

I need to rewrite expressions of the type $(\partial_\nu A_\mu)(\partial^\mu A^\nu)$ from the "covariant" form, to non-covariant form (so with roman indices). Here the greek indices run from ...
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How to calculate the variance on the ratio of 2 angular power spectra?

In the context of Survey of Dark energy stage IV, I need to evaluate the error on a new observable called "O" which is equal to : $$ O=\left(\frac{C_{\ell, \mathrm{gal}, \mathrm{sp}}^{\prime}...
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What is meaning of the phrase "The laws of physics are the same in all inertial frames of reference."? [duplicate]

One of the postulates special relativity is that "The laws of physics are the same in all inertial frames of reference." What is meaning of this statement? If it is talking about physical ...
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How to compute the covariance error term in cosmology context?

Below the error on photometric galaxy clustering under the form of covariance : $$ \Delta C_{i j}^{A B}(\ell)=\sqrt{\frac{2}{(2 \ell+1) f_{\mathrm{sky}} \Delta \ell}}\left[C_{i j}^{A B}(\ell)+N_{i j}^{...
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Transformations in General Theory of Relativity

In our lecture we dicussed that certain elements in GTR transform as certain other things, i.e. Christoffel symbol transforms as an affine connection covariant derivative of a contravariant vector ...
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1answer
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Why do basis vectors transform covariantly?

I have some trouble understanding transformation rules of basis vectors. My question/goal is to obtain a mathematical derivation to see why basis vectors transform covariantly and other vectors (...
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1answer
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Curve is geodesic iff $\nabla_k g(V,V)$ vanishes

Let $V$ be a Killing vector field and let $s \longmapsto x^i(s)$ be a curve such that $$\dot{x} \enspace \equiv \enspace \frac{dx^i}{dx}(s) \enspace = \enspace V^i\big(x(s)\big)$$ Show that $x^i(s)$ ...
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Solving for $\Gamma^c_{ab}$ Christoffel symbols from the metric $g_{ab}$

I am trying to compute the Christoffel symbols $\Gamma^c_{ab}$ to have a metric-compatible covariant derivative $\nabla_a g_{bc}=0$. I worked out $$\nabla_a g_{bd}=\partial_a g_{bd}-\Gamma^{c}_{ab}Y_{...
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Contravariant tensor definition must be incorrect?

Both my textbooks (Schaums Tensor calculus and Neuenschwander's Tensor calc for physics) define a contravariant tensor of order one (which they also state is synonymous with a simple vector) as ...
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Lagrange's equation is form invariant under ANY coordinate transformation. Hamilton's equations are not under ANY phase space transformation. Why?

When we make an arbitrary invertible, differentiable coordinate transformation $$s_i=s_i(q_1,q_2,...q_n,t),\forall i,$$ the Lagrange's equation in terms of old coordinates $$\frac{d}{dt}\left(\frac{\...
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Must the action be a coordinate scalar?

I know that an action must be locally-Lorentz invariant based on physical reasons, but is there any requirement for it to be a coordinate pseudo-scalar (up to surface terms)? In particular, would an ...
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Torsion tensor and affine connection symbols

I have read tons of questions about this topic but I think my particular issue is not solved. If so, please let me know. So I want to prove that the torsion tensor $\mathcal{T}$ actually transforms ...
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Definition of an equation to be Lorentz invariant

What is the precise mathematical definition of an equation to be Lorentz invariant? Is it the same as being invariant under the maps $x \mapsto \Lambda x$, with $\Lambda$ being a given Lorentz ...
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Transformation rules for quantities

When we formulate transformation laws for vectors and tensors, the transformation rule for $x^\mu$ is calculated via arguments from total derivatives considering $x^\mu=x^\mu(x^{'\nu})$ that in turn ...
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Invariance of Euler-Lagrange equation [duplicate]

How exactly does the variational approach make it clear that the Euler-Lagrange equations of motion will have exactly the same form no matter what coordinate system we use as long as it is done with ...
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Clarification on showing covariance of density expressions under Lorentz boost

For a collection of point charges, the charge density is defined as \begin{equation} \rho(\textbf{x},t) = \sum_{k=1}^{N} q_k \delta^{(3)}\left[\textbf{x}-\textbf{x}_k(t)\right] \end{equation} while ...
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Why does $k_\mu k_\nu \to k^2 \eta_{\mu\nu}/d$ work in QFT calculations?

When doing calculations of Feynman diagrams in QFT I've seen a trick used that goes something like this $$\int \frac{d^dk}{(2\pi)^d} \frac{k_\mu k_\nu}{f(k^2)}\quad\longrightarrow\quad\int \frac{d^dk}{...
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Understanding tensor transformations

I am trying to learn how tensors transform under coordinate transformations. For an example, under a transformation from the coordinate system $x^\mu \longrightarrow x'^\mu$ a covariant tensor is ...
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306 views

Covariant derivative of a one-form

I understood that the covariant derivative of a vector field is $$ \nabla_{i}v^j=\frac{\partial v^j}{\partial u^{i}}+\Gamma^j_{~ik}v^k $$ Then why is the covariant derivative of a covector field $$ \...
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How does Equivalence Principle imply a Curved Space-Time?

I am a bit confused as to how the Equivalence Principle implies a curved spacetime. Or if it doesn’t imply a curved spacetime, then what exactly makes it necessary to have a curved space time? I could ...
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Advise on fitting spatio-temporal covariance in a likelihood framework based on Gaussian Random Field Method

I'm working on a spatial-temporal model as described in https://www.sciencedirect.com/science/article/pii/S1090780714002134. I have modified this model such that I have an event at time ...
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1answer
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Electromagnetic waves in tensor notation

I'm trying to derive the wave equations for the electric and magnetic fields in covariant (tensor) formulation. Starting with Gauss-Ampere law, $$ \partial_\alpha F^{\alpha\beta}=\frac{4\pi}{c}J^\beta ...

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