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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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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127 views

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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3answers
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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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Scalar, pseudoscalar, vector, axial vector [closed]

What is the difference between scalar, pseudoscalar, vector and axial vector particles or interactions? Are they all genuine things or are some of them theoretical? These words, called “bilinear ...
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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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99 views

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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1answer
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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
66 views

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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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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1answer
71 views

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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1answer
76 views

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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37 views

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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81 views

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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25 views

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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24 views

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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1answer
48 views

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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1answer
80 views

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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2answers
193 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
49 views

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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1answer
95 views

How does the Lagrangian transform when coordinates are changed?

I'll talk of single particle Lagrangian in $n$ dimensions. Suppose in a given coordinate system with the coordinates $(q_i)_{i=1}^n$, the Lagrangian is given by $L(\mathbb{q, \dot q}, t)$. Suppose I ...
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1answer
77 views

Understanding tensor and covariance

I'm really struggling to understand the use of tensors when we want to have a covariant equation. From what I understand, if we write an equation using tensors only, then the physics behind it will be ...
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Confusion about covariance of physical laws and tensors

In classical mechanics physical laws must be covariant in inertial frames i.e. mathematically the laws will be such that two inertial observers are indistinguishable, for example the dependence of ...
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1answer
164 views

$ \mathbf{e}^i = g^{ij} \mathbf{e}_j$ interpretation

I've problems in the interpretation of the expression: $$ \mathbf{e}^i = g^{ij} \mathbf{e}_j$$ that can be found, by example, in this wiki chapter. Also here. Step by step of my erroneous logic: The ...
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2answers
131 views

Contravariant/covariant vectors or components?

Related: Covariant vs contravariant vectors Note: by vector I try to refer to the physics entity, not to the list of components. From wikipedia page and others: $$ \mathbf{v} = q_i \mathbf{e^i} = q^i \...
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1answer
57 views

Tensor transformation Formula Proof

Ok so basically I am trying to prove that the following expression: Can be written using matrices like this: Any suggestions on how to approach this?
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Which postulate of GR requires that $ds^2=g_{\mu\nu}(x)dx^\mu dx^\nu$ must be invariant? [duplicate]

The postulate that the velocity of light in a vacuum is the same in all inertial frames, can be exploited to establish that the spacetime interval between two events $x^\mu=(ct,{\bf r})$ and $x^\mu+dx^...
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58 views

Understanding where all the equations from a covariant derivative of a tensor come from

Suppose I have a situation where i know that $\nabla_i T^{ik}=0$ where $ T^{ik}$ is a tensor of rank 2, which is diagonal, such as the perfect fluid energy-momentum tensor. We are dealing in a ...
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63 views

Can one quantize Chern-Simons theory in the covariant phase space formalism?

The covariant phase space, which coincides with the space of solutions to the equations of motion, gives a notion of phase space which does not rely on a decomposition of spacetime of the form $M=\...
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Symmetries in Relativistic mechanics and Field Theory and Lorentz Invariance

In Non-Relativistic Lagrangian mechanics usually we didn't impose any constraint on the Action/Lagrangian, rather than to be respectively a functional/function (or 3 form on Manifolds). In QFT and ...
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101 views

Understanding the four-dimensional volume form in Action of Lagrangian

Into the following part below, I don't understand what is precisely a "four-dimensional volume form" implied in the integral below: For comparison, the ...
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45 views

Dirac delta and covariance [duplicate]

Is there a covariant form of the Dirac delta function? And how to build a covariant form of an identity that contains Dirac delta? To be more precise, what I am looking for is Some distribution that ...
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1answer
95 views

General covariance of the Maxwell equations in 4-tensor form

Are the Maxwell equations written with the derivatives of the EM field strength tensor not generally covariant? I can't seem to prove that is. The Maxwell equations in 4-tensor form: $\partial_{\mu}F_{...
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How did the Lagrangian and Hamiltonian theories of motion inspire the idea that forces should be treated as one-forms instead of vectors?

On page-5 of this paper1 by E. Minguzzi titled "A geometrical introduction to screw theory", he writes: Who adopts this point of view argues that it should also be adopted for forces in ...
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39 views

Rosen coordinates

I would like to express this spacetime interval for a spacetime with a monochromatic electromagnetic wave for which I believe $|E|=\frac{q}{\omega}\sin(u)$ expressed in Rosen coordinates: $ds^2=-2dudv+...
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1answer
162 views

Raised index of partial derivative

I am having a really hard time wrapping my head around component notation for tensor fields. For example, I do not know exactly what the following expression means $$\partial_\mu\partial^\nu \phi, \...

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