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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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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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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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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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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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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2 answers
173 views

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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3 answers
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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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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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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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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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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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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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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 EVERY coordinate transformation. Hamilton's equations are not under EVERY 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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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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1 vote
2 answers
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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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Why is the covariant derivative of a one-form $\nabla_{i}v_j=\frac{\partial v_j}{\partial u^{i}}-\Gamma^k_{~ij}v_k$?

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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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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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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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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