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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Covariant eigenvalue equation

I was working with a rank-2 tensor $G^{\mu \nu}$ and exploring the effect of covariant differentiation on it, until I found that it satisfied a "covariant eigenvalue equation" of the form: $$\...
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Testing the Lorentz invariance of tensors

Often one encounters statements like, "We know $X$ has to be $Y$ because it is the only Lorentz invariant object that exists." What is the most expeditious way to demonstrate that a tensor object is, ...
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Matrix multiplication and tensorial summation convention

I'm reading this introduction to tensors: https://arxiv.org/abs/math/0403252, specifically rules concerning summation convention (ref. page 13): Rule 1. In correctly written tensorial formulas free ...
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Difference in covariant/contravariant indexation order in Tensors

Warning: You might be wondering why this isn’t in Math Stack Exchange. In fact, it is. I asked the same question there a few days ago but got no answer, and since I think that this question is more ...
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A three rank Levi-Civita tensor in four dimensional spacetime

Is it possible to construct a Lorentz invariant, three rank Levi-Civita tensor in Minkowski Spacetime? If not, why so? I am talking about something like this $\epsilon_{\alpha\beta\gamma}$ or $\...
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The covariant derivative from the contravariant derivative

I know that the following is true: $$V^{\mu}_{~~~~~;\nu} = \frac{\partial V^{\mu}}{\partial x^{\nu}} +\Gamma^{\mu}_{~\sigma\nu} V^{\sigma}.$$ Also, by definition, we have that $V_{\rho} = g_{\rho\nu}...
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Different forms of the covariant derivative

I am working through lecture notes on General Relativity and I am at the section on covariant derivatives. I know that the following is true: $$V_{\mu;\nu} = \frac{\partial V_{\mu}}{\partial x^{\nu}}...
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Is there any example of a physical theory which isn't invariant under translations?

Isn't it trivial that all physical theories in spacetime are invariant under local translations? Is there an example of a theory which isn't invariant under translations? Please, take note that I'm ...
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Does Haag's theorem say covariant transformation of interacting field is not possible?

In https://www.physicsforums.com/threads/haags-theorem-perturbation-existence-and-qft.177865/#post-1384425 #2 post by meopemuk (Eugene) say that Haag's theorem says: $$U(\Lambda)\Phi(x) U^{-1}(\...
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Can cut-off regularisation cause a Poincaré anomaly?

Momentum cut-off regularisation leads to non-covariant results, i.e., it breaks the Poincaré covariance of the theory. Is there any guarantee that Poincaré covariance is always restored when we remove ...
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For a Levi-Civita connection $\nabla$, what does $\nabla^a \nabla_a$ mean? [closed]

Here $\nabla$ is the levi-civita connection of the given metric $g$. I am stuck at the last equality (g). What on earth does $$\nabla^a \nabla_a$$ mean? Isn't it just $$g^{ab}\nabla_a \nabla_b~?$$ But,...
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What is the mathematical formulation of the universality of physics in spacetime?

Consider a general spacetime manifold $\mathcal{M}$ of a given dimension (usually $D = 4$). I call two physical constraints that should be imposed on any reasonable classical theory of physics : ...
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Tensor relations on a manifold

In Hawking and Ellis, "The Large-Scale Structure of Spacetime" the following interesting remark appears: "...the only relations defined by a manifold structure are tensor relations..." Why is the ...
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Christoffel symbol derivation in book by Wald

In chapter 3 of Wald's General Relativity he starts by defining a covariant derivative $\nabla$ as a map on a manifold M from tensor fields $\mathscr{T}(k,l) \to \mathscr{T}(k,l+1)$ plus some required ...
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Difference between contravariant and covariant vector multiplication

I am having trouble in distinguishing the difference between two types of multiplication. Basically if we have $X^{\mu}=(ct,\textbf{x})^{\mu}$, what is the difference between $$\textbf{j}_{\mu}~X^{\...
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Showing that the angular momentum transforms as a vector

I define a vector as any object $(a_i,a_j, a_k)$ such that it transforms the same way as the coordinates themselves. That is if $x'_i = R_{ij}x_j$, then $a'_i = R_{ij}a_j$. Please correct me if this ...
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Showing path integral formalism is Lorentz-invariant without resorting to Hamiltonian formalism

I think people typically say that path integral formalism is manifestly Lorentz-invariant, because Lagrangian density is Lorentz-invariant. However, path formalism is typically defined with time ...
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Covariant Formulation of E&M

Can anybody explain me what does mean the "covariant formulation of electrodynamics"? What does the covariant here mean? Invariance of Maxwell equations under Lorentz Transformations? In what way? ...
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How does effective potential transform under coordinate transformation?

Let us say we have an equation of motion of the following form, $$\ddot{x}=g\tag{1}$$ For this system an effective potential can be defined as, $$\ddot{x}=-\dfrac{d}{dx}U_\text{eff}$$ $$U_\text{...
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Can an equation containing a specific tensor be Lorentz invariant?

Let $A_u$ be a vector field in spacetime. If we restrict to a $2+1$ spacetime, and define the Levi-Civita tensor $\epsilon^{uvp}$ by $\epsilon^{123}=1$, then is the following equation Lorentz ...
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A fundamental question about tensors and vectors [closed]

Studying relativity, I am deeply confused with the fundamental concept of vectors and tensors. Are they some specific "realities" that "exist" independently of coordinates? If so, given a vector $\...
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Action in curved space

I am reading Carroll's book on GR and am confused about the generalization of the action principle to curved space. Please refer to the snippet from the book below. After writing equation 4.47, we ...
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584 views

What is invariance of an equation?

I'm confused. Suppose we have a Schrodinger equation with a time-independent Hamiltonian: \begin{align} i\frac{\partial}{\partial t}\psi(x, t) = H\psi(x, t). \tag{1} \end{align} Under time reversal ...
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A vector that transforms like a four-vector

What is the criterion or what's the meaning when saying that a vector transforms like a four-vector?
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What is the difference between $T^\mu{}_\nu$ and $T_\nu{}^\mu$?

I do understand why the horizontal order matters for indices on the same vertical position, e.g.: $$T\left(V_{(1)},V_{(2)}\right) = T_\color{red}{\mu\nu}V^\mu_{(1)}V^\nu_{(2)} \neq T_\color{red}{\nu\...
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Transformation law for Christoffel's first kind

I don't understand this particular part in this image. I am following schaum's series book on "vector analysis". I didn't find any explanation for it. I also tried searching in Internet and somewhere ...
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Literature advice: Covariant formulation of classical physics [duplicate]

I am looking for a literature advice about the following. I'ld like to review classical physics (basically all undergrad / grad stuff) under the aspects of a modern covariant formulation with exterior ...
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Derivation of the infinitesimal Special Conformal Transformation

This question is somewhat connected to my last question on special conformal transformations. I'm considering the derivation of a special conformal transformation. Namely, the quadratic translation $$...
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Computing the derivatives of a Lagrangian on a Riemannian manifold

Consider a $Riemannian\space n-dimensional\space manifold$ with coordinates {$x^i$} $(i=1,...,n)$. Let the arc length parameter be $t$. So that $\frac{d}{dt}x^i(t)\equiv\dot{x}^i(t)$ is the usual ...
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Why is stress a scalar quantity, even though mathematically, it is (Internal restoring force/Area)?

Basically, what I need to know guys is that when we divide a vector by a scalar, we get a vector. Then what is different in the case of stress?? I mean, WHY IS IT STILL A SCALAR?
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Can Einstein-Hilbert action be derived from symmetry considerations?

The action of a free relativistic classical field theory can be derived from Poincare invariance, locality, and retaining terms quadratic in fields. Is there a similar set of symmetry principles which ...
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Expanding a summation of covariant derivatives

I hope this is not a silly question but I am trying to understand how this part of the equation works: $$ \nabla_{\lambda} \left( \nabla_{\mu}(R_{\nu \lambda}) + \nabla_{\nu}(R_{\mu \lambda}) \right) ...
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51 views

Question about the derivative of a contravariant tensor

I need to show that $\frac{\partial T^i}{\partial x^j}$ is not a tensor. How I proceeded and what I got was: $\dfrac{\partial T^{i'}}{\partial x^{j'}} = \dfrac{\partial}{\partial x^{j'}} \left ( T^i ...
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103 views

Prove that two four-vectors and a four-tensor yield a scalar

I have to do a task for my special relativity class: Prove that: If $\Gamma_{\mu\nu}a^{\mu}b^{\nu}$ is a scalar for arbitrary four-vectors $a^{\nu}$, $b^{\nu}$, then $\Gamma_{\mu\nu}$ is a four-...
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Uncertainty of interpolation when covariance matrix is given

Assume that there is a given covariance matrix of an evaluated quantity ( in my case it's a rection cross section $\sigma = f(E)$, where $\sigma$ is the cross section and $E$ is each energy point ...
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468 views

Question about the the velocity and acceleration in tensor notation

When computing the volicty of a particle moving along a curve parametrized by $Z^i(t)$ for each component i, the components of the velocity $V^i$ are given by $$V^i = (d/dt)Z^i$$ and the components fo ...
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627 views

What is the significance of the conformal invariance of electrodynamics in a covariant formulation?

I am a confused about the role of symmetry transformations in a covariant formulation. Maxwell's equations can be shown to be invariant under conformal transformations. See e.g. here: https://arxiv....
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What does covariance/non-covariance mean in QFT?

I'm studying QFT using the book of Mandl and Shaw. In the first chapter they start by quantising the electromagnetic field, but in a "non-covariant" way. What do they mean by that? They have a ...
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Partial derivatives of Wilson like integrals

I have a one-form field on Euclidean space. Suppose we integrate it over a loop around the specific point $x$. $$I(x)=\int_xU.$$ I want to calculate the partial derivatives of this integral respect to ...
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What does it mean to go from a co-variant vector to a contravariant vector?

In most presentations of general-relativity I see the following statement, We can change from a covariant vector to a contravariant vector by using the metric as follows, ${ A }^{ \mu }={ g }^{ \...
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Usefulness of Curl and Divergence as Multilinear Maps

Early in differential geometry, texts typically reformalize our usual gradient, divergence and curl operators as covariant tensors rather than vectors. This is primarily motivated by the observation ...
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Since all frames of reference are equal, can we treat the Earth as fixed?

Since Einsteins GR tells us all the frames of reference are equal, is there anything invalid about treating the Earth as unmoving and the universe itself rotating? Other than the fact that the ...
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Lorentz force in curved spacetime

I am trying to derive the equation for Lorentz force mentioned in the following Wikipedia article - https://en.wikipedia.org/wiki/Maxwell%27s_equations_in_curved_spacetime viz., $$ \frac{d p_{\alpha}...
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Geometrical picture of change of coordinates in case of Lagrangian

I was reading the part that Euler-Lagrange equation holds even on changing the coordinates. In the book by David Morin, the author talks of geometrical picture of the change of coordinates. He makes ...
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What Lorentz covariance has to do with Lorentz invariance? [duplicate]

Does saying that the Dirac equation is invariant under Lorentz transformations is the same as saying that it is Lorentz covariant?
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Problem in deriving elements of Lorentz Group

I am following a lecture series for Classical Field Theory in which they the lecturer uses the invariance of the length of a vector during a Lorentz transformation to derive the equation $$O^{T}\eta~O=...
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Does the covariant derivative still give divergence in skew coordinates?

I stumbled upon the formula $\,\,div \, \vec{F}=F^{\mu}_{\,\,\,;\mu}$. Does this still hold true in skew coordinates? I can picture it working geometrically in orthogonal coordinates, but in skew ...
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Why is covariant derivative a tensor?

I am trying to prove that the covariant derivative is a tensor (ie it transforms well under a change of coordinates) but I can't succeed to it. Here is the definition of the covariant derivative : $$...
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Why do you have to include the Jacobian for every coordinate system, but the Cartesian?

In physics and engineering it is common to convert between different coordinate systems - spherical, polar, Cartesian, e.t.c. - depending on the problem. Physically, they are all clearly equivalent ...
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General covariance from the equivalence principle

Einstein's equivalence principle (EEP) tells us that there is no way in principle to locally distinguish between inertial acceleration and the effects of a gravitational field by carrying out any non-...

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