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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519 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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332 views

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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238 views

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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317 views

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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2answers
280 views

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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185 views

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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94 views

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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353 views

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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4answers
997 views

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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580 views

Covariant derivative of the variation of the metric

Say you are varying some function (it could be a tensor, or a tensor density, scalar, etc.) that depends on the metric (the metric is what is varying), so you have $\delta f = \frac{\partial f}{\...
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681 views

Co and contravariant: tensors or components?

I am learning Special Relativity and have a question: given a four vector $\vec{x}$ whose contravariant components are $x^\mu$, do the covariant components $x_\mu = g_{\mu\nu}x^\nu$ make reference to ...
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361 views

Covariant derivative of unit vector/Kronecker delta

Consider: $\nabla_b \delta^a_c=\partial_b\delta^a_c-\Gamma^d_{bc}\delta^a_d+\Gamma^a_{bd}\delta^d_c=0-\Gamma^a_{bc}+\Gamma^a_{bc}=0$ But if I define: $T^a_{bc}=\nabla_b\delta^a_c$ Then I should have:...
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675 views

Covariant derivative of a Dirac spinor and Kosmann lift

In [1] I have found a definition of the covariant derivative of a Dirac field with a general connection $\omega_{\mu a}{}^{b}$ (with torsion and non-metricity) [see eq. (29)]: $$\nabla_{\mu}\psi=\...
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Lagrangian transformation [closed]

I want to prove that the Lagrangian transformation is covariant for $x_{i}\rightarrow q_{i}(x)$ and $x_{i}\rightarrow q_{i}(x,t)$. So far I've proven that it holds for the first transformation as ...
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631 views

Why are Maxwell's equations not Galilean invariant? [closed]

So i am writing an essay on the conflict between galilean invarience and maxwell's electromagnetism. I am struggling to come up with 3 evidences that they conflict because I have a mediocre ...
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1answer
482 views

Why do we raise and lower indices of tensors of various groups with the invariants of that group?

If $T_{ij}$ is tensor that transforms under $SO(N)$ then apparently (according to what I have been told) it does not matter whether we put the indices up or down. If we instead have a tensor that ...
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A postulate in the beginning of special relativity

There’s a postulate in special relativity as following: Physics laws are identical in all inertial reference frames. I’m a math student, recently when I reviewed special relativity before learning ...
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Symmetry under Lorentz transformation: precise definition

I am studying QFT but I need to fill some gaps in my comprehension of special relativity (I didn't study it very well and I know I still misunderstand things in S.R). In my book it is written: " A ...
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Tensor in different coordinate system

I have the tensors $F_{\mu\nu}$, $F^{\mu\nu}$ in coordinate system $(t,x,y,z)$ and want to transform these to coordinate system $(t',x',y',z')$ just by multiplicating matrices. My idea was to ...
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State amplitude and field operator covariance in QFT

I'm studying QFT on Bogoliubov-Shirkov's "Introduction to the theory of quantized fields" (3d edition). In $§9.3$ they discuss transformation properties of quantum states and operators in QFT. Given ...
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Question about the physical meaning of tensors: why are they used in physics? [closed]

I know that tensors are object we use in general relativity to describe phenomenon. They have the property to have the same expression in various coordinates systems. For example if I take : $\...
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Commutator relation of EM Field Covariant?

I read that for quantization of the EM-Field, you demand the canonical equal-time commutation relations: $$[A^\mu(\vec{x},t), \pi^\nu(\vec{y},t)] = i \hbar g^{\mu \nu} \delta^3(\vec{x} - \vec{y}). $$ ...
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Why is the inner product not invariant under general coordinate transformations?

This came up in some of my reading (Introduction to Tensor Calculus by Kees Dullemond & Kasper Peeters, page 15). Why is the inner product not invariant under general coordinate transformations? ...
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658 views

Why is there an emphasis on tensor equations in GR?

In my understanding the purpose of using tensor equations in GR is to ensure that they are true in all coordinate systems. I understand that writing equations tensorially ensures this will be the case;...
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387 views

Raising and lowering covariant and contravariant bases

The vector $\textbf{a} = a_{i}\textbf{e}^{i}$ in terms of covariant components. In terms of contravariant components, $\textbf{a} = a^{i}\textbf{e}_{i} = a^{j}\textbf{e}_j$. Thus, $a_{i}\textbf{e}^{i} ...
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1answer
260 views

What is the commutator of a Poisson bracket and the covariant derivative?

Consider a classical vector field $V^\mu$ on a curved background. We make a 3+1 split of coordinates into $t,x^i$, where $x^i$ are coordinates on spatial hypersurfaces $\Sigma$ and $t$ the parameter ...
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192 views

Confusion regarding the $\partial_{\mu}$ operator

I've been confused about the $\partial_{\mu}$ operator. Peskin and Schroeder defines it as $\partial_{\mu} = \frac{\partial}{\partial x^{\mu}}$ For example, the Euler Lagrange equation of motion is ...
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Do contravariant and covariant partial derivatives commute in GR?

I'm considering something like this: $\partial_{\mu}\partial^{\nu}A$ . I feel like we should be able to commute the derivatives so: $\partial_{\mu}\partial^{\nu}A = \partial^{\nu}\partial_{\mu}A$. ...
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4answers
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Double covariant derivative of tensor

Consider the covariant derivative of a type $(0,2)$ tensor given in terms of the connection: $$ h_{ab;c} \equiv \partial_c h_{ab} - \Gamma^d_{ca} h_{db} - \Gamma^d_{cb} h_{ad} $$ What would the term $...
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Is Electromagnetism Generally Covariant?

I'm sure there's a good explanation for the issues leading to my question so please read on: Classically, we can represent Electromagnetism using tensorial quantities such as the Faraday tensor $F^{\...
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4answers
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Which is the difference between the delta tensor $\delta_{a}^{b}$ and the metric tensor $g_{ab}$?

I don't understand which is the difference between the delta of Kronecker $\delta_{a}^{b}$ and the metric tensor $g_{ab}$. They looks to have the same effect when raising and lowering the indices, but ...
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1answer
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PDG review says “The flavour of a given neutrino is Lorentz invariant.”

This the starting paragraph of section 14.1 PDG review (PDF) asserts: The flavour of a given neutrino is Lorentz invariant. What does this really mean? A neutrino of a given flavour $\alpha$, i.e.,...
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1answer
130 views

Are vector expressions and vector operator expressions independent of coordinates

We encounter expressions for vectors and tensors in Euclidean space, such as $$\vec{F}=\vec{A}+\nabla\phi,$$ or $$\vec{H} = \nabla\vec{u}\cdot\vec{n}+\nabla\times(\nabla\times\vec{B}) + \frac{\...
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SUSY chiral covariant derivatives under change of coordinates

Reading Martin's SUSY Primer, section 4.4 on Chiral Superfields, he makes the statement that the SUSY chiral covariant derivatives $$D_\alpha=\dfrac{\partial}{\partial\theta^\alpha}-i(\sigma^\mu\...
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Why does the analogy between electromagnetism and general relativity differ if you consider them as gauge theories or fiber bundles?

Electromagnetism and general relativity can both be thought of as gauge theories, in which case there is a natural analogy between them: (Strictly speaking, the gauge symmetry of diffeomorphism ...
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Inverting Operators, and Propagators on Curved Spacetime

I am a bit confused about inverting operators, and calculating propagators on a curved spacetime. Consider the following example: If I have a Lagrangian for a charged scalar field on a curved ...
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3answers
533 views

Does it make sense to ask how the covariant derivative act on the partial derivative $\nabla_\mu ( \partial_\sigma)$? If so, what is the answer?

I want to find out how the covariant derivative acts on terms containing a partial derivative, e.g. $ \nabla_\mu(k^\sigma\partial_\sigma l_\nu)$. But I don't know how to evaluate the terms of the form ...
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397 views

Is potential energy a scalar operator?

If a scalar operator $\hat{S}$ is defined as an operator that is invariant under rotations, i.e $$U^\dagger S U = S,\,\,\,\,\,\,\, U=e^{-i\theta\hat{\mathbf{J}}\cdot{\mathbf{n}}}$$ which is ...
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Inconsistency with partial derivatives as basis vectors?

I have been trying to convince myself that it is consistent to replace basis vectors $\hat{e}_\mu$ with partial derivatives $\partial_\mu$. After some thought, I came to the conclusion that the basis ...
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1answer
912 views

Transformation of generalized coordinates

One of the advantages of Lagrangian formulation is that the equations of motion have the same form regardless of the choice of generalized coordinates. Suppose that a system has $s$ degrees of freedom,...
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How is the expression for the Stress-energy tensor in Cosmology a covariant expression?

Consider the energy-momentum tensor $$T_{\mu\nu}=(p+\rho)u_\mu u_\nu+pg_{\mu\nu}$$ used in Cosmology. I have a problem with this equation. Since this a tensor equation the RHS should transform in the ...
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888 views

Deriving Maxwell Equations in their covariant form

Mawell Equations, in a particular unit system, are: \begin{eqnarray} \nabla \cdot \vec{E} &=& \rho &(1)\\ \nabla \times \vec{B} &=& \frac{\partial \vec{E}}{\partial t} + \vec{J}&...
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$\Delta x^{\alpha}\equiv x_{2}^{\alpha}-x_{1}^{\alpha}$, the difference between two points, is not a vector

I want to show $\Delta x^{\alpha}\equiv x_{2}^{\alpha}-x_{1}^{\alpha}$, the difference between two points, is not a vector. By definition, if $\Delta x^{\alpha}\equiv x_{2}^{\alpha}-x_{1}^{\alpha}$ ...
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1answer
628 views

In flat spacetime, what is the mixed (invariant?) form of the metric tensor?

In flat space, the metric tensor is (in one of the two conventions) $$\eta^\mu{} ^\nu = \begin{bmatrix}1&0&0&0\\0&-1&0&0\\0&0&-1&0\\0&0&0&-1\end{...
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General relativity: Proof of Lorentz transformation validity for Linearized gravity

Wikipedia says that Lorentz transformation is only correct for inertial coordinates. However, I was flipping though Gravitation by Misner et al. On page 439, it says that for linearized gravity $$ \...

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