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.
460
questions
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1answer
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....
4
votes
1answer
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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0answers
163 views
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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2answers
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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1answer
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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2answers
206 views
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.,
$$
\frac{d p_{\alpha}...
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2answers
185 views
Geometrical picture of change of coordinates in case of Lagrangian
I was reading the part that Euler-Lagrange equation holds even on changing the coordinates. In the book by David Morin, the author talks of geometrical picture of the change of coordinates. He makes ...
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0answers
228 views
What Lorentz covariance has to do with Lorentz invariance? [duplicate]
Does saying that the Dirac equation is invariant under Lorentz transformations is the same as saying that it is Lorentz covariant?
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2answers
94 views
Problem in deriving elements of Lorentz Group
I am following a lecture series for Classical Field Theory in which they the lecturer uses the invariance of the length of a vector during a Lorentz transformation to derive the equation $$O^{T}\eta~O=...
2
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1answer
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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2answers
2k views
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 :
$$...
2
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4answers
997 views
Why do you have to include the Jacobian for every coordinate system, but the Cartesian?
In physics and engineering it is common to convert between different coordinate systems - spherical, polar, Cartesian, e.t.c. - depending on the problem. Physically, they are all clearly equivalent ...
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2answers
950 views
General covariance from the equivalence principle
Einstein's equivalence principle (EEP) tells us that there is no way in principle to locally distinguish between inertial acceleration and the effects of a gravitational field by carrying out any non-...
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0answers
580 views
Covariant derivative of the variation of the metric
Say you are varying some function (it could be a tensor, or a tensor density, scalar, etc.) that depends on the metric (the metric is what is varying), so you have
$\delta f = \frac{\partial f}{\...
4
votes
3answers
681 views
Co and contravariant: tensors or components?
I am learning Special Relativity and have a question: given a four vector $\vec{x}$ whose contravariant components are $x^\mu$, do the covariant components $x_\mu = g_{\mu\nu}x^\nu$ make reference to ...
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2answers
361 views
Covariant derivative of unit vector/Kronecker delta
Consider:
$\nabla_b \delta^a_c=\partial_b\delta^a_c-\Gamma^d_{bc}\delta^a_d+\Gamma^a_{bd}\delta^d_c=0-\Gamma^a_{bc}+\Gamma^a_{bc}=0$
But if I define:
$T^a_{bc}=\nabla_b\delta^a_c$
Then I should have:...
4
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1answer
675 views
Covariant derivative of a Dirac spinor and Kosmann lift
In [1] I have found a definition of the covariant derivative of a Dirac field with a general connection $\omega_{\mu a}{}^{b}$ (with torsion and non-metricity) [see eq. (29)]:
$$\nabla_{\mu}\psi=\...
2
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0answers
162 views
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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1answer
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 ...
4
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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 ...
3
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3answers
166 views
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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2answers
448 views
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 ...
1
vote
2answers
89 views
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 ...
5
votes
2answers
223 views
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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2answers
236 views
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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0answers
72 views
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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3answers
706 views
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?
...
4
votes
3answers
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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2answers
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} ...
3
votes
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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1answer
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
...
12
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2answers
2k views
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: $\partial_{\mu}\partial^{\nu}A = \partial^{\nu}\partial_{\mu}A$.
...
3
votes
4answers
4k views
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
$...
9
votes
1answer
330 views
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
1k views
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 ...
3
votes
1answer
86 views
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
$$\vec{H} = \nabla\vec{u}\cdot\vec{n}+\nabla\times(\nabla\times\vec{B}) + \frac{\...
0
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1answer
310 views
SUSY chiral covariant derivatives under change of coordinates
Reading Martin's SUSY Primer, section 4.4 on Chiral Superfields, he makes the statement that the SUSY chiral covariant derivatives
$$D_\alpha=\dfrac{\partial}{\partial\theta^\alpha}-i(\sigma^\mu\...
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3answers
1k views
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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1answer
213 views
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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0answers
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 ...
12
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4answers
2k views
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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1answer
208 views
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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1answer
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}&...
0
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1answer
138 views
$\Delta x^{\alpha}\equiv x_{2}^{\alpha}-x_{1}^{\alpha}$, 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 $\Delta x^{\alpha}\equiv x_{2}^{\alpha}-x_{1}^{\alpha}$ ...
3
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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)
$$\eta^\mu{} ^\nu = \begin{bmatrix}1&0&0&0\\0&-1&0&0\\0&0&-1&0\\0&0&0&-1\end{...
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160 views
General relativity: Proof of Lorentz transformation validity for Linearized gravity
Wikipedia says that Lorentz transformation is only correct for inertial coordinates. However, I was flipping though Gravitation by Misner et al.
On page 439, it says that for linearized gravity
$$
\...
