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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784 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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821 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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493 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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3answers
638 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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232 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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1answer
492 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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155 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
452 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
265 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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3answers
159 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
315 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 ...
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2answers
76 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 ...
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158 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
205 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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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
544 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? ...
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523 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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350 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
228 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
173 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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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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287 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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863 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 ...
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1answer
82 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
113 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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1answer
265 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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763 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
160 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
397 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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341 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
657 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
197 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
678 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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1answer
134 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 $\Delta x^{\alpha}\equiv x_{2}^{\alpha}-x_{1}^{\alpha}$ ...
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1answer
395 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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0answers
157 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 $$ \...
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2answers
575 views

Lorentz transformation of the dual tensor

I am trying to lorentz-transform the dual electromagnetic tensor $G^{\mu \nu}:= \frac{1}{2} \epsilon ^{\mu \nu \alpha \beta} F_{\alpha \beta}$ and also show (perhaps by using that last result) that $G^...
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1answer
181 views

Minimal coupling in general relativity

Consider the Einstein-Maxwell action (setting units $G_{N}=1$), $$S = \frac{1}{16\pi}\int d^{4}x\sqrt{-g}\ (R-F^{\mu\nu}F_{\mu\nu})$$ where $$F_{\mu\nu} = \nabla_{\mu}A_{\nu}-\nabla_{\nu}A_{\mu} = \...
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1answer
41 views

Differences in calculations should be due to differences in environment?

Lisa Randall in her book Warped Passages writes, "A very reasonable thing to expect from physical laws is that they should be the same for everyone. No one could blame us for questioning their ...
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1answer
852 views

Covariant formulation of electrodynamics

IMO 'covariant formulation' of electrodynamics means that the equations should remain invariant across different Lorentz frames. Now there are broadly two ways to write electrodynamics equations. ...
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1answer
1k views

How to check if a tensor transform a tensor?

Suppose $A^{\mu_1 \cdots \mu_n}_{\nu_{n+1}\cdots \nu_m}$ is a tensor. That means it transforms a tensor. How do I show that it transforms as a tensor? How do I see that $\cos (A^{\mu_1 \cdots \mu_n}_{\...
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113 views

What is not a tensor? [duplicate]

I've been taking GR, and all of a sudden I am not sure that I know the necessary and sufficient requirements of the tensor coefficients. This is because the lecturer asked me to prove that that "there ...
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1answer
175 views

How to prove $\nabla\vec{V}$ is a tensor without transformation properties?

In A First Course in General Relativity, Schutz asks the reader to prove that $\nabla \vec{V}$ is a $(1,1)$-tensor, where $$(\nabla\vec{V})^\alpha_{\ \ \ \ \beta} \equiv V^\alpha_{\ \ \ \ ;\beta} \...
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1answer
791 views

Lorentz force and charged partice's motion of equation in cylindrical coordinates

Is the Lorentz force $$\textbf{F} = q(\textbf{E} + \textbf{v} \times \textbf{B})$$ same in Descartes and in cylindrical coordinates? Moreover do the motion of equation of a charged partice with $\...
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1answer
193 views

How does a two-tensor transform under an infinitesimal shift?

This is a follow-up to this question I posted yesterday: How does a vector field transform under an infinitesimal coordinate transformation? If I have an infinitesimal coordinate shift of the form $x^...
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0answers
171 views

Determinant of a mixed rank-2 tensor

Often dispersion relations in plasmas are found by setting the determinant of some quantity to equal zero. My question is, how does one do this when working covariantly with tensors (e.g. Broderick, A....
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1answer
1k views

How does a vector field transform under an infinitesimal coordinate transformation?

If I have a vector $X^{\mu}(x)$, and then I consider an infinitesimal coordinate transformation of the form $x^{\mu} \to x^{\mu} + v^{\mu}(x)$, then how does my vector $X^{\mu}(x)$ transform? From ...
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3answers
269 views

Do the equations of general relativity apply to all coordinate systems?

I was inspired to ask the question, after seeing this: http://www.mathpages.com/home/kmath588/kmath588.htm A short passage from the paper relating to the question above: It’s possible to take a ...

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