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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1answer
207 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
157 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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2answers
1k 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: $\partial_{\mu}\partial^{\nu}A = \partial^{\nu}\partial_{\mu}A$. ...
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4answers
3k 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 $...
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1answer
277 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
772 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
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1answer
81 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
112 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
240 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
672 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
144 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
340 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
321 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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4answers
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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
550 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
190 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
614 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
131 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
338 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
152 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
493 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
166 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
40 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
805 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
946 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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0answers
112 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
150 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
689 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
168 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
154 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
253 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 ...
2
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2answers
248 views

The Klein-Gordon field

The Klein-Gordon field is said to be "scalar". But when we write an expansion of it in terms of creation and annihilation operators or when it and its conjugate field satisfy a commutation relation, ...
7
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1answer
275 views

Is the distinction between covariant and contravariant objects purely for the convenience of mathematical manipulation?

Two kinds of indices, covariant and contravariant, are introduced in special relativity. This, as far as I understand, is solely for mathematical luxury, i.e. write expressions in a concise, self-...
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1answer
304 views

General form of the matrix element of a Lorentz invariant rank two tensor operator?

Why can the matrix element of a traceless, symmetric tensor $Q_{\alpha \beta}$ that does not commute with Lorentz transformations, for a one particle state be written as $$ \langle p | Q_{\alpha \...
2
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1answer
219 views

Canonical momentum combinations in covariant Hamiltonian formalism

I am interested in a statement of the following paper (arxiv:hep-th/9802115), but I will describe the simplest case. I am interested in a free scalar Lagrangian with mostly plus signature (the above ...
3
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1answer
625 views

Is spacetime symmetry a gauge symmetry?

In previous questions of mine here and here it was established that Special Relativity, as a special case of General Relativity, can be considered as the theory of a (smooth) Lorentz manifold $(M,g)$ ...
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1answer
124 views

Analogy for covariant and contravariant tensors [closed]

I have been trying to grasp the difference between covariant and contravariant tensors in a somewhat qualitative way. This analogy popped into my mind and I wanted to check whether I'm on the right ...
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1answer
243 views

Tensors and derivatives

I am a maths student taking a module in (the mathematics of) Relativity so I get quite confused when looking for stuff that may help me understand where I go wrong in certain questions as I'm not ...
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2answers
120 views

Regarding the Dirac Hamiltonian's use of summation notation:

Einstein summation notation, as I understand it: By writing $A_i B^i$ one implicitly means a sum over elements of the rank 1 tensors A and B. The key is the contraction of an "up" and a "down" index. ...
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2answers
294 views

Global symmetries of spacetime and general covariance

I am self learning GR. This is a rather long post but I needed to clarify few things about the effect of general coordinate transformations on the global symmetries of metric. Any comments, insights ...
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1answer
154 views

Raising index of variation

I know how to prove e.g. $$A^{ik}B_{lk}=A_{k}^iB^{k}_l.\tag{1}$$ (Raising and Lowering Indices Question). Today in a book, I find: $$g^{ik}\delta g_{lk}=-g_{kl}\delta g^{ki}.\tag{2}$$ $g^{ik}$ is ...
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2answers
852 views

What exactly does it mean for a scalar function to be Lorentz invariant?

If I have a function $\ f(x)$, what does it mean for it to be Lorentz invariant? I believe it is that $\ f( \Lambda^{-1}x ) = f(x)$, but I think I'm missing something here. Furthermore, if $g(x,y)$ ...
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3answers
1k views

Confusions about Covariant and Contravariant vectors

I am trying to connect the concepts I learned from special relativity, to those of general relativity. Take a look at this example from wikipedia. They find a transformation matrix from the ...
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0answers
362 views

Notation for vectors and covectors

This is probably a very simple question, and I think I know the answer, but I cannot find a place to solidly confirm this. So if I want to write a vector $\mathbf{V}$ in terms of its contravariant (...
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1answer
201 views

How are the *constant vectors* different from *vector fields* in terms of their respective transfomation properties?

How does one distinguish between the transformation properties of a scalar field $\phi(\textbf{r})$ or vector field $\textbf{A}(\textbf{r})$ (more generally, the tensor fields) from the transformation ...
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0answers
66 views

Where is a proof that string field theory is generally covariant?

Given a space-time coordinate of a string $X^\mu(\sigma)$ dependent on the position $\sigma$ around the string. And a string field functional $\Phi[X]$, is there a proof that the equations of motion (...
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2answers
1k views

Tensor vs. Tensor Densities

Currently I'm reading through Sean Carroll's Spacetime and Geometry: an Introduction to General Relativity. According to Carroll, the symbol $$dx^0 \wedge dx^1 \wedge \cdots \wedge dx^{n-1},$$ ...
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2answers
476 views

Covariant and contravariant permutation tensor

I have been reading up on the permutation tensor, and have come across the following expression (in 'Generalized Calculus with Applications to Matter and Forces' by L.M.B.C Campos page 709): $$e_{i_1,\...
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1answer
152 views

Function form of Lorentz invariant functions

In QFT, Green function of gauge field is Lorentz invariant(i.e. $\forall \Lambda \in SO(3,1), f(\Lambda p)=\Lambda f(p)$).And according to the textbook I'm reading, The form of such functions is ...