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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Momentum operator in QM - scalar or vector?

The momentum operator for one spatial dimension is $-i \hbar d/dx$ (which isn't a vector operator) but for 3 spatial dimensions is $-i\hbar\nabla$ which is a vector operator. So is it a vector or a ...
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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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35 views

General relativity: Principle of minimal coupling computations

I have a question about computations in general relativity and transition from a Lorentz frame to a general fame just by substituting the flat metric with a general one and ordinary derivatives with ...
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Are Lagrange's equations physical laws?

Well, I studied that a physical law is an equation between tensors that are function of position and time because when the frame is changed tensors change in order to leave the equation true: $$T_1(\...
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Covariance of Euler-Lagrange equations under change of generalized coordinates

Suppose I have an inertial frame with coordinate $\{q\}$. Now I define another reference frame with coordinate $\{q'(q,\dot q,t)\}$. I obtain the equation of motion in $\{q'\}$ in two different ways: ...
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114 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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1answer
148 views

Dual space and Metric tensor

So I know that the dual space is the set of all linear transformations that map a vector from a vector space to the field of the space itself (the real number line, complex, quaternions). From YouTube ...
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37 views

Covariant derivative with respect to commutator

I have some confusion with the notion of $\nabla_{[A, B]}\bf{v}$, that expression, with a commutator of vector fields as the subindex of the connection appears for instance in the definition of the ...
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166 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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Identifying Lorentz Covariant Equations

Statement: $\phi , A^{\mu}, T^{\mu \nu}$ are a Lorentz scalar, vector, and tensor. Which of the following equations are Lorentz covariant. a. $\phi = A_{0}$ b. $\phi = A^{\mu}A_{\mu}$ c. $\phi = ...
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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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178 views

Second covariant derivative, computation problem

I am having a question on the wikipedia article http://en.wikipedia.org/wiki/Second_covariant_derivative Using the notation therein I don't get why $(\nabla_{u}\nabla_{v}w )^a=u^c\nabla_{c}v^b\nabla_{...
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Is there any meaning of tensor contraction?

Is there any meaning behind tensor contraction. Or is it just randomly getting rid of some components by only selecting those with same index and sum them up? For example, I know tensor is ...
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Parametric and covariant expressions for the acceleration vector

I am reading S. Neil Rasband book about Classical Dynamics. In the first chapter, there are two different forms of the acceleration: What he calls the "intrinsic". Given a trajectory with parameter $...
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Which are the Physical (Not Mathematical) Axioms of General Relativity? [duplicate]

I have an idea on what the correct answer may be. The first axiom is the axiom from Special Relativity. The laws of Physics apply identically to inertial frames of reference. This axiom contains the ...
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Signal-to-noise (SNR) in a quantum network

I'm studying dynamics of a quantum network of coupled oscillators driven by an external force. Am I doing right, if I calculate signal-to-noise ratio by dividing the expectation value of oscillator ...
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2answers
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In which sense equations of motion are covariant?

I read lots questions about what covariance is and I found out that, according to this topic Lorentz invariance of the Minkowski metric, we say an object is covariant if it doesn't take the same value ...
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Raising & lowering indices of 3-pseudovectors?

Now, let space tmie metric is $$\eta_{\mu\nu}=\text{diag}(+,-,-,-)$$ now $$x_{\mu}=(x^0,-\mathbf{x})$$ and $$x^{\mu}=(x^0,\mathbf{x})$$ and $$x^{\mu}=\eta^{\mu\nu}x_{\nu}$$ also $$\partial_\mu=(\...
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Choice of metric breaks diffeomorphism invariance?

In Weinberg's paper on the cosmological constant problem (CCP), he states that diffeomorphism invariance is always broken by the presence of any given metric $g_{\mu\nu}$. He then goes on to say that ...
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Transformation of $g^{\mu\nu}\partial_\mu f \partial_\nu f$

I have the expression $$g^{\mu\nu}\partial_\mu f \partial_\nu f$$ e.g. inside a Lagrange density, where $g$ is a metric tensor and I want to transform this expression to a new set of coordinates. Do ...
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Why does the factor $\sqrt{-g}$ make the volume element invariant?

My question is an extension on this and this question. The question is, how or "in what sense" does the factor $\sqrt{-g}$ make the measure invariant? Suppose, I do not add this factor to the measure....
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Divergence of a tensor

On pg.70 of Dalarsson's "Tensors, Relativity and Cosmology" For a mixed tensor of contravariant order 2 and covariant order 1 $(T^{mn}_{p,m})$, the divergence with respect to m is defined as:$$T^{...
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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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Einstein notation: can a free index be upper in one term and lower in another term?

Consider a linear combination of terms written using Einstein notation. Consider one free index in the linear combination: is it necessary that the index is upper in all terms or lower in all terms, ...
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Why do we need invariants to represent real life quantities?

Often it is said that one of the most useful properties of eigenvalues of a matrix is that they are invariant under change of basis. This in turn is said to be useful in physics because real, physical ...
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Indices of the Riemann Tensor of the first kind

When establishing the identity $V^i_{,kl}-V^i_{,lk}=-R^i_{tkl}V^t$ (, denotes covariant differentiation), one of the steps involves raising one of the indices of the Riemann Tensor of the first kind . ...
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Recommendation for contra-variant and contra-variant vectors in the context of special relativity [duplicate]

I have just started studying covariant and contra-variant vectors in special relativity. I am finding difficulty understanding this topic. Can anyone recommend a book (with a detailed chapter) or ...
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Variations of tensors are tensors?

Recently I posted a question about variation of metric. I thought I understood it and talked with my friend about it. After that he said he's not convinced because he can't prove variation of metric ...
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Why is the partial derivative a contravariant 4-vector?

The contravariant partial derivative is defined as following: $$\partial ^\mu = \frac{\partial}{\partial x_\mu}$$ where the index $\mu$ runs from 0 to 3. A contravariant vector under Lorentz ...
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How to determine if a tensor is covariant or contravariant?

In special relativity, the coordenates of a event are in general written using a 4-vector: $$x^{\mu} = \binom{ct}{\textbf{x}}$$ where $\textbf{x} = (x,y,z)$ are the spacial coordenates. This is a ...
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234 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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Coordinate transformation of basis vectors

The question Let $e_a$ be the coordinate basis vectors in a manifold described by coordinate system $x^a$. The vector displacement between two nearby points is given by \begin{equation} ds=dx^ae_a=dx'...
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Do Bianchi identities hold in all coordinates?

I understand by expanding out the Riemann tensor, that the Bianchi identities can be derived within a local inertial frame (LIF) by taking the partial derivatives of the Riemann tensor relations in a ...
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4answers
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Covariant and contravariant 4-vector in special relativity

I've just learned about contra- and covariant vector in the context of special relativity (in electrodynamic) and I'm struggling with some concept. From what I found, an intuitive definition of ...
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1answer
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Confusion Einstein notation polar coordinates

I'm having issues using Einstein notation in polar coordinates in flat space, I must be missing something basic. Consider the following example. Take the following metric on a 2+1 spacetime; $ds^2 = ...
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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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275 views

Relation between differentiation of one-form basis and Christoffel Symbols

If I want to covariantly differentiate a one form then I can write: $\nabla_\beta \tilde p = \dfrac{\partial p_\alpha}{\partial x^\beta} \tilde \omega^\alpha + p_\alpha \dfrac{\partial \tilde \omega^...
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General covariance and a running cosmological constant

A running cosmological constant $\Lambda(t)$ can always be included in the perfect fluid source tensor. By the transformation properties of tensors Einstein's field equation is still independent of ...
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3answers
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Origin of $\sqrt{-g}$ in the integral of action $S$

I have a question that might (and probably will) be stupid: I do not understand where does the factor $\sqrt{-g}$ (i.e. $\sqrt{-\det\left(g_{\mu\nu}\right)}$) come from in the action integral S when ...
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54 views

Is the addition of a Christoffel symbol and the partial derivative of a vector a tensor?

The partial derivative of a vector $V^\lambda , _\nu$ is not a tensor. Neither is a Christoffel symbol $\Gamma^\lambda _{\mu \nu}$. Is the addition of these two objects a tensor? If they were ...
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Generally Covariant Dirac equation: The spin connection

Wikipedia, an answer on stackexchange and a few papers in the Arxiv I've found all have different definitions of the spin connection found in the Dirac equation. Can anyone please tell me what the ...
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Gradient is covariant or contravariant?

I read somewhere people write gradient in covariant form because of their proposes. I think gradient expanded in covariant basis $i$, $j$, $k$, so by invariance nature of vectors, component of ...
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1answer
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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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199 views

Physical significance of one-form in a velocity field

Still tentatively feeling my way through this stuff, so please go easy. The velocity of a fluid at a point P are the components $V^{a}$ of a contravariant vector:$$v^{x},v^{y},v^{z}\equiv\frac{dx}{dt}...
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131 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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General Covariance, what does Einstein mean?

I have read the papers by Einstein and I am convinced I understand what he means completely. Given there are controversies, maybe I over understood it: It is I am convinced, can be said in two ...
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Covariant derivative for spinor fields

scalars (spin-0) derivatives is expressed as: $$\nabla_{i} \phi = \frac{\partial \phi}{ \partial x_{i}}.$$ vector (spin-1) derivatives are expressed as: $$\nabla_{i} V^{k} = \frac{\partial V^{k}}{ \...
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Derivation of Covariant derivative for fermionic fields

I've been reading about the Dirac equation in curved spacetime and understand the nature of the verbien, but am wondering what the relationship is between the two definitions of the Fermionic ...
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1answer
131 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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Re-Writing the Dirac Equation in True Covariant Form

This is a rather brief inquiry, but to get to the point it's always frustrated me that in non-relativistic and relativistic quantum mechanics spin matrices are written as a "vector of matrices" ...