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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68 views

Understanding the four-dimensional volume form in Action of Lagrangian

Into the following part below, I don't understand what is precisely a "four-dimensional volume form" implied in the integral below: For comparison, the ...
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Dirac delta and covariance [duplicate]

Is there a covariant form of the Dirac delta function? And how to build a covariant form of an identity that contains Dirac delta? To be more precise, what I am looking for is Some distribution that ...
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1answer
63 views

General covariance of the Maxwell equations in 4-tensor form

Are the Maxwell equations written with the derivatives of the EM field strength tensor not generally covariant? I can't seem to prove that is. The Maxwell equations in 4-tensor form: $\partial_{\mu}F_{...
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How did the Lagrangian and Hamiltonian theories of motion inspire the idea that forces should be treated as one-forms instead of vectors?

On page-5 of this paper1 by E. Minguzzi titled "A geometrical introduction to screw theory", he writes: Who adopts this point of view argues that it should also be adopted for forces in ...
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Rosen coordinates

I would like to express this spacetime interval for a spacetime with a monochromatic electromagnetic wave for which I believe $|E|=\frac{q}{\omega}\sin(u)$ expressed in Rosen coordinates: $ds^2=-2dudv+...
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1answer
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Raised index of partial derivative

I am having a really hard time wrapping my head around component notation for tensor fields. For example, I do not know exactly what the following expression means $$\partial_\mu\partial^\nu \phi, \...
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1answer
60 views

Do separation vectors exist in general relativity?

Deriving the equation of geodesic deviation one looks at two test masses on positions $x^\mu$ und $\tilde{x}^\mu$ and defines the separation vector $\boldsymbol{\chi}$ as $$\tilde{x}^\mu=x^\mu+\chi^\...
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1answer
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Is Einstein Equivalence principle a consequence of weak equivalence principle + covariance principle?

I have been doing some thinking about the Einstein Equivalence Principle (EEP) and its formulation, namely: The outcome of any local non-gravitational experiment in a freely falling laboratory ...
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Are 4-vectors covariant under general coordinate transformations?

In the context of special relativity, it is well known that 4-vectors are covariant under Lorentz transformations (which is a linear transformation in space-time), however are they covariant under ...
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1answer
66 views

How is d'Alembertian operator is defined in differential geometry?

Which general formula for the box operator is correct, $\Box=g^{ij}\partial_i\partial_j$ or $\Box=\frac{1}{\sqrt{g}}\partial_i(\sqrt{g}g^{ij}\partial_j)$? I have seen both the definition being used ...
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1answer
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In a classical scalar field theory, is the Hamiltonian Lorentz-invariant? How about the Lagrangian?

I encountered a statement that "while Lorentz invariance is apparent in the Lagrangian formulation, it is not so in the Hamiltonian formulation of a classical field." I do not completely ...
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Invariance of differential operators

How do we prove that del operator is invariant under any kind of change of coordinates, specifically under galilean transformations? I am getting an extra term containing the relative velocity of two ...
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Transformation of the Levi Civita symbol - Carroll

Background As per convention, Carroll defines the Levi-Civita symbol $\tilde{\epsilon}_{\mu_1 \mu_2 \dots \mu_n}$ as the sign of the permutation of $01\dots(n-1)$ on page 82. He states the Levi-Civita ...
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Why is the path integral Lorentz invariant?

It is often said one of the benefits of the path integral, $$\int D\phi \; e^{iS[\phi]}$$ is that it is manifestly Lorentz covariant if $S[\phi]$ is Lorentz covariant. However, this is not clear to me....
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What's with all the index notation in General Relativity?

I am self-studying General Relativity with Leonard Susskind's lectures from Stanford. The thing that is bothering me is the notation of GR, specifically, the index notation. In simple layman terms ...
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Is Newton's law really form-invariant w.r.t transformation from one inertial frame to another?

Newton's second law, $\vec{F}=m\vec{a}$, is form-invariant only under Galilean transformations but not under Lorentz transformations. Then why do we say that Newton's law is valid and form-invariant ...
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1answer
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Explanation of an equation in special relativity

$$ {\partial (0.5 (\partial_{\mu} A^{\mu})^2) \over \partial(\partial_{\mu} A_{\nu})} = {(\partial_{\rho} A^{\rho}) g^{\mu \nu} } $$ Can somebody explain why this is true?
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When can we use normal coordinates for a “proof”?

So I'm trying to find the equations of motion of a field in a particular metric. I know what the equations of motion of the field in flat space look like and how they simplify. I think it's always ...
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Question on four-velocity condition for timelike observers

I am a bit rusty on GR, I have the condition $u_{\mu}u^{\mu} = -1 $, in some notes I am given that we can obtain: $$-1 = u^{2}_{t} [g_{tt} +2 \Omega g_{t \phi} + \Omega^{2} g_{\phi\phi}]. \tag{1} $$ ...
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Are $v^ie_{i}$ and $v^iv_{i}$ (where $v$ are the components and $e$ the basis vectors) both tensors? Or only the second one?

I am studying the math of tensors, I have an understanding of the concepts of covariance, contravariance, dual spaces, Einstein notation and so on. I am a bit confused about the notation though. My ...
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Help in proof: Lorentz invariant Lagrangians lead to Lorentz covariant equations

I think that the following statement is true, but can't seem to prove it. Suppose we have a scalar field whose Lagrangian density$^1$ $\mathcal L(\phi, ~\partial_\mu\phi, ~X^\mu)$ is Lorentz invariant....
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3answers
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Covariance in special and general relativity

I am self-studying SR and GR and need to make sense of the covariance principle. I understand the idea that physical principles should have no preference in coordinates and therefore must be expressed ...
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1answer
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Transformation Properties of Connection Coefficients

This question is about pages 95 and 96 of Carroll's book: Spacetime and Geometry. We have the formula for the covariant derivate: $$\nabla _\mu V^\nu=\partial _\mu V^\nu + \Gamma _{\mu\lambda}^\nu V^\...
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1answer
50 views

Covariance of Noether's charge in GR

If we consider a theory of GR (the standard Einstein-Hilbert action) and a complex scalar field, we can easily see that we have a global $U(1)$ symmetry for the scalar field. Now, via Noether's ...
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Maxwell Equation: Definition of Invariance

Knowing Lorentz Transformation and knowing the differential formulation of Maxwell Equations: Precisely, what is the meaning of the statement: "Maxwell equation are invariant under Lorentz ...
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1answer
53 views

Raising and lowering indices: is it a convention? [duplicate]

We can raise and lower indices of any tensor with a non-zero rank by applying the metric tensor with indices properly located. My question is: why is the metric tensor the tool we use for such ...
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2answers
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Lorentz invariance of the Lorentz force law

I'm self-studying Friedman and Susskind's book Special Relativity and Classical Field Theory. The following question popped up while reading section 6.3.4 Lorentz Invariant Equations. In this Lecture, ...
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1answer
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How to be sure that a law is invariant under Lorentz's Transformation?

For starters let's talk about Maxwell's Equations; we know that Maxwell's Equations are invariant under Lorentz's Transformation, after all this is why all the relativity business got started. To ...
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1answer
67 views

Coordinate transformation. From differentials to contravariant vectors

I have a very silly doubt. If I have two coordinate systems and I want to calculate the coordinate differentials for the second one, I need to use the chain rule for the derivatives so that I obtain $$...
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1answer
53 views

Is Newton's second law covariant under a change from Cartesian to polar coordinates?

I'm aware that Newton's 2nd Law is covariant under a Galilean transformation or under any other linear transformation that's not parameterized by relative velocity of frames. But what about non-linear ...
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2answers
51 views

Confusion over the various equations of energy in basic special relativity

I have a fairly longstanding confusion over several different equations involving energy in special relativity and I struggle to see how they relate to each other. $E=mc^2$ $E=\gamma mc^2$ $E=mc^2+\...
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Energy-Momentum Tensor not traceless?

In my university, we have a different approach for deriving the energy-momentum tensor for electromagnetic field in vacuum. The result is: $T_i {_j} = \epsilon_0 \ (\frac{1}{2}E^2 \delta_i {_j}\ - ...
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1answer
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Covariant formulation of electrodynamics homogenous Maxwell eq

It is know that $$\epsilon{^\mu} {^\nu} {^\rho} {^\sigma} \partial_{\nu} F_{\rho} {_\sigma} = 0$$ How can one deduce from this equation that $$ \partial_{\mu}F_{\nu} {_\lambda} + \partial_{\lambda}F_{\...
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Poisson bracket properties for tensor densities

I am doing some constraint analysis in an extended theory of gravity, and I am confused about Poisson brackets. The standard PB relations are for example $\{ab,c\} = a\{b,c\} + \{a,c\}b$ etc. But I am ...
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Critique on tensor notation

I am studying tensor algebra for an introductory course on General Relativity and I have stumbled upon an ambiguity in tensor notation that I truly dislike. But I am not sure if I am understanding the ...
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1answer
96 views

Is the Covariant Derivative a phenomenological attempt?

I am trying to self study QFT and i am very confused about the covariant derivative. When we require our theory to be invariant under local gauge transformations we kind of "guess" that we ...
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1answer
67 views

How is tensor calculus applied to Einstein's field equations? [closed]

What is the relation between tensor calculus and Einstein's field equations? or What is the contribution of tensor calculus to Einstein’s field equations?
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1answer
84 views

Covariant form of Maxwell Equations in curved spacetime

The real world doesn't care about our choice of coordinate to describe nature. Maxwell equations in vectorial form are written with respect to an Inertial frame of reference as: \begin{align} \vec\...
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1answer
97 views

Is the Dirac equation really covariant under Lorentz transfromations or do we just “make” it covariant?

I often read that the Dirac equation is covariant under Lorentz transformations and that this property makes it the right equation and in some sense beautiful. The thing is, the equation $$ \left(i\...
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1answer
58 views

Physically measure the covariant and contravariant components of a vector?

I'm just wondering if there is a way to physically measure the covariant and contravariant components of a vector without prior knowledge of the metric. Suppose I have a speedometer of some sort to ...
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28 views

Constructing unimodular metrics

Can I construct a $d$-dimensional unimodular metric from $g_{\mu\nu}$ just by dividing it by some appropriate power of the determinant i.e. $$\tilde{g}_{\mu\nu} = \frac{g_{\mu\nu}}{g^{1/d}}$$ where $g$...
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Invariance of Lagrangian under Poincaré group transformations implies covariant Lagrange equations? [duplicate]

I'm taking a class on classical fields and I came across a statement that I can't think about an argument to show that its true. It says that Invariance of a Lagrangian under transformations on ...
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1answer
42 views

Commutator of covariant derivatives acting on a vector density

Let $\mathfrak n^\alpha$ be a vector density of weight 1. Define the covariant derivative $\nabla$ such that under a coordinate transformation $x^\mu \to \bar x^\mu$ $$ \nabla_\rho \mathfrak n^\alpha ...
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1answer
42 views

Velocity-dependent potential under generalized coordinates transformation

If under some generalized coordinates the force can be written as: $$Q_j =-\frac{\partial U}{\partial q_j}+\frac{d}{dt}\left(\frac{\partial U}{\partial \dot{q}_j}\right).$$ Then can the force always ...
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1answer
59 views

Confusion regarding concept of covariance of electrodynamics

I was reading Jackson(p. 556-557 3rd edition), where I got confused about covariance of electrodynamics. The equation of electrodynamics are written in 'contravariant' tensor then why we call them ...
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1answer
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Correlations vs. negligence of correlations in covariance matrix

Suppose I have a model composed of two parameters $(a,b)$ that I want to describe a set of data points with. In CASE A, I fit the model taking into consideration the correlations between the data ...
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2answers
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What is the real Physical Meaning of tensor?

I have read something about tensor calculus from Arfken and Corvet but all of them are more some mathematical algebra for tensors. what happens in reality and nature when we use tensors in our ...
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2answers
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Contravariant - covariant notations

Working in curvilinear coordinates, one can define basis vectors corresponding to those coordinates. In the figure below (taken from here), $\{\mathbf{g_i}\}$ are base vectors corresponding to the ...
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2answers
145 views

Covariant derivative of four-position

May someone confirm or deny that covariant derivative of four-position is just metric tensor? I mean: $\nabla_{\gamma}X_{\alpha} = g_{\gamma \alpha}$ When I try to rewrite it with base vectors it ...
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2answers
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Covariant vs contravariant vectors

I understand that, in curvilinear coordinates, one can define a covariant basis and a contravariant basis. It seems to me that any vector can be decomposed in either of those basis, thus one can have ...

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