# 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.

450 questions
Filter by
Sorted by
Tagged with
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 ...
39 views

### 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 ...
63 views

97 views

58 views

### 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 ...
53 views

### 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 ...
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 ...
64 views

### 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 ...
64 views

### 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 ...
113 views

### 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 ...
139 views

### 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....
147 views

### 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 ...
77 views

### 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 ...
56 views

### 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?
62 views

### 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 ...
52 views

### 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: $$-1 = u^{2}_{t} [g_{tt} +2 \Omega g_{t \phi} + \Omega^{2} g_{\phi\phi}]. \tag{1}$$ ...
81 views

### 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 ...
31 views

### 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....
150 views

### 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 ...
36 views

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 ...
51 views

52 views

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 ...
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 $g$...
38 views

### 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 ...
42 views

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 ...
37 views

### 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 ...
83 views

### 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 ...
71 views

### 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 ...
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 ...