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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23 views

Invariance of Euler-Lagrange equation [duplicate]

How exactly does the variational approach make it clear that the Euler-Lagrange equations of motion will have exactly the same form no matter what coordinate system we use as long as it is done with ...
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32 views

Clarification on showing covariance of density expressions under Lorentz boost

For a collection of point charges, the charge density is defined as \begin{equation} \rho(\textbf{x},t) = \sum_{k=1}^{N} q_k \delta^{(3)}\left[\textbf{x}-\textbf{x}_k(t)\right] \end{equation} while ...
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Why does $k_\mu k_\nu \to k^2 \eta_{\mu\nu}/d$ work in QFT calculations?

When doing calculations of Feynman diagrams in QFT I've seen a trick used that goes something like this $$\int \frac{d^dk}{(2\pi)^d} \frac{k_\mu k_\nu}{f(k^2)}\quad\longrightarrow\quad\int \frac{d^dk}{...
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Understanding tensor transformations

I am trying to learn how tensors transform under coordinate transformations. For an example, under a transformation from the coordinate system $x^\mu \longrightarrow x'^\mu$ a covariant tensor is ...
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75 views

Covariant derivative of a one-form

I understood that the covariant derivative of a vector field is $$ \nabla_{i}v^j=\frac{\partial v^j}{\partial u^{i}}+\Gamma^j_{~ik}v^k $$ Then why is the covariant derivative of a covector field $$ \...
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How does Equivalence Principle imply a Curved Space-Time?

I am a bit confused as to how the Equivalence Principle implies a curved spacetime. Or if it doesn’t imply a curved spacetime, then what exactly makes it necessary to have a curved space time? I could ...
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Advise on fitting spatio-temporal covariance in a likelihood framework based on Gaussian Random Field Method

I'm working on a spatial-temporal model as described in https://www.sciencedirect.com/science/article/pii/S1090780714002134. I have modified this model such that I have an event at time ...
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46 views

Electromagnetic waves in tensor notation

I'm trying to derive the wave equations for the electric and magnetic fields in covariant (tensor) formulation. Starting with Gauss-Ampere law, $$ \partial_\alpha F^{\alpha\beta}=\frac{4\pi}{c}J^\beta ...
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How does the Lagrangian transform when coordinates are changed?

I'll talk of single particle Lagrangian in $n$ dimensions. Suppose in a given coordinate system with the coordinates $(q_i)_{i=1}^n$, the Lagrangian is given by $L(\mathbb{q, \dot q}, t)$. Suppose I ...
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Understanding tensor and covariance

I'm really struggling to understand the use of tensors when we want to have a covariant equation. From what I understand, if we write an equation using tensors only, then the physics behind it will be ...
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Confusion about covariance of physical laws and tensors

In classical mechanics physical laws must be covariant in inertial frames i.e. mathematically the laws will be such that two inertial observers are indistinguishable, for example the dependence of ...
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$ \mathbf{e}^i = g^{ij} \mathbf{e}_j$ interpretation

I've problems in the interpretation of the expression: $$ \mathbf{e}^i = g^{ij} \mathbf{e}_j$$ that can be found, by example, in this wiki chapter. Also here. Step by step of my erroneous logic: The ...
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103 views

Contravariant/covariant vectors or components?

Related: Covariant vs contravariant vectors Note: by vector I try to refer to the physics entity, not to the list of components. From wikipedia page and others: $$ \mathbf{v} = q_i \mathbf{e^i} = q^i \...
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Tensor transformation Formula Proof

Ok so basically I am trying to prove that the following expression: Can be written using matrices like this: Any suggestions on how to approach this?
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Which postulate of GR requires that $ds^2=g_{\mu\nu}(x)dx^\mu dx^\nu$ must be invariant? [duplicate]

The postulate that the velocity of light in a vacuum is the same in all inertial frames, can be exploited to establish that the spacetime interval between two events $x^\mu=(ct,{\bf r})$ and $x^\mu+dx^...
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50 views

Understanding where all the equations from a covariant derivative of a tensor come from

Suppose I have a situation where i know that $\nabla_i T^{ik}=0$ where $ T^{ik}$ is a tensor of rank 2, which is diagonal, such as the perfect fluid energy-momentum tensor. We are dealing in a ...
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Can one quantize Chern-Simons theory in the covariant phase space formalism?

The covariant phase space, which coincides with the space of solutions to the equations of motion, gives a notion of phase space which does not rely on a decomposition of spacetime of the form $M=\...
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Symmetries in Relativistic mechanics and Field Theory and Lorentz Invariance

In Non-Relativistic Lagrangian mechanics usually we didn't impose any constraint on the Action/Lagrangian, rather than to be respectively a functional/function (or 3 form on Manifolds). In QFT and ...
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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
81 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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127 views

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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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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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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78 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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78 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 ...
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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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370 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 ...
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156 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....
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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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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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172 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 ...
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2answers
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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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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
63 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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148 views

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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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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77 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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74 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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