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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Covariant Formulation of E&M

Can anybody explain me what does mean the "covariant formulation of electrodynamics"? What does the covariant here mean? Invariance of Maxwell equations under Lorentz Transformations? In what way? ...
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63 views

How does effective potential transform under coordinate transformation?

Let us say we have an equation of motion of the following form, $$\ddot{x}=g\tag{1}$$ For this system an effective potential can be defined as, $$\ddot{x}=-\dfrac{d}{dx}U_\text{eff}$$ $$U_\text{...
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115 views

Can an equation containing a specific tensor be Lorentz invariant?

Let $A_u$ be a vector field in spacetime. If we restrict to a $2+1$ spacetime, and define the Levi-Civita tensor $\epsilon^{uvp}$ by $\epsilon^{123}=1$, then is the following equation Lorentz ...
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4answers
203 views

A fundamental question about tensors and vectors [closed]

Studying relativity, I am deeply confused with the fundamental concept of vectors and tensors. Are they some specific "realities" that "exist" independently of coordinates? If so, given a vector $\...
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1answer
87 views

Action in curved space

I am reading Carroll's book on GR and am confused about the generalization of the action principle to curved space. Please refer to the snippet from the book below. After writing equation 4.47, we ...
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1answer
128 views

What is invariance of an equation?

I'm confused. Suppose we have a Schrodinger equation with a time-independent Hamiltonian: \begin{align} i\frac{\partial}{\partial t}\psi(x, t) = H\psi(x, t). \tag{1} \end{align} Under time reversal ...
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2answers
85 views

A vector that transforms like a four-vector

What is the criterion or what's the meaning when saying that a vector transforms like a four-vector?
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138 views

What is the difference between $T^\mu{}_\nu$ and $T_\nu{}^\mu$?

I do understand why the horizontal order matters for indices on the same vertical position, e.g.: $$T\left(V_{(1)},V_{(2)}\right) = T_\color{red}{\mu\nu}V^\mu_{(1)}V^\nu_{(2)} \neq T_\color{red}{\nu\...
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259 views

Transformation law for Christoffel's first kind

I don't understand this particular part in this image. I am following schaum's series book on "vector analysis". I didn't find any explanation for it. I also tried searching in Internet and somewhere ...
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20 views

Literature advice: Covariant formulation of classical physics [duplicate]

I am looking for a literature advice about the following. I'ld like to review classical physics (basically all undergrad / grad stuff) under the aspects of a modern covariant formulation with exterior ...
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163 views

Derivation of the infinitesimal Special Conformal Transformation

This question is somewhat connected to my last question on special conformal transformations. I'm considering the derivation of a special conformal transformation. Namely, the quadratic translation $$...
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1answer
60 views

Computing the derivatives of a Lagrangian on a Riemannian manifold

Consider a $Riemannian\space n-dimensional\space manifold$ with coordinates {$x^i$} $(i=1,...,n)$. Let the arc length parameter be $t$. So that $\frac{d}{dt}x^i(t)\equiv\dot{x}^i(t)$ is the usual ...
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Why is stress a scalar quantity, even though mathematically, it is (Internal restoring force/Area)?

Basically, what I need to know guys is that when we divide a vector by a scalar, we get a vector. Then what is different in the case of stress?? I mean, WHY IS IT STILL A SCALAR?
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277 views

Can Einstein-Hilbert action be derived from symmetry considerations?

The action of a free relativistic classical field theory can be derived from Poincare invariance, locality, and retaining terms quadratic in fields. Is there a similar set of symmetry principles which ...
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1answer
108 views

Expanding a summation of covariant derivatives

I hope this is not a silly question but I am trying to understand how this part of the equation works: $$ \nabla_{\lambda} \left( \nabla_{\mu}(R_{\nu \lambda}) + \nabla_{\nu}(R_{\mu \lambda}) \right) ...
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49 views

Question about the derivative of a contravariant tensor

I need to show that $\frac{\partial T^i}{\partial x^j}$ is not a tensor. How I proceeded and what I got was: $\dfrac{\partial T^{i'}}{\partial x^{j'}} = \dfrac{\partial}{\partial x^{j'}} \left ( T^i ...
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86 views

Prove that two four-vectors and a four-tensor yield a scalar

I have to do a task for my special relativity class: Prove that: If $\Gamma_{\mu\nu}a^{\mu}b^{\nu}$ is a scalar for arbitrary four-vectors $a^{\nu}$, $b^{\nu}$, then $\Gamma_{\mu\nu}$ is a four-...
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63 views

Uncertainty of interpolation when covariance matrix is given

Assume that there is a given covariance matrix of an evaluated quantity ( in my case it's a rection cross section $\sigma = f(E)$, where $\sigma$ is the cross section and $E$ is each energy point ...
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204 views

Question about the the velocity and acceleration in tensor notation

When computing the volicty of a particle moving along a curve parametrized by $Z^i(t)$ for each component i, the components of the velocity $V^i$ are given by $$V^i = (d/dt)Z^i$$ and the components fo ...
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305 views

What is the significance of the conformal invariance of electrodynamics in a covariant formulation?

I am a confused about the role of symmetry transformations in a covariant formulation. Maxwell's equations can be shown to be invariant under conformal transformations. See e.g. here: https://arxiv....
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1answer
170 views

What does covariance/non-covariance mean in QFT?

I'm studying QFT using the book of Mandl and Shaw. In the first chapter they start by quantising the electromagnetic field, but in a "non-covariant" way. What do they mean by that? They have a ...
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144 views

Partial derivatives of Wilson like integrals

I have a one-form field on Euclidean space. Suppose we integrate it over a loop around the specific point $x$. $$I(x)=\int_xU.$$ I want to calculate the partial derivatives of this integral respect to ...
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2answers
177 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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177 views

Usefulness of Curl and Divergence as Multilinear Maps

Early in differential geometry, texts typically reformalize our usual gradient, divergence and curl operators as covariant tensors rather than vectors. This is primarily motivated by the observation ...
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160 views

Since all frames of reference are equal, can we treat the Earth as fixed?

Since Einsteins GR tells us all the frames of reference are equal, is there anything invalid about treating the Earth as unmoving and the universe itself rotating? Other than the fact that the ...
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2answers
205 views

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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124 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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165 views

What Lorentz covariance has to do with Lorentz invariance? [duplicate]

Does saying that the Dirac equation is invariant under Lorentz transformations is the same as saying that it is Lorentz covariant?
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85 views

Problem in deriving elements of Lorentz Group

I am following a lecture series for Classical Field Theory in which they the lecturer uses the invariance of the length of a vector during a Lorentz transformation to derive the equation $$O^{T}\eta~O=...
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1answer
255 views

Does the covariant derivative still give divergence in skew coordinates?

I stumbled upon the formula $\,\,div \, \vec{F}=F^{\mu}_{\,\,\,;\mu}$. Does this still hold true in skew coordinates? I can picture it working geometrically in orthogonal coordinates, but in skew ...
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1answer
890 views

Why is covariant derivative a tensor?

I am trying to prove that the covariant derivative is a tensor (ie it transforms well under a change of coordinates) but I can't succeed to it. Here is the definition of the covariant derivative : $$...
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4answers
683 views

Why do you have to include the Jacobian for every coordinate system, but the Cartesian?

In physics and engineering it is common to convert between different coordinate systems - spherical, polar, Cartesian, e.t.c. - depending on the problem. Physically, they are all clearly equivalent ...
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757 views

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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439 views

Covariant derivative of the variation of the metric

Say you are varying some function (it could be a tensor, or a tensor density, scalar, etc.) that depends on the metric (the metric is what is varying), so you have $\delta f = \frac{\partial f}{\...
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609 views

Co and contravariant: tensors or components?

I am learning Special Relativity and have a question: given a four vector $\vec{x}$ whose contravariant components are $x^\mu$, do the covariant components $x_\mu = g_{\mu\nu}x^\nu$ make reference to ...
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2answers
193 views

Covariant derivative of unit vector/Kronecker delta

Consider: $\nabla_b \delta^a_c=\partial_b\delta^a_c-\Gamma^d_{bc}\delta^a_d+\Gamma^a_{bd}\delta^d_c=0-\Gamma^a_{bc}+\Gamma^a_{bc}=0$ But if I define: $T^a_{bc}=\nabla_b\delta^a_c$ Then I should have:...
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1answer
418 views

Covariant derivative of a Dirac spinor and Kosmann lift

In [1] I have found a definition of the covariant derivative of a Dirac field with a general connection $\omega_{\mu a}{}^{b}$ (with torsion and non-metricity) [see eq. (29)]: $$\nabla_{\mu}\psi=\...
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0answers
144 views

Lagrangian transformation [closed]

I want to prove that the Lagrangian transformation is covariant for $x_{i}\rightarrow q_{i}(x)$ and $x_{i}\rightarrow q_{i}(x,t)$. So far I've proven that it holds for the first transformation as ...
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1answer
399 views

Why are Maxwell's equations not Galilean invariant? [closed]

So i am writing an essay on the conflict between galilean invarience and maxwell's electromagnetism. I am struggling to come up with 3 evidences that they conflict because I have a mediocre ...
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1answer
228 views

Why do we raise and lower indices of tensors of various groups with the invariants of that group?

If $T_{ij}$ is tensor that transforms under $SO(N)$ then apparently (according to what I have been told) it does not matter whether we put the indices up or down. If we instead have a tensor that ...
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3answers
153 views

A postulate in the beginning of special relativity

There’s a postulate in special relativity as following: Physics laws are identical in all inertial reference frames. I’m a math student, recently when I reviewed special relativity before learning ...
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2answers
264 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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2answers
70 views

Tensor in different coordinate system

I have the tensors $F_{\mu\nu}$, $F^{\mu\nu}$ in coordinate system $(t,x,y,z)$ and want to transform these to coordinate system $(t',x',y',z')$ just by multiplicating matrices. My idea was to ...
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2answers
150 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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2answers
189 views

Question about the physical meaning of tensors: why are they used in physics? [closed]

I know that tensors are object we use in general relativity to describe phenomenon. They have the property to have the same expression in various coordinates systems. For example if I take : $\...
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0answers
63 views

Commutator relation of EM Field Covariant?

I read that for quantization of the EM-Field, you demand the canonical equal-time commutation relations: $$[A^\mu(\vec{x},t), \pi^\nu(\vec{y},t)] = i \hbar g^{\mu \nu} \delta^3(\vec{x} - \vec{y}). $$ ...
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482 views

Why is the inner product not invariant under general coordinate transformations?

This came up in some of my reading (Introduction to Tensor Calculus by Kees Dullemond & Kasper Peeters, page 15). Why is the inner product not invariant under general coordinate transformations? ...
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469 views

Why is there an emphasis on tensor equations in GR?

In my understanding the purpose of using tensor equations in GR is to ensure that they are true in all coordinate systems. I understand that writing equations tensorially ensures this will be the case;...
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337 views

Raising and lowering covariant and contravariant bases

The vector $\textbf{a} = a_{i}\textbf{e}^{i}$ in terms of covariant components. In terms of contravariant components, $\textbf{a} = a^{i}\textbf{e}_{i} = a^{j}\textbf{e}_j$. Thus, $a_{i}\textbf{e}^{i} ...
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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 ...