# 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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### Why is $\frac{d^2x^{\mu}}{d\lambda^2}=0$ not a tensorial equation?

In flat space, the motion of freely falling particles given by the parametrized path $x^\mu(\lambda)$ is given by the geodesic equation $$\frac{d^2x^{\mu}}{d\lambda^2}=0.$$ Why is this not a tensorial ...
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### Covariant Maxwell equations

As we know, the covariant form of Maxwell's equations (there are 2 equations in this formulation) are covariant under Lorentz transformation. Are these equations covariant under general transformation，...
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### Question on the Einstein-Hilbert action

Does it make sense to write that the Einstein-Hilbert action as \begin{equation} S=\int\mathcal{L}\left(g^{\mu\nu},\partial^{\lambda}g^{\mu\nu}\right)\sqrt{-g}\,\mathrm{d}^4x=\frac{1}{2\kappa}\int R\,\...
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### What physical properties are invariant under relativistic transformation?

Most of the familiar physical properties vary according to the relativistic observer's reference frame - speed, mass, energy, time, length, etc. Which properties remain invariant, so everybody will ...
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### Autocorrelation and variance: can the fluctuation-dissipation theorem actually be written in terms of fluctuations?

I am considering the theorem in a statistical mechanics context, but I suppose the question could be extended to other fields where it applies as well. If we have a system with property $A$ and apply ...
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### Why do the author define energy using one-form? [duplicate]

I was reading the book First course on GR from Schurz. In the latter chapters the author is going to calculate how does the motion of a photon is affected by a spherical symmetric metric. See define ...
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### How do we know that solving Euler-Lagrange equation gives us the correct equation of motion in any coordinate?

As far as I understand it: If we defined a quantity called the Lagrangian as the difference between the kinetic energy and potential energy for a particle in one dimension in Cartesian coordinates, ...
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### Co-ordinate invariance in Lagrangian form of equations

I have read that in his Mecanique Analytique , Lagrange sought a “coordinate invariant expression for mass times acceleration”. The discussion regarding this is given in 'Marsden and Ratiu [15, ...
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### Has the Klein-Gordon equation in curved spacetimes the same form as in flat ones? [duplicate]

The KG equation in curved geometries has the following form: $$\frac{1}{\sqrt{-g}}\partial_\mu(\sqrt{-g}~g^{\mu \nu}\partial_\nu\phi) + m^2\phi = 0,$$ where $g$ is the determinant of the metric tensor ...
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### Is there a difference between base vectors and projection method of calculating coordinates? [duplicate]

I'm trying to grasp the idea of co- and contra-variance. During my study I meet something like that: Let's say we have a vector $\vec{v}$ and want to calculate its coordinates in base $\hat{b}$. I've ...
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### Vectors as functions?

In my study of general relativity, I came across tensors. First, I realized that vectors (and covectors and tensors) are objects whose components transform in a certain way (such that the underlying ...
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### Change of coordinate vs change of reference axes

Does basis vectors change opposite to coordinate scaling ? For example, suppose I have some oblique coordinate system, and I decide to scale up both 'axes' by a factor of $a$ and $b$ respectively. The ...
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### Covariant or contravariant nature of Gradient

I've been having this confusion regarding the gradient being a covariant vector. Intuitively I seem to have understood the concept. However, mathematically, I'm unable to show this, in a single ...
255 views

### Geometrical representation of Contravariant and covariant vectors

After cruising through a lot of material online, and answers over here, my understanding of contravariant and covariant vectors are, in a finite-dimensional vector space, suppose we have a vector, ...