Non-coordinate Basis: Explained

[Silviu]

Hello! I read about this in several place, but I haven't found a really satisfying answer, so here I am. As far as I understand, non-coordinate basis are mainly obtained from coordinate basis, by making the system orthonormal. For example the unit vector in polar coordinates in the direction of $\theta$ gets a factor of $1/r$ to make it orthonormal. I am a bit confused about the difference between the 2 types of basis. To me they seem equivalent in the sense that you can easily go from one to another by doing the right transformation. So beside the fact that one is more useful than the other, depending on the calculations, what is the fundamental difference between them? I am sure I am missing something, but I can't really see why they are so different conceptually.

 

[strangerep]

Non-coordinate bases are also called "non-holonomic" (or "anholonomic") bases.
See https://en.wikipedia.org/wiki/Holonomic_basis

The main conceptual difference is that the tetrad vector field constituting a non-holonomic basis is non-integrable, because the respective vector fields in the tetrad do not commute.

Also try googling "object of anholonomicity".

 

[pervect]

Hello! I read about this in several place, but I haven't found a really satisfying answer, so here I am. As far as I understand, non-coordinate basis are mainly obtained from coordinate basis, by making the system orthonormal. For example the unit vector in polar coordinates in the direction of $\theta$ gets a factor of $1/r$ to make it orthonormal. I am a bit confused about the difference between the 2 types of basis. To me they seem equivalent in the sense that you can easily go from one to another by doing the right transformation. So beside the fact that one is more useful than the other, depending on the calculations, what is the fundamental difference between them? I am sure I am missing something, but I can't really see why they are so different conceptually.

I don't see anything wrong with your understanding. I'd just add that coordinate basis are useful for calculations, especially of covariant derivatives. Non-coordinate bases are useful for physical understanding. So typically one will do calculations that need them in a coordinate basis, then transform to a non-coordinate orthonormal basis for ease of interpreting the physical significance of the calculations.

 

[Orodruin]

I would like to add that a non-holonomic basis need not be orthonormal or even orthogonal. It is just not a holonomic basis. Also, given a basis of the tangent space at any point, it can be extended to a holonomic basis in a neighborhood of that point. Being holonomic or not is a property of a set of vector fields (sections of the tangent bundle) that form a basis of the tangent space at every point. In general, you can choose this set of fields as you would like and your choice may be holonomic or not, it really just determines the basis you work with at each point. As has already been stated, some computations turn out to be simpler if you choose your basis to be holonomic, but there is a priori nothing stopping you from using a nonholonomic, nonorthogonal basis (you just might regret it).

It should also be mentioned that, even if spacetime is, some manifolds are not equipped with a metric. Obviously, orthogonality has no meaning in such a manifold but you can still consider holonomic and nonholonomic bases to your heart’s content.

 

[Demystifier]

Non-coordinate bases are also called "non-holonomic" (or "anholonomic") bases.
See https://en.wikipedia.org/wiki/Holonomic_basis

The main conceptual difference is that the tetrad vector field constituting a non-holonomic basis is non-integrable, because the respective vector fields in the tetrad do not commute.

Also try googling "object of anholonomicity".

I wonder, can the apparent paradox in https://www.physicsforums.com/threads/schwarzschild-from-minkowski.949984/ be resolved in terms of a non-coordinate basis?

Let $\mu,\nu$ be spacetime indices and $a,b$ indices in the tangent space. With tetrads, one usually has
$$\eta^{ab}=e^a_{\mu}e^b_{\nu}g^{\mu\nu}$$
Perhaps the apparent paradox above can be interpreted as a somewhat unusual tetrad formalism in which one has
$$g^{ab}=e^a_{\mu}e^b_{\nu}\eta^{\mu\nu}$$
where $g^{ab}$ are components of the Schwarzschild tensor.

The rational for such an unusual tetrad formalism is this. Given a point in spacetime, one can always choose coordinates in which metric looks like Minkowski near that point. But there is nothing special about Minkowski metric. One can choose coordinates in which metric looks like any metric with signature $(+,-,-,-)$ near that point.

 

[Silviu]

I don't see anything wrong with your understanding. I'd just add that coordinate basis are useful for calculations, especially of covariant derivatives. Non-coordinate bases are useful for physical understanding. So typically one will do calculations that need them in a coordinate basis, then transform to a non-coordinate orthonormal basis for ease of interpreting the physical significance of the calculations.

Thank you for this. Ok, that makes sense. I can see why coordinate basis are useful for calculations but could you please elaborate a bit more about why non-coordinate bases are more useful for physical understanding than the coordinate ones (or maybe point me towards some articles)? I am not sure I can easily understand this.

 

[Orodruin]

Thank you for this. Ok, that makes sense. I can see why coordinate basis are useful for calculations but could you please elaborate a bit more about why non-coordinate bases are more useful for physical understanding than the coordinate ones (or maybe point me towards some articles)? I am not sure I can easily understand this.

Well, even if I adamantly insisted that nonholonomic bases are not necessarily orthonormal, you can choose an orthonormal nonholonomic basis. It is then much easier to interpret the resulting vector and tensor components physically.

 

[Demystifier]

why non-coordinate bases are more useful for physical understanding than the coordinate ones

Suppose, for instance, that you define positions in space by positions relative to a set of satellites that move independently of each other. Each satellite can be thought of as a physical object that defines a local tetrad. This set of satellites is a physical model for a non-coordinate basis.

For a physical model of a coordinate basis you would need a large physical mesh, different parts of which do not move independently. In practice, that would be much more difficult to achieve.