What is the significance of the nonholonomity condition in General Relativity?

[samson]

Hi everyone! Please what are the conditions necessary for space and time to be nonholonomic?

 

[PeterDonis]

Holonomic is a condition on a set of basis vectors, not spacetime. See here:

https://en.wikipedia.org/wiki/Holonomic_basis

 

[pervect]

To give a specific example. In polar coordinates, $r, \theta$, a vector field that consists of unit radial vectors (often written as $\hat{r}$), and orthogonal tangential vectors (often written as $\hat{\theta}$) would be a non-holonomic basis. It might be more familiar in terms of the dual vectors (though the terminology might not be familiar, perhaps). dr and d$\theta$ can be interpreted as dual vectors. Then dr and d$\theta$ would be holonomic, and not normal (because the vectors don't have unit lengths). However, dr and $r \, d\theta$ would be non-holonmic, but normal, since the (dual) vectors do have unit length.

The notion of vectors as partial derivative operators and dual vectors gets some mention in Sean Caroll's lecture notes <<link>>, see for instance

In spacetime the simplest example of a dual vector is the gradient of a scalar function, the set of partial derivatives with respect to the spacetime coordinates, which we denote by “d”:

It's also worth looking up what a coordinate basis is. Note that "coordinate basis" is more or less just another term for "holonomic basis". (Perhaps some more mathematical sort will correct me if there is some tiny difference). If you look up "coordinate basis" in Caroll's lecture notes, you'll find that it's identified as being equiavalent to partial derivatives with respect to the coordaintes. So in our example a coordinate basis of vectors could be identified with partial derivative operators, such as $\partial_r$ or in different notation $\frac{\partial}{\partial r}$. The identification of vectors as partial derivative operators is not very intuitive, but an important definition for differential geometry.

 

[Ben Niehoff]

A holonomic basis is a basis where all of the basis vectors commute. Given a holonomic basis, it is possible to choose coordinates $x^\mu$ such that the basis vectors are the set of partial derivative operators $\partial/\partial x^\mu$.

 

[samson]

Hi everyone! Please what are the conditions necessary for space and time to be nonholonomic?

This is very helpful to me! Thanks for your time sir!

 

[samson]

A holonomic basis is a basis where all of the basis vectors commute. Given a holonomic basis, it is possible to choose coordinates $x^\mu$ such that the basis vectors are the set of partial derivative operators $\partial/\partial x^\mu$.

Thanks sir