Holonomic basis

In mathematics and mathematical physics, a coordinate basis or holonomic basis for a differentiable manifold $M$ is a set of basis vector fields ${e 1, ..., e n}$ defined at every point $P$ of a region of the manifold as

\mathbf {e} _{\alpha }=\lim _{\delta x^{\alpha }\to 0}{\frac {\delta \mathbf {s} }{\delta x^{\alpha }}},

where $δ s$ is the infinitesimal displacement vector between the point $P$ and a nearby point $Q$ whose coordinate separation from $P$ is $δx α$ along the coordinate curve $x α$ (i.e. the curve on the manifold through $P$ for which the local coordinate $x α$ varies and all other coordinates are constant).[1]

It is possible to make an association between such a basis and directional derivative operators. Given a parameterized curve $C$ on the manifold defined by $x α (λ)$ with the tangent vector $u = u α e α$ , where $u α = dx α / dλ$ , and a function $f (x α)$ defined in a neighbourhood of $C$ , the variation of $f$ along $C$ can be written as

{\frac {df}{d\lambda }}={\frac {dx^{\alpha }}{d\lambda }}{\frac {\partial f}{\partial x^{\alpha }}}=u^{\alpha }{\frac {\partial }{\partial x^{\alpha }}}f.

Since we have that $u = u α e α$ , the identification is often made between a coordinate basis vector $e α$ and the partial derivative operator $\partial / \partial x α$ , under the interpretation of vectors as operators acting on scalar quantities.[2]

A local condition for a basis ${e 1, ..., e n}$ to be holonomic is that all mutual Lie derivatives vanish:[3]

\left[\mathbf {e} _{\alpha },\mathbf {e} _{\beta }\right]={\mathcal {L}}_{\mathbf {e} _{\alpha }}\mathbf {e} _{\beta }=0.

A basis that is not holonomic is called a non-holonomic or non-coordinate basis.

Given a metric tensor $g$ on a manifold $M$ , it is in general not possible to find a coordinate basis that is orthonormal in any open region $U$ of $M$ .[4] An obvious exception is when $M$ is the real coordinate space $R n$ considered as a manifold with $g$ being the Euclidean metric $δ ij e i \otimes e j$ at every point.

References

M. P. Hobson; G. P. Efstathiou; A. N. Lasenby (2006), General Relativity: An Introduction for Physicists, Cambridge University Press, p. 57
T. Padmanabhan (2010), Gravitation: Foundations and Frontiers, Cambridge University Press, p. 25
Roger Penrose; Wolfgang Rindler, Spinors and Space–Time: Volume 1, Two-Spinor Calculus and Relativistic Fields, Cambridge University Press, pp. 197–199
Bernard F. Schutz (1980), Geometrical Methods of Mathematical Physics, Cambridge University Press, pp. 47–49, ISBN 9780521298872

Holonomic basis

References

See also