Holonomic basis

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

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

where {{math|δs}} is the displacement vector between the point {{math|P}} and a nearby point

{{math|Q}} whose coordinate separation from {{math|P}} is {{math|δx{{sup|α}}}} along the coordinate curve {{math|x{{sup|α}}}} (i.e. the curve on the manifold through {{math|P}} for which the local coordinate {{math|x{{sup|α}}}} varies and all other coordinates are constant).{{refn|{{citation |author1=M. P. Hobson |author2=G. P. Efstathiou |author3=A. N. Lasenby |year=2006 |title=General Relativity: An Introduction for Physicists |publisher=Cambridge University Press |page=57 }}}}

It is possible to make an association between such a basis and directional derivative operators. Given a parameterized curve {{math|C}} on the manifold defined by {{math|x{{sup|α}}(λ)}} with the tangent vector {{math|1=u = u{{sup|α}}e{{sub|α}}}}, where {{math|1=u{{sup|α}} = {{sfrac|dx{{sup|α}}|dλ}}}}, and a function {{math|f(x{{sup|α}})}} defined in a neighbourhood of {{math|C}}, the variation of {{math|f}} along {{math|C}} can be written as

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

Since we have that {{math|1=u = u{{sup|α}}e{{sub|α}}}}, the identification is often made between a coordinate basis vector {{math|e{{sub|α}}}} and the partial derivative operator {{math|{{sfrac|∂|∂x{{sup|α}}}}}}, under the interpretation of vectors as operators acting on functions.{{refn|{{citation |author=T. Padmanabhan |year=2010 |title=Gravitation: Foundations and Frontiers |publisher=Cambridge University Press |page=25}}}}

A local condition for a basis {{math|{e{{sub|1}}, ..., e{{sub|n}}}{{null}}}} to be holonomic is that all mutual Lie derivatives vanish:{{refn|{{citation |author1=Roger Penrose|author2=Wolfgang Rindler |title=Spinors and Space–Time: Volume 1, Two-Spinor Calculus and Relativistic Fields |publisher=Cambridge University Press |pages=197–199 }}}}

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

A basis that is not holonomic is called an anholonomic,{{refn|{{citation |author=Charles W. Misner |author2=Kip S. Thorne |author3=John Archibald Wheeler |year=1970 |title=Gravitation |page=210 }}}} non-holonomic or non-coordinate basis.

Given a metric tensor {{math|g}} on a manifold {{math|M}}, it is in general not possible to find a coordinate basis that is orthonormal in any open region {{math|U}} of {{math|M}}.{{refn|{{citation |author=Bernard F. Schutz |year=1980 |title=Geometrical Methods of Mathematical Physics |publisher=Cambridge University Press |pages=47–49 |isbn=978-0-521-29887-2 }}}} An obvious exception is when {{math|M}} is the real coordinate space {{math|R{{sup|n}}}} considered as a manifold with {{math|g}} being the Euclidean metric {{math|δ{{sub|ij }}e{{sup|i}} ⊗ e{{i sup|j}}}} at every point.