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 {e1, ..., en} defined at every point P of a region of the manifold as
- [math]\displaystyle{ \mathbf{e}_{\alpha} = \lim_{\delta x^{\alpha} \to 0} \frac{\delta \mathbf{s}}{\delta x^{\alpha}} , }[/math]
where δs is the displacement vector between the point P and a nearby point Q whose coordinate separation from P is δxα along the coordinate curve xα (i.e. the curve on the manifold through P for which the local coordinate xα 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 xα(λ) with the tangent vector u = uαeα, where uα = dxα/dλ, and a function f(xα) defined in a neighbourhood of C, the variation of f along C can be written as
- [math]\displaystyle{ \frac{df}{d\lambda} = \frac{dx^{\alpha}}{d\lambda}\frac{\partial f}{\partial x^{\alpha}} = u^{\alpha} \frac{\partial }{\partial x^{\alpha}} f . }[/math]
Since we have that u = uαeα, the identification is often made between a coordinate basis vector eα and the partial derivative operator ∂/∂xα, under the interpretation of vectors as operators acting on functions.[2]
A local condition for a basis {e1, ..., en} to be holonomic is that all mutual Lie derivatives vanish:[3]
- [math]\displaystyle{ \left[ \mathbf{e}_{\alpha} , \mathbf{e}_{\beta} \right] = {\mathcal{L}}_{\mathbf{e}_{\alpha}} \mathbf{e}_{\beta} = 0 . }[/math]
A basis that is not holonomic is called an anholonomic,[4] 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.[5] An obvious exception is when M is the real coordinate space Rn considered as a manifold with g being the Euclidean metric δij ei ⊗ ej 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
- ↑ Charles W. Misner; Kip S. Thorne; John Archibald Wheeler (1970), Gravitation, p. 210
- ↑ Bernard F. Schutz (1980), Geometrical Methods of Mathematical Physics, Cambridge University Press, pp. 47–49, ISBN 978-0-521-29887-2
See also
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