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

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

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

d f d λ = d x α d λ ∂ f ∂ x α = u α ∂ ∂ x α f . {\displaystyle {\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 = 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]

[ e α , e β ] = L e α e β = 0. {\displaystyle \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 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

  1. ^ M. P. Hobson; G. P. Efstathiou; A. N. Lasenby (2006), General Relativity: An Introduction for Physicists, Cambridge University Press, p. 57
  2. ^ T. Padmanabhan (2010), Gravitation: Foundations and Frontiers, Cambridge University Press, p. 25
  3. ^ Roger Penrose; Wolfgang Rindler, Spinors and Space–Time: Volume 1, Two-Spinor Calculus and Relativistic Fields, Cambridge University Press, pp. 197–199
  4. ^ Charles W. Misner; Kip S. Thorne; John Archibald Wheeler (1970), Gravitation, p. 210
  5. ^ Bernard F. Schutz (1980), Geometrical Methods of Mathematical Physics, Cambridge University Press, pp. 47–49, ISBN 978-0-521-29887-2

See also

  • Jet bundle
  • Tetrad formalism
  • Ricci calculus


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