Tensor[HodgeStar] - apply the Hodge star operator to a differential form
Calling Sequences
HodgeStar(g, omega)
Parameters
g - a metric tensor
omega - a differential form
option - (optional) the keyword argument detmetric
Description
Examples
The Hodge star operator is customarily denoted by ∗ and maps p-forms on an n-dimensional manifold to n−p-forms. To define the Hodge star operator, let ω1,ω2,...,ωn be an orthonormal basis of 1-forms with respect to the metric g. Pick integers i1,i2,...,ir with 1≤i1<i2...<ir≤n and let 1≤j1<j2<...<js≤n be the complementary set of integers r=n−s. Let ω=ωi1∧ωi2∧ ... ∧ωir. Then
∗ω=±ωj1∧ωj2∧ ... ∧ωjs,
where the sign±is chosen so that ω∧∗ω=ω1∧ω2∧ ... ∧ωn.
The command HodgeStar(g, omega) returns the Hodge star of the form omega with respect to the metric g.
The command HodgeStar(g, omega) calculates the Hodge star of the form omega as follows. [i] Convert omega to skew-symmetric, covariant rank r tensor T. [ii] Fully contract T with the contravariant permutation symbol and multiply by the weight 1 scalar density defined by the metric g. The result is a skew-symmetric contravariant rank n-r tensor S. [iii] Lower all the indices of S with the metric g and convert the resulting skew-symmetric, contravariant rank n-r tensor to a differential form.
By default, it is assumed that the metric g has positive determinant. To calculate the Hodge star operation with respect to a metric with negative determinant, include the keyword argument detmetric = -1.
This command is part of the DifferentialGeometry:-Tensor package, and so can be used in the form HodgeStar(...) only after executing the command with(DifferentialGeometry) and with(Tensor) in that order. It can always be used in the long form DifferentialGeometry:-Tensor:-HodgeStar.
withDifferentialGeometry:withTensor:
Example 1.
First create a 5-dimensional manifold M and define a metric tensor g on the tangent space of M.
DGsetupx1,x2,x3,x4,x5,M1:
g≔evalDGdx1&tdx1+dx2&tdx2+dx3&tdx3+dx4&tdx4+dx5&tdx5
g:=dx1dx1+dx2dx2+dx3dx3+dx4dx4+dx5dx5
The standard basis dx1,dx2, ...,dx5 is an orthonormal basis for g and therefore the Hodge star is easily computed.
HodgeStarg,dx1
dx2⋀dx3⋀dx4⋀dx5
HodgeStarg,dx2
−dx1⋀dx3⋀dx4⋀dx5
HodgeStarg,dx2&wdx3
dx1⋀dx4⋀dx5
HodgeStarg,dx2&wdx4
−dx1⋀dx3⋀dx5
HodgeStarg,dx2&wdx3&wdx4
−dx1⋀dx5
Example 2.
To show the dependence of the Hodge star upon the metric, we consider a general metric g on a 2-dimensional manifold.
DGsetupx,y,M2:
g≔evalDGadx&tdx+bdx&tdy+dy&tdx+cdy&tdy
g:=adxdx+bdxdy+bdydx+cdydy
HodgeStarg,dx
1ac−b2bdx+1ac−b2cdy
HodgeStarg,dy
−1ac−b2adx−1ac−b2bdy
f≔HodgeStarg,dx&wdy
f:=1ac−b2
HodgeStarg,f
dx⋀dy
Example 3.
The Laplacian of a function with respect to a metric g can be calculated using the exterior derivative operation and the Hodge star operator.
To illustrate this result, we use the Euclidean metric in polar coordinates r,ϑ.
DGsetupr,θ,M3:
g≔evalDGdr&tdr+r2dtheta&tdtheta
g:=drdr+r2dthetadtheta
To simplify the definition of the Laplacian, we define the Hodge operator with g fixed.
Hodge≔f↦HodgeStarg,fassuming0<r
Hodge:=f→DifferentialGeometry:-Tensor:-HodgeStarg,fassuming0<r
To display the Laplacian of φ in compact form we invoke the PDEtools[declare] command.
PDEtoolsdeclareφr,θ
φr,θwill now be displayed asφ
Here is the formula for the Laplacian in terms of HodgeStar and ExteriorDerivative. Recall that @ is the composition of functions.
Δ≔Hodge@ExteriorDerivative@Hodge@ExteriorDerivativeφr,θ
Δ:=rφr+r2φr,r+φθ,θr2
Example 4.
The HodgeStar program also works in the more general context of a vector bundle E→M.
DGsetupx,y,u,v,w,E
frame name: E
g≔evalDGdu&tdu+dv&tdv+dw&tdw
g:=dudu+dvdv+dwdw
HodgeStarg,du&wdv−3du&wdw+2dv&wdw
2du+3dv+dw
Example 5.
The HodgeStar operation can also be performed using an indefinite metric. The keyword argument detmetric = -1 must be used when the metric has negative determinant.
DGsetupx1,x2,x3,x4,M5:
g≔evalDGdx1&tdx1+dx2&tdx2+dx3&tdx3−dx4&tdx4
g:=dx1dx1+dx2dx2+dx3dx3−dx4dx4
HodgeStarg,dx1,detmetric=−1
dx2⋀dx3⋀dx4
HodgeStarg,dx3&wdx4,detmetric=−1
−dx1⋀dx2
See Also
DifferentialGeometry
Tensor
DGinfo
ExteriorDerivative
MetricDensity
PermutationSymbol
RaiseLowerIndices
Download Help Document
