Hodge star operator

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Short description: Exterior algebraic map taking tensors from p forms to n-p forms

In mathematics, the Hodge star operator or Hodge star is a linear map defined on the exterior algebra of a finite-dimensional oriented vector space endowed with a nondegenerate symmetric bilinear form. Applying the operator to an element of the algebra produces the Hodge dual of the element. This map was introduced by W. V. D. Hodge.

For example, in an oriented 3-dimensional Euclidean space, an oriented plane can be represented by the exterior product of two basis vectors, and its Hodge dual is the normal vector given by their cross product; conversely, any vector is dual to the oriented plane perpendicular to it, endowed with a suitable bivector. Generalizing this to an n-dimensional vector space, the Hodge star is a one-to-one mapping of k-vectors to (n – k)-vectors; the dimensions of these spaces are the binomial coefficients (nk)=(nnk).

The naturalness of the star operator means it can play a role in differential geometry, when applied to the cotangent bundle of a pseudo-Riemannian manifold, and hence to differential k-forms. This allows the definition of the codifferential as the Hodge adjoint of the exterior derivative, leading to the Laplace–de Rham operator. This generalizes the case of 3-dimensional Euclidean space, in which divergence of a vector field may be realized as the codifferential opposite to the gradient operator, and the Laplace operator on a function is the divergence of its gradient. An important application is the Hodge decomposition of differential forms on a closed Riemannian manifold.

Formal definition for k-vectors

Let V be an n-dimensional oriented vector space with a nondegenerate symmetric bilinear form ,, referred to here as an inner product. (In more general contexts such as pseudo-Riemannian manifolds and Minkowski space, the bilinear form may not be positive.) This induces an inner product on k-vectors α,βkV, for 0kn, by defining it on decomposable k-vectors α=α1αk and β=β1βk to equal the Gram determinant[1]:14

α,β=det(αi,βji,j=1k)

extended to kV through linearity.

The unit n-vector ωnV is defined in terms of an oriented orthonormal basis {e1,,en} of V as:

ω:=e1en.

(Note: In the general pseudo-Riemannian case, orthonormality means ei,ej{δij,δij} for all pairs of basis vectors.) The Hodge star operator is a linear operator on the exterior algebra of V, mapping k-vectors to (nk)-vectors, for 0kn. It has the following property, which defines it completely:[1]:15

α(β)=α,βω for all k-vectors α,βkV.

Dually, in the space nV*of n-forms (alternating n-multilinear functions on Vn), the dual to ω is the volume form det, the function whose value on v1vn is the determinant of the n×n matrix assembled from the column vectors of vi in ei-coordinates. Applying det to the above equation, we obtain the dual definition:

det(αβ)=α,β for all k-vectors α,βkV.

Equivalently, taking α=α1αk, β=β1βk, and β=β1βnk:

det(α1αkβ1βnk) = det(αi,βj).

This means that, writing an orthonormal basis of k-vectors as eI = ei1eik over all subsets I={i1<<ik} of [n]={1,,n}, the Hodge dual is the (n – k)-vector corresponding to the complementary set I¯=[n]I={i¯1<<i¯nk}:

eI=steI¯,

where s{1,1} is the sign of the permutation i1iki¯1i¯nk and t{1,1} is the product ei1,ei1eik,eik. In the Riemannian case, t=1.

Since Hodge star takes an orthonormal basis to an orthonormal basis, it is an isometry on the exterior algebra V.

Geometric explanation

The Hodge star is motivated by the correspondence between a subspace W of V and its orthogonal subspace (with respect to the inner product), where each space is endowed with an orientation and a numerical scaling factor. Specifically, a non-zero decomposable k-vector w1wkkV corresponds by the Plücker embedding to the subspace W with oriented basis w1,,wk, endowed with a scaling factor equal to the k-dimensional volume of the parallelepiped spanned by this basis (equal to the Gramian, the determinant of the matrix of inner products wi,wj). The Hodge star acting on a decomposable vector can be written as a decomposable (nk)-vector:

(w1wk)=u1unk,

where u1,,unk form an oriented basis of the orthogonal space U=W. Furthermore, the (nk)-volume of the ui-parallelepiped must equal the k-volume of the wi-parallelepiped, and w1,,wk,u1,,unk must form an oriented basis of V.

A general k-vector is a linear combination of decomposable k-vectors, and the definition of Hodge star is extended to general k-vectors by defining it as being linear.

Examples

Two dimensions

In two dimensions with the normalized Euclidean metric and orientation given by the ordering (x, y), the Hodge star on k-forms is given by 1=dxdydx=dydy=dx(dxdy)=1.

On the complex plane regarded as a real vector space with the standard sesquilinear form as the metric, the Hodge star has the remarkable property that it is invariant under holomorphic changes of coordinate. If z = x + iy is a holomorphic function of w = u + iv, then by the Cauchy–Riemann equations we have that x/u = y/v and y/u = −x/v. In the new coordinates α = pdx+qdy = (pxu+qyu)du+(pxv+qyv)dv = p1du+q1dv, so that α=q1du+p1dv=(pxv+qyv)du+(pxu+qyu)dv=q(xudu+xvdv)+p(yudu+yvdv)=qdx+pdy, proving the claimed invariance.

Three dimensions

A common example of the Hodge star operator is the case n = 3, when it can be taken as the correspondence between vectors and bivectors. Specifically, for Euclidean R3 with the basis dx,dy,dz of one-forms often used in vector calculus, one finds that dx=dydzdy=dzdxdz=dxdy.

The Hodge star relates the exterior and cross product in three dimensions:[2] (𝐮𝐯)=𝐮×𝐯(𝐮×𝐯)=𝐮𝐯. Applied to three dimensions, the Hodge star provides an isomorphism between axial vectors and bivectors, so each axial vector a is associated with a bivector A and vice versa, that is:[2] 𝐀=𝐚,  𝐚=𝐀.

The Hodge star can also be interpreted as a form of the geometric correspondence between an axis of rotation and an infinitesimal rotation (see also: 3D rotation group#Lie algebra) around the axis, with speed equal to the length of the axis of rotation. An inner product on a vector space V gives an isomorphism VV* identifying V with its dual space, and the vector space L(V,V) is naturally isomorphic to the tensor product V*VVV. Thus for V=3, the star mapping :V2VVV takes each vector 𝐯 to a bivector 𝐯VV, which corresponds to a linear operator L𝐯:VV. Specifically, L𝐯 is a skew-symmetric operator, which corresponds to an infinitesimal rotation: that is, the macroscopic rotations around the axis 𝕧 are given by the matrix exponential exp(tL𝐯). With respect to the basis dx,dy,dz of 3, the tensor dxdy corresponds to a coordinate matrix with 1 in the dx row and dy column, etc., and the wedge dxdy=dxdydydx is the skew-symmetric matrix [010100000], etc. That is, we may interpret the star operator as: 𝐯=adx+bdy+cdz𝐯  L𝐯 =[0cbc0aba0]. Under this correspondence, cross product of vectors corresponds to the commutator Lie bracket of linear operators: L𝐮×𝐯=L𝐮L𝐯L𝐯L𝐮.

Four dimensions

In case n=4, the Hodge star acts as an endomorphism of the second exterior power (i.e. it maps 2-forms to 2-forms, since 4 − 2 = 2). If the signature of the metric tensor is all positive, i.e. on a Riemannian manifold, then the Hodge star is an involution. If the signature is mixed, i.e., pseudo-Riemannian, then applying the operator twice will return the argument up to a sign – see § Duality below. This particular endomorphism property of 2-forms in four dimensions makes self-dual and anti-self-dual two-forms natural geometric objects to study. That is, one can describe the space of 2-forms in four dimensions with a basis that "diagonalizes" the Hodge star operator with eigenvalues ±1 (or ±i, depending on the signature).

For concreteness, we discuss the Hodge star operator in Minkowski spacetime where n=4 with metric signature (− + + +) and coordinates (t,x,y,z). The volume form is oriented as ε0123=1. For one-forms, dt=dxdydz,dx=dtdydz,dy=dtdxdz,dz=dtdxdy, while for 2-forms, (dtdx)=dydz,(dtdy)=dzdx,(dtdz)=dxdy,(dxdy)=dtdz,(dxdz)=dtdy,(dydz)=dtdx.

These are summarized in the index notation as (dxμ)=ημλελνρσ13!dxνdxρdxσ,(dxμdxν)=ημκηνλεκλρσ12!dxρdxσ.

Hodge dual of three- and four-forms can be easily deduced from the fact that, in the Lorentzian signature, ()2=1 for odd-rank forms and ()2=1 for even-rank forms. An easy rule to remember for these Hodge operations is that given a form α, its Hodge dual α may be obtained by writing the components not involved in α in an order such that α(α)=dtdxdydz.[verification needed] An extra minus sign will enter only if α contains dt. (For (+ − − −), one puts in a minus sign only if α involves an odd number of the space-associated forms dx, dy and dz.)

Note that the combinations (dxμdxν)±:=12(dxμdxνi(dxμdxν)) take ±i as the eigenvalue for Hodge star operator, i.e., (dxμdxν)±=±i(dxμdxν)±, and hence deserve the name self-dual and anti-self-dual two-forms. Understanding the geometry, or kinematics, of Minkowski spacetime in self-dual and anti-self-dual sectors turns out to be insightful in both mathematical and physical perspectives, making contacts to the use of the two-spinor language in modern physics such as spinor-helicity formalism or twistor theory.

Conformal invariance

The Hodge star is conformally invariant on n forms on a 2n dimensional vector space V, i.e. if g is a metric on V and λ>0, then the induced Hodge stars g,λg:ΛnVΛnV are the same.

Example: Derivatives in three dimensions

The combination of the operator and the exterior derivative d generates the classical operators grad, curl, and div on vector fields in three-dimensional Euclidean space. This works out as follows: d takes a 0-form (a function) to a 1-form, a 1-form to a 2-form, and a 2-form to a 3-form (and takes a 3-form to zero). For a 0-form f=f(x,y,z), the first case written out in components gives: df=fxdx+fydy+fzdz.

The inner product identifies 1-forms with vector fields as dx(1,0,0), etc., so that df becomes gradf=(fx,fy,fz).

In the second case, a vector field 𝐅=(A,B,C) corresponds to the 1-form φ=Adx+Bdy+Cdz, which has exterior derivative: dφ=(CyBz)dydz+(CxAz)dxdz+(BxAy)dxdy.

Applying the Hodge star gives the 1-form: dφ=(CyBz)dx(CxAz)dy+(BxAy)dz, which becomes the vector field curl𝐅=(CyBz,Cx+Az,BxAy).

In the third case, 𝐅=(A,B,C) again corresponds to φ=Adx+Bdy+Cdz. Applying Hodge star, exterior derivative, and Hodge star again: φ=AdydzBdxdz+Cdxdy,dφ=(Ax+By+Cz)dxdydz,dφ=Ax+By+Cz=div𝐅.

One advantage of this expression is that the identity d2 = 0, which is true in all cases, has as special cases two other identities: 1) curl grad f = 0, and 2) div curl F = 0. In particular, Maxwell's equations take on a particularly simple and elegant form, when expressed in terms of the exterior derivative and the Hodge star. The expression d (multiplied by an appropriate power of -1) is called the codifferential; it is defined in full generality, for any dimension, further in the article below.

One can also obtain the Laplacian Δf = div grad f in terms of the above operations: Δf=ddf=2fx2+2fy2+2fz2.

The Laplacian can also be seen as a special case of the more general Laplace–deRham operator Δ=dδ+δd where δ=(1)kd is the codifferential for k-forms. Any function f is a 0-form, and δf=0 and so this reduces to the ordinary Laplacian. For the 1-form φ above, the codifferential is δ=d and after some straightforward calculations one obtains the Laplacian acting on φ.

Duality

Applying the Hodge star twice leaves a k-vector unchanged except possibly for its sign: for ηkV in an n-dimensional space V, one has

η=(1)k(nk)sη,

where s is the parity of the signature of the inner product on V, that is, the sign of the determinant of the matrix of the inner product with respect to any basis. For example, if n = 4 and the signature of the inner product is either (+ − − −) or (− + + +) then s = −1. For Riemannian manifolds (including Euclidean spaces), we always have s = 1.

The above identity implies that the inverse of can be given as

1:kVnkVη(1)k(nk)sη

If n is odd then k(nk) is even for any k, whereas if n is even then k(nk) has the parity of k. Therefore:

1={sn is odd(1)ksn is even

where k is the degree of the element operated on.

On manifolds

For an n-dimensional oriented pseudo-Riemannian manifold M, we apply the construction above to each cotangent space Tp*M and its exterior powers kTp*M, and hence to the differential k-forms ζΩk(M)=Γ(kT*M), the global sections of the bundle kT*MM. The Riemannian metric induces an inner product on kTp*M at each point pM. We define the Hodge dual of a k-form ζ, defining ζ as the unique (nk)-form satisfying ηζ = η,ζω for every k-form η, where η,ζ is a real-valued function on M, and the volume form ω is induced by the Riemannian metric. Integrating this equation over M, the right side becomes the L2 (square-integrable) inner product on k-forms, and we obtain: Mηζ = Mη,ζ ω.

More generally, if M is non-orientable, one can define the Hodge star of a k-form as a (nk)-pseudo differential form; that is, a differential form with values in the canonical line bundle.

Computation in index notation

We compute in terms of tensor index notation with respect to a (not necessarily orthonormal) basis {x1,,xn} in a tangent space V=TpM and its dual basis {dx1,,dxn} in V*=Tp*M, having the metric matrix (gij)=(xi,xj) and its inverse matrix (gij)=(dxi,dxj). The Hodge dual of a decomposable k-form is: (dxi1dxik) = |det[gij]|(nk)!gi1j1gikjkεj1jndxjk+1dxjn.

Here εj1jn is the Levi-Civita symbol with ε1n=1, and we implicitly take the sum over all values of the repeated indices j1,,jn. The factorial (nk)! accounts for double counting, and is not present if the summation indices are restricted so that jk+1<<jn. The absolute value of the determinant is necessary since it may be negative, as for tangent spaces to Lorentzian manifolds.

An arbitrary differential form can be written as follows: α = 1k!αi1,,ikdxi1dxik = i1<<ikαi1,,ikdxi1dxik.

The factorial k! is again included to account for double counting when we allow non-increasing indices. We would like to define the dual of the component αi1,,ik so that the Hodge dual of the form is given by α=1(nk)!(α)ik+1,,indxik+1dxin.

Using the above expression for the Hodge dual of dxi1dxik, we find:[3] (α)ik+1,,in=|det[gab]|k!αi1,,ikεi1,,in.

Although one can apply this expression to any tensor α, the result is antisymmetric, since contraction with the completely anti-symmetric Levi-Civita symbol cancels all but the totally antisymmetric part of the tensor. It is thus equivalent to antisymmetrization followed by applying the Hodge star.

The unit volume form ω=1nV* is given by: ω=|det[gij]|dx1dxn.

Codifferential

The most important application of the Hodge star on manifolds is to define the codifferential δ on k-forms. Let δ=(1)n(k1)+1s d=(1)k1d where d is the exterior derivative or differential, and s=1 for Riemannian manifolds. Then d:Ωk(M)Ωk+1(M) while δ:Ωk(M)Ωk1(M).

The codifferential is not an antiderivation on the exterior algebra, in contrast to the exterior derivative.

The codifferential is the adjoint of the exterior derivative with respect to the square-integrable inner product: η,δζ = dη,ζ, where ζ is a (k + 1)-form and η a k-form. This property is useful as it can be used to define the codifferential even when the manifold is non-orientable (and the Hodge star operator not defined). The identity can be proved from Stokes' theorem for smooth forms: 0 = Md(ηζ) = M(dηζη(1)k+11dζ) = dη,ζη,δζ, provided M has empty boundary, or η or ζ has zero boundary values. (The proper definition of the above requires specifying a topological vector space that is closed and complete on the space of smooth forms. The Sobolev space is conventionally used; it allows the convergent sequence of forms ζiζ (as i) to be interchanged with the combined differential and integral operations, so that η,δζiη,δζ and likewise for sequences converging to η.)

Since the differential satisfies d2=0, the codifferential has the corresponding property δ2=s2dd=(1)k(nk)s3d2=0.

The Laplace–deRham operator is given by Δ=(δ+d)2=δd+dδ and lies at the heart of Hodge theory. It is symmetric: Δζ,η=ζ,Δη and non-negative: Δη,η0.

The Hodge star sends harmonic forms to harmonic forms. As a consequence of Hodge theory, the de Rham cohomology is naturally isomorphic to the space of harmonic k-forms, and so the Hodge star induces an isomorphism of cohomology groups :HΔk(M)HΔnk(M), which in turn gives canonical identifications via Poincaré duality of H k(M) with its dual space.

In coordinates, with notation as above, the codifferential of the form α may be written as δα= 1k!gml(xlαm,i1,,ik1Γmljαj,i1,,ik1)dxi1dxik1, where here Γmlj denotes the Christoffel symbols of {x1,,xn}.

Poincare lemma for codifferential

In analogy to the Poincare lemma for exterior derivative, one can define its version for codifferential, which reads[4]

If δω=0 for ωΛk(U), where U is a star domain on a manifold, then there is αΛk+1(U) such that ω=δα.

A practical way of finding α is to use cohomotopy operator h, that is a local inverse of δ. One has to define a homotopy operator[4]

Hβ=01𝒦β|F(t,x)tkdt,

where F(t,x)=x0+t(xx0) is the linear homotopy between its center x0U and a point xU, and the (Euler) vector 𝒦=i=1n(xx0)ixi for n=dim(U) is inserted into the form βΛ*(U). We can then define cohomotopy operator as[4]

h:Λ(U)Λ(U),h:=η1H,

where ηβ=(1)kβ for βΛk(U).

The cohomotopy operator fulfills (co)homotopy invariance formula[4]

δh+hδ=ISx0,

where Sx0=1sx0*, and sx0*is the pullback along the constant map sx0:xx0.

Therefore, if we want to solve the equation δω=0, applying cohomotopy invariance formula we get

ω=δhω+Sx0ω, where hωΛk+1(U) is a differential form we are looking for, and 'constant of integration' Sx0ω vanishes unless ω is a top form.

Cohomotopy operator fulfills the following properties:[4] h2=0,δhδ=δ,hδh=h. They make it possible to use it to define[4] anticoexact forms on U by 𝒴(U)={ωΛ(U)|ω=hδω}, which together with exact forms 𝒞(U)={ωΛ(U)|ω=δhω} make a direct sum decomposition[4]

Λ(U)=𝒞(U)𝒴(U).

This direct sum is another way of saying that the cohomotopy invariance formula is a decomposition of unity, and the projector operators on the summands fulfills idempotence formulas:[4] (hδ)2=hδ,(δh)2=δh.

These results are extension of similar results for exterior derivative.[5]

Citations

  1. 1.0 1.1 Harley Flanders (1963) Differential Forms with Applications to the Physical Sciences, Academic Press
  2. 2.0 2.1 Pertti Lounesto (2001). "§3.6 The Hodge dual". Clifford Algebras and Spinors, Volume 286 of London Mathematical Society Lecture Note Series (2nd ed.). Cambridge University Press. p. 39. ISBN 0-521-00551-5. https://books.google.com/books?id=E_xvJuA4M7QC&pg=PA39. 
  3. Frankel, T. (2012). The Geometry of Physics (3rd ed.). Cambridge University Press. ISBN 978-1-107-60260-1. 
  4. 4.0 4.1 4.2 4.3 4.4 4.5 4.6 4.7 Kycia, Radosław Antoni (2022-07-29). "The Poincare Lemma for Codifferential, Anticoexact Forms, and Applications to Physics" (in en). Results in Mathematics 77 (5): 182. doi:10.1007/s00025-022-01646-z. ISSN 1420-9012. https://doi.org/10.1007/s00025-022-01646-z. 
  5. Edelen, Dominic G. B. (2005). Applied exterior calculus (Revised ed.). Mineola, N.Y.. ISBN 978-0-486-43871-9. OCLC 56347718. https://www.worldcat.org/oclc/56347718. 

References

  • David Bleecker (1981) Gauge Theory and Variational Principles. Addison-Wesley Publishing. ISBN 0-201-10096-7. Chpt. 0 contains a condensed review of non-Riemannian differential geometry.
  • Riemannian Geometry and Geometric Analysis. Springer-Verlag. 2002. ISBN 3-540-42627-2. 
  • Charles W. Misner, Kip S. Thorne, John Archibald Wheeler (1970) Gravitation. W.H. Freeman. ISBN 0-7167-0344-0. A basic review of differential geometry in the special case of four-dimensional spacetime.
  • Steven Rosenberg (1997) The Laplacian on a Riemannian manifold. Cambridge University Press. ISBN 0-521-46831-0. An introduction to the heat equation and the Atiyah–Singer theorem.
  • Tevian Dray (1999) The Hodge Dual Operator. A thorough overview of the definition and properties of the Hodge star operator.