Hodge's Dual

In mathematics, the Hodge star operator on the vector space V is a linear operator on the exterior algebra of V, interchanging the subspaces of k-vectors and that of n−k-vectors where n = dim V, for 0 ≤ k ≤ n.

Definition

Informally it is defined "distributing the volume shape" ω, thought of as n standard basis vectors multiplied externally such that:

α α ∧ ∧ ↓ ↓ α α =ω ω {displaystyle alpha wedge ;*alpha =omega }

Except sign, provided α is an external product of some standard base vectors. Given a measure on a variety n dimensional expressable as a n-form μ (not all measures are in this way, for example, the "function" delta of Dirac), the Dual Hodge of the p-form A defined as contraction μ μ ! ! ,A {displaystyle langle {bar {mu }},mathbf {A} rangle } where μ μ ! ! {displaystyle {bar {mu}}} is the dual n-vector. See sign convention.

Formal definition

Formally in a variety of riemannian or pseudoriemannian dimension n We need to define the Hodge dual of a p-form β β {displaystyle beta ,} like the (n-p)-form ↓ ↓ β β {displaystyle}beta ,} such that:

Русский Русский α α :α α ∧ ∧ ↓ ↓ β β = α α ,β β ω ω {displaystyle forall alpha:alpha wedge *beta =langle alphabeta rangle omega }

with α α 日本語β β {displaystyle langle alpha Δbeta rangle } product scale of forms and

ω ω =ε ε 日本語g日本語dx1∧ ∧ ...∧ ∧ dxn{displaystyle omega =varepsilon {sqrt}dx^{1}land dx^{n}{n}}

is the n-form of volume, where g is the determinant of the metric tensor and ε = sgn(g). Hence the relationship:

↓ ↓ ↓ ↓ α α =(− − 1)k(n− − k)ε ε α α {displaystyle ;**;alpha =(-1)^{k(n-k)}varepsilon alpha }

in particular ε = 1 in a variety of Riemann and ε=-1 in a variety Lorentz-Minkowski (n− − 1,1){displaystyle (n-1,1);}.

Uses of the Hodge Dual in physics

Euclidean space endowed with vector product

A common example of the star operator is the three-dimensional Euclidean space endowed with the ordinary metric. In fact, the cross product is nothing more than the Hodge dual of the exterior product of two differential forms built from the vectors. Formally, the cross product turns out to be:

a→ → × × b→ → =↓ ↓ (φ φ a→ → ∧ ∧ φ φ b→ → ){displaystyle {vec {a}}times {vec {b}}=*(phi _{vec {a}}wedge phi _{vec {b}}}}}}}})}

To explain this construction we need to introduce the isomorphism between vectors of three-dimensional space and 1-forms of the same space:

a→ → =(ax,aand,az) φ φ a→ → =axdx+aanddand+azdz{displaystyle {vec {a}}=(a_{x},a_{y},a_{z})mapsto phi _{vec {a}}=a_{x}dx+a_{y}dy+a_{z}dz}

Now it is worth noting that in 3 dimensions the dual of a 1-form is an antisymmetric 2-form, and the dual of a 2-form is a 1-form. This allows us to construct another isomorphism between 1-forms and 2-forms, precisely this isomorphism is the Hodge dual. To clarify how this isomorphism works, we are going to interpret the 2-forms as 3x3 antisymmetric matrices as follows:

F=Fijdxi dxj [chuckles]0+F12− − F13− − F210+F23+F31− − F320]{displaystyle mathbf {F} =F_{ij}dx^{i}otimes dx^{jmapsto {begin{bmatrix0 fake+F_{12}{12}{12}{F_{13}-F_{21}{b_{23}+F_{+F_{31}{32matrix}{b}{end}{x}{x}{x}{x}{x}{b_{x}{x}{b_}{b_}{b_}}}{b_{b_{b_}}}}}{b_}{b-F_}{b-F_}{b-f}}}{b-f}}{b-f}{b-f}{b-f}{b-f}{b-f}{b-f}}{b-f}{21}{21}{21}{21}}}{

We can see that this matrix has only three independent components that can be interpreted as a vector given by the dual Hodge operator:

F ↓ ↓ F=F23dx+F31dand+F12dz{displaystyle mathbf {F} mapsto *mathbf {F} =F_{23}dx+F_{31}dy+F_{12}dz}

That is, in three-dimensional Euclidean space, there is a correspondence between vectors and antisymmetric 3x3 matrices. So let's review the steps:

1. Any three-dimensional euclid space vector with the ordinary metric can be interpreted naturally as a 1-form of that space.
2. The product of two 1-forms is a 2-form, which in the three-dimensional euclide space can be matched with a vector, thanks to the isomorphism associated with the dual operator of Hodge.
3. The vector product is nothing else an axial vector given by the Hodge dual of the external product of the two 1-forms naturally associated with the two vectors from which it was split.

• Also for all 1-form in R3{displaystyle mathbb {R} ^{3} is fulfilled that **α = α.
• Another interesting point related to physics, is that all axial vector is actually the dual Hodge of an antisimetric matrix. In fact, when relativistic quantities associated with Newtonian quantities are constructed, since the space of relativity theory is four-dimensional, isomorphism between 1-forms and 2-forms disappears. That implies that the axial vectors of Newtonian mechanics should be treated as part of antismetric tensions in relativity theory.

Minkowski space

Another fundamental application of the Hodge dual operator in physics appears in the Minkowski space of the special theory of relativity. Given the dimension n = 4 of the Minkoski space and given the metric there is a fundamental isomorphism between:

• 0-forms and 4-forms.
• 1-forms and 3-forms.
• 2-forms and 2-forms (endomorphism).

Furthermore, the following fundamental relation results for every 2-form:

↓ ↓ ↓ ↓ F=− − F{displaystyle **F=-F,}

This relation can be used to very briefly formulate Maxwell's equations of electromagnetism, taking into account that the electromagnetic field is given by a 2-form or antisymmetric tensor, which in Cartesian components is:

F=[chuckles]0Ex/cEand/cEz/c− − Ex/c0Bz− − Band− − Eand/c− − Bz0Bx− − Ez/cBand− − Bx0]↓ ↓ F=[chuckles]0BxBandBz− − Bx0Ez/c− − Eand/c− − Band− − Ez/c0Ex/c− − BzEand/c− − Ex/c0]♪ I don't know ♪

Maxwell's equations can be written in terms of the 2-form electromagnetic field and Hodge dual operator as simply (cgs system):

dF=0↓ ↓ d(↓ ↓ F)=4π π cJ{displaystyle dmathbf {F} =0qquad *d(*mathbf {F})={frac {4pi }{c}}mathbf {J} }

Where J{displaystyle mathbf {J} } is the 1-form naturally associated with the current density quadrant.