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HW #3

  1. Hodge Dual in Minkowski Space

    Minkowski 4-space has orthonormal, oriented basis $ \{dx,dy,dz,dt\} $ with volume element (choice of orientation) $ \omega=dx\wedge dy \wedge dz \wedge dt $ and inner product $ g(,) $ defined by

    \begin{align*}<br /> g(dx,dx)&=1 \\<br /> g(dy,dy)&=1 \\<br /> g(dz,dz)&=1 \\<br /> g(dt,dt)&=-1 \\<br /> g(d\alpha_i,d\alpha_{j\ne i})&=0 \\<br /> \end{align*}

    1. Determine the Hodge dual operator * on all forms by computing its action on basis forms at each rank.
      In general, the dual can be computed as (Equation 20, http://oregonstate.edu/~drayt/Courses/MTH434/2007/dual.pdf):

      \[ \ast(\sigma^1\ldots\sigma^p)=g(\sigma^1,\sigma^1)\ldots g(\sigma^p,\sigma^p)\sigma^{p+1}\wedge\ldots\wedge\sigma^n \]

      where $ \sigma^1\ldots\sigma^n=\omega $ is the volume element.
      In particular:

      \begin{align*}<br /> \ast 1=\omega<br /> \end{align*}
      \begin{align*}<br /> \ast dx &=dy \wedge dz \wedge dt \\<br /> \ast dy &=(-dx \wedge dz \wedge dt) \\<br /> \ast dz &=dx \wedge dy \wedge dt \\<br /> \ast dt &=(-1)(-dx \wedge dy \wedge dz) \\<br /> \end{align*}
      \begin{align*}<br /> \ast dx\wedge dy &=dz \wedge dt) \\<br /> \ast dx\wedge dz &=(-dy \wedge dt) \\<br /> \ast dx\wedge dt &=(-1)(-dy \wedge dz) \\<br /> \ast dy\wedge dz &=dx \wedge dt) \\<br /> \ast dy\wedge dt &=(-1)(-dx \wedge dz) \\<br /> \ast dz\wedge dt &=(-1)(dx \wedge dy) \\<br /> \end{align*}
      \begin{align*}<br /> \ast dx \wedge dy \wedge dz &=dt \\<br /> \ast dx \wedge dy \wedge dt &=(-1)(-dt) \\<br /> \ast dx \wedge dz \wedge dt &=(-1)dt \\<br /> \ast dy \wedge dz \wedge dt &=(-1)(-dt) \\<br /> \end{align*}
      \begin{align*}<br /> \ast \omega = -1<br /> \end{align*}
    2. How does your answer change if the opposite orientation is chosen, namely $ \omega = dt\wedge dx \wedge dy \wedge dz $?
      The sign of each dual would change, due to the additional odd number (3) of transpositions to the volume element. The sign of the number of transpositions required to transform one volume element into another is an invariant (independent of a particular permutation).

  2. Spherical Coordinates

    Spherical coords in $ E^3 $ with $ \omega = r^2\sin{<br /> \theta}dr\wedge d\theta \wedge d\phi $

    1. Calculate the action of * on the basis elements of each rank.
    2. Compute the dot- and cross-products of 2 arbitrary "vector fields" (1-forms) using the expressions:
      \begin{align*}<br /> \alpha\cdot\beta&=\ast(\alpha\wedge\ast\beta) \\<br /> \alpha\times\beta&=\ast(\alpha\wedge\beta) \\<br /> \end{align*}

Tue, 2007-02-13 23:35 | eric