nLab Hodge star operator on Minkowski spacetime -- section

Hodge star operator on Minkowski spacetime

We spell out component expressions for the Hodge star operator on $D = d+1$-dimensional Minkowski spacetime.

Conventions

We use Einstein summation convention throughout. With this convention, a generic differential p-form reads

$$\alpha \;=\; \tfrac{1}{p!} \alpha_{ \color{green} \mu_1 \cdots \mu_p } d x^{ \color{green} \mu_1} \wedge \cdots \wedge d x^{\color{green} \mu_p} \,.$$

Here $p! \coloneqq 1 \cdot 2 \cdot 3 \cdots p \,\in \mathbb{N} \subset \mathbb{R}$ denotes the factorial of $p \in \mathbb{N}$.

We take the Minkowski metric to be the $D \times D$ diagonal matrix of the form

$$\eta \;=\; (\eta_{\mu \nu}) \;=\; (\eta^{\mu \nu}) \;\coloneqq\; diag(-1,+1, +1 , \cdots , +1) \,.$$

We normalize the Levi-Civita symbol as

(1)$$\epsilon_{0 1 2 \cdots d} \;\coloneqq\; + 1$$

which means that

(2)$$\epsilon^{0 1 2 \cdots d} \;=\; - 1 \,.$$

We normalize the sign of the volume form as

(3)$$\begin{aligned} dvol & \coloneqq\; d x^0 \wedge d x^1 \wedge \cdots \wedge d x^d \\ & = \tfrac{1}{D!} \epsilon_{ \color{green} \mu_1 \cdots \mu_D } d x^{\color{green}\mu_1} \wedge \cdots \wedge d x^{\color{green}\mu_D} \end{aligned}$$

We write

(4)$$\delta^{ \mu_1 \cdots \mu_p }_{ \nu_1 \cdots \nu_p } \;\coloneqq\; \left\{ \array{ sgn(\sigma) &\vert& \underset{ \sigma \in Sym(p) }{\exists} \left( \underset{1 \leq i \leq p}{\forall} \left( \nu_{\sigma(i)} = \mu_i \right) \right) \\ 0 &\vert& \text{otherwise} } \right.$$

for the generalized Kronecker delta, whose value is the signature of the permutation that takes the upper indices to the lower indices, if any such exists, and zero otherwise.

This appears whenever the Levi-Civita symbol is contracted with itself:

(5)$$\epsilon_{ { \color{green} \mu_1 \cdots \mu_p } {\color{blue} \mu_{p+1} \cdots \mu_{D} } } \epsilon^{ { \color{orange} \nu_1 \cdots \nu_p } { \color{blue} \mu_{p+1} \cdots \mu_D } } \;=\; { \color{magenta} - } (D-p)! \; \delta_{ \color{green} \mu_1 \cdots \mu_p }^{ \color{orange} \nu_1 \cdots \nu_p }$$

Notice the minus sign in (5), which comes, via (2), from the Minkowski signature.

Definition

We write $\iota_\mu$ for the operator of contraction of differential forms with the vector field $d/d x^\mu$, hence the linear operator on differential forms with anticommutator

$$\big\{ \iota_\mu, d x^\nu \wedge \big\} \;=\; \delta_\mu^\nu$$

With the volume form as in (3) it follows that (notice the reversion of the index ordering in the contraction operators $\iota$)

(6)$$\alpha^{ \color{green} \mu_1 \cdots \mu_p } \iota_{\color{green} \mu_p} \cdots \iota_{ \color{green} \mu_1} dvol \;=\; \epsilon_{ { \color{green} \mu_1 \cdots \mu_p } { \color{orange} \nu_1 \cdots \nu_{(D-p)} } } d x^{\color{orange} \nu_1} \wedge \cdots \wedge d x^{\color{orange} \nu_{(D-p)}}$$

Definition

For a differential p-form

$$\alpha \;\coloneqq\; \tfrac{1}{ \color{green} p! } \alpha_{ \color{green} \mu_1 \cdots \mu_p} d x^{ \color{green} \mu_1 } \wedge \cdots \wedge d x^{ \color{green} \mu_p }$$

its Hodge dual is:

(7)$$\begin{aligned} \star \alpha & \coloneqq \tfrac{1}{ { \color{green} p! } { \color{orange} (D-p)! } } \, \alpha^{ \color{green} \mu_1 \cdots \mu_p } \iota_{ \color{green} \mu_p } \cdots \iota_{ \color{green} \mu_1 } \, dvol \\ & = \tfrac{1}{ { \color{green} p! } { \color{orange} (D-p)! } } \, \alpha^{ \color{green} \mu_1 \cdots \mu_p } \epsilon_{ { \color{green} \mu_1 \cdots \mu_p } { \color{orange} \mu_{p+1} \cdots \mu_D } } \, d x^{ \color{orange} \mu_{p+1} } \wedge \cdots \wedge d x^{ \color{orange} \mu_D } \,, \end{aligned}$$

where in the second line we used (6).

Properties

Proposition

(Hodge pairing)

For a differential p-form $\alpha \;\coloneqq\; \tfrac{1}{p!} \alpha_{\mu_1 \cdots \mu_p} d x^{\mu_1} \wedge \cdots \wedge d x^{\mu_p}$ on $D$-dimensional Minkowski spacetime its wedge product with its Hodge dual (7) is

(8)$$\alpha \wedge \star \alpha \;=\; \tfrac{ \color{magenta} -1 }{ { p! } } \alpha_{ \mu_1 \cdots \mu_p } \alpha^{ \mu_1 \cdots \mu_p } \, dvol \,.$$

Proof

We compute as follows:

$$\begin{aligned} \alpha \wedge \star \alpha & = \tfrac{1}{ { \color{green} p! } { \color{orange} p! } { \color{blue} (D-p)! } } \alpha_{ \color{green} \mu_1 \cdots \mu_p } d x^{ \color{green} \mu_1 } \wedge \cdots \wedge d x^{ \color{green} \mu_p } \wedge \alpha^{ \color{orange} \nu_1 \cdots \nu_p } \iota_{ \color{orange} \nu_p } \cdots \iota_{ \color{orange} \nu_1 } dvol \\ & = \tfrac{1}{ { \color{green} p! } { \color{orange} p! } { \color{blue} (D-p)! } } \alpha_{ \color{green} \mu_1 \cdots \mu_p } \alpha^{ \color{orange} \nu_1 \cdots \nu_p } \epsilon_{ { \color{orange} \nu_1 \cdots \nu_p } { \color{blue} \nu_{p+1} \cdots \nu_D } } d x^{ \color{green} \mu_p } \wedge \cdots \wedge d x^{ \color{green} \mu_1 } \wedge d x^{ \color{blue} \nu_{p+1} } \wedge \cdots d x^{ \color{blue} \nu_{D} } \\ & = \tfrac{1}{ { \color{green} p! } { \color{orange} p! } { \color{blue} (D-p)! } } \alpha_{ \color{green} \mu_1 \cdots \mu_p } \alpha^{ \color{orange} \nu_1 \cdots \nu_p } \epsilon_{ { \color{orange} \nu_1 \cdots \nu_p } { \color{blue} \nu_{p+1} \cdots \nu_D } } \epsilon^{ { \color{green} \mu_p \cdots \mu_1 } { \color{blue} \nu_{p+1} \cdots \nu_{D} } } \, dvol \\ & = \tfrac{ \color{magenta} -1 }{ { \color{green} p! } { \color{orange} p! } } \alpha_{ \color{green} \mu_1 \cdots \mu_p } \alpha^{ \color{orange} \nu_1 \cdots \nu_p } \delta^{ \color{green} \mu_1 \cdots \mu_p }_{ \color{orange} \nu_1 \cdots \nu_p } \, dvol \\ & = \tfrac{ \color{magenta} -1 }{ { \color{green} p! } } \alpha_{ \color{green} \mu_1 \cdots \mu_p } \alpha^{ \color{green} \mu_1 \cdots \mu_p } \, dvol \end{aligned}$$

Here the sign in the last lines arises from the Minkowski signature via (5).

Proposition

(double Hodge dual)

For a differential p-form $\alpha \;=\; \tfrac{1}{p!} \alpha_{\mu_1 \cdots \mu_p} d x^{\mu_1} \wedge \cdots \wedge d x^{\mu_p}$ on $D$-dimensional Minkowski spacetime, its double Hodge dual (7) is

(9)$$\star \star \alpha \;=\; {\color{magenta} -} (-1)^{ p (D - p) } \, \alpha \,.$$

Proof

We compute as follows:

$$\begin{aligned} & \star \star \tfrac{1}{ \color{green} p! } \alpha_{ \color{green} \mu_1 \cdots \mu_p} d x^{ \color{green} \mu_1 } \wedge \cdots \wedge d x^{ \color{green} \mu_p } \\ & = \star \tfrac{1}{ { \color{green} p! } { \color{orange} (D-p)! } } \alpha^{ \color{green} \mu_1 \cdots \mu_p} \iota_{ \color{green} \mu_p} \cdots \iota_{ \color{green} \mu_1} dvol \\ & = \star \tfrac{1}{ { \color{green} p! } { \color{orange} (D-p)! } } \alpha^{ \color{green} \mu_1 \cdots \mu_p } \epsilon_{ { \color{green} \mu_1 \cdots \mu_p } { \color{orange} \mu_{p+1} \cdots \mu_D } } d x^{\color{orange} \mu_{p+1}} \wedge \cdots d x^{ \color{orange} \mu_d} \\ & = \tfrac{1}{ { \color{green} p! } { \color{orange} (D-p)! } { \color{blue} p! } } \alpha_{ \color{green} \mu_1 \cdots \mu_p } \epsilon^{ { \color{green} \mu_1 \cdots \mu_p } { \color{orange} \mu_{p+1} \cdots \mu_D } } \epsilon_{ { \color{orange} \mu_{p+1} \cdots \mu_D } { \color{blue} \nu_1 \cdots \nu_p } } \, d x^{ \color{blue} \nu_1} \wedge \cdots \wedge d x^{ \color{blue} \nu_D } \\ \\ & = \tfrac{ (-1)^{ {\color{green} p} { \color{orange} (D-p) } } }{ { \color{green} p! } { \color{orange} (D-p)! } { \color{blue} p! } } \alpha_{ \color{green} \mu_1 \cdots \mu_p } \epsilon^{ { \color{green} \mu_1 \cdots \mu_p } { \color{orange} \mu_{p+1} \cdots \mu_D } } \epsilon_{ { \color{blue} \nu_1 \cdots \nu_p } { \color{orange} \mu_{p+1} \cdots \mu_D } } \, d x^{ \color{blue} \nu_1} \wedge \cdots \wedge d x^{ \color{blue} \nu_D } \\ & = {\color{magenta} -} \tfrac{ (-1)^{ {\color{green}p} {\color{orange} (D-p) } } }{ { \color{green} p! } { \color{blue} p! } } \alpha_{ \color{green} \mu_1 \cdots \mu_p } \delta^{ { \color{green} \mu_1 \cdots \mu_p } }_{ { \color{blue} \nu_1 \cdots \nu_p } } \, d x^{ \color{blue} \nu_1} \wedge \cdots \wedge d x^{ \color{blue} \nu_D } \\ & = {\color{magenta} -} (-1)^{ {\color{green}p} {\color{orange} (D-p) } } \, \alpha \end{aligned}$$

Here the sign in the last lines arises from the Minkowski signature via (5).

Proposition

(Laplace operator/wave operator)

Let $\alpha = \tfrac{1}{p!} \alpha_{\mu_1 \cdots \mu_p} d x^{\mu_1} \wedge \cdots \wedge d x^{\mu_p}$ be a differential p-form on $D$-dimensional Minkowski spacetime such that

$$\partial^\nu \alpha_{\nu \mu_1 \cdots \mu_{p-1}} \;=\; 0$$

(i.e. Lorenz gauge).

Then the Laplace-Beltrami operator

$$\star d \star d \alpha \;=\; { \color{magenta} - } \partial^\nu \partial_\nu \alpha$$

is the wave operator acting on the components of $\alpha$.

Proof

We compute as follows:

$$\begin{aligned} & \star d \star d \tfrac{1}{ \color{green} p! } \alpha_{ \color{green} \mu_1 \cdots \mu_p } d x^{ \color{green} \mu_1 } \wedge \cdots \wedge d x^{ \color{green} \mu_p } \\ & = \star d \star \tfrac{1}{ \color{green} p! } \partial_{ \color{magenta} \nu } \alpha_{ \color{green} \mu_1 \cdots \mu_p } d x^{ \color{magenta} \nu } \wedge d x^{ \color{green} \mu_1 } \wedge \cdots \wedge d x^{ \color{green} \mu_p } \\ & = \star d \tfrac{1}{ { \color{green} p! } { \color{orange} (D-(p+1))! } } \partial^{ \color{magenta} \nu } \alpha^{ \color{green} \mu_1 \cdots \mu_p } \epsilon_{ { \color{magenta} \nu } { \color{green} \mu_1 \cdots \mu_{p} } { \color{orange} \mu_{p+2} \cdots \mu_D } } \, d x^{ \color{orange} \mu_{p+2} } \wedge \cdots \wedge d x^{ \color{orange} \mu_D } \\ & = \star \tfrac{1}{ { \color{green} p! } { \color{orange} (D-(p+1))! } } \partial_{ \color{red} \nu' } \partial^{ \color{magenta} \nu } \alpha^{ \color{green} \mu_1 \cdots \mu_p } \epsilon_{ { \color{magenta} \nu } { \color{green} \mu_1 \cdots \mu_p } { \color{orange} \mu_{p+2} \cdots \mu_D } } \, d x^{ \color{red} \nu' } \wedge d x^{ \color{orange} \mu_{p+2} } \wedge \cdots \wedge d x^{ \color{orange} \mu_D } \\ & = \tfrac{1}{ { \color{green} p! } { \color{orange} (D-(p+1))! } { \color{blue} p! } } \partial^{ \color{red} \nu' } \partial_{ \color{magenta} \nu } \alpha_{ \color{green} \mu_1 \cdots \mu_p } \epsilon^{ { \color{magenta} \nu } { \color{green} \mu_1 \cdots \mu_p } { \color{orange} \mu_{p+2} \cdots \mu_D } } \epsilon_{ { \color{red} \nu' } { \color{orange} \mu_{p+2} \cdots \mu_D } { \color{blue} \kappa_1 \cdots \kappa_p } } \, d x^{\color{blue} \kappa_1} \wedge \cdots d x^{\color{blue}\kappa_p} \\ & = { \color{magenta} - } \tfrac{1}{ { \color{green} p! } { \color{blue} p! } } \partial^{ \color{red} \nu' } \partial_{ \color{magenta} \nu } \alpha_{ \color{green} \mu_1 \cdots \mu_p } \delta^{ { \color{magenta} \nu } { \color{green} \mu_1 \cdots \mu_p } } _{ { \color{red} \nu' } { \color{blue} \kappa_1 \cdots \kappa_p } } \, d x^{\color{blue} \kappa_1} \wedge \cdots d x^{\color{blue}\kappa_p} \\ & = { \color{magenta} - } \tfrac{1}{ { \color{green} p! } { \color{blue} p! } } \partial^{ \color{magenta} \nu } \partial_{ \color{magenta} \nu } \alpha_{ \color{green} \mu_1 \cdots \mu_p } \delta^{ { \color{green} \mu_1 \cdots \mu_p } } _{ { \color{blue} \kappa_1 \cdots \kappa_p } } \, d x^{\color{blue} \kappa_1} \wedge \cdots d x^{\color{blue}\kappa_p} \\ & = { \color{magenta} - } \partial^{ \color{magenta} \nu } \partial_{ \color{magenta} \nu } \alpha \end{aligned}$$

Here the sign in the last lines arises from the Minkowski signature via (5).