nLab Minkowski space

Definition

For $d-1 \in \mathbb{N}$, $d$-dimensional Minkowski space is the Lorentzian manifold whose underlying smooth manifold is the Cartesian space $\mathbb{R}^d$ and whose pseudo-Riemannian metric is at each point the Minkowski metric.

This is naturally a spacetime.

Properties

Isometries

The isometry group of Minkowski space is the Poincaré group. The study of Minkowski spacetime with its isometries is also called Lorentzian geometry. This is the context of the theory of special relativity.

Gravitational stability

Theorem

Minkowski spacetimes is a stable? solution of the vacuum Einstein equations.

This is due to (ChristodoulouKlainerman 1993).

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).

Related entries

• Minkowski metric

• spacetime

  • super Minkowski space

  • Schwarzschild spacetime

• light cone, celestial sphere

• twistor space

• piecewise flat spacetime

• flat space holography

References

The notion of Minkowski spacetime originates with

• Hermann Minkowski, Die Grundgleichungen für die elektromagnetischen Vorgänge in bewegten Körpern, Math. Ann. (1910) 68: 472, reprinted from: Nachrichten der Kgl. Ges. d. Wiss. zu Göttingen, Math.-phys. Kl., Sitzung vom 21. Dezember 1907 (doi:10.1007/BF01455871)

The views of space and time that I wish to lay before you have sprung from the soil of experimental physics, and therein lies their strength. They are radical. Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of both will retain an independent reality.

(Address to the 80th Assembly of German Natural Scientists and Physicians, (Sep 21, 1908), see WikiQuote)

On the early history of the idea:

• Galina Weinstein: Max Born, Albert Einstein and Hermann Minkowski’s Space-Time Formalism of Special Relativity [arXiv:1210.6929]

See also:

• Wikipedia, Minkowski space

With attention to the conformal boundary:

• Juan A. Valiente Kroon: Chapter 16 of: Conformal Methods in General Relativity, Cambridge University Press (2017, 2022) [doi:10.1017/9781009291309, oapen:20.500.12657/59217, pdf]

Gravitational stability of Minkowski spacetime:

• Demetrios Christodoulou, Sergiu Klainerman, The global nonlinear stability of the Minkowski space Princeton University Press (1993)