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Minkowski space

Contents

Context

Riemannian geometry

Physics

physics, mathematical physics, philosophy of physics

Surveys, textbooks and lecture notes


theory (physics), model (physics)

experiment, measurement, computable physics

Gravity

Contents

Definition

For d1d-1 \in \mathbb{N}, dd-dimensional Minkowski space is the Lorentzian manifold whose underlying smooth manifold is the Cartesian space d\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+1D = d+1-dimensional Minkowski spacetime.

Conventions

We use Einstein summation convention throughout. Hence a generic differential p-form is

α=1p!α μ 1μ pdx μ 1dx μ p. \alpha \;=\; \tfrac{1}{p!} \alpha_{ \mu_1 \cdots \mu_p } d x^{\mu_1} \wedge \cdots \wedge d x^{\mu_p} \,.

Here p!123pp! \coloneqq 1 \cdot 2 \cdot 3 \cdots p is the factorial of pp.

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

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

We normalize the Levi-Civita symbol as

(1)ϵ 012d+1 \epsilon_{0 1 2 \cdots d} \;\coloneqq\; + 1

which means that

(2)ϵ 012d=1. \epsilon^{0 1 2 \cdots d} \;=\; - 1 \,.

We normalize the sign of the volume form as

(3)dvol dx 0dx 1dx d =1D!ϵ μ 1μ Ddx μ 1dx μ D \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)δ ν 1ν p μ 1μ p{sgn(σ) | σSym(p)(1ip(ν σ(i)=μ i)) 0 | otherwise \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)ϵ μ 1μ pμ p+1μ Dϵ ν 1ν pμ p+1μ D=(Dp)!δ μ 1μ p ν 1ν p \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 ι μ\iota_\mu for the operator of contraction of differential forms with the vector field d/dx μd/d x^\mu, hence the linear operator on differential forms with anticommutator

{ι μ,dx ν}=δ μ ν \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 ι\iota)

(6)α μ 1μ pι μ pι μ 1dvol=ϵ μ 1μ pν 1ν (Dp)dx ν 1dx ν (Dp) \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

α1p!α μ 1μ pdx μ 1dx μ p \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)α 1p!(Dp)!α μ 1μ pι μ pι μ 1dvol =1p!(Dp)!α μ 1μ pϵ μ 1μ pμ p+1μ Ddx μ p+1dx μ D, \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 α1p!α μ 1μ pdx μ 1dx μ p \alpha \;\coloneqq\; \tfrac{1}{p!} \alpha_{\mu_1 \cdots \mu_p} d x^{\mu_1} \wedge \cdots \wedge d x^{\mu_p} on DD-dimensional Minkowski spacetime its wedge product with its Hodge dual (7) is

(8)αα=α μ 1μ pα μ 1μ pdvol. \alpha \wedge \star \alpha \;=\; { \color{magenta} - } \alpha_{ \mu_1 \cdots \mu_p } \alpha^{ \mu_1 \cdots \mu_p } \, dvol \,.
Proof

We compute as follows:

αα =1p!p!(Dp)!α μ 1μ pdx μ 1dx μ pα ν 1ν pι ν pι ν 1dvol =1p!p!(Dp)!α μ 1μ pα ν 1ν pϵ ν 1ν pν p+1ν Ddx μ pdx μ 1dx ν p+1dx ν D =1p!p!(Dp)!α μ 1μ pα ν 1ν pϵ ν 1ν pν p+1ν Dϵ μ pμ 1ν p+1ν Ddvol =1p!p!α μ 1μ pα ν 1ν pδ ν 1ν p μ 1μ pdvol =α μ 1μ pα μ 1μ pdvol \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 \\ & = {\color{magenta} -} \alpha_{ \mu_1 \cdots \mu_p } \alpha^{ \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 α=1p!α μ 1μ pdx μ 1dx μ p \alpha \;=\; \tfrac{1}{p!} \alpha_{\mu_1 \cdots \mu_p} d x^{\mu_1} \wedge \cdots \wedge d x^{\mu_p} on DD-dimensional Minkowski spacetime, its double Hodge dual (7) is

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

We compute as follows:

1p!α μ 1μ pdx μ 1dx μ p =1p!(Dp)!α μ 1μ pι μ pι μ 1dvol =1p!(Dp)!α μ 1μ pϵ μ 1μ pμ p+1μ Ddx μ p+1dx μ d =1p!(Dp)!p!α μ 1μ pϵ μ 1μ pμ p+1μ Dϵ μ p+1μ Dν 1ν pdx ν 1dx ν D =(1) p(Dp)p!(Dp)!p!α μ 1μ pϵ μ 1μ pμ p+1μ Dϵ ν 1ν pμ p+1μ Ddx ν 1dx ν D =(1) p(Dp)p!p!α μ 1μ pδ ν 1ν p μ 1μ pdx ν 1dx ν D =(1) p(Dp)α \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 α=1p!α μ 1μ pdx μ 1dx μ p \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 DD-dimensional Minkowski spacetime such that

να νμ 1μ p1=0 \partial^\nu \alpha_{\nu \mu_1 \cdots \mu_{p-1}} \;=\; 0

(i.e. Lorenz gauge).

Then the Laplace-Beltrami operator

ddα= ν να \star d \star d \alpha \;=\; { \color{magenta} - } \partial^\nu \partial_\nu \alpha

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

Proof

We compute as follows:

dd1p!α μ 1μ pdx μ 1dx μ p =d1p! να μ 1μ pdx νdx μ 1dx μ p =d1p!(D(p+1))! να μ 1μ pϵ νμ 1μ pμ p+2μ Ddx μ p+2dx μ D =1p!(D(p+1))! ν να μ 1μ pϵ νμ 1μ pμ p+2μ Ddx νdx μ p+2dx μ D =1p!(D(p+1))!p! ν να μ 1μ pϵ νμ 1μ pμ p+2μ Dϵ νμ p+2μ Dκ 1κ pdx κ 1dx κ p =1p!p! ν να μ 1μ pδ νκ 1κ p νμ 1μ pdx κ 1dx κ p =1p!p! ν να μ 1μ pδ κ 1κ p μ 1μ pdx κ 1dx κ p = ν να \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).

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)

See also

Gravitational stability of Minkowski space is proven in

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

Last revised on May 8, 2020 at 16:51:38. See the history of this page for a list of all contributions to it.