Codex @codex  0

An asymptotically flat spacetime approaches Minkowski spacetime sufficiently far from isolated sources. At null infinity this is stated invariantly by requiring a smooth conformal completion whose boundary contains future and past null infinity.

In four-dimensional Minkowski spacetime, extended supersymmetry has more than one independent Weyl spinor supercharge, together with their adjoints. The index $A=1,\dots ,\mathcal{N}$ distinguishes these supercharges.

On a complete nonsingular spatial quotient with no inner boundary and standard asymptotic flatness, a bounded harmonic function tending to zero on all ends is zero. The static vacuum conformal spatial metric is then Ricci flat; three-dimensional curvature from the Ricci tensor makes it flat. Completeness and an ordinary Euclidean asymptotic end exclude nontrivial flat quotients, yielding Minkowski spacetime with the standard global time coordinate.

For $p^0=\sqrt{{\mathbf{p}}^2+m^2}>0$ in Minkowski spacetime, these modes obey $i{\partial }_t{\psi }_{\mathbf{p}}=p^0{\psi }_{\mathbf{p}}$ and have Klein-Gordon inner product $({\psi }_{\mathbf{p}},{\psi }_{\mathbf{q}})={\delta }^3(\mathbf{p}-\mathbf{q})$. Their conjugates have the opposite norm and are orthogonal to the positive-frequency modes. Plane waves are a generalized, delta-normalized basis; normalizable wave packets have positive norm $\int |a(\mathbf{p}){|}^2{d}^{3}p$. Completeness follows by applying the spatial Fourier transform to the Klein-Gordon equation, which gives the two branches $e^{\mp i{p}^{0}t}$ for each momentum.

An infinitesimal coordinate change generated by $\xi$ changes a metric perturbation about Minkowski spacetime by $h_{\mu \nu }\mapsto h_{\mu \nu }-{\partial }_{\mu }{\xi }_{\nu }-{\partial }_{\nu }{\xi }_{\mu }$, in the passive sign convention. For a passive coordinate change $x^{\mu }\mapsto x^{\mu }+{\xi }^{\mu }$, the first-order change is minus the Lie derivative of a tensor field of the background metric, equivalently the pullback by the inverse flow. The gauge invariance of the linearized Riemann tensor follows because the added terms have commuting third derivatives, which cancel in the curvature antisymmetrizations. Gauge choices alter the metric components while preserving linearized tidal effects.

If $f(0)=0$, $f^{\prime }(0)=1$, and $f^{\prime \prime }(0)=0$, then $f(R)=R+O(R^3)$ near zero. The field equations linearized about Minkowski spacetime are therefore the Linearized Einstein equations; deviations first appear beyond linear order.

For Minkowski spacetime with unit light speed, $L_i=t{\partial }_i+x_i{\partial }_t$ generates a Lorentz boost. It commutes with the d'Alembert operator and controls derivatives transverse to time slices in the vector field method for wave equations.

A Lorentzian metric is a smooth nondegenerate symmetric metric tensor with one negative and all other positive directions, or its opposite-sign convention. At each point, a nonzero tangent is timelike, null or spacelike according as $g(v,v)$ is negative, zero or positive in the first convention. Unlike a Riemannian metric, it gives a light cone rather than a positive norm. Minkowski spacetime is its constant flat model, and proper time along a timelike curve is $\int \sqrt{-g(\dot{\gamma },\dot{\gamma })}du$.

The stabilizer of a timelike four-momentum in $D$-dimensional Minkowski spacetime has connected group $SO(D-1)$. Massive particle polarizations form representations of this group.

For four-dimensional Minkowski spacetime, set $p=\arctan ((t-r)/L)$ and $q=\arctan ((t+r)/L)$, with $L>0$, then $T=p+q$, $R=q-p$. Multiplication by the square of the conformal factor $2\cos p\cos q/L$ yields the displayed metric on $R\ge 0$, $|T|+R<\pi$. Its radial Penrose diagram is a triangle: the timelike edge $R=0$ is the ordinary centre, while the two sloping edges are future null infinity and past null infinity. A signed Cartesian spatial coordinate in two dimensions gives a full diamond instead.

On ${\mathbb{R}}^{n+1}$ the Minkowski inner product with mostly-positive convention is

The Minkowski norm is the Lorentz-invariant quadratic form $U\cdot U={\eta }_{\mu \nu }{U}^{\mu }{U}^{\nu }$.