Charts, tetrads, observer congruences and adapted charts in affine Minkowski space

Work in the oriented time-oriented affine Minkowski space $(A,V,\eta ,C^{+})$ (Oriented time-oriented affine Minkowski space (the spacetime primitives)) with its inertial atlas (The affine atlas makes the Minkowski model a smooth 4-manifold diffeomorphic to R^4) and the canonical identification of tangent spaces with $V$ (The tangent space of the affine Minkowski model at any event is canonically V). The objects defined below are, in general, different objects, and the distinctions between them are part of the definition.

(a) Chart and inertial chart. A chart of the model is a diffeomorphism $u:U\to u(U)$ from a nonempty open subset $U\subseteq A$ onto an open subset of ${\mathbb{R}}^4$ (Diffeomorphisms and local diffeomorphisms of manifolds, Smooth manifolds and their smooth charts), written with components $(u^0,u^1,u^2,u^3)$. An inertial chart is the global chart $x(p)=E^{-1}(p-o)$ of an inertial frame $(o;e_0,e_1,e_2,e_3)$ (Inertial frames and inertial charts of the affine Minkowski model). A general chart need not be inertial: if $x$ is an inertial chart and $N:x(U)\to {\mathbb{R}}^4$ is a diffeomorphism onto its image which is not the restriction of an affine map, then $u:=N\circ x$ is a chart that is not inertial, because if $u$ were the inertial chart of a frame then $x\circ u^{-1}=N^{-1}$ would be affine by Inertial charts are related by proper orthochronous affine Lorentz maps, and every such map is a chart transition and hence $N=(N^{-1}{)}^{-1}$ would be affine as the inverse of an affine bijection, a contradiction; the map $N(y)=(y^0,y^1+(y^0{)}^2,y^2,y^3)$ on ${\mathbb{R}}^4$ is an explicit example.

(b) Tetrad at an event and tetrad field. A tetrad at $p\in A$ is an ordered basis $(e_0,e_1,e_2,e_3)$ of $V$ with $\eta (e_a,e_b)=\mathrm{diag}(-1,1,1,1{)}_{ab}$ and $e_0$ future, $e_0\in C^{+}$ (Timelike, null, spacelike and causal vectors of a Lorentz vector space, Time orientation, future and past timelike/null/causal vectors, and the closed cones): this is the basis of an inertial frame without an origin and without the orientation condition. A tetrad field on a nonempty open subset $\mathrm{\Omega }\subseteq A$ is a tuple $(E_0,E_1,E_2,E_3)$ of smooth vector fields on $\mathrm{\Omega }$ (C^k tensor fields on affine Minkowski space and their inertial components, type $(1,0)$) whose values at every $p\in \mathrm{\Omega }$ form a tetrad at $p$. A tetrad field is observer-adapted to a prescribed congruence $n$ when $E_0=n$. Along a prescribed $C^1$ future timelike worldline $x:I\to A$, observer adaptation means $E_0(x(s))=x^{\mathrm{\prime }}(s)\mathrm{/}\sqrt{-\eta (x^{\mathrm{\prime }}(s),x^{\mathrm{\prime }}(s))}$ whenever $x(s)\in \mathrm{\Omega }$; this is its normalized future unit tangent (Curves, regular causal curves and future timelike worldlines in affine Minkowski space).

(c) Observer congruence. An observer congruence on a nonempty open subset $\mathrm{\Omega }\subseteq A$ is a smooth vector field $n:\mathrm{\Omega }\to V$ with $\eta (n,n)=-1$ and $n(p)\in C^{+}$ for every $p\in \mathrm{\Omega }$. Its observers are its maximal integral curves (Integral curves of a vector field, Through each point there is a unique maximal integral curve). Each observer is a future timelike worldline (Curves, regular causal curves and future timelike worldlines in affine Minkowski space): its tangent at a parameter value $s$ is $n(x(s))$, which is future timelike, and it is regular because $\eta (n,n)=-1$ makes $n$ nowhere zero. Completeness of the integral curves, a global product structure of $\mathrm{\Omega }$, and global synchronization are not asserted by this definition.

(d) Adapted chart. For a congruence $n$ on $\mathrm{\Omega }$, an $n$-adapted chart is a chart $(u^0,u^1,u^2,u^3):U\to {\mathbb{R}}^4$ with $U$ a nonempty open subset of $\mathrm{\Omega }$ whose spatial coordinates $u^1,u^2,u^3$ are constant along every integral curve of $n$ contained in $U$. No global adapted chart is asserted, and $u^0$ is not required to be affine.

(e) Distinctions. A chart need not be inertial, as in (a); a chart need not be $n$-adapted for any congruence. A tetrad field is not in general the coordinate frame of a chart: if $(E_0,E_1,E_2,E_3)$ is the coordinate frame of a chart $(v^0,v^1,v^2,v^3)$, that is, $E_a=\mathrm{\partial }\mathrm{/}\mathrm{\partial }{v}^a$, then $[E_a,E_b]=0$ for all $a,b$ (The Lie bracket of smooth vector fields, Coordinate formula for the Lie bracket), so a tetrad field whose members do not commute pairwise is not the coordinate frame of a chart; no converse is claimed here. A congruence need not be the flow of a coordinate, and its pointwise rest spaces need not assemble into hypersurfaces; that integrability question is decided later on this page. In an inertial chart $x$ of $(o;e_0,e_1,e_2,e_3)$, the coordinate vector fields $\mathrm{\partial }\mathrm{/}\mathrm{\partial }{x}^a=e_a$ form a constant tetrad field, with values equal to the frame's ordered basis under $T_{p}A=V$. Its time vector field $n=\mathrm{\partial }\mathrm{/}\mathrm{\partial }{x}^0=e_0$ alone defines the congruence of stationary observers, whose integral curves are $s\mapsto x^{-1}(s+s_0,r^1,r^2,r^3)$ for fixed $s_0,r^1,r^2,r^3$. The chart, the ordered tetrad and the single vector field $n$ remain distinct objects.

• Non-inertial charts exist, and the example is checked. The map $N(y)=(y^0,y^1+(y^0{)}^2,y^2,y^3)$ is a diffeomorphism of ${\mathbb{R}}^4$ with the smooth two-sided inverse $z\mapsto (z^0,z^1-(z^0{)}^2,z^2,z^3)$. It is not the restriction of an affine map: $N(0)=0$ and $N(2e_0)=(2,4,0,0)\mathrm{\ne }2N(e_0)=(2,2,0,0)$ for $e_0=(1,0,0,0)$, so $N$ is not linear, and with $N(0)=0$ it is therefore not affine. If $N\circ x$ were an inertial chart, the transition $x\circ (N\circ x{)}^{-1}=N^{-1}$ between inertial charts would be affine by Inertial charts are related by proper orthochronous affine Lorentz maps, and every such map is a chart transition; the inverse of an affine bijection $z\mapsto Az+b$ of ${\mathbb{R}}^4$ is again affine, $w\mapsto A^{-1}w-A^{-1}b$ (Linear and affine Lorentz maps, and the Lorentz group), so $N$ would be affine, a contradiction. Hence $N\circ x$ is a chart that is not inertial, and being $C^{\mathrm{\infty }}$ it belongs to the model's smooth structure.

• Tetrads exist at every event, and tetrad fields are smooth by components. Every Lorentz form admits an ordered basis with matrix diag(-1,1,1,1) supplies an $\eta$-orthonormal basis $(f_0,f_1,f_2,f_3)$ of $V$; since $\eta (f_0,f_0)=-1<0$ the vector $f_0$ is timelike, so exactly one of $f_0$ and $-f_0$ lies in $C^{+}$ because the timelike set is the disjoint union of the two future and past cones (Time orientation, future and past timelike/null/causal vectors, and the closed cones, Timelike, null, spacelike and causal vectors of a Lorentz vector space), and negating it preserves $\eta$-orthonormality; no orientation condition is imposed on a tetrad, so this already produces a tetrad at any prescribed event. The formulation with the additional positive-orientation condition, and with an arbitrary origin, is the inertial-frame existence statement of Existence of future-oriented inertial frames at every event. A tetrad field is componentwise smooth in an inertial chart by the definition of $C^k$ tensor fields on the affine model (C^k tensor fields on affine Minkowski space and their inertial components, The tangent space of the affine Minkowski model at any event is canonically V).

• Congruences have well-defined observers. A smooth unit field $n$ on $\mathrm{\Omega }$ is a smooth map $\mathrm{\Omega }\to V$ with values in the future timelike cone, and Through each point there is a unique maximal integral curve supplies through every point a unique maximal integral curve of $n$ (Integral curves of a vector field); along such a curve the identity $\eta (n,n)=-1$ gives $\eta (x^{\mathrm{\prime }},x^{\mathrm{\prime }})=-1$ at every parameter value, so each observer is a future timelike worldline with future unit tangent, in particular regular (Curves, regular causal curves and future timelike worldlines in affine Minkowski space). Nothing about completeness of the maximal domains is asserted.

• The commutation condition is only necessary. If $(E_0,\dots ,E_3)$ is the coordinate frame of a chart $(v^0,\dots ,v^3)$, then $[E_a,E_b]=0$ by the coordinate formula for the Lie bracket; a tetrad field with $[E_a,E_b]\mathrm{\ne }0$ for some $a,b$ therefore cannot be a coordinate frame. The converse (a commuting tetrad field is locally a coordinate frame) is a Frobenius-type statement and is not asserted by this definition.