Local frame fields, coordinate frames and tetrad fields

Let $(M,g,N)$ be a time-oriented smooth Lorentzian four-manifold and let $U\subseteq M$ be a nonempty open subset (Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model). Write $\eta :=\mathrm{diag}(-1,1,1,1)$ for the four-by-four diagonal matrix with diagonal entries $-1,1,1,1$, and use capital indices $A,B=0,1,2,3$ for frame members.

(a) Local frame fields. A local frame field on $U$ is a tuple $E=(E_0,E_1,E_2,E_3)$ of smooth vector fields on $U$ (A smooth vector field is a smooth section of the tangent bundle) such that for every $q\in U$ the vectors $E_0(q),E_1(q),E_2(q),E_3(q)$ form an ordered basis of $T_{q}M$, equivalently a pointwise independent tuple of four vector fields.

(b) Coordinate frames. If $(U,x)$ is a smooth chart, the tuple $(\mathrm{\partial }\mathrm{/}\mathrm{\partial }{x}^0,\mathrm{\partial }\mathrm{/}\mathrm{\partial }{x}^1,\mathrm{\partial }\mathrm{/}\mathrm{\partial }{x}^2,\mathrm{\partial }\mathrm{/}\mathrm{\partial }{x}^3)$ of coordinate fields is a local frame field on $U$, its coordinate frame. Two coordinate fields of one chart commute: by the coordinate formula for the Lie bracket (Coordinate formula for the Lie bracket, The Lie bracket of smooth vector fields) applied to constant coordinate components, $[\mathrm{\partial }\mathrm{/}\mathrm{\partial }{x}^a,\mathrm{\partial }\mathrm{/}\mathrm{\partial }{x}^b]=0$ for all $a,b$. Consequently a local frame field $(E_0,\dots ,E_3)$ with $[E_A,E_B]\mathrm{\ne }0$ for some $A,B$ is not the coordinate frame of any chart; this necessary condition is the only one asserted here, and no converse is claimed by this definition.

(c) Tetrad fields. A tetrad field (or $g$-orthonormal frame field) on $U$ is a local frame field $(E_0,E_1,E_2,E_3)$ on $U$ such that

$g_q(E_A(q),E_B(q))={\eta }_{AB}\text{for all }A,B\text{ and all }q\in U\mathrm{.}$

A tetrad field is future-directed when in addition $E_0(q)$ is future-directed for every $q\in U$ (Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model); then each $E_0(q)$ is an observer at $q$ (Observers, rest spaces and the spatial projection at an event), so $E_0$ is an observer congruence on $U$ (Observer congruences, their observer curves and adapted charts), while $E_1,E_2,E_3$ are pointwise an orthonormal basis of the rest space of that observer. Neither existence of a tetrad field on a prescribed open set, nor global existence on $M$, is asserted by this definition; local existence near every point is proved in the existence theorem on this page.

(d) Observer-adapted tetrads. Given an observer $n$ at a point $p$, a tetrad at $p$ is $n$-adapted when its first member is $n$. Given an observer congruence $n$ on $U$, a tetrad field on $U$ is $n$-adapted when $E_0=n$; its remaining members are then $g$-orthonormal fields in the moving rest spaces of the congruence.

(e) Scope. A tetrad field is not assumed to be a coordinate frame, and a frame field is not assumed to be orthonormal. Components of a tensor field in a tetrad and components in a chart are different objects; the transformation rules for the two are distinguished later on this page. No relation to the physical interpretation of a reference frame is asserted here beyond the geometric data listed above.

Choice convention. Smooth vector fields are tangent-bundle sections, whose canonical smooth structure assumes the axiom ${\mathrm{A}\mathrm{C}}_{\omega }$ of The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$) (A smooth vector field is a smooth section of the tangent bundle); this definition inherits that assumption and uses no further choice principle.