Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model

Let $M$ be a smooth four-manifold without boundary, so that $M$ is Hausdorff and second countable and carries a maximal smooth atlas (Smooth manifolds and their smooth charts). Let $g$ be a smooth covariant $2$-tensor field on $M$ in the sense of Smooth covariant 2-tensor fields on a smooth manifold. Then $g$ assigns to every $p\in M$ a bilinear form $g_p:T_{p}M\times T_{p}M\to \mathbb{R}$ (Derivations at a point and the tangent space).

(a) Lorentzian metric. The field $g$ is a Lorentzian metric of signature $(-,+,+,+)$ on $M$ when it is symmetric and nondegenerate, and at every $p\in M$ the pair $(T_{p}M,g_p)$ is a Lorentz vector space of signature $(-,+,+,+)$ in the sense of Lorentz forms of inertia (3,1,0) on a 4-dimensional real vector space: that is, $g_p$ has inertia $(3,1,0)$, so that some — equivalently every — basis of $T_{p}M$ in which the matrix of $g_p$ is diagonal has exactly one negative and three positive diagonal entries. The pair $(M,g)$ is a smooth Lorentzian four-manifold (or Lorentzian manifold of signature $(-,+,+,+)$). In a smooth chart $(U,x)$ the components of $g$ are the $C^{\mathrm{\infty }}$ functions $g_{ab}(q)=g_q({\mathrm{\partial }}_a{\mathrm{\mid }}_q,{\mathrm{\partial }}_b{\mathrm{\mid }}_q)$ of Smooth covariant 2-tensor fields on a smooth manifold, and $g_{ab}=g_{ba}$ by symmetry. We use indices $a,b,c,d=0,1,2,3$ and the Einstein summation convention, with components in a chart written as lower-case indexed symbols and equations between indexed fields understood basis-independently. These conventions are fixed once for this library and the physical reading of the symbols is recorded with the framework postulates, separately from this mathematical definition.

(b) Pointwise causal structure. For every $p\in M$ the vocabulary of Timelike, null, spacelike and causal vectors of a Lorentz vector space applies to the Lorentz vector space $(T_{p}M,g_p)$: a tangent vector $v\in T_{p}M$ is timelike when $g_p(v,v)<0$, null when $v\mathrm{\ne }0$ and $g_p(v,v)=0$, spacelike when $g_p(v,v)>0$, and causal when it is timelike or null. Every nonzero $v\in T_{p}M$ has exactly one causal character. The timelike set $T_p:=\{v\in T_{p}M:g_p(v,v)<0\}$ has exactly two connected components, each nonempty, open, convex and path-connected; each $g_p$-orthonormal basis $e_0,e_1,e_2,e_3$ of $T_{p}M$ satisfies $g_p(e_a,e_b)=\mathrm{diag}(-1,1,1,1{)}_{ab}$ for some ordering, so in particular nonzero orthonormal vectors exist at every point (Every Lorentz form admits an ordered basis with matrix diag(-1,1,1,1), The timelike set has exactly two convex components, described in orthonormal coordinates).

(c) Time orientation. A time orientation of $(M,g)$, also called a choice of future cone, is a smooth vector field $N$ on $M$ (A smooth vector field is a smooth section of the tangent bundle, whose tangent-bundle smooth structure uses the axiom ${\mathrm{A}\mathrm{C}}_{\omega }$ of The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$); this is the only use of that axiom in this definition) such that

$g_p(N_p,N_p)<0\text{for every }p\in M\mathrm{.}$

The function $p\mapsto g_p(N_p,N_p)$ is $C^{\mathrm{\infty }}$ because $g$ is a smooth covariant $2$-tensor field and $N$ a smooth field, so the condition is a pointwise strict inequality between a smooth function and $0$; since $g_p(N_p,N_p)<0$ implies $N_p\mathrm{\ne }0$, a time orientation is nowhere vanishing. A time-oriented smooth Lorentzian four-manifold is a triple $(M,g,N)$ consisting of a smooth Lorentzian four-manifold and a time orientation $N$.

Given a time orientation $N$ and $p\in M$, the vector $N_p$ is timelike, and we define a nonzero causal vector $v\in T_{p}M$ to be future-directed when $g_p(v,N_p)<0$ and past-directed when $g_p(v,N_p)>0$. The classification of nonzero causal vectors into exactly these two classes, and its agreement with the two components of the timelike set (the future component is the component $C_p^{+}$ containing $N_p$, and the future causal vectors form $\mathrm{cl}(C_p^{+})\setminus \{0\}$), is proved in Classification of future and past causal vectors by a timelike field ↗; in particular no nonzero causal $v$ satisfies $g_p(v,N_p)=0$, so the definition is exhaustive and mutually exclusive. Replacing $N$ by $-N$ reverses every future/past label, and two time orientations $N,N^{\mathrm{\prime }}$ determine the same future-directed causal vectors at every point where $g_p(N_p,N_p^{\mathrm{\prime }})<0$, while they determine opposite ones where $g_p(N_p,N_p^{\mathrm{\prime }})>0$, by Two time orientations agree or are opposite on each connected set ↗; that lemma also shows the sign of $p\mapsto g_p(N_p,N_p^{\mathrm{\prime }})$ is constant on every connected component of $M$.

(d) The inverse metric. Because $g_p$ is nondegenerate, it induces at each $p$ an isomorphism ${\mathrm{♭}}_p:T_{p}M\to T_p^{\ast }M$, $v\mapsto g_p(v,\cdot )$, and an inverse metric $g_p^{-1}$ on the dual space; the inverse metric, the musical isomorphisms and the components $g^{ab}$ are defined separately on this page, where the smoothness of those components is also proved. Neither construction uses any choice principle beyond the one already recorded in (c).

(e) Smooth timelike curves and their scalar normalization. Before using these quantities in the physical conventions, fix a positive real scaling parameter $c$ (with no physical meaning or dimension in this mathematical item). A smooth future timelike curve is a smooth map $\gamma :I\to M$ from a nonempty open real interval $I$ (Intervals of $\mathbb{R}$: the nine order-convex forms, nondegeneracy, and length, $C^r$ and smooth maps between smooth manifolds) whose velocity derivation ${\gamma }^{\mathrm{\prime }}(s)\in T_{\gamma (s)}M$ (The velocity derivation of a smooth curve) satisfies $g({\gamma }^{\mathrm{\prime }},{\gamma }^{\mathrm{\prime }})<0$ and $g({\gamma }^{\mathrm{\prime }},N)<0$ at every $s\in I$. In a chart its components are $d(x^a\circ \gamma )\mathrm{/}ds$, $a=0,1,2,3$. For a supplied base point $s_0\in I$ and additive constant ${\tau }_0\in \mathbb{R}$ define

$h(s):=\sqrt{-g_{\gamma (s)}({\gamma }^{\mathrm{\prime }}(s),{\gamma }^{\mathrm{\prime }}(s))},\tau (s):={\tau }_0+\frac{1}{c}{\int }_{s_0}^{s}h(r)dr,U(s):=\frac{c{\gamma }^{\mathrm{\prime }}(s)}{h(s)}\mathrm{.}$

The integral is oriented when $s<s_0$. The function $\tau$ is called proper time for this scaling and $U$ the four-velocity of the smooth curve; its increment times $c$ is the Lorentzian length. The well-definedness proof below shows ${\tau }^{\mathrm{\prime }}>0$, that $\tau :I\to \tau (I)$ is a smooth diffeomorphism, and that $U$ is the velocity of $\gamma \circ {\tau }^{-1}$. Thus $g(U,U)=-c^2$ and $U$ is future timelike. This introduces the smooth case before any physical unit reading; the later worldline definition extends it to its stated $C^k$, endpoint and piecewise-$C^1$ classes. It asserts no clock coupling, motion law or physical value of $c$.

Convention and scope. The physical reading of a smooth time-oriented Lorentzian four-manifold as a classical spacetime model, and the domain in which that reading is adopted, are recorded in the framework postulate post-gr-spacetime-and-classical-domain; the words spacetime and event carry no mathematical content beyond this definition, and this item asserts no claim about nature. The dimension $4$ and the absence of boundary are part of the data; nothing in the definition assumes that $M$ is nonempty or connected, and no completeness, extendibility or maximality property of $(M,g)$ is asserted.