Observers, rest spaces and the spatial projection at an event

Let $(M,g,N)$ be a time-oriented smooth Lorentzian four-manifold and let $p\in M$ (Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model).

(a) Observers. An observer at $p$ is a tangent vector $n\in T_{p}M$ such that $g_p(n,n)=-1$ and $n$ is future-directed in the sense of Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model, that is, $g_p(n,N_p)<0$. Because $g_p(n,n)=-1\mathrm{\ne }0$, an observer is timelike and in particular nonzero (Timelike, null, spacelike and causal vectors of a Lorentz vector space). Observers at $p$ exist: the vector $N_p$ is timelike, and $n:=N_p\mathrm{/}\sqrt{-g_p(N_p,N_p)}$ satisfies $g_p(n,n)=-1$ and $g_p(n,N_p)=-\sqrt{-g_p(N_p,N_p)}<0$, where the square root is the nonnegative root of Rational powers $a^r$ of a positive base; then $-n$ is a unit timelike vector with $g_p(-n,N_p)>0$, so it is not an observer, and the two vectors $n,-n$ point in opposite time directions.

(b) Rest space. For an observer $n$ at $p$, the rest space of $n$ is the $g_p$-orthogonal complement of the line $\mathbb{R}n$,

$n^{\perp }=\{v\in T_{p}M:g_p(v,n)=0\},$

in the sense of eta-orthogonality and orthogonal complements in a Lorentz vector space. It is a linear subspace of $T_{p}M$. The further facts that $\dim n^{\perp }=3$ and that the restriction of $g_p$ to $n^{\perp }$ is positive definite are proved in The observer rest-space decomposition and the spatial projection ↗; they are consequences, not part of the definition.

(c) Spatial projection. The spatial projection of the observer $n$ is the map

$h_n:T_{p}M⟶T_{p}M,h_n(v):=v+g_p(n,v)n\mathrm{.}$

The map is linear because $g_p$ is bilinear, its image is contained in $n^{\perp }$, and it fixes $n^{\perp }$ pointwise; that it is the $g_p$-orthogonal projection onto $n^{\perp }$ with kernel $\mathbb{R}n$ is proved in The observer rest-space decomposition and the spatial projection ↗.

(d) Scope. An observer at $p$ is a single tangent vector. It is not an ordered basis or frame of $T_{p}M$, not a chart, not a field of observers such as the congruences defined later on this page, and not a curve or a worldline: no parametrized curve, clock reading, proper time or measurement is attached to $n$ by this definition. The physical reading of an observer as the direction of an ideal clock's worldline, and the measurement interpretation of the observer-relative quantities introduced later on this page, belong to the physical postulates and unit conventions of the framework and are not asserted here.

Both conditions are used. The condition $g_p(n,n)=-1$ makes $n$ a unit timelike vector of a fixed normalization and forces $n\mathrm{\ne }0$; the condition $g_p(n,N_p)<0$ selects one of the two time directions and is not implied by the first, since replacing $n$ by $-n$ preserves $g_p(n,n)=-1$ and reverses the sign of $g_p(n,N_p)$ (Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model). No positivity of any observer-relative energy is asserted or used in this definition.