Particle spacetime, observers and worldlines

Use the signature $(-,+,+,+)$ throughout. In the flat case, Minkowski spacetime means the affine model $(A,V,\eta ,C^{+})$ of Oriented time-oriented affine Minkowski space (the spacetime primitives): $A$ is an affine space, $\eta$ is a nondegenerate Lorentz form on its four-dimensional displacement space, and $C^{+}$ selects the future cone. In the curved case, a background spacetime means a prescribed smooth time-oriented Lorentzian four-manifold $(M,g,N)$ of Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model; the metric and time orientation are fixed data, not dynamical unknowns here. No field equation or completeness assumption is included.

A massive worldline on an open interval $I\subseteq \mathbb{R}$ is a $C^2$ curve $x:I\to M$ with future-directed timelike tangent $T=dx\mathrm{/}ds$ at every parameter value. This is the $C^2$ subclass of Timelike worldlines, proper time, length and four-velocity on a Lorentzian four-manifold. A null worldline is a $C^2$ curve with nonzero future-directed null tangent everywhere. Its tangent has no proper-time normalization: $g(T,T)=0$, so the massive proper-time formula does not apply. In the flat model these are $C^2$ curves $x:I\to A$, with tangent defined by differentiating the displacement $x(s)-o$ for any fixed $o\in A$, lying in the selected future timelike or nonzero null cone of Oriented time-oriented affine Minkowski space (the spacetime primitives). The derivative is independent of $o$ because a change of origin adds a constant displacement. No affine-line or geodesic condition is imposed.

An observer at an event $p\in M$ is a future unit timelike vector $n\in T_{p}M$ with $g_p(n,n)=-1$, as in Observers, rest spaces and the spatial projection at an event. Its instantaneous rest space is $n^{\perp }=\{v:g_p(v,n)=0\}$. This is a pointwise vector-space construction; the definition does not assign a clock reading or a moving frame to a lone observer.

An observer congruence on an open set $U\subseteq M$ is a smooth field of future unit timelike vectors on $U$, as in Observer congruences, their observer curves and adapted charts; its observer curves are its integral curves. A tetrad on a nonempty open set $U$ means a smooth ordered frame field $(E_0,E_1,E_2,E_3)$ with $g(E_a,E_b)=\mathrm{diag}(-1,1,1,1{)}_{ab}$ for $a,b=0,1,2,3$, as in Local frame fields, coordinate frames and tetrad fields. A tetrad is future-directed when additionally $E_0$ is future-directed everywhere: $g(E_0,N)<0$ in the curved case, or $E_0(p)\in C^{+}$ in the flat case with $g=\eta$. In flat spacetime this future-directed subclass is the tetrad-field convention of Charts, tetrads, observer congruences and adapted charts in affine Minkowski space; the unqualified tetrad here retains the general GR convention. A chart is a coordinate map; an adapted chart is a chart whose spatial coordinates are constant along the congruence's integral curves. These objects are distinct: a chart need not be inertial, a tetrad need not be coordinate induced, and a congruence is a field of observers rather than a chart or a frame. This distinction agrees with Charts, tetrads, observer congruences and adapted charts in affine Minkowski space.

The coordinates and units used later are SI: in an inertial chart $x^0=ct$ and $x^i$ are metres, $t$ is seconds, $c>0$ is metres per second, and proper time is measured in seconds. In a general chart, coordinate components inherit their coordinate units; invariant equations are tensorial. No global inertial chart, global tetrad, or global observer congruence is asserted to exist.

Let $(M,g,N)$ be a smooth time-oriented Lorentzian four-manifold and let $m,c>0$. Let $\mathcal{V}\subset TM$ be an open set of future timelike vectors and let $f$ be a smooth vector field along the bundle projection $\mathcal{V}\to M$, with $f(p,V)\in T_{p}M$ and $g_p(V,f(p,V))=0$ for every $(p,V)\in \mathcal{V}$. Fix an initial parameter $s_0\in \mathbb{R}$. For every initial datum $(p_0,U_0)\in \mathcal{V}$ satisfying $g_{p_0}(U_0,U_0)=-c^2$, there is an open interval $J\ni s_0$ and a unique local solution consisting of a $C^2$ curve $x:J\to M$ and its $C^1$ tangent field $U=dx\mathrm{/}ds$, with $x(s_0)=p_0$, $U(s_0)=U_0$, $dx\mathrm{/}ds=U$, and $D_{s}U=f(x,U)\mathrm{/}m$. The solution depends smoothly on its initial data and on any finite-dimensional parameters on which $f$ depends smoothly, on a common sufficiently short compact interval containing $s_0$ near the base datum. It remains on the future unit timelike shell. There is a unique maximal solution on an open interval containing $s_0$, but this interval need not be all of $\mathbb{R}$.

Let $x:I\to M$ be a $C^2$ future timelike curve on a nonempty open interval $I$, with parameter $s$ and tangent $T=dx\mathrm{/}ds$. Fix $c>0$, $s_0\in I$, and ${\tau }_0\in \mathbb{R}$, and define $\mathrm{\ell }(s)=\sqrt{-g(T,T)}$ and $\tau (s)={\tau }_0+c^{-1}{\int }_{s_0}^s\mathrm{\ell }(r)dr$. Then ${\tau }^{\mathrm{\prime }}(s)=\mathrm{\ell }(s)\mathrm{/}c>0$, so $\tau$ is an increasing $C^2$ coordinate on its open image $J=\tau (I)$. The tangent with respect to this parameter is $U=dx\mathrm{/}d\tau =cT\mathrm{/}\mathrm{\ell }$ and satisfies $g(U,U)=-c^2$. This is a mathematical reparametrization result; by itself it does not identify $\tau$ with a physical clock reading.

Adopted convention. On a prescribed background $(M,g,N)$ from Particle spacetime, observers and worldlines, let $x:[a,b]\to M$, $a<b$, be a continuous piecewise-$C^1$ future timelike curve in the sense of Timelike worldlines, proper time, length and four-velocity on a Lorentzian four-manifold: it has a finite partition $a=s_0<\cdots <s_k=b$, each segment is $C^1$ up to its endpoints with future timelike one-sided tangents there, and the two tangents at a corner need not agree. If $s$ denotes its parameter and $T_i=dx\mathrm{/}ds$ on the $i$th segment, an ideal clock carried along it records the increment

$\mathrm{\Delta }\tau =\frac{1}{c}{\sum }_{i=1}^k{\int }_{s_{i-1}}^{s_i}\sqrt{-g(T_i,T_i)}ds\mathrm{.}$

The increment is independent of the admissible finite partition, by additivity of the integral, and is additive across corners. On each smooth segment this is the proper-time parameter with $d\tau =c^{-1}\sqrt{-g(dx\mathrm{/}ds,dx\mathrm{/}ds)}ds$. The flat-spacetime specialization is inherited from Adopted ideal-clock and ideal-vacuum-light interpretation; extending that convention to the curved prescribed background is a separate adoption of this framework.

When the parameter $s$ is measured in seconds, $x^0=ct$ is in metres, $c$ is in metres per second, and $\tau$ is in seconds. This is a model postulate, not a theorem or a reported measurement. It does not assert that a physical clock realizes the idealization exactly, that the background solves gravitational field equations, or that the test body affects the background.

Let $(M,g,N)$ be a prescribed smooth time-oriented Lorentzian four-manifold, $a<b$, and $x:[a,b]\to M$ a $C^2$ future timelike curve. Put $T=dx\mathrm{/}ds$, $\mathrm{\ell }=\sqrt{-g(T,T)}>0$, and for a constant $m>0$ and $c>0$ define the length action $S[x]=-mc{\int }_a^b\mathrm{\ell }(s)ds$. Let $H(ϵ,s)$ be a $C^2$ fixed-endpoint variation through future timelike curves for $ϵ$ near $0$, with $H(0,s)=x(s)$, $H(ϵ,a)=x(a)$, $H(ϵ,b)=x(b)$, and variation field $J={\mathrm{\partial }}_{ϵ}H{\mathrm{\mid }}_0$. Then

$\delta S={\frac{d}{dϵ}S[H(ϵ,\cdot )]\mid }_0=-mc{\int }_a^{b}g\text{\hspace{0.17em}⁣}(J,D_s\text{\hspace{0.17em}⁣}\left(\frac{T}{\mathrm{\ell }}\right))ds\mathrm{.}$

Consequently $x$ is stationary under all such variations exactly when $D_s(T\mathrm{/}\mathrm{\ell })=0$, equivalently $D_{\tau }U=0$ for the proper-time parameter of Proper-time parameter and normalized tangent. The length integral has the same value under every orientation-preserving $C^1$ change of variable $s=\varphi (r)$ with $\varphi :[a^{\mathrm{\prime }},b^{\mathrm{\prime }}]\to [a,b]$ a diffeomorphism and ${\varphi }^{\mathrm{\prime }}>0$, namely ${\int }_{a^{\mathrm{\prime }}}^{b^{\mathrm{\prime }}}{\mathrm{\ell }}_r(r)dr={\int }_a^b\mathrm{\ell }(s)ds$ for $T_r(r)={\varphi }^{\mathrm{\prime }}(r)T(\varphi (r))$ and ${\mathrm{\ell }}_r(r)=\sqrt{-g(T_r(r),T_r(r))}$. As a transformation of the stated $C^2$ curve and variation class, require instead that $\varphi$ be an orientation-preserving $C^2$ diffeomorphism; then the stationarity equation is covariant, since $D_r(T_r\mathrm{/}{\mathrm{\ell }}_r)(r)={\varphi }^{\mathrm{\prime }}(r)(D_s(T\mathrm{/}\mathrm{\ell }))(\varphi (r))$. This is a local variational statement; it makes no claim about global proper-time maximality, and the square-root action is not defined for a nonzero null tangent.

Let $a,b\in \mathbb{R}$ with $a<b$, and let $x:[a,b]\to M$ be a $C^2$ future timelike curve on the prescribed background of Ideal clocks on a prescribed Lorentzian background. Let $H:(-{ϵ}_0,{ϵ}_0)\times [a,b]\to M$ be a $C^2$ variation, for some ${ϵ}_0>0$, such that each $H(ϵ,\cdot )$ is future timelike, $H(0,s)=x(s)$, and $H(ϵ,a)=x(a)$ and $H(ϵ,b)=x(b)$. For a spinless, structureless massive test body of constant primitive mass $m>0$, measured in kilograms, and the SI conversion constant $c>0$ metres per second, adopt the action on this finite segment

$S_m[x]=-mc{\int }_a^b\sqrt{-g(dx\mathrm{/}ds,dx\mathrm{/}ds)}ds=-m{c}^2\mathrm{\Delta }\tau ,$

where $\mathrm{\Delta }\tau :=c^{-1}{\int }_a^b\sqrt{-g(dx\mathrm{/}ds,dx\mathrm{/}ds)}ds$ is the ideal-clock increment specified by Ideal clocks on a prescribed Lorentzian background on this smooth segment. The model's free trajectories are precisely the curves stationary under the fixed-endpoint $C^2$ variations just specified; this is the local variation class in Timelike length action: arbitrary-parameter first variation. When $s$ is measured in seconds, $T$ has units metres per second, and the action has units joule-seconds. This is an adopted model premise, not a consequence of the metric or clock postulate. It excludes self-gravity, radiation reaction, spin, finite-size structure, particle creation, and quantum effects; it does not assert empirical adequacy for any particular body.

Let $a<b$, let $x:[a,b]\to M$ be a $C^2$ curve with parameter $s$ and $T=dx\mathrm{/}ds$, fix $c>0$, let $e(s)>0$ be smooth, and let $m\ge 0$ be constant. Take independent $C^2$ fixed-endpoint variations of $x$ and smooth interior variations of $e$. For the action

$S_e[x,e]=\frac{1}{2}{\int }_a^b\text{\hspace{0.17em}⁣}(e^{-1}g(T,T)-e{m}^2{c}^2)ds,$

independent fixed-endpoint variations in $x$ and interior variations in $e$ give

$g(T,T)+e^2{m}^2{c}^2=0,D_s(T\mathrm{/}e)=0.$

On the future timelike massive branch $m>0$, this constraint uniquely gives $e=\mathrm{\ell }\mathrm{/}(mc)$, $T\mathrm{/}e=mU$, and the action reduces to $-mc\int \mathrm{\ell }ds$; in proper-time gauge $s=\tau$, $e=1\mathrm{/}m$. On the nonzero future null branch $m=0$, $P:=T\mathrm{/}e$ is nonzero future null and $D_{s}P=0$; if $d\lambda =eds$, then $dx\mathrm{/}d\lambda =P$ and $D_{\lambda }P=0$. There is no null proper-time gauge, and the scale of $P$ remains initial data. For finite intervals $[a^{\mathrm{\prime }},b^{\mathrm{\prime }}]$ and $[a,b]$, an orientation-preserving $C^{\mathrm{\infty }}$ diffeomorphism $s=\varphi (r):[a^{\mathrm{\prime }},b^{\mathrm{\prime }}]\to [a,b]$ with ${\varphi }^{\mathrm{\prime }}(r)>0$ gives $T_r(r)={\varphi }^{\mathrm{\prime }}(r)T(\varphi (r))$ and $e_r(r)={\varphi }^{\mathrm{\prime }}(r)e(\varphi (r))$; the einbein action is invariant, and this transformation preserves the stated $C^2$ curve and smooth-einbein configuration class.

Let $a,b\in \mathbb{R}$ with $a<b$. A configuration on $[a,b]$ consists of a $C^2$ curve $x:[a,b]\to M$ and a smooth positive function $e:[a,b]\to (0,\mathrm{\infty })$ on the prescribed background of Ideal clocks on a prescribed Lorentzian background. Vary $x$ and $e$ independently: $x$ through $C^2$ variations $H:(-{ϵ}_0,{ϵ}_0)\times [a,b]\to M$ for some ${ϵ}_0>0$, with $H(0,s)=x(s)$ and $H(ϵ,a)=x(a)$, $H(ϵ,b)=x(b)$, and no causal restriction on the varied curves; and $e$ through jointly smooth positive variations $e_{ϵ}:[a,b]\to (0,\mathrm{\infty })$ whose variation field $\delta e:={\mathrm{\partial }}_{ϵ}{e}_{ϵ}{\mathrm{\mid }}_{ϵ=0}$ is compactly supported in $(a,b)$. Adopt a distinct massless test-particle sector with primitive action

$S_0[x,e]=\frac{1}{2}{\int }_a^{b}e(s{)}^{-1}{g}_{x(s)}(T(s),T(s))ds,T(s)=dx\mathrm{/}ds,$

and stationary under all such independent variations. The variational equations and their null branch follow from Positive-einbein variation and massive/null sector equations: the constraint is $g(T,T)=0$, and the physical branch requires $T$ to be nonzero and future-directed. The associated momentum $P=T\mathrm{/}e$ is nonzero future null; its positive scale is primitive initial data, not $mU$ for any $m=0$ substitution. Null curves have no proper-time parameter. With $s$ in seconds, $T$ in metres per second, and $e$ in kg${}^{-1}$, the integral has units joule-seconds. For finite intervals $[a^{\mathrm{\prime }},b^{\mathrm{\prime }}]$ and $[a,b]$, an orientation-preserving $C^{\mathrm{\infty }}$ diffeomorphism $s=\varphi (r):[a^{\mathrm{\prime }},b^{\mathrm{\prime }}]\to [a,b]$ with ${\varphi }^{\mathrm{\prime }}>0$ transforms $T_r(r)={\varphi }^{\mathrm{\prime }}(r)T(\varphi (r))$ and $e_r(r)={\varphi }^{\mathrm{\prime }}(r)e(\varphi (r))$; this leaves the functional invariant by Positive-einbein variation and massive/null sector equations and preserves the stated configuration class. Neither the scale nor a clock reading is inferred from the ray image alone.

This is an adopted classical model formulation, not an empirical observation or a theorem. Its scope excludes mass generation, interactions, and quantum field effects.

Use the massive and null sectors as separately adopted in Massive spinless test-particle action and Massless test-particle action and momentum scale. For a massive body of constant mass $m>0$, its mechanical four-momentum is $P=mU$, where along an admissible $C^2$ future timelike trajectory $x:I\to M$, with $I\subseteq \mathbb{R}$ an interval with nonempty interior and parameter $s$, put $T=dx\mathrm{/}ds$, and $U:=cT\mathrm{/}\sqrt{-g(T,T)}$ is its future tangent normalized by $g(U,U)=-c^2$. The clock convention in the massive-sector adoption gives $d\tau \mathrm{/}ds=\sqrt{-g(T,T)}\mathrm{/}c$, so $U=dx\mathrm{/}d\tau$. In the null sector, the mechanical four-momentum is the separately specified nonzero future-null vector $P$ from the einbein formulation; no null proper time or $P=mU$ is defined.

At an event, for any observer $n$ of Observers, rest spaces and the spatial projection at an event, define the spatial momentum and observer energy by

$q_n(P):=P+g(P,n)n\in n^{\perp },E_n(P):=-cg(P,n)\mathrm{.}$

The projection identity and positive-definiteness of $n^{\perp }$ are supplied by The observer rest-space decomposition and the spatial projection. With SI units, $P$ has units kg m s${}^{-1}$, $q_n$ has the same units, and $E_n$ has units joules. This definition is pointwise and does not assert conservation, positivity, or an energy interpretation for an arbitrary vector field that is not an observer.

On a prescribed smooth time-oriented Lorentzian background, stationarity for the massive action of Massive spinless test-particle action is defined on a finite segment $[a,b]$ with the stated fixed-endpoint $C^2$ variations. A stationary massive path on that segment satisfies $D_{\tau }U=0$ in proper-time parameter, with $P=mU$ and $m>0$ constant. In the separately adopted null sector of Massless test-particle action and momentum scale, stationarity for the einbein action is likewise defined on a finite segment, using independent fixed-endpoint $C^2$ variations of the curve and smooth interior variations of the einbein. Its nonzero future-null branch has momentum $P$ satisfying $D_{\lambda }P=0$ in an affine parameter $\lambda$ with $dx\mathrm{/}d\lambda =P$. These equations hold locally on the interiors of the finite segments; their ODE solutions extend uniquely to maximal solutions on open intervals, with no action integrated over an infinite interval. For any ${\tau }_0\in \mathbb{R}$ and $p_0\in M$, an initial vector $U_0\in T_{p_0}M$ that is future timelike and satisfies $g_{p_0}(U_0,U_0)=-c^2$ determines a unique local massive solution with $x({\tau }_0)=p_0$, $U({\tau }_0)=U_0$, and a unique maximal solution on an open interval containing ${\tau }_0$. For any ${\lambda }_0\in \mathbb{R}$ and $p_0\in M$, a nonzero future-null vector $P_0\in T_{p_0}M$ determines a unique local null solution with $x({\lambda }_0)=p_0$, $P({\lambda }_0)=P_0$, and a unique maximal solution on an open interval containing ${\lambda }_0$. Neither maximal interval is asserted to be all of $\mathbb{R}$.

More explicitly, there are open intervals $J\ni {\tau }_0$ and $J^{\mathrm{\prime }}\ni {\lambda }_0$ such that the massive local solution is a $C^2$ curve $x:J\to M$ with $U=dx\mathrm{/}d\tau$, and the null local solution is a $C^2$ curve $x:J^{\mathrm{\prime }}\to M$ with $P=dx\mathrm{/}d\lambda$; their stated initial values are imposed at the indicated parameter values.

For a named massive test body with constant primitive mass $m>0$ measured in kilograms, SI conversion constant $c>0$ in metres per second, and momentum $P=mU$ as defined in Mechanical momentum and observer energy, define the future unit timelike shell by

${\mathrm{\Sigma }}_c^{+}:=\{(p,V)\in TM:V\text{ future timelike},g_p(V,V)=-c^2\}\mathrm{.}$

Let $\mathcal{V}\subset TM$ be an open neighborhood of ${\mathrm{\Sigma }}_c^{+}$ contained in the future timelike velocity cone. For a massive body trajectory $x:I\to M$ that is $C^2$ and parameterized by proper time $\tau$, with $I\subseteq \mathbb{R}$ a nonempty open interval, put $U=dx\mathrm{/}d\tau$ and require $(x(\tau ),U(\tau ))\in {\mathrm{\Sigma }}_c^{+}\subseteq \mathcal{V}$ for every $\tau \in I$. Specify a smooth map $f:\mathcal{V}\to TM$ over the bundle projection $\pi :TM\to M$, meaning $\pi (f(p,V))=p$ (equivalently, $f(p,V)\in T_{p}M$), and require $g_p(V,f(p,V))=0$ for every $(p,V)\in \mathcal{V}$. One may adopt this force map and the equation

$D_{\tau }P=f(x,U)\mathrm{.}$

The orthogonality condition is part of this model's compatibility assumptions, imposed throughout $\mathcal{V}$; it preserves $g(U,U)=-c^2$ for constant $m$. Since $P=mU$, this equation is equivalent to $D_{\tau }U=f(x,U)\mathrm{/}m$ on the stated domain. It is not a law selecting a force for every particle or interaction. A local IVP follows only for data in $\mathcal{V}$, as in Smooth shell-tangent force initial-value problem. The force has SI units newtons when $\tau$ is in seconds and $P$ has units kg m s${}^{-1}$. No empirical force measurements, universal coupling, or global existence are asserted.

Let $\mathrm{\Omega }\subseteq M$ be an open region of the prescribed background and let $K$ be a smooth Killing field on $\mathrm{\Omega }$. For a massive force-postulate trajectory $x:I\to M$, parameterized by proper time on a nonempty open interval $I\subseteq \mathbb{R}$ with $x(I)\subseteq \mathrm{\Omega }$, constant $m>0$, $P=mU$, and $D_{\tau }P=f$, the scalar $g(K,P)$ obeys