Connections And Free Fall

Let $(M,g,N)$ be a time-oriented smooth Lorentzian four-manifold (Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model) and let $c>0$ be a fixed real constant. All curves below take values in $M$; the physical reading of $c$, of a worldline as the history of a clock or body, and of the parameter values as times is not part of this definition and is adopted separately in the postulates of the framework.

(a) Regularity classes. Let $I\subseteq \mathbb{R}$ be an interval with nonempty interior (Intervals of $\mathbb{R}$: the nine order-convex forms, nondegeneracy, and length) and let $\gamma :I\to M$ be a map. A chart representative of $\gamma$ is the map $x\circ \gamma$ for a smooth chart $(U,x)$ of $M$ (Smooth manifolds and their smooth charts).

• For $k\in {\mathbb{N}}_0\cup \{\mathrm{\infty }\}$ and an interior point $t_0\in I$, the map $\gamma$ is $C^k$ at $t_0$ when there is a chart representative $x\circ \gamma$ defined near $t_0$ that is $C^k$ there in the componentwise Euclidean sense. This condition is independent of the chart: on the overlap of two chart representatives the transition map is smooth and the chain rule applies (The chain rule for total derivatives: $D(g\circ f)(a)=Dg(f(a))\circ Df(a)$, $C^r$ and smooth maps between smooth manifolds). If $t_0\in I$ is an included endpoint of $I$, the coordinate representative and its derivatives through order $k$ are continuous up to $t_0$ on the relative interval $I\cap (t_0-\delta ,t_0+\delta )$ for some $\delta >0$, with derivatives at $t_0$ taken one-sided. For $k=\mathrm{\infty }$ this holds at every finite order. Merely possessing a one-sided secant tangent, as defined next, does not by itself make a curve $C^1$ at the endpoint. The map $\gamma$ is $C^k$ on $I$ when it is $C^k$ at every point of $I$.

• Let $b\in \mathbb{R}$ and $\epsilon >0$. A curve $\gamma :(b-\epsilon ,b]\to M$ has left tangent $v\in T_{\gamma (b)}M$ at $b$ when for one — equivalently every — chart $(U,x)$ around $\gamma (b)$, writing $v={\sum }_{a=0}^3{v}^a{\mathrm{\partial }}_a{\mathrm{\mid }}_{\gamma (b)}$ (Coordinate derivations form a basis of the tangent space), the difference quotients satisfy $(x^a(\gamma (t))-x^a(\gamma (b)))\mathrm{/}(t-b)\to v^a$ as $t\uparrow b$ for $a=0,1,2,3$. A right tangent at the left end of a curve is defined analogously with $t\downarrow b$. Chart independence is the one-sided chain-rule computation for the smooth transition maps (The chain rule for total derivatives: $D(g\circ f)(a)=Dg(f(a))\circ Df(a)$).

• A curve $\gamma :[a,b]\to M$ with $a<b$ is piecewise $C^1$ when there is a finite partition $a=s_0<s_1<\cdots <s_m=b$ such that for every $i$ the restriction of $\gamma$ to $[s_{i-1},s_i]$ is $C^1$ up to its included endpoints in the one-sided sense above: in every endpoint chart its component derivatives on the open piece have continuous one-sided limits, agreeing with its endpoint derivatives. Equivalently each piece has a $C^1$ extension to its closed parameter piece. In particular its right tangent at $s_{i-1}$ and left tangent at $s_i$ exist. Such a partition is called admissible for $\gamma$. The piece extensions give continuity at the corners; no agreement of the two one-sided tangents at a corner is required.

(b) Velocity. If $\gamma$ is $C^1$ at an interior point $t$, its velocity ${\gamma }^{\mathrm{\prime }}(t)\in T_{\gamma (t)}M$ is the derivation $[f]\mapsto (f\circ \gamma {)}^{\mathrm{\prime }}(t)$ on smooth germs at $\gamma (t)$, extending The velocity derivation of a smooth curve from smooth to $C^1$ curves. Equal germ representatives agree along the curve near $t$, and the one-variable product rule proves the derivation identity. In a chart around $\gamma (t)$ the velocity has the component expression

${\gamma }^{\mathrm{\prime }}(t)={\sum }_{a=0}^3\frac{d}{dt}(x^a\circ \gamma )(t){\mathrm{\partial }}_a{\mathrm{\mid }}_{\gamma (t)},$

the coordinate expression of the derivation (Coordinate derivations form a basis of the tangent space). At an included endpoint of the domain, ${\gamma }^{\mathrm{\prime }}$ means the corresponding one-sided tangent of clause (a).

(c) Timelike worldlines. For $k\in \{1,2,3,\dots \}\cup \{\mathrm{\infty }\}$, a $C^k$ future timelike worldline is a $C^k$ curve $\gamma :I\to M$ whose velocity ${\gamma }^{\mathrm{\prime }}(t)$ is future timelike at every $t\in I$, in the sense of Timelike, null, spacelike and causal vectors of a Lorentz vector space and Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model: $g_{\gamma (t)}({\gamma }^{\mathrm{\prime }}(t),{\gamma }^{\mathrm{\prime }}(t))<0$ and $g_{\gamma (t)}({\gamma }^{\mathrm{\prime }}(t),N_{\gamma (t)})<0$. A piecewise-$C^1$ future timelike worldline is a piecewise-$C^1$ curve $\gamma :[a,b]\to M$ whose velocity at every point of each open piece and whose one-sided tangents at the two ends of each piece are future timelike; the definition is applied to a map on an arbitrary interval $I$ with nonempty interior by requiring it on every compact subinterval of $I$. A future timelike velocity is in particular nonzero, so every future timelike worldline is regular.

(d) Proper time and length. Let $\gamma :[a,b]\to M$ be a piecewise-$C^1$ future timelike worldline and let $a=s_0<s_1<\cdots <s_m=b$ be an admissible partition. On each open piece $(s_{i-1},s_i)$ the function

$h_i(s):=\sqrt{-g_{\gamma (s)}({\gamma }^{\mathrm{\prime }}(s),{\gamma }^{\mathrm{\prime }}(s))}$

is continuous and strictly positive: it is a finite sum of products of continuous real functions in any chart around a point of the piece, as $g_{ab}\circ \gamma$ and the components of ${\gamma }^{\mathrm{\prime }}$ are continuous (A vector-valued function has a limit, or is continuous, if and only if each of its components does; with the algebra of continuous vector-valued functions, Sums, scalar multiples, products, absolute values, maxima, minima and quotients with nonvanishing denominator of continuous functions are continuous, as are constants, the identity and every polynomial function, Continuity of $f:A\to \mathbb{R}$ at a point of $A$ and on $A$: the $\epsilon$-$\delta$ condition, its agreement with ${\lim }_{x\to c}f(x)=f(c)$ at a limit point, and continuity at an isolated point), and the square root is continuous on $(0,\mathrm{\infty })$ (Rational powers $a^r$ of a positive base, Continuity and derivatives of positive-base real powers). The continuous one-sided limits of the component derivatives, required by the $C^1$ extension of each piece, and continuity of $g$ imply that $h_i$ tends to the norm of the corresponding endpoint tangent. The future timelike hypothesis on those tangents makes these limits strictly positive, and $h_i$ extends to a continuous function on $[s_{i-1},s_i]$; hence it is Darboux integrable there (The lower and upper Darboux integrals of a bounded $f$ on $[a,b]$ as ${\sup }_{P}L(f,P)$ and ${\inf }_{P}U(f,P)$, Darboux integrability as their equality, and the notation ${\int }_a^{b}f$, A continuous function on $[a,b]$ is Riemann integrable, by Heine-Cantor and Riemann's criterion). The proper-time increment of $\gamma$ over $[a,b]$ is

$\tau (b)-\tau (a):=\frac{1}{c}{\sum }_{i=1}^m{\int }_{s_{i-1}}^{s_i}{h}_i(s)ds,$

and the Lorentzian length of $\gamma$ over $[a,b]$ is $L_g(\gamma ):=c(\tau (b)-\tau (a))$. For a base point $s_0\in I$ the proper-time function $\tau :I\to \mathbb{R}$ of a future timelike worldline is the function with $\tau (s_0)=0$ and $\tau (s)$ equal to the increment from $s_0$ to $s$ when $s>s_0$, and minus the increment from $s$ to $s_0$ when $s<s_0$. That the increment is independent of the admissible partition, so that $\tau$ is well defined, and that it is invariant under increasing $C^1$ reparametrization with strictly positive derivative (including one-sided endpoint derivatives), comparing the increment over the actual image interval, are proved in Proper time is well defined and reparametrization invariant; the four-velocity is its proper-time tangent ↗; the display is the value for any admissible partition.

(e) Four-velocity. Let $\gamma$ be a piecewise-$C^1$ future timelike worldline and let $t$ be a parameter value at which $\gamma$ is differentiable (interior point of a piece, or an included endpoint with its one-sided tangent). The four-velocity of $\gamma$ at $t$ is

$U(t):=\frac{c{\gamma }^{\mathrm{\prime }}(t)}{\sqrt{-g_{\gamma (t)}({\gamma }^{\mathrm{\prime }}(t),{\gamma }^{\mathrm{\prime }}(t))}}\in T_{\gamma (t)}M\mathrm{.}$

By construction $g_{\gamma (t)}(U(t),U(t))=-c^2$, and ${\gamma }^{\mathrm{\prime }}(t)=\left(\sqrt{-g({\gamma }^{\mathrm{\prime }},{\gamma }^{\mathrm{\prime }})}\mathrm{/}c\right)U(t)$ with a strictly positive scalar factor. If $U$ is written as the derivative of $\gamma$ with respect to proper time, $U=d\gamma \mathrm{/}d\tau$, this is a statement proved in Proper time is well defined and reparametrization invariant; the four-velocity is its proper-time tangent ↗, not part of the present definition.

(f) Scope. Proper time, length and four-velocity are defined only for future timelike worldlines. A null or spacelike curve receives no proper time and no four-velocity from this definition, and no limiting or distributional extension of the square-root integrand to such curves is asserted. The parametrization of a worldline is part of the data and is arbitrary in this definition: no affine parameter, no geodesic equation, no completeness or extendability of the domain, and no measurement or clock reading is assumed. Null and spacelike regularity classes are not defined here beyond the general $C^k$ and piecewise-$C^1$ notions of clause (a), which make no causal requirement.

• Units and the constant $c$. This item treats $c$ as a positive real number and fixes no numerical value or dimension for it. The framework's physical reading of $c$ as the conversion constant with dimension $\mathrm{m}{\mathrm{s}}^{-1}$, of $\tau$ as a duration in seconds and of $g$ with dimensionless components in length coordinates belongs to def-gr-si-unit-and-coordinate-conventions, which is a physical item and is therefore not a prerequisite of this mathematical definition.

• Choice convention. The charts, tangent spaces and derivations used here are those of Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model and The velocity derivation of a smooth curve; no further choice principle occurs. The only inherited assumption is the ${\mathrm{A}\mathrm{C}}_{\omega }$ recorded with the smooth structure of $M$ in Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model.

Let $(M,g,N)$ be a time-oriented smooth Lorentzian four-manifold and let $c>0$ (Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model).

1. Well-definedness and additivity of proper time. Let $\gamma :[a,b]\to M$ be a piecewise-$C^1$ future timelike worldline (Timelike worldlines, proper time, length and four-velocity on a Lorentzian four-manifold). Then the sum defining $\tau (b)-\tau (a)$ is the same for every admissible partition of $[a,b]$; it is written $\frac{1}{c}{\int }_a^{b}h$, where $h$ is the piecewise-defined proper-time density. For every $u\in (a,b)$, $\tau (b)-\tau (a)=(\tau (u)-\tau (a))+(\tau (b)-\tau (u))$. If $a<b$, the increment is strictly positive.

2. Reparametrization and orientation, including subintervals. Let $a^{\mathrm{\prime }}<b^{\mathrm{\prime }}$ and let $\phi :[a^{\mathrm{\prime }},b^{\mathrm{\prime }}]\to [a,b]$ be $C^1$ up to its included endpoints with nonzero derivative everywhere, including one-sided endpoint derivatives. Let $\tilde{\gamma }:=\gamma \circ \phi$. Its derivative has constant sign and $\phi$ is a $C^1$ diffeomorphism onto its actual image interval. If ${\phi }^{\mathrm{\prime }}>0$, then $\tilde{\gamma }$ is a piecewise-$C^1$ future timelike worldline and $\tilde{\tau }(b^{\mathrm{\prime }})-\tilde{\tau }(a^{\mathrm{\prime }})=\tau (\phi (b^{\mathrm{\prime }}))-\tau (\phi (a^{\mathrm{\prime }}))\mathrm{.}$ This equals $\tau (b)-\tau (a)$ when $\phi (a^{\mathrm{\prime }})=a$ and $\phi (b^{\mathrm{\prime }})=b$. If ${\phi }^{\mathrm{\prime }}<0$, then $\tilde{\gamma }$ is a piecewise-$C^1$ past timelike traversal. Its unsigned density $\tilde{h}=\sqrt{-g({\tilde{\gamma }}^{\mathrm{\prime }},{\tilde{\gamma }}^{\mathrm{\prime }})}$ obeys $\frac{1}{c}{\int }_{a^{\mathrm{\prime }}}^{b^{\mathrm{\prime }}}\tilde{h}=\tau (\phi (a^{\mathrm{\prime }}))-\tau (\phi (b^{\mathrm{\prime }}))>0,$ whereas the oriented original difference $\tau (\phi (b^{\mathrm{\prime }}))-\tau (\phi (a^{\mathrm{\prime }}))$ is negative. For endpoint-reversing $\phi (a^{\mathrm{\prime }})=b$, $\phi (b^{\mathrm{\prime }})=a$, the latter equals $-(\tau (b)-\tau (a))$. This negative oriented difference is not elapsed proper time for a future curve; parameter reversal reverses the causal direction. Non-surjective maps compare increments over their actual image interval, not over the whole original interval.

3. Proper-time function and four-velocity. Let $\gamma :I\to M$ be a $C^1$ future timelike worldline on an interval with nonempty interior and let $\tau$ be its proper-time function based at $t_0\in I$ (Timelike worldlines, proper time, length and four-velocity on a Lorentzian four-manifold). Then $\tau$ is $C^1$ on $I$ with ${\tau }^{\mathrm{\prime }}(t)=\frac{1}{c}\sqrt{-g_{\gamma (t)}({\gamma }^{\mathrm{\prime }}(t),{\gamma }^{\mathrm{\prime }}(t))}>0,$ so $\tau$ is strictly increasing, and at every $t\in I$ the velocity, proper-time density and four-velocity satisfy ${\gamma }^{\mathrm{\prime }}(t)={\tau }^{\mathrm{\prime }}(t)U(t)$ and $g_{\gamma (t)}(U(t),U(t))=-c^2$. If $\tau$ is strictly increasing on $I$ with image $J$, then the reparametrized curve ${\gamma }_{\tau }:=\gamma \circ (\tau {\mathrm{\mid }}_I{)}^{-1}$ is a $C^1$ future timelike worldline on $J$ with ${\gamma }_{\tau }^{\mathrm{\prime }}(\sigma )=U({\tau }^{-1}(\sigma )),g({\gamma }_{\tau }^{\mathrm{\prime }}(\sigma ),{\gamma }_{\tau }^{\mathrm{\prime }}(\sigma ))=-c^2\mathrm{.}$ The same formulas hold piece by piece for a piecewise-$C^1$ future timelike worldline, on each open piece and at its endpoints one-sided, with the two corner velocities kept distinct.

• Contrapositive reading of additivity. Steps 2.1 and 3.1 imply that the increment over $[a,b]$ of a piecewise-$C^1$ future timelike worldline is the sum of the increments of its restrictions to any admissible subdivision; this is used without further comment in The Lorentzian Gauss lemma and the radial maximum of proper time and First variation of the length of a timelike curve: stationarity is the geodesic equation.

• No choice principle is used. All partitions inserted in the proof are finite and explicitly constructed from the given data, the inverse function theorem is applied at individual points, and the integral is the Darboux integral of continuous functions. The inherited ${\mathrm{A}\mathrm{C}}_{\omega }$ of the ambient smooth structure is the only choice assumption (Timelike worldlines, proper time, length and four-velocity on a Lorentzian four-manifold, Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model).

Let $M$ be a smooth four-manifold without boundary and let $g$ be a smooth symmetric covariant $2$-tensor field on $M$ such that $g_p$ is nondegenerate at every $p\in M$ (Smooth covariant 2-tensor fields on a smooth manifold). Assume ${\mathrm{A}\mathrm{C}}_{\omega }$ for the canonical tangent-bundle smooth structure (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$), A smooth vector field is a smooth section of the tangent bundle). For this general metric define ${\mathrm{♭}}_p(v):=g_p(v,\cdot )$, ${\mathrm{♯}}_p:={\mathrm{♭}}_p^{-1}$, and $g_p^{-1}(\alpha ,\beta ):=g_p({\mathrm{♯}}_p\alpha ,{\mathrm{♯}}_p\beta )$. Nondegeneracy makes ${\mathrm{♭}}_p$ injective and hence invertible between the four-dimensional tangent space and its dual. In a chart, $g^{ab}$ are the entries of the inverse matrix of $(g_{ab})$. These definitions agree with The inverse metric, musical isomorphisms and index raising in its Lorentzian scope. Here and below repeated upper/lower indices are summed over $0,1,2,3$. This holds in particular when $(M,g,N)$ is a time-oriented smooth Lorentzian four-manifold and $g$ is its metric (Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model). For smooth local vector fields $X,Y,Z$ put

$K(X,Y,Z):=X(g(Y,Z))+Y(g(Z,X))-Z(g(X,Y))+g([X,Y],Z)+g([Z,X],Y)-g([Y,Z],X),$

with brackets as in The Lie bracket of smooth vector fields.

(a) Existence and formulas. There is exactly one affine connection $\mathrm{\nabla }$ on $M$ (Affine connection on a smooth manifold) satisfying

$2g({\mathrm{\nabla }}_{X}Y,Z)=K(X,Y,Z)\text{for all smooth local }X,Y,Z\mathrm{.}$

It is torsion-free and $g$-compatible, so it is a metric-selected connection for $(M,g)$, meaning torsion-free and satisfying $Xg(Y,Z)=g({\mathrm{\nabla }}_{X}Y,Z)+g(Y,{\mathrm{\nabla }}_{X}Z)$ for smooth local fields (the conditions of Metric compatibility, torsion freedom and the metric-selected connection class extended here to nondegenerate $g$); in every smooth chart its Christoffel symbols are

${\mathrm{\Gamma }}^a{}_{bc}=\frac{1}{2}{g}^{ad}({\mathrm{\partial }}_b{g}_{dc}+{\mathrm{\partial }}_c{g}_{bd}-{\mathrm{\partial }}_d{g}_{bc}),$

with $g^{ad}$ the inverse-metric components defined above (Christoffel symbols of an affine connection).

(b) Uniqueness. If an affine connection ${\mathrm{\nabla }}^{\mathrm{\prime }}$ on $M$ is torsion-free and $g$-compatible, then ${\mathrm{\nabla }}^{\mathrm{\prime }}=\mathrm{\nabla }$. Consequently the class $\mathrm{LC}(M,g)$ of metric-selected connections has exactly one element; it is the Levi–Civita connection of $g$.

(c) Induced derivative on tensor fields. Let ${\mathrm{\nabla }}^{\ast }$ be the dual connection on $T^{\ast }M$ and let the product connection of Product connection on tensor and hom bundles act on $T_s^{r}M:=(TM{)}^{\otimes r}\otimes (T^{\ast }M{)}^{\otimes s}$, for integers $r,s\ge 0$, constructed by Finite tensor products of smooth vector bundles. A type $(r,s)$ tensor at $p$ is a multilinear map $(T_p^{\ast }M{)}^r\times (T_{p}M{)}^s\to \mathbb{R}$ (A type $(r,s)$ tensor on a finite-dimensional vector space); a smooth tensor field is a section with smooth coefficients in each local product frame. The empty product is the scalar line. Its chart components are $T^{a_1\cdots a_r}{}_{b_1\cdots b_s}=T(d{x}^{a_1},\dots ,d{x}^{a_r},{\mathrm{\partial }}_{b_1},\dots ,{\mathrm{\partial }}_{b_s})$, where $d{x}^a$ is dual to ${\mathrm{\partial }}_a$. This agrees with Tensor fields of type (r,s) on a Lorentzian four-manifold and their components in the Lorentzian case. Put $T^{a_1\cdots a_r}{}_{b_1\cdots b_s;c}:=({\mathrm{\nabla }}_{{\mathrm{\partial }}_c}T{)}^{a_1\cdots a_r}{}_{b_1\cdots b_s}$. Then, in every smooth chart, for every smooth tensor field $T$ of type $(r,s)$,

$T^{a_1\cdots a_r}{}_{b_1\cdots b_s;c}={\mathrm{\partial }}_c{T}^{a_1\cdots a_r}{}_{b_1\cdots b_s}+{\sum }_{i=1}^r{\mathrm{\Gamma }}^{a_i}{}_{dc}{T}^{a_1\cdots d\cdots a_r}{}_{b_1\cdots b_s}-{\sum }_{j=1}^s{\mathrm{\Gamma }}^d{}_{b_{j}c}{T}^{a_1\cdots a_r}{}_{b_1\cdots d\cdots b_s},$

where the slot marked $d$ replaces the named index; this derivative commutes with every contraction in the sense of The contraction of a mixed tensor, and it annihilates the metric and the inverse metric:

$g_{ab;c}=0,g^{ab}{}_{;c}=0.$