Connections And Free Fall — Examples

The connection $\mathrm{\nabla }$ has $\mathrm{\Gamma }\equiv 0$, so it is torsion-free; and the compatibility identity ${\mathrm{\partial }}_a{g}_{bc}={\sum }_d({\mathrm{\Gamma }}^d{}_{ab}{g}_{dc}+{\mathrm{\Gamma }}^d{}_{ac}{g}_{bd})$ holds because both sides are $0$ with constant metric components and vanishing coefficients. By [L2] it is the metric-selected connection of $g$.

Substituting $\mathrm{\Gamma }=0$ into [L3], the geodesic equation is ${\ddot{x}}^k=0$, whose solutions are exactly the affine lines $t\mapsto p+tv$ on parameter intervals; by [L5] the maximal geodesic through $(p,v)$ is this line extended to all of $\mathbb{R}$.

If $V={\sum }_a{V}^a{\mathrm{\partial }}_a$ along a curve, then by [L3] $D_{t}V=0$ if and only if ${\mathrm{\partial }}_t{V}^a=0$ for all $a$, that is, the components are constant; so parallel sections are exactly the constant-component fields along the curve and $P_{\gamma }$ is the identity identification of fibres in the identity chart. Hence transport preserves $g$ and is independent of the path (it depends only on the endpoints), by [L2] and the component computation.

Let $\gamma (t)=p+tv$ with $g(v,v)<0$ and $g(v,{\mathrm{\partial }}_0)<0$; then $\gamma$ is a future timelike worldline and its density is the constant $\sqrt{-g(v,v)}$. By [L4] the proper-time increment over $[t_0,t]$ is $\sqrt{-g(v,v)}(t-t_0)\mathrm{/}c$, and the four-velocity is the constant vector $U=cv\mathrm{/}\sqrt{-g(v,v)}$ with $g(U,U)=-c^2$. In particular, for the future timelike unit vector $u$ with $g(u,u)=-c^2$ the worldline $\tau \mapsto p+\tau u$ is parametrized by proper time up to its origin.

By step 2.1, the geodesic through $p$ with initial tangent $v$ is $t\mapsto p+tv$ on all of $\mathbb{R}$, so $1\in I_{p,v}$ for every $v$ and ${\exp }_p(v)=p+v$ on all of $T_{p}M$. Differentiating $v\mapsto p+v$ at $0$ gives the identity, and the chart $y:=x-x(p)$ centered at $p$ is the identity chart translated; in it $g_{ab}=\mathrm{diag}(-1,1,1,1)$ identically, so $g_{ab}(p)={\eta }_{ab}$ and ${\mathrm{\partial }}_a{g}_{bc}(p)=0$ hold exactly, not merely at first order.

The coefficients are smooth functions of $x$ because $\phi$ is smooth, and they satisfy ${\mathrm{\Gamma }}^a{}_{bc}-{\mathrm{\Gamma }}^a{}_{cb}=({\delta }^a{}_b{\mathrm{\partial }}_c\phi +{\delta }^a{}_c{\mathrm{\partial }}_b\phi )-({\delta }^a{}_c{\mathrm{\partial }}_b\phi +{\delta }^a{}_b{\mathrm{\partial }}_c\phi )-({\eta }_{bc}-{\eta }_{cb}){\eta }^{ad}{\mathrm{\partial }}_d\phi =0$ by symmetry of $\eta$. Hence the connection they define is torsion-free by [L2].

For the compatibility identity, multiply the candidate by the metric: ${\sum }_d{\mathrm{\Gamma }}^d{}_{ab}{g}_{dc}=e^{2\phi }{\sum }_d{\mathrm{\Gamma }}^d{}_{ab}{\eta }_{dc}=e^{2\phi }({\mathrm{\partial }}_b\phi {\eta }_{ac}+{\mathrm{\partial }}_a\phi {\eta }_{bc}-{\eta }_{ab}{\mathrm{\partial }}_c\phi )$, using ${\delta }^d{}_a{\eta }_{dc}={\eta }_{ac}$, ${\delta }^d{}_b{\eta }_{dc}={\eta }_{bc}$ and ${\eta }^{de}{\eta }_{dc}={\delta }^e{}_c$. Adding the analogous expression with $b$ and $c$ interchanged gives $e^{2\phi }(2{\mathrm{\partial }}_a\phi {\eta }_{bc}+{\mathrm{\partial }}_b\phi {\eta }_{ac}-{\mathrm{\partial }}_b\phi {\eta }_{ac}+{\eta }_{ab}{\mathrm{\partial }}_c\phi -{\eta }_{ab}{\mathrm{\partial }}_c\phi )=2{\mathrm{\partial }}_a\phi e^{2\phi }{\eta }_{bc}={\mathrm{\partial }}_a\left(e^{2\phi }{\eta }_{bc}\right)={\mathrm{\partial }}_a{g}_{bc}$, since ${\mathrm{\partial }}_a{\eta }_{bc}=0$ and ${\mathrm{\partial }}_a{e}^{2\phi }=2{\mathrm{\partial }}_a\phi e^{2\phi }$.

Steps 1.1 and 2.1 verify the two hypotheses of [L3] for the connection with coefficients $\mathrm{\Gamma }$; hence this connection is the unique metric-selected connection of $g$ on the chart domain, and since the identity chart covers $U$ it is that connection on $U$.

(Finite check for $\phi =\lambda x^1$.) Here ${\mathrm{\partial }}_d\phi =\lambda {\delta }^1{}_d$, so ${\mathrm{\Gamma }}^a{}_{bc}=\lambda ({\delta }^a{}_b{\delta }^1{}_c+{\delta }^a{}_c{\delta }^1{}_b-{\eta }_{bc}{\eta }^{a1})$. Substitution gives exactly the symbols listed in the Example: the first two arise from the Kronecker-delta terms with $a=0$; for $a=1$ the diagonal cases $(b,c)=(0,0),(1,1),(2,2),(3,3)$ give respectively $\lambda ,\lambda ,-\lambda ,-\lambda$; and for $a=2,3$ the delta terms give respectively the symmetric pairs $(1,2),(1,3)$ with value $\lambda$. All other cases vanish by the displayed formula. If $\lambda =0$ every coefficient is zero. For $(a,b,c)=(1,0,0)$ the identity of claim 2 reads ${\mathrm{\partial }}_1{g}_{00}=2{\mathrm{\Gamma }}^0{}_{10}{g}_{00}=2\lambda (-e^{2\lambda x^1})=-2\lambda e^{2\lambda x^1}$, which equals ${\mathrm{\partial }}_1(-e^{2\lambda x^1})$; for $(a,b,c)=(0,0,0)$ both sides vanish.

The curve $\gamma$ is geodesic by [L1] because its components are affine; on its parameter interval $(0,\mathrm{\infty })$ its image is the positive $x^0$-axis, and $\beta =\gamma \circ h$ with $h(s)=s^2$, whose image is the same half-axis.

By [L2], $D_s{\beta }^{\mathrm{\prime }}(s)={\mathrm{\partial }}_s(2s,0,0,0)=(2,0,0,0)\mathrm{\ne }0$ for every $s>0$; by [L4], $\beta$ is therefore not an affinely parametrized geodesic, so its parameter is not affine. This is consistent with $h^{\mathrm{\prime }\mathrm{\prime }}(s)=2\mathrm{\ne }0$ in the criterion of [L3].

(General affine reparametrization.) For the regular curve of clause 3, [L5] gives $F^{\mathrm{\prime }}=f$, $a^{\mathrm{\prime }}=fa$ and ${\sigma }^{\mathrm{\prime }}=a>0$. Thus $\sigma$ is $C^2$, strictly increasing, and its image is an open interval: near every point [L5] gives a local inverse, and these agree by injectivity. Its global inverse $h:={\sigma }^{-1}$ is $C^1$ with $h^{\mathrm{\prime }}=1\mathrm{/}(a\circ h)$; since $a$ is $C^1$, this formula makes $h^{\mathrm{\prime }}$ $C^1$, so $h$ is $C^2$ and $h^{\mathrm{\prime }\mathrm{\prime }}=-f(h)\mathrm{/}(a(h){)}^2$. Applying the component formula [L2] in any local chart gives $D_{\sigma }{\widehat{\gamma }}^{\mathrm{\prime }}=(h^{\mathrm{\prime }}{)}^2(D_s{\beta }^{\mathrm{\prime }})\circ h+h^{\mathrm{\prime }\mathrm{\prime }}({\beta }^{\mathrm{\prime }}\circ h)=((h^{\mathrm{\prime }}{)}^{2}f(h)+h^{\mathrm{\prime }\mathrm{\prime }})({\beta }^{\mathrm{\prime }}\circ h)=0.$ Hence $\widehat{\gamma }$ is an affinely parametrized geodesic by [L4], and its image is exactly $\beta (I)$ because $h$ maps $\sigma (I)$ onto $I$. All integrals here are over compact segments inside $I$; no endpoint convergence or completeness is required.

Directly from [L3], $D_s{\beta }^{\mathrm{\prime }}(s)=h^{\mathrm{\prime }\mathrm{\prime }}(s)({\gamma }^{\mathrm{\prime }}\circ h)(s)=2(1,0,0,0)$, which matches the computation of step 1.2; dividing by $h^{\mathrm{\prime }}(s)=2s$ gives the proportionality $D_s{\beta }^{\mathrm{\prime }}=f{\beta }^{\mathrm{\prime }}$ with $f(s)=1\mathrm{/}s$.

For $f(s)=1\mathrm{/}s$ and $s_0=1$, the positive function $a(s)=s$ satisfies $a^{\mathrm{\prime }}=fa$, $a(1)=1$, so it equals $\exp (F(s))$ (differentiate $a\exp (-F)$). Integrating gives $\sigma (s)=(s^2-1)\mathrm{/}2$. Thus the displayed curve has a geodesic image but is not affine in $s$, and the general converse of clause 3 is verified by step 1.3.

The coefficients are symmetric in the lower indices, ${\mathrm{\Gamma }}^{\mathrm{\prime }a}{}_{bc}={\mathrm{\Gamma }}^{\mathrm{\prime }a}{}_{cb}$, so ${\mathrm{\nabla }}^{\mathrm{\prime }}$ is torsion-free by [L2]. It is not the zero connection: for instance ${\mathrm{\Gamma }}^{\mathrm{\prime }0}{}_{00}={\delta }^0{}_0{\delta }^0{}_0+{\delta }^0{}_0{\delta }^0{}_0=2\mathrm{\ne }0$, so ${\mathrm{\nabla }}^{\mathrm{\prime }}\mathrm{\ne }{\mathrm{\nabla }}_0$.

(Local ${\mathrm{\nabla }}^{\mathrm{\prime }}$-geodesic reparametrizations of straight lines.) Let $\gamma (t)=p+tv$ be a regular straight line and fix $t_0$ in its open parameter interval. For any $s_0$, choose an open interval $J$ about $s_0$ small enough that $1+2v^0(s-s_0)>0$ on $J$. If $v^0=0$, put $h(s)=t_0+(s-s_0)$. If $v^0\mathrm{\ne }0$, put $h(s):=t_0+\ln (1+2v^0(s-s_0))\mathrm{/}(2v^0)$. In either case, shrink $J$ if necessary so that $h(J)$ lies in the parameter interval of $\gamma$. Then $h$ is a $C^2$ diffeomorphism from $J$ onto its open image, with $h(s_0)=t_0$ and $h^{\mathrm{\prime }}\mathrm{\ne }0$; in the second case $h^{\mathrm{\prime }}=1\mathrm{/}(1+2v^0(s-s_0))$ and $h^{\mathrm{\prime }\mathrm{\prime }}=-2v^0(h^{\mathrm{\prime }}{)}^2$. The ${\mathrm{\nabla }}^{\mathrm{\prime }}$-geodesic equation for $\gamma \circ h$ is $(h^{\mathrm{\prime }\mathrm{\prime }}+2v^0(h^{\mathrm{\prime }}{)}^2)v^a=0$, so it is satisfied. This proves that every regular straight-line geodesic of ${\mathrm{\nabla }}_0$ is locally a reparametrized ${\mathrm{\nabla }}^{\mathrm{\prime }}$-geodesic.

(Every regular ${\mathrm{\nabla }}^{\mathrm{\prime }}$-geodesic is locally a reparametrized straight line.) Let $\sigma \mapsto x(\sigma )$ solve the ${\mathrm{\nabla }}^{\mathrm{\prime }}$-equation ${\ddot{x}}^a+2{\dot{x}}^0{\dot{x}}^a=0$ with $\dot{x}\mathrm{\ne }0$ on an interval. For $a=0$ this is ${\ddot{x}}^0=-2({\dot{x}}^0{)}^2$, so $y:={\dot{x}}^0$ solves $y^{\mathrm{\prime }}=-2y^2$. The set $A=\{\sigma :y(\sigma )=0\}$ is closed by continuity and open by uniqueness of solutions through $y({\sigma }_0)=0$ (the zero function is a solution, and [L4] gives uniqueness); hence on the connected parameter interval either $y\equiv 0$ or $y$ is nowhere zero. If $y\equiv 0$, the equations give ${\ddot{x}}^a=0$ for every $a$, so $x$ is an affine regular line. Otherwise, for $a=1,2,3$, $\frac{d}{d\sigma }\left({\dot{x}}^a\mathrm{/}{\dot{x}}^0\right)=({\ddot{x}}^a{\dot{x}}^0-{\dot{x}}^a{\ddot{x}}^0)\mathrm{/}({\dot{x}}^0{)}^2=0$, so $x^a=c^a{x}^0+d^a$ for constants $c^a,d^a$. Since $h(\sigma ):=x^0(\sigma )$ has nowhere-zero derivative, it is a $C^2$ local diffeomorphism; in the parameter $h$, the curve is the affine line $(h,c^{1}h+d^1,c^{2}h+d^2,c^{3}h+d^3)$, a regular ${\mathrm{\nabla }}_0$-geodesic by [L1]. Thus every regular ${\mathrm{\nabla }}^{\mathrm{\prime }}$-geodesic is locally a regular reparametrization of a ${\mathrm{\nabla }}_0$-geodesic.

${\mathrm{\nabla }}^{\mathrm{\prime }}$ is not $g$-compatible. For $(a,b,c)=(0,0,0)$ the identity of [L3] would read ${\mathrm{\partial }}_0{g}_{00}=2{\mathrm{\Gamma }}^{\mathrm{\prime }0}{}_{00}{g}_{00}$; the left side is $0$ because $g_{00}=-1$ is constant, while the right side is $2\cdot 2\cdot (-1)=-4\mathrm{\ne }0$.

Steps 1.2 and 1.3 show that ${\mathrm{\nabla }}_0$ and ${\mathrm{\nabla }}^{\mathrm{\prime }}$ have the same local unparametrized geodesic paths, while steps 1.1 and 2.1 show that ${\mathrm{\nabla }}^{\mathrm{\prime }}$ is a torsion-free connection distinct from ${\mathrm{\nabla }}_0$ and not $g$-compatible. Hence the displayed statement is refuted: torsion freedom plus equality of the local unparametrized paths do not determine the metric-selected connection, and metric compatibility is an independent condition.

By [L2], ${\alpha }_a$ is a piecewise smooth (indeed smooth) future timelike worldline with image in the exponential domain ${\mathbb{R}}^4$ of $p$, and its length is $L_g({\alpha }_a)={\int }_0^1\sqrt{c^2-a^2{\pi }^2{\cos }^2(\pi t)}dt$.

For $a=0$ the integrand is the constant $c$, so $L_g({\alpha }_0)=c$; and ${\alpha }_0(t)=(ct,0,0,0)$ is the straight affine line through $p$ with direction $(c,0,0,0)=v_q$, which by [L1] is the radial geodesic from $p$ to $q$. Thus equality holds in the sense of [L3].

For $a\mathrm{\ne }0$ the integrand $f(t)=\sqrt{c^2-a^2{\pi }^2{\cos }^2(\pi t)}$ is continuous and satisfies $f(t)\le c$, with strict inequality $f(t)<c$ at every $t$ with $\cos (\pi t)\mathrm{\ne }0$; such points form a nondegenerate subinterval of $[0,1]$, for instance around $t=0$. By [L4], ${\int }_0^{1}fdt<c$, so $L_g({\alpha }_a)<\sqrt{-g(v_q,v_q)}=c$; moreover ${\alpha }_a$ is not a reparametrization of the straight segment because its second coordinate $a\sin (\pi t)$ is not identically zero.

Steps 2.1 and 2.2 verify both the inequality and the equality case of The Lorentzian Gauss lemma and the radial maximum of proper time on the family $\{{\alpha }_a{\}}_{0\le a<c\mathrm{/}\pi }$: the length is maximized exactly by the radial geodesic, and every nonradial member of the family is strictly shorter, in agreement with the theorem's proper-time statement after division by $c$.