Convexity of the squared Euclidean radius along small contained geodesics: segments with endpoints in a small ball stay in the ball

Let $(M,g,N)$ be a time-oriented smooth Lorentzian four-manifold with metric-selected connection $\mathrm{\nabla }$ (Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model, Metric compatibility, torsion freedom and the metric-selected connection class) and let geodesics be those of The geodesic initial-value problem and the exponential map of the metric-selected connection. Let $(U,x)$ be a smooth chart of $M$ and write $x=(x^0,x^1,x^2,x^3)$ with its Euclidean norm $\mathrm{\mid }\cdot {\mathrm{\mid }}_{\mathrm{e}}$ on $x(U)\subseteq {\mathbb{R}}^4$; write $B_s:=\{y\in {\mathbb{R}}^4:\mathrm{\mid }y{\mathrm{\mid }}_{\mathrm{e}}<s\}$ and, for a compact subset $K\subseteq U$, let $\mathrm{\parallel }\mathrm{\Gamma }{\mathrm{\parallel }}_K:={\sup }_{y\in x(K)}{\sup }_{\mathrm{\mid }u{\mathrm{\mid }}_{\mathrm{e}}=\mathrm{\mid }v{\mathrm{\mid }}_{\mathrm{e}}=1}\mathrm{\mid }\mathrm{\Gamma }(y)[u,v]{\mathrm{\mid }}_{\mathrm{e}},$ where $\mathrm{\Gamma }(y)[u,v{]}^a={\sum }_{b,c=0}^3{\mathrm{\Gamma }}^a{}_{bc}(y)u^b{v}^c$. Thus this is the Euclidean bilinear operator norm, not the maximum of the individual coefficients; set $\mathrm{\parallel }\mathrm{\Gamma }{\mathrm{\parallel }}_{\mathrm{\varnothing }}=0$. The Christoffel symbols are the smooth chart functions of Christoffel symbols of an affine connection.

(a) Second-derivative inequality. Let $\gamma :J\to U$ be a geodesic and let $f:J\to \mathbb{R}$, $f(t):=\mathrm{\mid }x(\gamma (t)){\mathrm{\mid }}_{\mathrm{e}}^2$, where the norm is the Euclidean norm of the chart image. At every parameter $t\in J$, $f^{\mathrm{\prime }\mathrm{\prime }}(t)=2\mid \dot{x}(t){\mid }_{\mathrm{e}}^2-2x(t)\cdot \mathrm{\Gamma }(x(t))[\dot{x}(t),\dot{x}(t)],$ where $\mathrm{\Gamma }(x)[\dot{x},\dot{x}]$ is the vector with components ${\sum }_{i,j}{\mathrm{\Gamma }}^a{}_{ij}(x){\dot{x}}^i{\dot{x}}^j$ and $\cdot$ is the Euclidean scalar product. Consequently, along every geodesic whose image lies in a fixed compact $K\subseteq U$, $f^{\mathrm{\prime }\mathrm{\prime }}\ge 2(1-\mathrm{\parallel }\mathrm{\Gamma }{\mathrm{\parallel }}_K\mathrm{\mid }x(\gamma )\mathrm{\mid })\mid \dot{x}{\mid }_{\mathrm{e}}^2\mathrm{.}$ In particular, if $x(K)\subseteq B_{r_1}$ and $\mathrm{\parallel }\mathrm{\Gamma }{\mathrm{\parallel }}_K{r}_1\le \frac{1}{2}$ then $f^{\mathrm{\prime }\mathrm{\prime }}\ge \mathrm{\mid }\dot{x}{\mathrm{\mid }}_{\mathrm{e}}^2\ge 0$ at every parameter of such a geodesic.

(b) Maximum principle. Let $\gamma :[0,1]\to U$ be a geodesic whose image is contained in a compact $K$ with $x(K)\subseteq B_{r_1}$ and $\mathrm{\parallel }\mathrm{\Gamma }{\mathrm{\parallel }}_K{r}_1\le \frac{1}{2}$, and suppose $\mathrm{\mid }x(\gamma (0)){\mathrm{\mid }}_{\mathrm{e}}\le r_2$ and $\mathrm{\mid }x(\gamma (1)){\mathrm{\mid }}_{\mathrm{e}}\le r_2$ for some $r_2<r_1$. Then $\mathrm{\mid }x(\gamma (t)){\mathrm{\mid }}_{\mathrm{e}}\le r_2\text{for every }t\in [0,1]\mathrm{.}$

• Only one chart. The estimate is a chart statement; it is used in Speed bounds along geodesics in a chart, and small initial velocity of every geodesic segment contained in a small ball and Existence of strictly convex normal neighborhoods: small coordinate balls are normal neighborhoods of each of their points to control contained geodesics in a single coordinate ball, and no covariance is claimed for the Euclidean radius function.

• Why convexity and not distance to the boundary. The function $\mathrm{\mid }x{\mathrm{\mid }}_{\mathrm{e}}^2$ is $C^{\mathrm{\infty }}$ and its second derivative along geodesics is computable from the geodesic equation alone; it is exactly the function used in the classical proof that small coordinate balls are geodesically convex, and it does not use the metric's signature.