Global viscosity solutions for eikonal equations on class A Lorentzian 2-tori

Abstract

On the Abelian cover $({\mathbb {R}}^{2},g)$ of a class A Lorentzian 2-torus $({\mathbb {T}}^{2},g)$, we showed the existence of global viscosity solutions to the eikonal equation

$$\begin{aligned} g(\nabla u,\nabla u)=-1 \end{aligned}$$

associated to those homologies in the interior of the stable time cone. Some other related dynamical properties are also considered. As an application of the main results, we study the differentiability of the unit sphere of the stable time separation associated to the class A Lorentzian 2-torus.

Appendix

In this section, we collect part of elementary concepts and results in global Lorentzian geometry which are frequently used in this article. For a comprehensive introduction to this topic, see standard textbooks [6, 17]. Our presentation relies on Suhr’s papers [20, 22, 23].

First, we shall state a result concerning a closed Riemannian manifold $(M,g_{R})$.

Theorem 8.1

[7] Let $(M,g_{R})$ be a compact Riemannian manifold. Then there exists a unique norm $\Vert \cdot \Vert :H_{1}(M,{\mathbb {R}})\rightarrow {\mathbb {R}}$ and a constant std$(g_{R})<\infty$ such that

$$\begin{aligned} |\text {dist}(x,y)-\Vert x-y\Vert |\le \text {std}(g_{R}) \end{aligned}$$

for any $x,y\in \overline{M}$. Here $\Vert \cdot \Vert$ is called to be the stable norm of $g_{R}$ on $H_{1}(M,{\mathbb {R}})$.

In Riemannian geometry, there are some important concepts that relate the Riemannian structure to the metric or topology structure. The same concepts also lie at the foundation of Loretzian geometry. But on the contrary, they are far from being well-known. Only several general properties are proved.

Definition 8.2

Let (M, g) be a spacetime and $\gamma :[a,b]\rightarrow M$ a causal curve, the Lorentzian length of $\gamma$ is defined by

$$\begin{aligned} L^{g}(\gamma ):=\int _{a}^{b}\sqrt{-g(\dot{\gamma }(t),\dot{\gamma }(t))}dt. \end{aligned}$$

Since causal curves always admit Lipschitz parametrization, the length is well defined.

Proposition 8.3

If a sequence of causal curves $\gamma _{n}:[a,b]\rightarrow M$, parameterized by the arclength w.r.t. $g_{R}$, converges uniformly to a causal curve $\gamma \,{:}\,[a,b]\rightarrow M$, then

$$\begin{aligned} L^{g}(\gamma )\ge \limsup _{n\rightarrow \infty }L^{g}(\gamma _{n}). \end{aligned}$$

Definition 8.4

Let (M, g) be a spacetime, we define the time separation or the Lorentzian distance function as $d(p,q):=sup\{L^{g}(\gamma ):\gamma \in C^{+}(p,q)\}$, where $C^{+}(p,q)$ denotes the set of future-directed causal curves connecting p with q. If $C^{+}(p,q)=\emptyset$, then set $d(p,q):=0$.

Proposition 8.5

For general spacetime (M, g), the time separation is only lower semicontinuous on $M\times M$. If (M, g) is globally hyperbolic, the time separation is continuous and there exists a maximal causal geodesic connecting p with q for all $q\in J^{+}(p)$.

The following definition gives a picture of what the time looks like in the theory of general relativity.

Definition 8.6

Let (M, g) be a spacetime. A function $\tau \,{:}\,M\rightarrow {\mathbb {R}}$ is called

• (1) a time function if it is continuous and strictly increasing on each future-directed causal curve in (M, g);

• (2) a temporal function if it is $C^{1}$ and has a past-directed timelike gradient at every point on M.

Remark 8.7

Any temporal function on (M, g) must be a time function, and if it satisfies the Lorentzian eikonal equation $g(\nabla \tau ,\nabla \tau )=-1$, it is then a global viscosity (in fact, classical) solution to the Lorentzian eikonal equation. The latter case could happen only when M is noncompact.

Finally, we recall two important properties of the so called class $A_{1}$ spacetimes defined in [23, Sections 2 and 3]. As S. Suhr showed in [23], class A Lorentzian 2-tori are also class $A_{1}$, so these two properties apply to our case. They are crucial for the proof of our main results.

The first concerns how the global behavior of maximal curves effects their local property.

Proposition 8.8

[23, Proposition 4.1] Let (M, g) be of class $A_{1}$. Then for any $\epsilon >0$ there exist $\delta >0$ and $K<\infty$ such that

$$\begin{aligned} \dot{\gamma }(t)\in \text {Time}^{\epsilon }(M^{2},[g])_{\gamma (t)} \end{aligned}$$

for all maximizers $\gamma \,{:}\,[a,b]\rightarrow M$ with $\gamma (b)-\gamma (a)\in \mathfrak {T}^{\delta }{\setminus } B_{K}(0)$ and all $t\in [a,b]$.

The second contains an answer of a problem which lies at the foundation of the global Lorentzian geometry. The problem is to find some appropriate spacetime on which the time separation (or Lorentzian distance function) is locally Lipschitz w.r.t some Riemannian structure.

Proposition 8.9

[23, Theorem 4.3] Let (M, g) be of class $A_{1}$. Then for any $\epsilon >0$ there exist constants $K(\epsilon ),L(\epsilon )<\infty$ such that $(x,y)\mapsto d(x,y)$ is $L(\epsilon )$-Lipschitz on $\{(x,y)\in \overline{M}\times \overline{M}|y-x\in \mathfrak {T}^{\epsilon }{\setminus } B_{K(\epsilon )}(0)\}$.

A basic lemma in convex analysis is needed in the proof of Lemma 7.4. For completeness, we give the proof here.

Lemma 8.10

Let $f\,{:}\,{\mathbb {R}}^{n}\rightarrow {\mathbb {R}}$ be a concave function. Suppose f is differentiable at x along straight lines $L_{i}$, where $L_{i}:=\{x+tV_{i}|t\in {\mathbb {R}}\}, V_{1},V_{2},\ldots ,V_{n}$ are n linearly independent vectors, then f is differentiable at x.

Proof

Fix a point $x\in {\mathbb {R}}^{n}$, denote the set of super-gradients of f at x by $D^{+}f(x)$ (namely, $D^{+}f(x)=\{p\in T_{x}{\mathbb {R}}^{n}|f(y)-f(x)-\langle p,y-x\rangle \le 0,\text {for any }y\in {\mathbb {R}}^{n}\}$) and the directional derivative of f at x along V by $D_{V}f(x)$. Since f is differentiable along straight lines $L_{i}$, by using the concavity of f, we have

$$\begin{aligned} -\langle P,-V_{i}\rangle \le -D_{-V_{i}}f(x)=D_{V_{i}}f(x)\le \langle P,V_{i}\rangle \end{aligned}$$

(8.1)

for any $P\in D^{+}f(x)$. So $\langle P,V_{i}\rangle =D_{V_{i}}f(x)$ for any $P\in D^{+}f(x)$. Now let $P,P^{\prime }$ be two elements in $D^{+}f(x)$, then we have

$$\begin{aligned} \langle P-P^{\prime },V_{i}\rangle =0. \end{aligned}$$

(8.2)

Since $V_{i}(i=1,\ldots ,n)$ are linearly independent, we must have $P-P^{\prime }=0$. Thus the set of super-gradients of f at x degenerates to a singleton and f is differentiable at x. $\square$