The existence of smooth solutions in q-models

Abstract

The q-models are scenarios that may explain the smallness of the cosmological constant (Klinkhamer and Volovik in Phys Rev D 77:085015, 2008; Phys Rev D 78:063528, 2008; JETP Lett 88:289, 2008; Mod Phys Lett A 31(28):1650160, 2016; JETP Lett 91:259, 2010; Phys Rev D 79:063527, 2009; J Phys Conf Ser 314:012004, 2011). The vacuum in these theories is presented as a self-sustainable medium and include a new degree of freedom, the q-variable, which establishes the equilibrium of the quantum vacuum. In the present work, the Cauchy formulation for these models is studied in detail. It is known that there exist some limits in which these theories are described by an F(R) gravity model, and these models posses a well posed Cauchy problem. This paper shows that the Cauchy problem is well posed even not reaching this limit. By use of some mathematical theorems about second order non linear systems, it is shown that these scenarios admit a smooth solution for at least a finite time when some specific type of initial conditions are imposed. Some technical conditions of Ringstrom (The Cauchy problem in general relativity, European Mathematical Society, Warsaw, 2000) play an important role in this discussion.

The existence of smooth solutions

In the present work the Eq. (5.1) played a fundamental role. The left hand side of (5.1) involves the inverse metric $g^{\mu \nu }$, which is a function of the metric $g_{\mu \nu }$ to be solved. The problem is that the obtained solution $g_{\mu \nu }$, $\psi$ and $\eta$ are of x-compact support. A true space metric is not expected to be of this form. In the book [11] a procedure in order to avoid this problem was presented. The idea is to multiply the equations and the initial conditions by certain suitable functions, which makes the system of compact x-support. After this, by a suitable gluing process, an smooth space time metric which is not of compact support may be constructed. Here we give some more or less detailed account of this procedure, applied to the present case, but further details may be found in the chapters 9 and 14 of the original reference [11].

1.1 The local form of the metric

The first task is to define an achronal surface where the initial conditions will be settled. Recall that a given an space time $(M, g_{\mu \nu })$ there exists the so called synchronous reference system for which the metric may be expressed as follows

$$\begin{aligned} g=-dt^2+g_{ij}(t,x) dx^i dx^j. \end{aligned}$$

The coordinate t represents the proper time and the spatial metric $g_{ij}(x_i,t)$ depends on t as a parameter. This system of coordinates exists locally, that is, it exists in a subset of the space time of the form $U\times I\subseteq M$ with $I=[0, t_1)$ and $t_1>0$. Consider an achronal initial Cauchy surface $\Sigma$, in such a way that the metric is regular on this surface. Given a subset $U\subseteq \Sigma$ there exist a neighbourhood O with synchronous coordinates $(x_i,t)$ such that the subset U is given by the surface $t=0$. Then, one may adapt the arguments given in chapter 9 of [11] to the present situation as follows.

Consider a subset $V\subseteq U$ such that its closure $\overline{V}\subseteq U$. Take a function $\widetilde{g}_{00}$ such that its range is bounded in $[-2,-1/4]$ and such that $\widetilde{g}_{00}=g_{00}$ when the value of $g_{00}$ is in the range $[-3/2,-1/2]$. In addition, define $\widetilde{g}_{0i}$ such that its range is bounded in $[-2, 2]$ and such that $\widetilde{g}_{0i}=g_{0i}$ when the value of $g_{0i}$ is in the range $[-1,1]$. There is nothing special about this choice of intervals, and a continuum of other choices are possible. The important point is however that the interval on which $\widetilde{g}_{0i}=g_{0i}$ should contain 0. Moreover, the range of $\widetilde{g}_{0i}$ should contain the interval on which $\widetilde{g}_{0i}=g_{0i}$, with a margin. Finally consider an open set R of symmetric $3\times 3$ matrices such that the values of $g_{ij}(x)$ with $x\in \overline{V}$ are in R and the closure of R in the set of $3\times 3$ matrices is compact and included in the set of positive defined ones. Then, one defines $\widetilde{g}_{ij}=g_{ij}$ when the value of $g_{ij}$ is in U and it is assumed that $\widetilde{g}_{ij}$ has a positive lower bound and a positive upper one. Furthermore, it is assumed that the derivatives of $\widetilde{g}_{\mu \nu }$ with respect to the metric is are of compact support. Then replace $g^{\mu \nu }$ by $\widetilde{g}^{\mu \nu }$ in the system (5.1). Also replace the quantity $f^\mu$ by $c f^\mu$ with $c\in C_0^{\infty }[(-1,1)\times U]$ such that c takes values $[-1/2, 1/2]\times \overline{V}$. The system becomes The general form of the system (4.4) is the following

$$\begin{aligned} \widetilde{g}^{\mu \nu }(x,t,u)\partial _\mu \partial _\nu u(x,t)=c f^\mu (x,t,u, \partial _\alpha u). \end{aligned}$$

(A.1)

In order to apply the Proposition 1 of the text, one must modify the initial conditions by multiplying them by a function $h(x_i)$ which is $C^\infty$ and of compact support, and such that $h(q)=1$ for $q\in V$.

$$\begin{aligned} u(x,T_0)=h U_0,\qquad \partial _t u(x, T_0)=h U_1. \end{aligned}$$

(A.2)

In these terms the Proposition 1 applies and one obtains a local solution. This implies that, for a given point p in $\Sigma$, there exists an open neighbourhood O such that there exist a solution for which $g_{\mu \nu }=\widetilde{g}_{\mu \nu }$ and $h=1$. This is the local form of the metric we were looking for, and it is of x-compact support.

1.2 Gluing the local metric to a global one

1.2.1 Preliminary lemmas

The main task now is to glue the local solution described in the previous paragraph to a global one. But before doing that, it is necessary to state some results about sequence of points. Note that situation described above corresponds to a globally hyperbolic space time which admits an smooth Cauchy hypersurface $\Sigma$ and there exist a metric defined in an open set $U\subseteq M$.

Recall that the $J^{-}(p)$ is the causal past of the point p, which is composed for all the points x that they causally precede p, that is

$$\begin{aligned} J^{-}(p)=\{x|\;\;\; x<<p\}. \end{aligned}$$

Analogous definition holds for $J^{+}(p)$. The chronological past and future of p, namely $I^{\pm }(p)$, is defined by changing the word causally by chronologically in the previous definition. For a subset S of M one defines

$$\begin{aligned} J^{\pm }(S)=\cup _{x\in S} J^{\pm }(x), \end{aligned}$$

and the analogous hold for $I^{\pm }(S)$. The future Cauchy development of S, $D^{+}(S)$ is the set of all points x for which every past directed inextendible causal curve through x intersects S at least once. Similarly for the past Cauchy development. The Cauchy development is the union of the future and past Cauchy developments. An space is globally hyperbolic if there exists a surface $\Sigma$ such that $D(\Sigma )=D^+(\Sigma )\cup D^{-}(\Sigma )=M$. The surface $\Sigma$ is known as a Cauchy surface of M and if there is one, there is a continuum of them. In these terms, the following two lemmas apply.

Lemma A.1

Given a Cauchy surface $\Sigma$ in a globally hyperbolic space time (M, g), denote its Cauchy development by $D(\Sigma )=D^+(\Sigma )\cup D^{-}(\Sigma )=M$. For any point $p\in$ Int $D(\Sigma )-I^{-}(\Sigma )$ the set $J^{-}(p)\cap J^{+}(\Sigma )$ is compact.

The statement of Lemma A.1 is intuitive by analyzing it in the case of a Minkowski space time, as in this case, the resulting set $J^{-}(p)\cap J^{+}(\Sigma )$ is the intersection of two compact spaces. However, the proof of this statement is not that straightforward for generic globally hyperbolic space times, as it requires to understand the infinite dimensional space of causal curves $C(\Sigma , p)$ connecting $\Sigma$ with p with a given second countable Haussdorf topology and, in particular, to show that it is compact. Details are given in the books [49,50,51,52].

Lemma A.2

For a given globally hyperbolic space time (M, g) with a smooth Cauchy hypersurface $\Sigma$, consider an open set $U\in M$ and a point q such that $J^+(S)\cap J^-(q)\in U$. Given a sequence $q_i\rightarrow q$ then $J^+(S)\cap J^-(q_i)\subset U$ for $i\ge i_0$.

Proof

Consider a point q which belongs to the causal future $J^{+}(\Sigma )$ of $\Sigma$ and such that $J^+(\Sigma )\cap J^-(q)\subseteq U$. Then, if there is a future directed time like curve $\gamma$ which connects q with another generic point, say p, it follows that q is in the interior of $J^{-}(p)$. Furthermore lemma 1 shows that $J^{-}(p)\cap J^+(\Sigma )$ is a compact set, and this will be exploited to prove the assertion. Consider a sequence of points $q_i$ in $J^+(\Sigma )$ such that $q_i\rightarrow q$, then it is not difficult to see that $J^{-}(q_i)\subseteq J^{-}(p)$ when $i>i_0$. This follows from the fact that q is the accumulation point of the sequence, and for i large enough, these points will be in the causal past of p, as both p and q are connected by a time like curve. It is also intuitive that $J^{-}(q_i)\cap J^{+}(\Sigma )\subseteq U$ when $i>i_1$, since we are assuming that $J^+(\Sigma )\cap J^-(q)\subseteq U$. In fact, suppose that there were a subsequence of points $q_l$ such that the corresponding set $J^{-}(q_l)\cap J^{+}(\Sigma )$ contains point $r_l$ which are outside U even for l large. Then these points $r_l$ are located in $J^{-}(p)\cap J^{+}(\Sigma )-U$, which is a compact set as it is a compact space with a deleted space open space U. Thus $J^{-}(p)\cap J^{+}(\Sigma )-U$ contains its accumulation points and the sequence $r_l$ converges to a point r. Every point $r_l\in J^{-}(q_l)\cap J^{+}(\Sigma )-U$ and therefore $r\in J^{-}(q)\cap J^{+}(\Sigma )-U$. But $J^+(\Sigma )\cap J^-(q)\subseteq U$ and thus, the last statement is inconsistent. This contradiction shows that, in fact, $J^+(S)\cap J^-(q_i)\subset U$ for $i\ge i_0$ and this is precisely the assertion we wanted to prove. $\square$

1.2.2 The gluing process

Equipped with this lemma, let us return now to the gluing process. The following is an adaptation of some arguments of [11] to the present situation. For this, let $W_p$ an open neighbour of p such that its closure is W. Then consider the manifold $M=\cup _{p}W_p$. Given two sets $W_p$ and $W_q$ there are two fields $u_1$ and $u_2$ which are solutions of the corresponding equations. The harmonic coordinate equation is satisfied in both systems. The initial data also coincide in both systems. The main task is to show that the solutions coincide in $\overline{W}_p\cap \overline{W}_q$. For this one let us define the time interval $I\in [0,\infty )$ for which both solutions coincide in

$$\begin{aligned} S_t=[0,t]\times \Sigma \cap \overline{W}_p\cap \overline{W}_q, \end{aligned}$$

and also such that

$$\begin{aligned} J_p^{-}(x)\cap J_p^{+}(\Sigma )=J_q^{-}(x)\cap J_q^{+}(\Sigma ), \end{aligned}$$

(A.3)

for any $x\in S_t$. The strategy is to prove that I is open and closed, and non empty, thus it is the full interval where both solutions are defined. To prove that is not empty is immediate. The initial conditions coincide, then I contains the point $t=0$, which is enough to show that it is not empty. On the other hand, the set (A.3) is compact due to the lemma 1, and a bit of reasoning implies that I should be closed as well. Next, one may prove that I is open. For this, let $x=(t,r)$ be such that $J_p^{-}(x)\cap J_p^{+}(\Sigma )\subseteq W_p\cap W_q$. Then the lemma 2 can be applied with the open subset $W_p\cap W_q$ playing the role of U. From this lemma, it is concluded that for the point $x+\delta x=(t+\epsilon , r +\delta (\epsilon ))$ one has that $J_p^{-}(x+\delta x)\cap J_p^{+}(\Sigma )\subseteq W_p\cap W_q$ if $\epsilon$ is small enough. Thus, the space time extends to the point $x+\delta x$. However, this conclusion does not warrant that the two fields $u_1$ and $u_2$ coincide for $x+\delta x$. To prove that this is indeed the case, take the point t as a new initial condition. This is valid, since the solutions $u_1$ and $u_2$ are known to coincide by our assumption up to t. Then take the difference between the two solutions $u_1$ and $u_2$ at t. This difference $u_1-u_2$ is zero a t and their derivatives are also zero at t. Thus, by general results about quasi linear hyperbolic systems, it follows that $u_1-u_2=0$ up to the time where this solutions exist. Thus $u_1=u_2$ up to

$$\begin{aligned} J_p^{-}(t+\epsilon , r+\delta (\epsilon ))\cap J_p^{+}(\Sigma )=J_q^{-}(t+\epsilon , r+\delta (\epsilon ))\cap J_q^{+}(\Sigma ). \end{aligned}$$

This implies that, given a time $t\in I$, then $[t, t+\epsilon ]\subseteq I$ for $\epsilon$ small enough. This means that I is an open set.

From the paragraph given above, it is clear that I is empty, open and closed, thus $I=[0,\infty )$. This shows that the constructed metric glues properly on $M=\cup _{p}W_p$, up to a point where a singularity appears, or at all times if the universe is future eternal.