Non-singular space-times with a negative cosmological constant: V. Boson stars

Abstract

We prove existence of large families of solutions of Einstein-complex scalar field equations with a negative cosmological constant, with a stationary or static metric and a time-periodic complex scalar field.

Appendix A: Smooth dependence upon (V, g)

Let M be a non-compact smooth manifold (the Cauchy surface). Let ${\mathring{g}}$ be a smooth background Riemannian metric on M with good properties (at least complete, or with bounded geometry, for example). Let $\rho \in C^\infty (M,\mathbb R_{>0})$ be a fixed smooth positive function on M.

Let $E\rightarrow M$ be a tensor bundle like $S^2T^*M$. Then let $\Gamma _{C^{k,\alpha }({\mathring{g}})}(E)$ be the Banach space of all $C^{k,\alpha }$ sections f of E (for each smooth curve $c:\mathbb R\rightarrow M$ the composition $f\circ c$ is $C^k$ and its k-th derivative is locally Hölder of class $\alpha$) with norm (one of many equivalent conditions)

$$\begin{aligned} \Vert f\Vert _{C^{k,\alpha }} =&\sup _{x\in M}\Big ( |f(x)| + |\nabla ^{{\mathring{g}}}f(x)|_{{\mathring{g}}} + \cdots + |(\nabla ^{{\mathring{g}}} )^kf(x)|_{{\mathring{g}}}\Big ) \nonumber \\&+ \sup _{0<{\text {dist}}^{{\mathring{g}}}(x,y)\le \varepsilon } \frac{\Vert (\nabla ^{{\mathring{g}}})^kf(x)-{\text {Pt}}_{y,x}((\nabla ^{{\mathring{g}}})^kf(y))\Vert _{{\mathring{g}}}}{{\text {dist}}^{{\mathring{g}}}(x,y)^\alpha }, \end{aligned}$$

(A.1)

where we used the geodesic distance on M induced by ${\mathring{g}}$, and the fibre metric on $\otimes ^kT^*M\otimes E$ induced by ${\mathring{g}}$, and the parallel transport ${\text {Pt}}_{y,x}$ from y to x along the short geodesic from x to y; here $\varepsilon$ is smaller than the injectivity radius of $(M,{\mathring{g}})$.

Below we will meet mappings on $\Gamma _{C^{k,\alpha }({\mathring{g}})}(E)$ which are real analytic; since we are on a Banach space, these are given by convergent power series of bounded multilinear homogeneous expressions. See [24, Sections 10 and 11]. These mappings will be visibly real analytic, since they will involve only differentiations, multiplications and inversion of matrices.

Moreover, we let $\Gamma _{C^{k,\alpha }_r({\mathring{g}})}(E) = \{s: \rho ^{-r}s \in \Gamma _{C^{k,\alpha }({\mathring{g}})}(E)\}$. If E is the trivial line bundle, we just write $C^{k,\alpha }_r({\mathring{g}})(M)$ instead of $\Gamma _{C^{k,\alpha }_r({\mathring{g}})}(M\times \mathbb R)$.

Let ${\mathring{V}}$ be a smooth positive function on M and let $\mathcal V$ be the space of all functions $V\in C^{\infty }(M,\mathbb R)$ such that $V-{\mathring{V}}\in C^{k+1,\alpha }_1({\mathring{g}})(M)$ and $V>0$ everywhere on M. Then $\mathcal V$ is an open set in an affine space modelled on a Banach space.

Let $\mathcal M$ be the space of all Riemannian metrics g on M such that $g-{\mathring{g}}\in \Gamma _{C^{k+2,\alpha }_2({\mathring{g}})}(S^2T^*M)$. It then follows that also $g^{-1}-{\mathring{g}}^{-1}\in \Gamma _{C^{k+2,\alpha }_2({\mathring{g}})}(S^2T^*M)$. Note that $\mathcal M$ is an open set in an affine space modelled on a Banach space of tensor fields on M.

(1) For each $g\in \mathcal M$ the volume density ${\text {vol}}(g)$ satisfies ${\text {vol}}(g) = F(g){\text {vol}}({\mathring{g}})$ for a function $F(g)= \sqrt{\frac{\det (g_{ij})}{\det ({\mathring{g}}_{ij})}}\in C^{k+2,\alpha }({\mathring{g}})(M)$ with $F(g)>0$ and $\frac{1}{F(g)}\in C^{k+2,\alpha }({\mathring{g}})(M)$. Moreover, F(g) visibly depends real analytically on g.

(2) For each $g\in \mathcal M$ the Hilbert space $L^2(\rho ^{2\delta } {\text {vol}}(g))$ is isomorphic (but not isometric) to $L^2(\rho ^{2\delta } {\text {vol}}({\mathring{g}}))$ via the multiplication operator $F(g):L^2(\rho ^{2\delta } {\text {vol}}({\mathring{g}}))\rightarrow L^2(\rho ^{2\delta } {\text {vol}}(g))$. So we may consider just one Hilbert space $L^2(\rho ^{2\delta } {\text {vol}}({\mathring{g}}))$ with different inner products $\langle \alpha ,\beta \rangle _g = \langle F(g)^{-1}\alpha ,F(g)^{-1}\beta \rangle _{{\mathring{g}}}$.

We pass now to the description of our assumptions:

Assumption A.1

Let $U:= \mathcal V\times \mathcal M$, an open subset in an affine space modelled on a Banach space. For each $x \in U$, let A(x) be an unbounded closed operator on $L^2(\rho ^{2\delta } {\text {vol}}(g))$, such that the domains satisfy $\mathcal D(A(x)) = F(g)\mathcal D(A({\mathring{V}},{\mathring{g}}))$.

In the case of interest in this paper $x = (V,g)$.

The operators in this paper even have equality of all domains in $L^2(\rho ^{2\delta } {\text {vol}}({\mathring{g}}))$.

By replacing A(x) with $F(g)^{-1}A(x)F(g)$, we may assume that A is a map from an open subset $U:=\mathcal V\times \mathcal M$ in an affine space modelled on a Banach space to the set of unbounded closed operators on some fixed Hilbert space $H:=L^2(\rho ^{2\delta } {\text {vol}}({\mathring{g}}))$ with common domain ${\mathfrak {V}}=\mathcal D(A(x))\subseteq H$. Furthermore, we assume that A is real analytic in the sense, that for all vectors $v\in {\mathfrak {V}}$ and $w\in H$ the composite $x\mapsto \langle A(x)v,w\rangle$ is real analytic; see [24, Section 10] for more information. This weak definition suffices due to the real analytic uniform boundedness theorem [24, 11.12].

We emphasise that it is not assumed that the A(x)’s are self-adjoint or with compact resolvent.

Theorem A.2

Under the assumptions A.1, let $\lambda (x_0)$ be a simple isolated eigenvalue of $A(x_0)$ with eigenvector $v(x_0)\in {\mathfrak {V}}$, where $x_0 \in U$ is fixed.

Then one may extend $\lambda$ and v to locally defined real analytic mappings, such that $\lambda (x)$ is a simple isolated eigenvalue of A(x) with corresponding eigenvector v(x) for x near $x_0$ in U.

If $\lambda$ is real-valued and nonnegative (e.g. A(x) is symmetric for some inner product possibly different from the given one), then the nonnegative root $\omega (x)=\sqrt{\lambda (x)}$ is locally Lipschitz in x. On the subset of those x for which $\lambda (x)>0$, the function $\omega (x)$ depends real analytically on x.

It is in general not possible to have a differentiable function $\omega (x)$ such that $\omega (x)^2 = \lambda (x)$, see e.g. [1, 5.2].

Proof

The following argument is adapted from [1, 7.4] and [23, Proof of resolvent lemma]: For each $x\in U$ consider the norm $\Vert u\Vert _x^2:=\Vert u\Vert _H^2+\Vert A(x)u\Vert _H^2$ on ${\mathfrak {V}}$. Since A(x) is closed, $({\mathfrak {V}},\Vert \quad \Vert _x)$ is also a Hilbert space with inner product $\langle u,v\rangle _x:=\langle u,v\rangle _H+\langle A(x)u,A(x)v\rangle _H$. Then $U\ni x\mapsto \langle u,v\rangle _x$ is real analytic for fixed $u,v\in {\mathfrak {V}}$, and by the multilinear uniform boundedness principle [24, 5.18 and 11.14], the mapping $x\mapsto \langle \;,\;\rangle _x$ is real analytic into the space of bounded bilinear forms on $({\mathfrak {V}}, \Vert \quad \Vert _{x_0})$. By the exponential law [24, 3.12 and 11.18] the mapping $(x,u)\mapsto \Vert u\Vert ^2_x$ is real analytic from $U \times (\mathfrak V,\Vert ~\Vert _{x_0})$ to $\mathbb R$ for each fixed $x_0$. Thus, all Hilbert norms $\Vert \quad \Vert _x$ are equivalent: for $B\subset U$ bounded, $\{\Vert u\Vert _x:x\in B,\Vert u\Vert _{x_0}\le 1 \}$ is bounded by $C_{B,x_0}$ in $\mathbb R$, so $\Vert u\Vert _x\le C_{B,x_0}\Vert u\Vert _{x_0}$ for all $x\in B$. Moreover, each A(x) is a globally defined operator $(\mathfrak V,\Vert ~\Vert _{x_0}) \rightarrow H$ with closed graph and is thus bounded, and by using again the (multi)linear uniform boundedness principle [24, 5.18 and 11.14] as above we see that $x\mapsto A(x)$ is real analytic $U \rightarrow L((\mathfrak V,\Vert ~\Vert _{x_0}),H)$.

We consider the global resolvent set

$$\begin{aligned} \mathcal R = \{(x,\mu )\in U\times \mathbb C: A(x)-\mu : ({\mathfrak {V}},\Vert \quad \Vert _{x_0}) \rightarrow H\text { is invertible}\} \end{aligned}$$

which is an open subset of $U\times \mathbb C$, since $(A(x)-\mu )\circ (A(x_0)-\mu _0)^{-1}\in L(H)$ and equals ${\text {Id}}$ for $(x,\mu ) = (x_0,\mu _0)$. By assumption, $\lambda (x_0)$ is a simple isolated eigenvalue of $A(x_0)$ with eigenvector $v(x_0)$. We choose a smooth positively oriented curve $\gamma$ in $\mathbb C$ which contains only $\lambda (x_0)$ in its interior and all other eigenvalues of $A(x_0)$ in the exterior; in particular, $\{x_0\}\times \gamma \subset \mathcal R$. Since $\gamma \subset \mathbb C$ is compact, we may cover $\{x_0\}\times \gamma$ by finitely many open sets of the form $W_i\times \tilde{W}_i$ contained in $\mathcal R$; for $U_1 = \bigcap W_i$ we then have $U_1\times \gamma \subset \mathcal R$ where $U_1$ is an open neighbourhood of $x_0$ in U. By [21, III.6.17], for ${x \in U}_1$ the spectrum of A(x) is separated into the two parts contained in the interior and in the exterior of $\gamma$ and the resolvent integral

$$\begin{aligned} P(x) = -\frac{1}{2\pi i}\int _\gamma (A(x) - \mu )^{-1}\,\mathrm{d}\mu : H\rightarrow {\mathfrak {V}}\subseteq H \end{aligned}$$

is a projection operator onto the sum of all generalised eigenspaces of all eigenvalues of A(x) in the interior of $\gamma$, for ${x \in U}_1$. We now argue as in the proof of [1, 7.8, Claim 1] (see also [24, 50.16, Claim 1]) as follows: By replacing A(x) by $A(x)-z_0$ if necessary we may assume that 0 is not in the interior of $\gamma$. Since $U_1\ni x\mapsto P(x)$ is a smooth (even real analytic) mapping into the space of bounded projections in L(H) with finite dimensional ranges, the rank of P(x) cannot fall locally, and it cannot increase locally since the distance in L(H) of P(x) to the subset of operators of rank 1 is continuous in x and is either 0 or $\ge 1$. See also [21, I.§4.6 and I.6.36].

So we conclude that for x in a (possibly smaller) open set $U_1$ there is only one (counted with multiplicity) eigenvalue (denoted $\lambda (x)$) of A(x) in the interior of $\gamma$ and hence P(x) is a projection on its eigenspace. See also [21, IV.§3.4-5].

Then $v(x): = P(x)v(x_0)$ is an eigenvector for A(x) depending real analytically on x near $x_0$. The corresponding eigenvalue is also real analytic, since

$$\begin{aligned} \lambda (x) = \frac{\langle A(x)v(x),v(x)\rangle _{{\mathring{g}}}}{\Vert v(x)\Vert _{{\mathring{g}}}^2}\,. \end{aligned}$$

Near positive $\lambda (x_0)$’s the square root is obviously also real analytic. If the smooth eigenvalue $\lambda (x)$ is always nonnegative, then the nonnegative square root $\omega (x)=\sqrt{\lambda (x)}$ is locally Lipschitz in x by [26]. $\square$

Remark A.3

The assumption that $\lambda (x_0)$ is a simple eigenvalue of $A(x_0)$ is quite essential in the above theorem. Near eigenvalues with higher multiplicity the situation becomes much more difficult. Real analytic curves of self-adjoint or normal unbounded operators with compact resolvent and common domain of definition admit real analytic choices of their eigenvalues and eigenvectors. However, if the parameter space is at least 2-dimensional, examples can be given with no differentiable choice for self-adjoint operators and no continuous choice for normal operators. If the parameter space is finite dimensional, then, locally, the eigenvalues and eigenvectors can be chosen real analytically after blowing up the parameter space. Even less can be said if the operators depend only smoothly on a parameter and distinct eigenvalues have infinite order of contact. Without normality, even real analytic curves of diagonalisable matrices need not admit smooth choices of the eigenvalues. All this can be found in [33] and the references therein. For the optimal (Sobolev) regularity of the eigenvalues of smooth curves of arbitrary quadratic matrices see [32].