Conformal scattering of the Maxwell-scalar field system
on de Sitter space
We prove small data energy estimates of all orders of differentiability between past null infinity and future null infinity of de Sitter space for the conformally invariant Maxwell-scalar field system and construct bounded invertible nonlinear scattering operators taking past asymptotic data to future asymptotic data. We also deduce exponential decay rates for solutions with data having at least two derivatives. The construction involves a carefully chosen complete gauge fixing condition which allows us to control all components of the Maxwell potential, and a nonlinear Grönwall inequality for higher order estimates.
Studies of scattering go back to the beginnings of physics. A famous modern mathematical treatment was developed in the 1960s by Lax and Phillips [31, 32], who succeeded in using functional analytic techniques to study scattering by an obstacle in flat space. In general relativity it is of interest to study metric scattering, that is the effects of curved space on the asymptotic behavior of fields. Around the same time as Lax and Phillips were developing their framework, Roger Penrose discovered a way to compactify certain spacetimes by conformally rescaling the metric and attaching a boundary, [41, 42]. He called the class of spacetimes admitting such a compactification asymptotically simple and the boundary so attached null infinity, for this was where all null geodesics ended up ‘at infinity’. This led to a brand new way of viewing the asymptotics of massless fields in general relativity: one works in Penrose’s conformally compactified spacetime and studies the regularity of fields on , and then translates the regularity in the conformally rescaled spacetime to fall-off conditions in the physical spacetime.
It was not until the work of Friedlander  in 1980 (see also the posthumously published work ), however, that it was understood that the approaches of Lax and Phillips on the one hand and Penrose on the other could be combined. Friedlander showed that, although one cannot perform the same analytically explicit constructions in curved space, one can make sense of the Lax–Phillips asymptotic profiles of fields by identifying them with suitably rescaled limits of fields going to infinity along null directions. These became known as Friedlander’s radiation fields. The ideas of such conformal scattering were taken up by Baez, Segal and Zhou [6, 7, 8, 9] to study a nonlinear wave equation and to some extent Yang–Mills equations on flat space, and later by Mason and Nicolas [33, 34] to study linear equations on a large class of asymptotically simple spacetimes constructed by Corvino, Schoen, Chruściel, Delay, Klainerman, Nicolò, Friedrich and others [12, 13, 15, 16, 28, 29]. This spurred a programme of constructing conformal scattering theories for various fields on a variety of backgrounds and since then a number of works have appeared, many focussing on conformal scattering on black hole spacetimes222See also [26, 51] for some results in interiors of black holes.[23, 25, 37, 39, 40]. It should be mentioned that there have been plenty of works studying relativistic scattering theory without employing the conformal method, notably by Dimock and Kay in the 1980s [17, 18] and later by Bachelot [3, 4] and collaborators Nicolas, Häfner, Daudé, and Melnyk, among many others, a programme which eventually led to rigorous proofs of the Hawking effect [5, 35].
The above programmes were concerned mainly with asymptotically flat spacetimes. However, astronomical observations indicate that the cosmological constant in our universe, though tiny, is positive [44, 45, 47, 49]. It is thus of interest to study scattering, especially of nonlinear fields, on de Sitter space. De Sitter space is the Lorentzian analogue of the sphere in Euclidean geometry and one of the three archetypal spacetimes as classified by the sign of the cosmological constant, with flat Euclidean space corresponding to Minkowski space () and hyperbolic space corresponding to anti-de Sitter space (). As such, de Sitter space differs from Minkowski space in several crucial aspects. Firstly, it is not asymptotically flat. Nonetheless, it is asymptotically simple in the sense of Penrose  and so admits a conformal compactification. Secondly, the positive cosmological constant, no matter how small, renders null infinity spacelike in de Sitter space, which has implications for conformal scattering. In the asymptotically flat case the constructions of Mason and Nicolas required the resolution of a global linear Goursat problem, which had been shown by Hörmander  to be solvable in some generality. In de Sitter space, however, a spacelike means that the construction of a scattering theory instead requires the resolution of a regular Cauchy problem. Thirdly, while obtaining flat space scattering and peeling results through conformal techniques is fine for linear fields, nonlinear fields generically possess so-called charges at spacelike infinity [46, 1, 14]. This is a major obstruction to constructing conformal scattering theories for nonlinear fields in flat space and is related to infrared divergences in quantum field theory [30, 38]. The problem is entirely absent in de Sitter space as it is spatially compact.
Finally, from a more physical perspective, de Sitter space has the peculiar feature that no single observer can ever observe the entire spacetime, in contrast to the Minkowski case where an observer’s past lightcone eventually contains the whole history of the universe. This is related to the existence of cosmological horizons, null hypersurfaces criss-crossing the Penrose diagram of de Sitter space. Their existence has implications for the definition of a classical scattering matrix: the construction of one requires a timelike Killing or conformally Killing vector field, and here one has a choice in de Sitter space. One might wish to use the Killing field provided by the standard static coordinates, i.e. the coordinates an observer at the south pole in de Sitter space might use for themselves, but this is problematic as it fails to be timelike and future pointing beyond the cosmological horizons. Another approach is to conformally compactify de Sitter space and embed it in the Einstein cylinder, where one has a natural globally timelike Killing field which becomes conformally Killing in physical de Sitter space. This can then be used to define an observer-oblivious classical scattering matrix in de Sitter space. We adopt the latter approach here. The importance of the construction of such scattering matrices for quantum gravity in de Sitter is explained well in  and the references therein.
From an analytic point of view, it has been known since the work of Friedrich  that de Sitter space is a stable solution of Einstein’s equations with a positive cosmological constant, so one expects scattering results on de Sitter space to fit into a larger host of stories on asymptotically de Sitter spacetimes. Results in this vein have been obtained by, for example, Vasy, Melrose and Sá Barreto, [52, 36].
This paper is organized as follows. In Section 3 we state the conventions and notation used in the paper, and in Section 4 we introduce the conformally invariant Maxwell-scalar field system that we subsequently study. In Section 5 we describe de Sitter space , a number of standard coordinate systems on , its conformal compactification, and our choice of energy-momentum tensor for the Maxwell-scalar field system on the conformally rescaled spacetime. In Section 6 we state the main results in detail. Sections 8 and 7 contain a detailed derivation of the required gauge fixing conditions, the formulation of the Cauchy problem for our system, and an existence theorem. Sections 11, 10 and 9 contain the inductive energy estimates on which our results rest. Sections 14, 13 and 12 finish off the proofs of the main results.
Acknowledgments. The author wishes to thank Lionel Mason, Qian Wang, Jan Sbierski, Jean-Philippe Nicolas and Mihalis Dafermos for useful guidance and discussions.
We prove small data energy estimates of all orders of differentiability between and of de Sitter space for the conformally invariant Maxwell-scalar field system and show the existence of small data scattering operators for all . Slightly more precisely, we may state the main theorem as follows. The full statements of the main theorems can be found in Section 6. Consider the Penrose diagram of de Sitter space and an initial surface ,
For any there exist bounded invertible forward and backward wave operators mapping small Maxwell-scalar field data on to small Maxwell-scalar field data on , and a bounded invertible scattering operator
mapping small Maxwell-scalar field data on to small Maxwell-scalar field data on .
As a corollary, our estimates imply exponential decay rates for the Maxwell-scalar field system on de Sitter space with small initial data. The decay rates are a partial extension of the results of Melrose, Sá Barreto and Vasy .
The scalar field and the components of the Maxwell potential decay exponentially in proper time along timelike geodesics approaching .
The asymptotic behaviour of solutions to the Einstein–Maxwell–Yang–Mills equations has previously been studied by Friedrich  by employing the machinery of symmetric hyperbolic systems. The estimates we prove here are finer and explicit, allowing us to define the sets of scattering data and read off precise decay rates.
Since the nonlinearities are of the same order, in principle there is no obstruction to extending our estimates to the Yang–Mills–Higgs system on de Sitter space. As a result, the same scattering and decay results should apply there.
We use the spacetime signature . Our main estimates will be performed on the Einstein cylinder with metric , where is the standard positive-definite metric on . We will use Penrose’s abstract index notation and use the Roman indices to refer to tensors on and contractions with respect to the full spacetime metric (or sometimes a general spacetime with metric ), and use the Greek indices to refer to tensors on and contractions with respect to the metric . At a certain point we will also use the indices , and to refer to a basis of vector fields on , but this will be made explicit at the time. We will use to denote the Levi–Civita connection of the full spacetime metric (or a general metric ), and to denote the Levi–Civita connection of . Thus, as , we shall have . We will use to denote the volume form of the full spacetime metric ( or ), and to denote the volume form of . In the case of we will thus have , being the coordinate on . For a -form on we will use to denote the projection of onto , to denote the component of along , and dot (as in ) to denote differentiation with respect to . The Lebesgue and Sobolev norms and of a scalar or vector will refer to and , unless specifically stated otherwise. Occasionally we shall use the symbol to denote equality on null infinity (see Section 5).
We will also adopt Penrose’s sign convention for the curvature tensors, meaning that the Riemann curvature tensor will satisfy
The Ricci tensor and the scalar curvature are then defined as usual,
so that in these conventions the scalar curvature of, for example, a -sphere with the positive-definite metric is negative, to be exact. However, since our metrics will be of signature , that will mean that a spacelike -sphere in our construction will have positive scalar curvature equal to .
4 The Conformally Invariant Maxwell-Scalar Field System
Let be a -dimensional Lorentzian manifold and consider the Lagrangian density
where is a real -form called the Maxwell field, is a real -form called the Maxwell potential, is a complex scalar field on , is the scalar curvature of , and , where is the Levi–Civita connection of . The differential operator is called the gauge covariant derivative. The Euler–Lagrange equations associated to (4.1) are
The Maxwell-scalar field system (4.1) is the simplest classical field theory exhibiting a non-trivial gauge dependence. Indeed, the -form is not uniquely determined by the -form , and any transformation of the form
leaves unchanged. This transforms
so that if one makes the corresponding transformation
The gauge covariant derivative acting on is a connection on a principal bundle over with fibre . This connection is represented by the real -form on in any trivialisation of , where the factor of in comes from . The scalar field is a section of a complex line bundle over associated to by the representation of .
Consider a conformal rescaling of ,
It turns out that in many cases it is possible to fully or partially compactify by choosing the conformal factor so that it compensates for the divergence of distances with respect to the physical metric and attach the boundary to ; this is Roger Penrose’s notion of asymptotically simple spacetimes first described around 1963 in  and . For our purposes it will be sufficient to assume that the spacetime is regular enough so that it may be compactified in this way to make a smooth compact manifold with boundary, , although weaker, partial compactifications leaving singularities at a finite number of points in the boundary are widely used to study, for example, black hole spacetimes [25, 33, 34, 37, 39, 40]. We equip with the rescaled (also called unphysical) metric and call the spacetime the rescaled spacetime.
It is possible to transport the fields into the rescaled spacetime by weighting them appropriately by the conformal factor so that the field equations (4.2) are preserved in . The correct choice of conformal weights for are ,
where , and using this one calculates that
Moreover, because in dimensions the scalar curvature transforms as (see , eq. (6.8.25))
Adding these together one sees that the Lagrangian transforms as
Now the volume form of is related to the volume form of by , so the action
In other words, is conformally invariant up to a boundary term. Since the Euler-Lagrange equations arise from a local variation of the action, this implies the conformal invariance of the field equations (4.2).
5 De Sitter Space
5.1 Global Coordinates and Conformal Compactification
The -dimensional de Sitter space is defined to be the hyperboloid
in -dimensional Minkowski space
where and is the standard metric on the -sphere . If we set
so that is a coordinate on , the metric descends to the metric on ,
This provides a global coordinate system on and is known as the closed slicing of de Sitter space. Note that the topology is manifest in these coordinates. The metric (5.1) can be visualized as a compact spacelike slice expanding in time , as depicted in fig. 2.
To conformally compactify , however, we need a further change of coordinates
In terms of the metric becomes
where . This makes it obvious as to what should be taken as the conformal factor to compactify , namely
and we define
In this conformal scale the hypersurfaces are regular, in contrast to the physical metric (5.2). In fact, the metric clearly extends smoothly for all , so one may consider the extended spacetime known as the Einstein cylinder. We thus identify compactified de Sitter space with the subset of the Einstein cylinder by attaching to (5.2) the boundary . This boundary is the union of two disjoint smooth surfaces
which we call future null infinity and past null infinity respectively. Note that are spacelike hypersurfaces of ; the name null infinity derives from the fact that is where all future (past) pointing null geodesics in de Sitter space end up at infinity. Note also that the vector field is a timelike Killing field in , and in particular it is automatically uniformly timelike since is spatially compact.
As a result, provides a uniformly spacelike foliation of by the level surfaces of the coordinate given explicitly by . Our energies will be defined with respect to .
The fact that is spacelike is, of course, a consequence of the fact that is a solution to Einstein’s equations with a positive cosmological constant ,
Indeed, in general the norm squared on of the normal to is
In the case of , so that . Note that corresponds to the Hubble constant in vacuum.
Writing the -sphere metric as for and quotienting by the symmetry group of we obtain the Penrose diagram for ,
The coordinate varies from to going from left to right, with the vertical lines and representing the North Pole and the South Pole respectively. The coordinate varies from to going up, with the horizontal lines and representing past and future null infinities , as remarked earlier. The dashed lines are the past and future horizons for an observer at the South Pole: a classical observer sitting at can never observe the region , and can never send a signal to the region . Thus region I is the region of communications for an observer at the South Pole, while region III is completely inaccessible.
5.2 Static Coordinates
A set of physical space coordinates on that exhibit an explicit future-pointing timelike Killing field in the region may be constructed by defining
for and . Then the unrescaled metric takes the form
where . In these coordinates the cosmological horizons represented by the dashed lines in fig. 4 are given by , are given by , the North and South Poles are at , and the four corners of the Penrose diagram are at . The vector field is manifestly a timelike Killing vector in the region , but becomes null on the cosmological horizon . It is future-pointing in the region , past-pointing in the region , and spacelike in the regions and . The arrows in fig. 5 represent the directions of the flow of .
5.3 Choice of Energy-Momentum Tensor on
From now on we denote by and the scalar field and Maxwell potential on the Einstein cylinder , and by and the conformally related physical fields on de Sitter space ,
We define the energy-momentum tensor for the system (4.2) on to be
One can check by direct calculation that, as a consequence of the field equations (4.2), is conserved,
so is suitable for defining a conserved energy for the system (4.2),
Since is Killing on , this clearly satisfies
if the field equations (4.2) are satisfied. We call (5.7) the geometric energy for the system (4.2). We also define the geometric energies for the individual sectors of the scalar field and the Maxwell potential ,
The sectorial geometric energies and are not conserved individually and can exchange energy throughout the evolution, but of course the total geometric energy is.
For we also define the Sobolev-type approximate energies
where . Furthermore, for brevity we will often simply write to mean .
5.4 Scaling of Initial Energies
We will consider initial data on the hypersurface and use the coordinate and the metric on the rescaled spacetime, and the coordinate and the metric (5.1) on the physical spacetime. By differentiating the relationship we find
so raising indices with , where is the metric (5.1), we find that and are related by
Furthermore, the conformal factor in the global coordinates (5.1) is given by
Consider the rescaled energies
On the initial surface the conformal factor is a constant and has vanishing derivative, , so the rescaled scalar field is related to the physical scalar field by
while their time derivatives are related by
Since the conformal factor is independent of the coordinates, , and the metric induced on by (5.1) is equivalent to , the rescaled and physical norms of the scalar field are equivalent,
where there is equality if . One similarly checks that
where , and are coordinates on . Thus
and also .
6 Main Theorems
Let be a Cauchy surface in and consider data for the Maxwell-scalar field system on the corresponding Cauchy surface in . We say the data
is admissible if it satisfies the strong Coulomb gauge333See Section 7.1. and solves the elliptic equation
Theorem 6.2 (Energy Estimates).
Let . For -small admissible data
on for the Maxwell-scalar field system on in strong Coulomb gauge one has
for all . In particular,
where is the future (past) null infinity of de Sitter space .
Theorem 6.3 (Conformal Scattering).
For let be the subset of of distributions of admissible data on and let be the subset of of distributions of admissible data on , all equipped with the natural norm . Denote by the open ball of radius in , and write and . Then for every there exist , and sets with such that
there exist bounded invertible nonlinear operators , called the forward and backward wave operators
such that is the forward (backward) Maxwell-scalar field development of on restricted to , and
there exists a bounded invertible nonlinear operator
called the scattering operator, given by
such that is the Maxwell-scalar field development of on restricted to and
for some constants .
Theorem 6.4 (Decay Rates).
Let and be the physical fields related to the conformally rescaled fields and by (5.5). Suppose is small initially. Then the Maxwell-scalar field development of this initial data decays exponentially in proper time along timelike geodesics in . Explicitly using the global timelike coordinate , one has the estimates
as . Furthermore, in the static coordinates (5.4)
as and is fixed. Moreover, if is small initially then there exists a constant such that
as , where is the eigenmode of the linear conformally invariant wave operator on .
7 Field Equations and Gauge Fixing
The field equations (4.2) can be written out in terms of the Maxwell potential ,
We will need to commute derivatives into these equations, so it will be useful to introduce the operators representing their left-hand sides. For any -form and any scalar field we set
The system (7.1) is then equivalent to
In the following sections we specialise to the case of the Einstein cylinder . As noted earlier, for ease of notation we will not hat any rescaled quantities on and instead denote the corresponding physical quantities on with a tilde, as in or . For the metric we compute
7.1 Strong Coulomb Gauge
We will work in the Coulomb gauge adapted to the foliation ,
but will also need to use the residual gauge freedom to fix the gauge fully. More precisely, given a solution to the Maxwell-scalar field system (7.1), a general gauge transformation sends and , and (7.3) is imposed by solving the elliptic equation
on for every fixed . This does not determine uniquely: there is still the residual gauge freedom of , where solves
on each . Because is compact, the kernel of the Laplacian is just the vector space of constant functions, i.e. those which satisfy , but the dependence in the is still arbitrary. Thus in the Coulomb gauge we have the residual gauge freedom
which allows one to choose