Rigidity of geodesic completeness in the Brinkmann class of gravitational wave spacetimes
aa r X i v : . [ m a t h - ph ] J un Rigidity of geodesic completeness in theBrinkmann class of gravitational wave spacetimes
I.P. Costa e Silva ∗ , J.L. Flores , and J. Herrera Department of Mathematics,, University of Miami, Coral Gables, FL 33124, USA. Departamento de ´Algebra, Geometr´ıa y Topolog´ıa,, Facultad de Ciencias,Universidad de M´alaga , Campus Teatinos, 29071 M´alaga, Spain. Department of Mathematics, Universidade Federal de Santa Catarina, 88.040-900Florian´opolis-SC, Brazil.
November 8, 2018
Abstract
We consider restrictions placed by geodesic completeness on spacetimespossessing a null parallel vector field, the so-called
Brinkmann spacetimes .This class of spacetimes includes important idealized gravitational wavemodels in General Relativity, namely the plane-fronted waves with parallelrays , or pp-waves , which in turn have been intensely and fruitfully studiedin the mathematical and physical literatures for over half a century. Moreconcretely, we prove a restricted version of a conjectural analogue forBrinkmann spacetimes of a rigidity result obtained by M.T. Andersonfor stationary spacetimes. We also highlight its relation with a long-standing 1962 conjecture by Ehlers and Kundt. Indeed, it turns out thatthe subclass of Brinkmann spacetimes we consider in our main theorem isenough to settle an important special case of the Ehlers-Kundt conjecturein terms of the well known class of Cahen-Wallach spaces.
In 2000, M.T. Anderson proved a remarkable rigidity theorem [1] establishingthat every geodesically complete, chronological, Ricci-flat -dimensional station-ary spacetime is isometric to (a quotient of ) Minkowski spacetime . (Recall thata spacetime, i.e., a connected time-oriented Lorentzian manifold, is said to be stationary if it admits a complete timelike Killing vector field.) The proof ofthis result is a powerful adaptation of the Cheeger-Gromov theory of sequencesof collapsing Riemannian manifolds (see also [10] for a much simpler proof of ∗ Visiting Scholar. Permanent address: Department of Mathematics, Universidade Federalde Santa Catarina, 88.040-900 Florian´opolis-SC, Brazil. singular-ity theorems of Mathematical Relativity [4, 18, 22, 24]. Since gravity is thoughtto be always attractive (at least when one disregards quantum effects), gravi-tational systems are often unstable, and one expects gravitational collapse tobe rather ubiquitous. This general idea led R. Geroch [17] to conjecture thatgeodesically complete solutions to the Einstein field equations should be rare (insome suitable sense). Anderson’s result can be viewed as one precise geometricrealization of this physical idea.Another such example appeared as early as 1962, in a separate development,when J. Ehlers and K. Kundt [12, Section 2-5.7] put forth the conjecture that every geodesically complete, Ricci-flat -dimensional pp-wave is a plane wave .A (standard) pp-wave is a spacetime of the form ( R n , g ), where the metric g isgiven in Cartesian coordinates ( u, v, x , . . . , x n − ) by g = 2 du ( dv + H ( u, x ) du ) + n − X i =1 ( dx i ) , (1)and H : R n → R is a smooth function independent of the v -coordinate. Thisclass of spacetimes has been intensely studied both in the mathematical andphysical literatures, since they give an idealized description of gravitationalwaves in General Relativity. A pp-wave where H is quadratic on the x -coordinates,i.e., where H ( u, x ) = n − X i,j =1 a ij ( u ) x i x j , is called a (standard) plane wave . Plane waves have a number of importantphysical and geometrical properties (see, e.g., Ch. 13 of [4] for some of these).The original version of the Ehlers-Kundt conjecture is still open, but there hasbeen progress in obtaining partial results [5, 15, 19, 21].Now, pp-waves are not stationary in general, but one can easily check from(1) that the vector field ∂ v is null and parallel (i.e., with ∇ ∂ v = 0; here and Observe that pp-waves and plane waves can be defined intrinsically (i.e., in a coordinate-independent fashion - see, e.g., [21, Definitions 1 and 2]). Therefore, we use the term standard when there exists a preferred global coordinate system which allows one express the metricin a concrete way. Throughout the present paper, however, all pp-waves and plane wavesconsidered will be standard in this sense, and so we shall omit this term unless there is riskof confusion. Nevertheless, this rule will not be applied to general Brinkmann spacetimes,because we will work simultaneously with general and standard ones. ∇ will denote the Levi-Civita connection of the underlying metrictensor). Therefore, pp-waves are distinguished representatives of the largerclass of Brinkmann spaces , that is, Lorentzian manifolds admitting a null par-allel vector field X . Such manifolds owe their name to H.W. Brinkmann, whodiscovered them in 1925 [8]. Brinkmann spaces have special Lorentzian holon-omy, which in turn gives rise to a number of interesting geometric properties(see, e.g., [2, 3, 16, 21] and references therein for recent results). Brinkmannspaces are always time-orientable [3], so there is no loss of generality in consider-ing only connected, time-oriented Brinkmann spaces, which we will call simplyBrinkmann spacetimes.A nice way of viewing Brinkmann spacetimes is as null analogues of station-ary spacetimes. One is then naturally led to consider the following null versionof the Anderson’s rigidity theorem, firstly conjectured in [11]. Conjecture 1.1.
Every strongly causal, Ricci-flat -dimensional Brinkmannspacetime satisfying certain completeness condition is isometric to (a quotientof ) a plane wave spacetime. Here, the phrase “certain completeness condition” parallels the condition,present in Anderson’s theorem, that spacetime be geodesically complete. It isnot clear to the authors if geodesic completeness alone would suffice in this con-text. Nevertheless, we do believe that geodesic completeness plus transversalcompleteness (see the definition just above Theorem 2.1) should be enough toget the conclusion of Conjecture 1.1, since Theorems 2.1 and 3.1 show that,under these hypotheses, Conjecture 1.1 reduces to the Ehlers-Kundt conjecture.On the other hand, the assumption that spacetime is strongly causal , i.e., hasno “almost closed” nonspacelike curves, replaces the condition in Anderson’stheorem that spacetime be chronological , that is, the absence of closed timelikecurves [11]. These causal conditions imply that the Killing vector field in eachcase will give rise to an isometric action without fixed points, which is in turnconvenient in taking quotients. While the exact analogue of chronology in thenull case would be to require that spacetime be only causal , i.e., has no closedcausal curves, it turns out, after a closer inspection, that a little more causalityis required to have the mentioned quotient behave well [11]. Since in particu-lar every plane wave spacetime is causally continuous [13], and hence stronglycausal, this assumption does not seem very restrictive.Our main goal in this paper is to present a proof of the following restrictedversion of Conjecture 1.1.
Theorem 1.2.
Let ( M, g ) be a geodesically complete, strongly causal, Ricci-flat -dimensional Brinkmann spacetime. If ( M, g ) is transversally Killing, then theuniversal covering ( M , g ) of ( M, g ) is isometric to a plane wave. If X ∈ Γ( T M ) denotes the null parallel vector field in the Brinkmann space-time (
M, g ), the extra condition “transversally Killing” means, by definition,that there exists a Killing vector field
Y ∈ Γ( T M ) such that g ( X , Y ) = 1 (seediscussion after Theorem 2.1 below). A concrete situation where this occurs is3hen the Brinkmann spacetime is an autonomous pp-wave (1), i.e., H does notdepend on the variable u . In this case Y := ∂ u will ensure that the pp-waveis indeed transversally Killing (with X = ∂ v ). Therefore, from the proof ofTheorem 1.2 we deduce the following version of the Ehlers-Kundt conjecture. Corollary 1.3.
Every geodesically complete, strongly causal, autonomous,
Ricci-flat, -dimensional pp-wave is a Cahen-Wallach space. Recall that a
Cahen-Wallach space is an indecomposable, solvable geodesi-cally complete symmetric Lorentzian manifold. These were classified by M.Cahen and N. Wallach in 1970 [9], who showed the universal covering of anyconnected component of such a manifold is isometric to ( R n , g λ ), where λ :=( λ , . . . , λ n − ) ∈ R \ { } and g λ = 2 dudv + n − X i =1 λ i ( x i ) du + n − X i =1 ( dx i ) , which we recognize to be a plane wave without u -dependence.There are several results related to Conjecture 1.1 in the literature. Forexample, Leistner and Schliebner [21] have recently shown that the universalcovering of any compact Ricci-flat Brinkmann spacetime is a plane wave . Theirresult holds in any dimension, and is geometrically quite interesting, but it isunclear to us how strong the assumption of compactness actually is for this classof spacetimes. For instance, for Brinkmann spacetimes it in particular impliesgeodesic completeness [21], which unlike the Riemannian case does not followautomatically from compactness alone. At any rate, compact spacetimes arenever even chronological [22], and therefore have arguably less physical interest.This, in part, has motivated our search for analogous results in the non-compactsetting.After finishing this paper, we became aware of another, much more generalcontext in which the assumptions in Theorem 1.2 are natural , namely in a localclassification scheme, recently carried out by M. Mars and J.M.M. Senovilla [20](see also [6]), for a class of algebraically special spacetimes which includes bothKerr and Brinkmann spacetimes (as well as generalizations of these). Morespecifically, in [20] the authors investigate 4-dimensional Einstein spacetimes( M, g ) endowed with a Killing vector field
Y ∈ Γ( T M ) and satisfying a certain“alignment” relation between the Weyl tensor and the curl of Y . In an importantspecial case (cf. Theorem 1 of [20]) of their classification scheme as applied toRicci-flat spacetimes, these authors show the global existence of a parallel nullvector field X ∈ Γ( T M ) for which g ( X , Y ) = 1, and therefore end up preciselywith what we call here a transversally Killing Brinkmann spacetime. Indeed,they show that these are locally isometric to autonomous pp-waves. (Mars andSenovilla call these stationary vacuum Brinkmann spacetimes .) Theorem 1.2can thus be viewed as a global rigidity result pertaining to such a subclass ofspacetimes. We are grateful to J.M.M. Senovilla for bringing this to our attention, and for pointingout Refs. [6, 20] to us. standard
Brinkmann spacetimes, and establish some of the terminology whichwe will need in the main proof. Brinkmann already knew that a 4-dimensionalRicci-flat Brinkmann spacetime is locally a pp-wave [8] (see also [16, 23]), but wewish to globalize this result here, which we do in Section 3. After some technicallemmas, Theorem 1.2 is proved in Section 4. We finish with a discussion at theend of this same section of a context in which our theorem implies that ourspacetime is of Cahen-Wallach type.
Let ( M n , g ) ( n ≥
3) be a Brinkmann spacetime, which, recall, is a smooth connected time-oriented Lorentzian manifold admitting a complete null parallelvector field X (i.e., with ∇X = 0). We will say that the Brinkmann spacetime( M, g ) is standard if M = R × Q for some ( n − Q which we shall call the spatial fiber , and the metric g can be expressed as g = du ⊗ ( dv + H du + Ω) + ( dv + H du + Ω) ⊗ du + γ, (2)where:(a) γ is a smooth (0 , R × Q whose radical at each p = ( v , u , x ) ∈ R × Q is span { ∂ v | p , ∂ u | p } , and so Q ∋ x γ ( v ,u ,x ) | T x Q × T x Q defines asmooth Riemannian metric on Q .(b) Ω is a smooth 1-form on R × Q with Ω( ∂ v ) = Ω( ∂ u ) = 0, and so Q ∋ x Ω ( v ,u ,x ) | T x Q defines a smooth 1-form on Q and(c) γ , Ω and H have no dependence on the v -coordinate as ∂ v = X is inparticular a Killing vector field.As discussed in the Introduction, if Ω = 0 and γ is the flat Euclidean metricon Q = R , we will say that ( M, g ) is a (standard) pp-wave . Moreover, astandard pp-wave where H is quadratic on the x -coordinates will be calleda (standard) plane wave . Finally, a standard Brinkmann spacetime will becalled autonomous if the quantities H, Ω and γ in (2) have no dependence onthe coordinate u . Therefore, in the autonomous standard Brinkmann case thevector field ∂ u is also a Killing vector field.In general, a Brinkmann spacetime need not be standard. However, as re-cently shown by two of us (IPCS and JLF) [11], it is possible to obtain mildconditions ensuring that a Brinkmann spacetime ( M, g ) the complete parallelvector field X can indeed be expressed in the standard form. This will hap-pen, for instance, if ( M, g ) is transversally complete , which means by definitionthat there exists a complete field
Y ∈ Γ( T M ) conjugate to X , in the sense that g ( X , Y ) = 1 and [ X , Y ] = 0. Concretely (see [11, Theorem V.11]), Here and hereafter, by smooth we always mean C ∞ . heorem 2.1. Let ( M, g ) be a strongly causal and transversally complete Brink-mann spacetime. Then, the universal covering ( M , g ) of ( M, g ) is isometric toa standard Brinkmann spacetime (2). The isometry can be chosen to be suchthat it associates the lift X of X to ∂ v and the lift Y of Y to ∂ u . We will say that a Brinkmann spacetime (
M, g ) is transversally Killing ifthere exists a (not necessarily complete) Killing field Y conjugate to the completeparallel vector field X of ( M, g ). Actually, as long as Y is Killing, we need onlyimpose either that [ X , Y ] = 0 on M and g ( X ( p ) , Y ( p )) = 1 at a single point p ∈ M , or that g ( X , Y ) = 1 on M in order to ensure that ( M, g ) is transversallyKilling:
Proposition 2.2.
Let ( M, g ) be a spacetime with a parallel vector field X anda Killing vector field Y . Then [ X , Y ] = 0 if and only if g ( X , Y ) is constantthroughout M .Proof. Given any vector field Z ∈ Γ( T M ), we have d ( g ( X , Y ))( Z ) = Zg ( X , Y ) = g ( X , ∇ Z Y ) = − g ( Z, ∇ X Y ) = g ( Z, [ Y , X ]) , where we have used that Y is Killing on the third equality and that X is parallelon the second and the last equalities. (cid:3) As a consequence of Theorem 2.1, we have the following.
Corollary 2.3.
Let ( M, g ) be a strongly causal, geodesically complete and transver-sally Killing Brinkmann spacetime. Then, the universal covering ( M , g ) of ( M, g ) is isometric to a standard autonomous Brinkmann spacetime.Proof. Note that since (
M, g ) is transversally Killing, there exists a Killingvector field Y such that g ( X , Y ) = 1 and [ X , Y ] = 0. Since ( M, g ) is geodesicallycomplete, Y is actually a complete vector field conjugate to X , and so ( M, g ) isalso transversally complete. Thus, we can apply Theorem 2.1 and obtain thatthe universal cover of (
M, g ) is isometric to a standard Brinkmann spacetime.Moreover, the isometry can be chosen to be such that ∂ u is associated to Y thelift of Y , and so ∂ u is Killing. In particular, the standard Brinkmann spacetimeis autonomous.Without any causality assumptions we have the following rigidity result,whose proof uses (a version of) Theorem 3 of [21]. Proposition 2.4.
Let ( M, g ) be a geodesically complete, Ricci-flat -dimensionalBrinkmann spacetime. If ( M, g ) is transversally Killing, then the universal cov-ering ( M , g ) of ( M, g ) is isometric to a standard pp-wave.Proof. Since (
M, g ) is Ricci-flat and 4-dimensional, it is locally a standard pp-wave (see, e.g., [16] or the proof of Theorem 3.1 below), and therefore we have[21] that R ( V, W ) = 0 , ∀ V, W ∈ X ⊥ . Y with g ( X , Y ) = 1,( M, g ) is a pp-wave (in the intrinsic way, see Footnote 1) and Theorem 3 of [21]yields the result. Remark 2.5.
We shall need to give below an alternative proof of Proposition2.4 which uses strong causality. The justification is that this proof has theadvantage of giving a very concrete form for the effect of the Killing vector fieldon the function H (cf. Remark 3.2 below). This will be crucial for our mainproof. Moreover, strong causality is needed elsewhere in the proof anyway, sothere is no real loss of generality in that causal assumption.We end this section with some comments regarding notation. A coordi-nate system on a standard Brinkmann spacetime will be often denoted by { u, v, x , . . . , x n − } , where { x , . . . , x n − } is a local coordinate system for Q .We will denote generic spatio-temporal indices by greek letters α, β, . . . , andindices on the spatial fiber by latin letters i, j, . . . . We will also make use of u, v for the corresponding indices, to avoid confusion with spatial fiber indices. Weuse throughout the Einstein’s summation convention. Finally, the superscript“ Q ” indicates (covariant or exterior) differentiation and/or geometric quantitieson the Riemannian manifold ( Q, γ ). Our first aim in this paper is to show that under conditions analogous tothose appearing in Anderson’s rigidity theorem, standard Brinkmann spacetimesare isometric to pp-waves. Concretely,
Theorem 3.1.
Let ( M n , g ) be a standard Brinkmann spacetime and assumethat:i) M is simply connected,ii) n = 4,iii) ( M, g ) is geodesically complete, andiv)
Ric = 0, i.e., (
M, g ) is Ricci-flat.Then (
M, g ) is isometric to a pp -wave. In fact, M = R and there exist coordi-nates { U, V, X, Y } with ( U, V, X, Y ) ∈ R , such that g = 2 dU ( dV + ˜ H ( U, X, Y ) dU ) + dX + dY , with ˜ H harmonic in X, Y . In [21] the authors assume in their Theorem 3 the existence of a complete null vector field Y such that g ( X , Y ) = 1, but actually the causal character of Y is not used anywhere in theproof. roof. Note, first of all, that (i) implies that Q is connected and simply con-nected. Pick local coordinates { x , . . . , x n − } on an open connected and simplyconnected patch U ⊆ Q , together with the given global coordinates ( u, v ) onthe R part. On the neighborhood covered by the coordinates u, v, x , . . . , x n − ,the metric (2) becomes g = 2 du ( dv + H ( u, x ) du + Ω i ( u, x ) dx i ) + γ ij ( u, x ) dx i dx j , (3)where x = ( x , . . . , x n − ). A direct computation of the Christoffel symbolsshows that the only non-zero ones areΓ iuu = − ( ∇ Q H ) i + γ ij ∂ Ω j ∂u , Γ iuk = Γ iku = − γ ij ( d Q Ω) jk + γ ij ∂γ jk ∂u , Γ ijk = (Γ Q ) ijk , Γ vku = ∂H∂x k + γ ij Ω i ( d Q Ω) jk − γ ij Ω i ∂γ jk ∂u , Γ vij = [( ∇ Q ) i Ω j + ( ∇ Q ) j Ω i − ∂γ ij ∂u ] Γ vuu = ∂H∂u + Ω i ( ∇ Q H ) i − γ ij Ω i ∂ Ω j ∂u . (4)Again, a direct calculation reveals that R ijkl = ( R Q ) ijkl , R vαvβ = 0 , R ujul = 0 , (5)so that 0 = ( Ric ) jl = ( Ric Q ) jl , i.e., ( Q, γ ) is Ricci-flat. Specializing to n = 4, wehave that dim Q =2, so Q is flat. Since Q is simply connected and bidimensional,it is diffeomorphic to R , and we may take U ≡ Q , which we do from now on.We may, therefore, select a posteriori coordinates x := x and y := x so that γ ij = δ ij , and thus Γ ijk = (Γ Q ) ijk = 0 globally .With these coordinates, we have0 = ( Ric ) uk = ∂ ( d Q Ω) ik ∂x i , or ∂∂x i (cid:18) ∂ Ω i ∂x k − ∂ Ω k ∂x i (cid:19) = 0 . ( k = 1 , α := 12 (cid:18) ∂ Ω ∂y − ∂ Ω ∂x (cid:19) (6)only depends on the parameter u , i.e., α ≡ α ( u ). We may therefore define anew 1-parameter family of closed (thus, exact ) 1-forms ˜Ω( u ) on Q by˜Ω := ( − αy + Ω ) dx + ( αx + Ω ) dy. Therefore, there exists some function f ∈ C ∞ ( R × Q ) such that ˜Ω = df , and so,Ω ( u, x, y ) = ∂f∂x ( u, x, y ) + α ( u ) y, Ω ( u, x, y ) = ∂f∂y ( u, x, y ) − α ( u ) x. Consider now the change of variable V = v + f ( u, x, y ) which transforms themetric (3) into 8 du (cid:0) dV + ˇ H ( u, x, y ) du + α ( u ) ( ydx − xdy ) (cid:1) + dx + dy , (7)where ˇ H ( u, x, y ) := H ( u, x, y ) − ∂f∂u ( u, x, y ) . (8)Then, all we need to do is remove the term α ( u ) ( ydx − xdy ) to obtain a pp-wave. In order to achieve this, let us consider the following coordinates X = cos( β ( u )) x + sin( β ( u )) y,Y = − sin( β ( u )) x + cos( β ( u )) y, (9)where β ( u ) = R u α ( s ) ds , and observe that dX + dY = dx + dy + α ( u ) (cid:0) x + y (cid:1) du + 2 α ( u ) du ( ydx − xdy ) . In conclusion, in the coordinates { U, V, X, Y } (with U := u ) we have thatthe metric (7) becomes g = 2 dU ( dV + ˜ H ( U, X, Y ) dU ) + dX + dY , (10)where ˜ H ( U, X, Y ) := H ( u, x, y ) − ∂f∂u ( u, x, y ) − α ( u )2 ( x + y ) . (11)The condition that ( M, g ) is Ricci-flat translates, in terms of these coordi-nates, into ∂ ˜ H∂X + ∂ ˜ H∂Y = 0 . (12)Now, U and V clearly have range R , and the form of the metric (10) impliesthe 2-dimensional submanifolds U, V = const. are totally geodesic, and hence(since ( M, g ) is geodesically complete) are isometric copies of the Euclidean2-dimensional spaces. Hence, the range of both X and Y is also R , whichconcludes the proof. Remark 3.2.
In the particular case when the standard Brinkmann spacetime(
M, g ) is autonomous, we deduce that the function α defined on (6) is actuallyconstant. Hence, the new coordinates X and Y can be viewed as arising from x, y , for each u , via a rotation of angle α · u . So, even if we start from anautonomous Brinkmann spacetime, the pp-wave obtained after the changes ofvariables discussed above is not necessarily autonomous. Note, however, that inthis case ˜ H has a very concrete dependence on U (:= u ), given precisely by thevariable change (9), and we deduce from (11) that˜ H ( U, X, Y ) = ˆ H ( x, y ) := H ( x, y ) − α x + y ) . (13)9n the other hand, if we also assume that span { ∂ v , ∂ u } ⊥ is integrable, thenΩ is closed, and thus exact (as we are in the universal cover). Therefore, thefunction α not only is constant, but actually equal to zero, and the change ofcoordinates in the spatial fiber (9) becomes trivial. We conclude that in thiscase, the arguments in Theorem 3.1 lead in fact to an autonomous pp-wave. In order to prove Theorem 1.2, we will need some preliminary lemmas anddefinitions. The following definition was introduced (in slightly different form)in Refs. [14, 15].
Definition 4.1.
A function F : R n → R is at most quadratic if there existnumbers a, b > such that F ( x ) ≤ a k x k + b, ∀ x ∈ R n . Remark 4.1.
Note that if a function F : R n → R is not at most quadratic,then there exists a sequence { x k } k in R n for which F ( x k ) > k k x k k + k, ∀ k ∈ N , and, in particular, k x k k → + ∞ as k → + ∞ . Clearly, if F remains boundedabove by a polynomial of degree at most 2 outside a compact subset of R n then F is at most quadratic.The importance of Definition 4.1 in our context arises from the followingresult, due to H.P. Boas and R.P. Boas (see [7], Thm. II). Lemma 4.2.
A harmonic function F : R n → R bounded from one side by apolynomial of degree m is also a polynomial of degree at most m . In particular,if F is at most quadratic, then there exist numbers a ij , b j ∈ R ( i, j ∈ { , . . . , n } )such that F ( x ) = n X i,j =1 a ij x i x j + n X j =1 b j x j + F (0) . (cid:3) The following two technical lemmas will also be instrumental in our proof.
Lemma 4.3.
Let Ω ⊆ C ≡ R be an open set containing , and let F : Ω → R be a harmonic function such that F (0) = 0 . Then, for each R > such that B R (0) ⊂ Ω , there exists a number θ R ∈ [0 , π ) for which Z R F ( re iθ R ) dr = 0 . roof. Fix one such
R >
0. Consider the continuous function I R : θ ∈ [0 , π ) I R ( θ ) ∈ R given by I R ( θ ) := Z R F ( re iθ ) dr, ∀ θ ∈ [0 , π ) . Integrating this function on the interval [0 , π ), we get R π I R ( θ ) dθ = R π R R F ( re iθ ) drdθ = R R (cid:16)R π F ( re iθ ) dθ (cid:17) dr = R R πr F (0) dr ≡ , where we have used the mean value theorem for harmonic functions in the thirdequality. Hence, for some θ R ∈ [0 , π ) , I R ( θ R ) = 0 as claimed. (cid:3) Lemma 4.4.
Let Ω ⊆ C ≡ R be an open set containing , and let F : Ω → R be a harmonic function such that F (0) = 0 . Then, for each R > such that B R (0) ⊂ Ω , and for each p ∈ ∂B R (0) , there exists a piecewise smooth curve z : [0 , → B R (0) such thati) z (0) = z (1) = 0 and z ( t ) = p for some t ∈ (0 , ,ii) R F ( z ( t )) dt ≥ F ( p ) , andiii) R k ˙ z ( t ) k dt ≤ π R .Proof. Fix one such
R >
0, and let θ R ∈ [0 , π ) be as in Lemma 4.3. Write p = Re iθ ≡ ( R cos θ , R sin θ ). We may assume θ ≥ θ R , since the case when θ ≤ θ R is entirely analogous.Assume first that θ = θ R . In this case, we define z ( t ) = tRe iθ R if t ∈ [0 , / p if t ∈ [2 / , / (1 − t ) Re iθ R if t ∈ [3 / , . (14)With this definition, appropriate changes of variables immediately show that Z / F ( z ( t )) dt = Z / F ( z ( t )) dt = (2 / R ) Z R F ( re iθ R ) dr = 0from the choice of θ R , and hence Z F ( z ( t )) dt = 15 F ( p ) . Moreover, Z k ˙ z ( t ) k dt = 5 R < π R .
11e now assume that θ > θ R . Consider the standard parametrization γ : t ∈ [0 , π ] Re it ∈ C of the circle of radius R . By the mean value theorem forthe harmonic function F we have0 = F (0) = 12 π Z θ R +2 πθ R F ( γ ( t )) k ˙ γ ( t ) k dt, whence we conclude that0 = Z θ R +2 πθ R F ( γ ( t )) dt = Z θ θ R F ( γ ( t )) dt + Z θ R +2 πθ F ( γ ( t )) dt. (15)We may consider two cases:( a ) Z θ θ R F ( γ ( t )) dt ≥ , or ( b ) Z θ R +2 πθ F ( γ ( t )) dt ≥ . For (a), consider the reparametrization β : t ∈ [1 / , / γ (5( θ − θ R ) t +2 θ R − θ ) of the curve γ . Then0 ≤ Z θ θ R F ( γ ( t )) dt = 15( θ − θ R ) Z / / F ( β ( s )) ds, and since we assume θ > θ R we conclude that Z / / F ( β ( s )) ds ≥ . (16)Also, note that β (1 /
5) = Re iθ R and β (2 /
5) = Re iθ = p , and Z / / k ˙ β ( t ) k dt = 5( θ − θ R ) R ≤ π R . We may therefore define z : [0 , → C by z ( t ) = tRe iθ R if t ∈ [0 , / β ( t ) if t ∈ [1 / , / p if t ∈ [2 / , / β (1 − t ) if t ∈ [3 / , / − t ) Re iθ R if t ∈ [4 / , . (17)Thus, Z F ( z ( t )) dt = Z / F (5 tRe iθ R ) dt + Z / / F ( β ( t )) dt + 15 F ( p ) (18)+ Z / / F ( β (1 − t )) dt + Z / F (5(1 − t ) Re iθ R ) dt = 25 R Z R F ( re iθ R ) dr ( ≡
0) + 2 Z / / F ( β ( t )) dt ( ≥
0) + 15 F ( p ) ≥ F ( p ) , Z k ˙ z ( t ) k dt = 2 Z / / k ˙ β ( t ) k dt + 10 R ≤ π R , which concludes case (a). The case (b) follows analogously, just interchanging β with a map ˜ β : [1 / , / → C defined by˜ β ( t ) = γ (5( θ R + 2 π − θ ) t + 2 θ − θ R − π )(compare with the definition of β ), and the result follows. (cid:3) Proof of Theorem 1.2 .Note that since (
M, g ) is transversally Killing, Corollary 2.3 ensures that theuniversal covering of (
M, g ) is a standard autonomous Brinkmann spacetime.We can then assume without loss of generality that (
M, g ) is a standard au-tonomous Brinkmann spacetime, and so, that M = R × Q and g is expressedas (2) where H , Ω and γ do not depend on the variable u .Observe that now we can apply Theorem 3.1, which ensures the existence ofcoordinates { U, V, X, Y } with ( U, V, X, Y ) ∈ R for which g has the expression g = 2 dU ( dV + ˜ H ( U, X, Y ) dU ) + dX + dY , with ˜ H harmonic in X, Y .Moreover, due the fact that (
M, g ) is autonomous, Remark 3.2 ensures that˜ H ( U, X, Y ) = ˆ H ( x, y ) (see (13)).We wish to show that ˜ H is quadratic in the coordinates X , Y . Now, sincethe coordinate transformations for X and Y are linear in x, y (cf. (9)), in orderto accomplish this it is enough to show that ˆ H is a quadratic function of x and y . Assume then, by way of contradiction, that ˆ H is not quadratic as a functionof x, y . Since − ˆ H is harmonic in x, y , due to Lemma 4.2 − ˆ H can not be atmost quadratic in these coordinates. Therefore (cf. Remark 4.1) we can pick asequence p k = ( x k , y k ) in R for which − ˆ H ( p k ) > k k p k k + k, ∀ k ∈ N (19)and R k := k p k k → + ∞ as k → + ∞ .Our strategy from now on is as follows. We will show the existence ofsome open set U containing the origin (0 , , ,
0) of M ≡ R and timelikecurve segments with endpoints arbitrarily close to the origin, such that theyare not contained in U , in violation of our assumption of strong causality for( M, g ). This contradiction then yields that ˆ H is indeed quadratic, which in turnestablishes the theorem.So, let us fix U be the open Euclidean ball in R centered at the originand with radius R >
0. Fix a number 0 < ∆ < R . We can assume that R k > R for all k ∈ N , and so, that any point ( u, v, p k ) / ∈ U . For each k ∈ N ,we may use Lemma 4.4 with F = − ˆ H and pick a piecewise smooth curve z k : t ∈ [0 , B R k (0) ⊂ C ≡ R such that13i) z k (0) = z k (1) = (0 ,
0) and z ( t k ) = p k for some t k ∈ (0 , R ˆ H ( z k ( t )) dt ≥ − ˆ H ( p k ), and(iii) R k ˙ z k ( t ) k dt ≤ π R k .Using these curves, we may define for each k ∈ N , the piecewise curve Z k : t ∈ [0 , C given by Z k ( t ) = e iα ∆ t z k ( t ) for each t ∈ [0 , α is definedin (6) and it is constant due the autonomous character of ( M, g ). Observethat, from construction, ˜ H ( α ∆ t, Z k ( t )) = ˆ H ( z k ( t )). Therefore, if we write z k ( t ) = x k ( t ) + iy k ( t ) we compute: k ˙ Z k k = ˙ Z k ˙ Z k = α ∆ k z k k + k ˙ z k k + iα ∆( z k ˙ z k − ˙ z k z k ) (20)= α ∆ R k + k ˙ z k k + 2 α ∆( x k ˙ y k − y k ˙ x k ) ≤ α ∆ R k + k ˙ z k k + 4 | α | ∆ R k k ˙ z k k≤ α ∆ R k + 3 k ˙ z k k . Joining the previous inequality with (iii) we conclude that Z k ˙ Z k ( t ) k dt ≤ C ( α, ∆) R k , (21)where C ( α, ∆) = 3 α ∆ + 150 π . Finally, for each number
E > k ∈ N , we can define the curveΓ Ek : [0 , → R given, for each t ∈ [0 , Ek ( t ) := ( V Ek ( t ) , ∆ t, Z k ( t )) , where V Ek ( t ) := − ∆ Z t ˜ H ( α ∆ s, Z k ( s )) ds − Z t k ˙ Z k ( s ) k ds − Et
2∆ (22)= − ∆ Z t ˆ H ( z k ( s )) ds − Z t k ˙ Z k ( s ) k ds − Et . It is easy to check, using the line element of g in the form (10), that each Γ Ek defines a timelike curve in ( M, g ).Now, consider the smooth functions h k : E ∈ (0 , + ∞ ) V Ek (1) ∈ R ( k ∈ N ).Clearly, for each k ∈ N , h k ( E ) < E . However, collecting ourestimates (ii), (19) and (21), we get from (22) h k (1) ≥ ( k − C ( α, ∆)2∆ ) R k + k − . (23)It is clear from inequality (23) that we can pick k ∈ N for which h k (1) > h k is continuous, there exists E > h k ( E ) = 0. Wethen conclude that Γ E k is a timelike curve such that Γ E k (0) = (0 , , ,
0) andΓ E k (1) = (0 , ∆ , , ∈ U but Γ E k ( t k ) = ( V E k ( t k ) , ∆ t k , p k ) / ∈ U , as desired;so the proof is complete. 14 In the Introduction, we have shown how Theorem 1.2 implies, via Corollary1.3, that a physically relevant but relatively restricted class of pp-waves actuallyfall under the important Cahen-Wallach subclass. Our final goal in this paperis to widen the scope of that result so as to encompass the larger class of thoseBrinkmann spacetimes envisaged in Theorem 1.2, and give a precise, concretegeometric characterization of when these are Cahen-Wallach spaces.In order to achieve this, let (
M, g ) be a Brinkmann spacetime in the condi-tions of Theorem 1.2. Corollary 2.3 allows us to assume without loss of generalitythat (
M, g ) is a standard autonomous Brinkmann spacetime. Now we assume,in addition, that span {X , Y} ⊥ is an integrable distribution, being X , Y thecorresponding parallel and Killing null fields. Then, by applying Theorem 3.1,and taking into account Remark 3.2, we deduce that ( M, g ) is isometric to anautonomous pp-wave, i.e. with H independent of u . So, if we take up again thearguments of the proof of Theorem 1.2, we conclude that H must be quadraticin the spatial coordinates, and thus a Cahen-Wallach space. Summarizing: Corollary 4.5.
Let ( M, g ) be a Brinkmann spacetime in the conditions of The-orem 1.2 and denote by X and Y , respectively, the corresponding parallel andKilling null fields. Then, the universal cover of ( M, g ) is a Cahen-Wallachspacetime if and only if span {X , Y} ⊥ is an integrable distribution. Acknowledgments
The authors are partially supported by the Spanish Grant MTM2013-47828-C2-2-P (MINECO and FEDER funds). JLF is also supported by the RegionalJ. Andaluc´ıa grant PP09-FQM-4496, with FEDER funds. IPCS wishes to ac-knowledge a research scholarship from CNPq, Brazil, Programa Ciˆencia semFronteiras, process number 200428/2015-2. JH would like to thank the Depart-ment of Algebra, Geometry and Topology of the University of M´alaga, JLFthe Department of Mathematics of the Universidade Federal de Santa Catarina,while IPCS extends his warm thanks to the faculty and staff members of the De-partment of Mathematics of the University of Miami, for their kind hospitalitywhile part of the work on this paper was being carried out.
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