J Bičák

M Scholtz
email: scholtzzz@gmail.com

P Tod

On asymptotically flat solutions of Einstein's equations periodic in time II. Spacetimes with scalar field sources

 to show that timeperiodic, asymptotically-flat solutions of the Einstein equations analytic at I, whose source is one of a range of scalar-field models, are necessarily stationary. We also show that, for some of these scalar-field sources, in stationary, asymptoticallyflat solutions analytic at I, the scalar field necessarily inherits the symmetry.

To prove these results we investigate miscellaneous properties of massless and conformal scalar fields coupled to gravity, in particular Bondi mass and its loss.

Introduction

In this article, we continue the study begun in [START_REF] Bičák | On asymptotically flat solutions of Einstein's equations periodic in time I. Vacuum and electrovacuum solutions[END_REF] (paper I) of asymptotically-flat solutions of Einstein's equations that are periodic in time. In [START_REF] Bičák | On asymptotically flat solutions of Einstein's equations periodic in time I. Vacuum and electrovacuum solutions[END_REF], we showed that such space-times, if either vacuum or electrovacuum and analytic at I, are necessarily stationary near I. Here we extend this result to space-times whose source is one of a range of scalar field models.

In [START_REF] Bičák | On asymptotically flat solutions of Einstein's equations periodic in time I. Vacuum and electrovacuum solutions[END_REF], we also considered the problem of inheritance of symmetry. This is the question of whether, if a space-time which is a solution of Einstein's equations with some matter source has a symmetry, the matter source necessarily has the same symmetry. For asymptotically-flat electrovacuum space-times which are analytic near I we showed that the symmetry is necessarily inherited. For scalar field sources, we now obtain the same result in some cases but not in others.

The scalar fields we consider fall into two broad classes. The first class includes the complex, massless Klein-Gordon (KG) field which satisfies the wave equation [START_REF] Barceló | Scalar field, energy conditions and transversable wormholes[END_REF] and has energy-momentum tensor as in [START_REF] Bekenstein | Exact solutions of Einstein-conformal scalar equations[END_REF]. Here we prove Confidential: not for distribution. Submitted to IOP Publishing for peer review 21 June 2010 Theorem 5.2: A weakly-asymptotically-simple time-periodic solution of the Einstein-massless-KG field equations which is analytic in a neighbourhood of I - necessarily has a Killing vector which is time-like in the interior and extends to a translation on I -.

It is also possible to include a potential for the scalar field, as in subsection (2.3), and therefore to include a mass-term, and the above result will continue to hold subject to a weak condition on the potential. Now one knows, for example from [START_REF] Bizoń | On existence of mini-boson stars[END_REF], that there exist boson-star solutions of the Einstein-massive-KG system for which the metric is static, spherically-symmetric and asymptotically-flat, while the complex scalar field takes the form f (r)e iωt : these solutions are genuinely periodic in time but not stationary, and the source does not inherit all the symmetries of the metric. However, it is easy to see that these solutions are not analytic near I, which is why they do not violate our result.

The other class of scalar fields contains what we shall call the conformal scalar field, that is, it satisfies the conformally-invariant wave equation (35). For simplicity, we shall take the field to be real, though the formalism allows a complex field. In the real case, there is a conserved energy-momentum tensor for such a source due originally to [START_REF] Newman | New Conservation Laws for Zero Rest-Mass Fields in Asymptotically Flat Space-Time[END_REF] (see also [START_REF] Callan | A new improved energy-momentum tensor Ann[END_REF] and [START_REF] Penrose | Spinors and space-time vol II Cambridge Monographs on Mathematical Physics[END_REF]), given in (36). This leads to a form of the Einstein equations (37) studied in [START_REF] Hübner | General relativistic scalar-field models and asymptotic flatness[END_REF], where it was shown that there is a well-posed initial value problem. With suitable data on hyperboloidal surfaces extending to I, there exist asymptotically-flat solutions even with a regular point at i + [START_REF] Hübner | General relativistic scalar-field models and asymptotic flatness[END_REF]. Explicit static spherically-symmetric solutions were earlier given in [START_REF] Bekenstein | Exact solutions of Einstein-conformal scalar equations[END_REF], [START_REF] Bekenstein | Black holes with scalar charge[END_REF]. Starting from the assumption of an asymptotically-flat solution of these Einstein equations, we may proceed as before, with the corresponding result: Theorem 5.3: A weakly-asymptotically-simple time-periodic solution of the Einstein-conformal-scalar field equations which is analytic in a neighbourhood of I - necessarily has a Killing vector which is time-like in the interior and extends to a translation on I -.

Turning to the question of inheritance, for the first class of fields we show (Theorem 6.1) that the only way a stationary symmetry can fail to be inherited in the class of space-times under consideration is if the (necessarily complex) scalar field has the form f (x i )e iωt in terms of comoving space-coordinates x i and time t. This is periodic and the previous result can be applied to deduce that the symmetry is in fact inherited. For a complex conformal scalar field, the same argument can be used to show that there are no non-inheriting fields of this form but this is only a partial result as we cannot characterise the non-inheriting fields in the same way.

The plan of the article is as follows: in section 2 we review the Einsteinmassless-KG equations and show how to formulate the conformal Einstein-massless-KG equations, by which we mean the equations formulated for an unphysical, rescaled metric which correspond to the physical Einstein-massless-KG equations. This enables the equations to be extended to I. In section 3, we do the same thing for the Einsteinconformal-scalar equations, using the conserved enery-momentum tensor proposed in [START_REF] Newman | New Conservation Laws for Zero Rest-Mass Fields in Asymptotically Flat Space-Time[END_REF] (see also [START_REF] Callan | A new improved energy-momentum tensor Ann[END_REF]). This energy-momentum tensor does not satisfy the Dominant Energy Condition, but we give some arguments why it might none-the-less lead to positive total energy. In section 4, we give expressions for the Bondi mass and Bondi mass-loss for both classes of scalar field sources. The Bondi mass-loss for the conformal scalar field is not manifestly positive (at I + ) but in the periodic case the average over a period is. In section 5, we recall the coordinate and null-tetrad system used in [START_REF] Bičák | On asymptotically flat solutions of Einstein's equations periodic in time I. Vacuum and electrovacuum solutions[END_REF] and prove Theorems 5.2 and 5.3, to show that, in this setting, periodic solutions are actually stationary. The proof is much as in [START_REF] Bičák | On asymptotically flat solutions of Einstein's equations periodic in time I. Vacuum and electrovacuum solutions[END_REF]: one shows inductively that all radial derivatives of all metric components at I -are v-independent so that K = ∂/∂v is a Killing vector. In section 6, we discuss inheritance and prove Theorem 6.1 to show that stationarity is necessarily inherited in an analytic, weakly-asymptotically-flat, Einstein-massless-KG solution.

In order to be able to follow clearly the arguments in the main text, in Appendix A we review all Newman-Penrose equations for a general source: that is, the commutation relations of the NP operators, and the Ricci and Bianchi identities. In Appendix B the conformally rescaled scalar wave equations and conformal Bianchi identities for the massless scalar field are written down in unphysical space in manifestly regular form. The regular conformal Bianchi identities for conformally invariant scalar fields follow in fact from the conformal Bianchi identities for any matter field for which the Ricci spinor behaves as O(Ω 2 ) at I. The projections of the Bianchi identities are given in section B4 of Appendix B. The asymptotic form of the solutions of the Einstein-massless-scalar field equations at future null infinity I + is discussed in Appendix C; this is used in section 4 in the derivation of the Bondi mass and the mass-loss formula for both massless-Klein-Gordon field and conformalscalar field. Finally, in Appendix D we give some examples of exact solutions of the Einstein-conformal-scalar equations and discuss the possible presence of singularities.

The massless KG field

Basic relations

First we investigate the complex scalar field which satisfies the massless Klein-Gordon equation in the physical spacetime, ˜ φ = 0.

(

) 1
We shall consistently use the tilde to indicate quantities in the physical space-time, untilded quantities referring to the rescaled, unphysical space-time. The energymomentum tensor whose conservation is implied by this equation is

T ab = 1 4π 2 ∇ (a φ ∇ b) φ -gab gcd ∇ c φ ∇ d φ . (2)
First we have to determine the conformal behaviour of the scalar field. Since the wave equation ( 1) is not conformally invariant, there is a priori no preferred choice. However, since near I -(r → ∞) the radiative part of the field behaves as

φ ∼ 1 r , (3)
and we wish to have a non-vanishing regular unphysical field on I -, we define φ = Ω φ.

In the following, we employ the notation

φAA = ∇ AA φ, ϕ AA = ∇ AA φ, s AA = ∇ a Ω, (5)
and using the NP formalism ‡ we denote the components of the fields s a and ϕ a by special symbols:

DΩ = s 00 = S 0 , δΩ = s 01 = S 1 , δΩ = s 10 = S1 = S1 , ∆Ω = s 11 = S 2 , Dφ = ϕ 00 = ϕ 0 , δφ = ϕ 11 = ϕ 1 , δφ = ϕ 01 = ϕ1, ∆φ = ϕ 10 = ϕ 2 , (6)
and correspondingly with tildes in the physical spacetime.

In this notation, the spinor form of Einstein's equations in the physical spacetime is

ΦABA B = 2 φ(A(A φB )B) , 6 Λ = -φc φc , (7)
the components of the Ricci spinor with respect to the spin basis are Φ00 = 2 φ0 φ0 ,

Φ01 = 2 φ(0 φ1) , Φ02 = 2 φ1 φ1 , Φ11 = φ(0 φ2) + φ(1 φ1 ) , Φ12 = 2 φ(1 φ2) , Φ22 = 2 φ2 φ2 , (8)
and the scalar curvature is

Λ = 1 3 -φ(0 φ2) + φ(1 φ1 ) . (9)

The conformal Einstein-massless-KG equations

In this subsection, we find a system of equations regular at I for all unphysical quantities. This system, by analogy with Friedrich's 'conformal Einstein equations' [START_REF] Friedrich | On the regular and the asymptotic characteristic initial value problem for Einstein's vacuum field equations[END_REF] we shall call the 'conformal Einstein-massless-KG equations'.

In [START_REF] Bičák | On asymptotically flat solutions of Einstein's equations periodic in time I. Vacuum and electrovacuum solutions[END_REF] we derived the physical Bianchi identities expressed in terms of the unphysical quantities as

Ω 2 ∇ D A ψ ABCD = Ω ∇ B (C Φ AB)A B + s B (C Φ AB)A B + ∇ B (C ∇ A(A s B )B) . ( 10
)
where s AA is given by ( 5), Φ ABA B is the Ricci spinor and ψ ABCD = Ω -1 Ψ ABCD is the rescaled Weyl spinor (see eq. ( 6) in I). Using the rule for the conformal transformation of the Ricci spinor, ‡ The explicit expressions for the NP tetrad, the corresponding spin basis, the NP operators, etc., in the coordinate system (v, r, θ, φ) used in the following are introduced in Section 3 and Appendix A of paper I and repeated in section 5 below.

∇ A(A s B )B = Ω ΦABA B -Ω Φ ABA B , (11)
we find

∇ D A ψ ABCD = Ω -2 s B (C ΦAB)A B + Ω -1 ∇ B (C ΦAB)A B . (12)
The right hand side of this equation is not manifestly regular on I, while the left hand side is regular by assumption of asymptotic flatness.

Next we express the physical Ricci spinor via the unphysical quantities,

ΦABA B = 2 Ω 2 ϕ (A(A φB )B) + 2 φ φ s (A(A s B )B) + 2 Ω φ ϕ (A(A s B )B) + 2 Ω φ φ(A(A s B )B) . (13)
and insert this expression into [START_REF] Hugget | An introduction to twistor theory[END_REF]. In order to simplify the resulting equations, we introduce following notation: let X a , Y a and Z a be arbitrary vector fields and define

(XY Z) = X B (C Y A(A Z B )B) . (14)
The expression (XY Z) is obviously symmetric in Y Z. It is straightforward to derive the relation

(XY Z) + (ZXY ) + (Y ZX) = 0, (15)
with special case (XXX) = 0. After inserting the Ricci spinor (13) into the Bianchi identities [START_REF] Hugget | An introduction to twistor theory[END_REF], we arrive at

∇ D A ψ ABCD = 2 Ω -1
This can be simplified using identity [START_REF] Penrose | Spinors and space-time vol II Cambridge Monographs on Mathematical Physics[END_REF]:

∇ D A ψ ABCD = 2 φ φ Ω -1 (∇ s s) + 2 Ω (∇ ϕ φ) + 4 (s ϕ φ) + 2 φ (∇ ϕ s) + 2 φ (∇ φ s) , (17)
where, e.g. (∇ ϕ s) = ∇ B (C ϕ A(A s B )B) , so ∇ acts on both ϕ and s. The last equation is still formally singular on I -because of the factor Ω -1 , but using [START_REF] Hübner | General relativistic scalar-field models and asymptotic flatness[END_REF] we finally obtain

∇ D A ψ ABCD = 2 φ φ s B (C Φ AB)A B + 4 (s ϕ φ) + 2 φ (∇ s φ) + 2 φ (∇ s ϕ) + 4 Ω 1 2 (∇ ϕ φ) -φ φ2 (s ϕ s) -φ φ 2 (s φ s) -4 Ω 2 φ φ (s ϕ φ) , (18)
which is manifestly smooth at I.

Next we wish to derive equations for the conformal factor. The commutator of covariant derivatives annihilates scalars, so contracting ∇ [a ∇ b] Ω = 0 with A B gives the relation

∇ A (A s A B) = 0. ( 19
)
By decomposing ∇ AA s BB into its symmetric and antisymmetric parts and using the equation above, we obtain

∇ AA s BB = ∇ (A(A s B )B) + 1 4 AB A B Ω. ( 20
)
The first term on the r.h.s. is given by [START_REF] Hübner | General relativistic scalar-field models and asymptotic flatness[END_REF]. We now define the quantity (cf. eq. ( 15) in I)

F = 1 2 Ω -1 g ab s a s b , (21)
which is regular on I. The rule for the conformal transformation of the scalar curvature can be written in the form (eq. ( 16) in I)

Ω = 4 Ω Λ -4 Ω -1 Λ + 4 F. ( 22
)
We thus have found an expression for the second derivatives of the conformal factor Ω:

∇ AA s BB = Ω ΦABA B -Ω Φ ABA B + AB A B ΩΛ -Ω -1 Λ + F . (23)
The last expression contains a term Ω -1 Λ which again seems to be singular on I. This is not the case, however, since by [START_REF] Callan | A new improved energy-momentum tensor Ann[END_REF], ( 4) and (21) we have

Λ = - 1 6 Ω 3 Ω ϕ c φc + φ φc s c + φ ϕ c s c + 2 φ φ F . ( 24
)
The physical scalar curvature is therefore manifestly at least O(Ω 3 ). The projections of the last equation are written down explicitly in Appendix B, eqs. (B14)-(B23). Now we wish to derive equations governing the quantity F . The contracted Ricci identities read

∇ a Ω -∇ b ∇ a s b = R d a s d . (25)
Using the spinor decomposition of the Ricci tensor

R ab = -2 Φ ABA B + 6 Λ AB A B . ( 26
)
and the expression (23), we find after some arrangements

∇ AA F = 1 3 Ω 1 ∇ BB ΦABA B + 1 3 s BB ΦABA B -s BB Φ ABA B + Λ s AA -Ω -2 Λ s AA + Ω -1 ∇ AA Λ. ( 27
)
The first term on the r.h.s. can be rewritten as

∇ BB ΦABA B = Ω -2 ∇BB ΦABA B + 2 Ω -1 ΦABA B s BB . ( 28
)
Now we employ the contracted physical Bianchi identities ∇AA ΦABA B = -3 ∇BB Λ and obtain

∇ AA F = s BB ΦABA B -s BB Φ ABA B + (Λ -Ω -2 Λ) s AA . (29)
The projections of this equation can be found in Appendix B, eqs. (B25)-(B27).

Finally we derive conformal equations for the field ϕ AA . The expression ( +4Λ)φ is conformally invariant with the conformal weight 3, so

φ = -4 (Λ -Ω -2 Λ) φ, (30)
where we used the wave equation ( 1) in the physical spacetime. The symmetric part of ∇ A A ϕ BA is zero and we find

∇ A A ϕ BA = - 1 2 AB φ. (31)
Combining the last two equations we arrive at

∇ A A ϕ BA = 2 Λ -Ω -2 Λ φ AB . (32)
To summarise: in the unphysical spacetime we have the following variables: {Ω, φ, s a , ϕ a , F, ψ ABCD , Φ ABA B , Λ}. The evolution of these quantities is given by eqs. ( 18), ( 23), ( 29) and (32), together with the contracted Bianchi identities

∇ AA Φ ABA B = -3 ∇ BB Λ.
(33)

Potentials

The massless KG equation ( 1) can be generalized to include self-interactions of the scalar field by adding a potential term to the energy-momentum tensor (2),

T ab → T ab + 1 4π gab V ( φ, φ),
so that the field equation acquires the form

φ + ∂V ∂ φ = 0.
Since the potential term in the energy-momentum tensor is proportional to the metric, it will contribute to the scalar curvature Λ, but not to the trace-free Ricci spinor. The new form of Einstein's equations ( 7) is therefore

ΦABA B = 2 φ(A(A φB )B) , 6 Λ = -φc φc + 2 V. (34)
For our proof we require φ = Ω -1 φ and Ω -3 Λ to be regular on I -, cf. ( 4), ( 29) and (24). From (34) we can see that this will be satisfied, if Ω -3 V is regular on I -. In this case the proof works without change. An example is massless φ 4 -theory, where

V = ( φφ ) 2 = O(Ω 4 ).
If there is a mass term m 2 φφ in V , the asymptotic behaviour of the unphysical field changes to

φ = O(e -mr ), so Ω -3 m 2 φ φ ∼ m 2 r e -2mr ,
which is regular. The field φ is now not analytic at I -, so our argument does not apply to this case, which is the class including the boson stars of [START_REF] Bizoń | On existence of mini-boson stars[END_REF]. Notice, however, that in general the asymptotic behaviour of massive fields at I is a subtle question which appears to be carefully analyzed only at the level of linearized theory [START_REF] Winicour | Massive fields at null infinity[END_REF].

The conformal-scalar field

Basic relations

Consider now the conformal-scalar field, by which we mean a scalar field satisfying the equation

+ 1 6 R φ ≡ + 4 Λ φ = 0. (35)
This is conformally invariant if φ transforms as (4), i.e. φ = Ω φ. For simplicity we assume the field φ to be real, but the procedure is easily generalized to complex φ.

The energy-momentum tensor conserved due to equation ( 35) is (see [START_REF] Newman | New Conservation Laws for Zero Rest-Mass Fields in Asymptotically Flat Space-Time[END_REF], [START_REF] Callan | A new improved energy-momentum tensor Ann[END_REF] or [START_REF] Penrose | Spinors and space-time vol II Cambridge Monographs on Mathematical Physics[END_REF], Volume II, page 125)

T ab = 1 4π 2 φA(A φB )B -φ ∇ A(A φB )B + φ2 ΦABA B . (36)
Furthermore, this energy momentum tensor also has good conformal behavior, rescaling as

T ab = Ω 2 T ab ,
but it will not satisfy any of the usual energy conditions. We shall return to this point. We take Einstein's equations to be, as usual, Φab + 3 Λg ab = 4π Tab , then we can solve to find

ΦABA B = 1 -φ2 -1 2 φ(A(A φB )B) -φ ∇(A(A φB )B) , Λ = 0 . ( 37
)
These equations are singular when φ2 = 1 but there are known solutions which avoid this singularity [START_REF] Bekenstein | Exact solutions of Einstein-conformal scalar equations[END_REF], [START_REF] Bekenstein | Black holes with scalar charge[END_REF] (for explicit examples, see Appendix D) and it is known that there is a well-posed initial value problem [START_REF] Hübner | General relativistic scalar-field models and asymptotic flatness[END_REF] which with suitable data extends to I + . We shall therefore assume that we have an asymptotically-flat solution, periodic in time, with φ tending to zero at infinity, so that φ2 < 1 everywhere.

In the absence of the Dominant Energy Condition, it isn't clear that any version of the Positive Mass Theorem holds but there is some reason to expect a positive global energy. To see this, integrate the energy density over an asymptotically-flat maximal space-like hypersurface Σ (assuming for the moment that one exists) with normal N a . Note from (36) and (37) that

T ab = 1 4π 1 1 -φ2 2 φA(A φB )B -φ ∇ A(A φB )B . (38)
Then a measure of total energy at Σ is

E : = T ab N a N b dΣ = 1 4π 1 1 -φ2 (N a φa ) 2 - 1 2 gab φa φb -φ N a N b ∇ a φb dΣ. ( 39
)
Now,

N a N b ∇ a φb = (h ab + gab ) ∇ a φb = h ij ∇ i φj = h ij D i φj + (N a φa )K,
where D i is the derivative operator associated with the three-dimensional metric h ij induced on Σ, K is the trace of the extrinsic curvature which vanishes for a maximal surface, and we have used φ = 0. Note also

gab φa φb = (N a φa ) 2 -h ij φi φj ,
and integrate by parts in (39) to find

E = 1 4π dΣ (1 -φ2 ) -1 3 2 (N a φa ) 2 + 1 2 3 + φ2 1 -φ2 h ij φi φj ,
which is manifestly non-negative. Thus on a maximal surface the global energy is positive without local positivity. Positive energy also holds for hyperplanes in Minkowski space with T ab as in (36) and we shall see something similar below, namely that, while the Bondi mass-loss is not necessarily positive at any particular cut, nonetheless the mass-loss integrated over a period in a periodic space-time is nonnegative.

The Ricci spinor written in terms of unphysical quantities reads

1 -Ω 2 φ 2 ΦABA B = 2 Ω 2 ϕ (A(A ϕ B )B) -Ω 2 φ ∇ (A(A ϕ B )B) -Ω φ 2 ∇ (A(A s B )B) . (40)
Let us define the "rescaled Ricci spinor"

φ ABA B = Ω -2 ΦABA B , (41)
which should be distinguished from the unphysical Ricci spinor. Substituting the rule for conformal transformation of the Ricci spinor [START_REF] Hübner | General relativistic scalar-field models and asymptotic flatness[END_REF] into (40) we arrive at the following simple expression for the rescaled Ricci spinor:

φ ABA B = 2 ϕ (A(A ϕ B )B) -φ ∇ (A(A ϕ B )B) + φ 2 Φ ABA B . ( 42
)
This spinor is regular on I -. Notice that we do not write the tilde over φ ABA B (as we wrote over ΦABA B in ( 7)), since we expect that the physical Ricci spinor has been already expressed in terms of the unphysical quantities and the following relations become simpler. The components of φ ABA B with respect to the spin basis are

φ 00 = 2ϕ 2 0 -φ [Dϕ 0 -(ε + ε)ϕ 0 + κϕ 1 + κϕ1] + φ 2 Φ 00 , φ 01 = 2ϕ 0 ϕ 1 - 1 2 φ [Dϕ 1 + δϕ 0 -(ᾱ + β + π)ϕ 0 + κϕ 2 + (ρ -ε + ε)ϕ 1 + σϕ1] + φ 2 Φ 01 , φ 02 = 2ϕ 2 1 -φ δϕ 1 -λϕ 0 + σϕ 2 + (ᾱ -β)ϕ 1 + φ 2 Φ 02 , φ 12 = 2ϕ 1 ϕ 2 - 1 2 φ ∆ϕ 1 + δϕ 2 -νϕ 0 + (β + τ + ᾱ)ϕ 2 + (γ -γ -µ)ϕ 1 -λϕ1 + φ 2 Φ 12 , φ 22 = 2ϕ 2 2 -φ [∆ϕ 2 + (γ + γ)ϕ 2 -νϕ 1 -νϕ1] + φ 2 Φ 22 , (43)
φ 11 = ϕ 0 ϕ 2 + ϕ 1 ϕ1 - 1 4 φ Dϕ 2 + ∆ϕ 0 + δϕ1 + δϕ 1 -(γ + γ + µ + μ)ϕ 0 - 1 4 φ (ρ + ρ + ε + ε)ϕ 2 + (τ -α + β -π)ϕ 1 + (τ -ᾱ + β -π)ϕ1 + φ 2 Φ 11 .

The conformal Einstein-conformal-scalar equations

Now, as in subsection 2.2, we obtain a system of conformal Einstein equations, regular in the unphysical, rescaled space-time and equivalent to the Einstein-conformal-scalar equations.

In order to derive the conformal Bianchi identities for the conformal-scalar field we return to the general physical Bianchi identities [START_REF] Hugget | An introduction to twistor theory[END_REF]. Using the rescaled Ricci spinor instead of ΦABA B the Bianchi identities become

∇ D A ψ ABCD = 3 s B (C φ AB)A B + Ω ∇ B (C φ AB)A B . (44)
Projections of these equations on the spin basis can be found in Appendix B, eqs. (B59)-(B66).

Next we turn to the contracted Bianchi identities in the physical spacetime

∇BB ΦABA B = -3 ∇AA Λ, (45)
where Λ = 0 by (37). Following the rules for conformal transformation of the covariant derivative we find that the left hand side transforms like

∇BB ΦABA B = Ω 2 ∇ BB ΦABA B -2 Ω s BB ΦABA B ( 46
)
or, using (41),

∇BB ΦABA B = Ω 4 ∇ BB φ ABA B , (47)
and thus the contracted Bianchi identities have a simple form, just as in the physical spacetime:

∇ BB φ ABA B = 0. ( 48
)
Projections of these equations on the spin basis can be obtained from the Bianchi identities (A8a)-(A8c) by deleting terms containing Λ and replacing Φ mn → φ mn .

Bondi mass

One of the necessary ingredients in this work is to find restrictions which the assumption of periodicity imposes on the Bondi mass. As long as the Bondi mass M B (u) on I + is a non-increasing function of retarded time u, it can be periodic only if it is constant. In [START_REF] Bičák | On asymptotically flat solutions of Einstein's equations periodic in time I. Vacuum and electrovacuum solutions[END_REF] we used the well-known formula for the Bondi mass of an electro-vacuum spacetime §,

M B (u) = - 1 2 √ π dS Ψ 0 2 + σ 0 σ0 , (49)
when its time decrease is given by the "mass-loss" formula

ṀB (u) = - 1 2 √ π dS σ0 σ0 + φ 0 2 φ0 2 . ( 50
)
This expression is manifestly non-positive. To achieve periodicity of the Bondi mass we thus had to set σ0 = 0 and φ 0 2 = 0. The loss of Bondi mass due to the gravitational radiation is described by the news function -σ0 and the electromagnetic contribution by the quantity φ 0 2 . Periodicity thus requires the absence of both gravitational and electromagnetic radiation.

In order to repeat this reasoning in the case of spacetimes with scalar fields, we need the appropriate formula for Bondi mass-loss. The gravitational contribution will again be expressed by the news function and there will be a contribution from the matter. The energy flux due to the matter is described by the energy-momentum tensor T ab (omitting tildes for clarity, in this subsection only). If we write, using the NP formalism,

T ab = A l a l b + B n (a l b) + C n a n b + • • • , (51)
then the component A = T ab n a n b is the energy radiated out of I + (recall that n a is tangential to I + , l a points into the space-time towards I -). In terms of the Ricci spinor NP component we get

T ab n a n b ∝ Φ 22 . ( 52
)
For the complex scalar field φ, Φ 22 ∝ φ φ, where dot means the derivative with respect to u. § In fact, in [START_REF] Bičák | On asymptotically flat solutions of Einstein's equations periodic in time I. Vacuum and electrovacuum solutions[END_REF] we constructed the proof -and the same will be done here -at I -where the Bondi mass is non-decreasing but it is straightforward to get one from the other. Since it is more common to work at I + , in this section we discuss the Bondi mass there.

For the scalar field we thus expect

ṀB (u) = - 1 2 √ π σ0 σ0 + k φ0 φ0 dS, ( 53
)
where k is a positive constant factor and φ 0 is the radiative part of the scalar field, i.e. φ = φ 0 r -1 + O(r -2 ). We shall now calculate the Bondi mass for the scalar field which will imply the exact formula for the mass-loss.

Massless-KG field

To compute the Bondi mass we use a method based on the asymptotic twistor equation as described in [START_REF] Stewart | Advanced general relativity[END_REF]. More detail can be found in [START_REF] Hugget | An introduction to twistor theory[END_REF] or [START_REF] Penrose | Spinors and space-time vol II Cambridge Monographs on Mathematical Physics[END_REF]. In this approach we have to find the asymptotic solution of the Einstein-massless-KG equations in the neighbourhood of I + (in the physical spacetime). We give enough of this for our present purposes in Appendix C. The Bondi mass is then given by the coefficient µ (2) , which is the O(Ω 2 ) term in the expansion of the spin coefficient µ. This term is given by (C71) and reads

µ (2) = -ððσ (0) -Ψ (0) 2 -2 Λ (0) -σ (0) σ(0) .
Since the term ððσ (0) vanishes on integration, we find the Bondi mass to be (with the normalization used in I)

M B (u) = - 1 2 √ π dS Ψ (0) 2 + 2 Λ (0) + σ (0) σ(0) . (54)
Using the expansion of Λ given by (C72) leads to the final expression

M B (u) = - 1 2 √ π dS Ψ (0) 2 + 1 3 ∂ u φ (0) φ(0) + σ (0) σ(0) . ( 55
)
To find the time derivative of the Bondi mass we use the leading term in the Bianchi identity (A7c):

Ψ(0) 2 + 2 Λ(0) = ðΨ (0) 3 + Φ (0) 22 + σ (0) Ψ (0) 4 . (56)
The term ðΨ

(0)
3 vanishes on integration. By (C72) we have Ψ (0) 4 = -σ(0) . The leading term of Φ 22 is found from (8) to be Φ (0) 22 = 2 φ(0) φ(0) , and the mass-loss formula thus acquires the form

Ṁ B (u) = - 1 2 √ π dS σ(0) σ(0) + 2 φ(0) φ(0) . ( 57
)
This expression is manifestly non-positive. If we demand the spacetime to be periodic, the Bondi mass must be constant, i.e. σ(0) = φ(0) = 0.

Conformal-scalar field

The same calculation can be repeated with minor changes in the case of the conformalscalar field. Now we obtain following expressions for the Bondi mass and its "loss":

M B (u) = - 1 2 √ π dS Ψ (0) 2 + σ (0) σ(0) , Ṁ B (u) = - 1 2 √ π dS σ(0) σ(0) + 2 φ(0) 2 -φ (0) φ(0) . (58)
Now the formula for the rate of change of the Bondi mass is not manifestly non-positive, so it can apparently increase as well as decrease. This seems to be a consequence of the fact, that the energy-momentum tensor (36) does not obey the energy condition T ab l a n b ≥ 0 for arbitrary future null vectors l a and n a .

However, if the Bondi mass is supposed to be periodic, its overall change ∆M B during one period T is non-positive. Indeed,

∆M B = - 1 2 √ π u+T u du dS σ(0) σ(0) + 3 φ(0) φ(0) + 1 2 √ π dS φ (0) φ(0) u+T u , (59)
where we have integrated the term containing φ(0) by parts. The second term in (59) vanishes because of periodicity and we arrive at a manifestly non-positive expression for the loss of mass during one period. Such an expression can be periodic only if it is constant, so we again obtain the condition σ(0) = φ(0) = 0.

Periodic solutions are necessarily stationary: proof of the theorems

The massless-KG field

In this section we prove that all periodic asympotically-flat Einstein-massless-KG spacetimes, analytic near I -in the coordinates we shall introduce, are necessarily stationary. First we set up a coordinate system, choose the null tetrad and fix the conformal gauge as in paper I, and the justification for the assertions below is given there. The coordinates are denoted x µ = (v, r, θ, φ). Here v is the affine parameter along the generators of I -and has the meaning of the advanced time. The coordinate r is an affine parameter along the null geodesics ingoing from I -with the property Ω = r + O(r 2 ), and (θ, φ) are standard spherical coordinates on the unit sphere. The NP operators D, ∆ and δ representing derivatives in the directions of the vectors l, n and m (constituting the null tetrad) can be expressed in coordinates x µ in the following way:

D = ∂ v -H ∂ r + C I ∂ I , ∆ = ∂ r , δ = P I ∂ I . (60)
The metric functions H, C I and P I are governed by the frame equations

∆ H = -(ε + ε), (61) δ H = -κ, (62)
∆ C I = -2 π P I -2 π P I , (63)
δ P I -δ P I = (α -β) P I -(ᾱ -β) P I , (64)
∆ P I = -(µ -γ + γ) P I -λ P I , (65) δ C I -D P I = -(ρ + ε -ε) P I -σ P I , (66)
which can be understood as determining the non-zero spin coefficients. We choose

P 2 = 1 √ 2 , P 3 = i √ 2 sin θ on I -. (67)
The metric functions H and C I vanish on I -by construction, so the operator D reduces to ∂ v there and we have:

H = C I = 0, DP I = 0 on I -. (68)
As a consequence of the choice of the coordinates and the tetrad we have

ρ -ρ = µ -μ = ν = π -α -β = τ -β -ᾱ = 0, everywhere, α = -β = - 1 2 √ 2 cot θ, κ = 0, on I -. (69)
Exploiting the tetrad gauge freedom corresponding to the rotation of (m, m) we achieve γ = 0 everywhere,

ε = 0 on I -. (70)
Using the conformal gauge freedom we set µ = 0 everywhere.

Recall from (24) that the physical scalar curvature is O(Ω 3 ). Eqs. (B14)-(B23) for the conformal factor then reveal that on I -

F = ρ = σ = π = τ = 0, ∆S 0 = ∆S 1 = ∆S 2 = DS 2 = 0. ( 72
)
Eqs. (B25) and (B26) for derivatives of F imply

Φ 00 = Φ 01 = 0 on I -. (73)
We saw in the previous section that the periodicity of the solution requires the constancy of the Bondi mass. This is expressed by the relations

ψ 0 = ∆Ψ 0 = 0, ϕ 0 = Dφ = 0, on I -. (74)
These equations also imply Dϕ 1 = Dϕ1 = 0 on I -, as can be seen from ( B10) and (A1). First we prove that, assuming periodicity, all NP quantities are time-independent on I -, i.e. independent of v. This follows immediately from the choices made above for all spin coefficients except for λ. The Ricci identity (A5g) and Bianchi identity (A7a) show

Dλ = Φ 20 , DΦ 02 = 0, on I -, (75)
and therefore

D 2 λ = 0. ( 76
)
By the same argument as in [START_REF] Gibbons | Absence of asymptotically flat solutions of Einstein's equations which are periodic and empty near infinity in Classical general relativity[END_REF] and [START_REF] Bičák | On asymptotically flat solutions of Einstein's equations periodic in time I. Vacuum and electrovacuum solutions[END_REF], we conclude

Dλ = 0 on I -, (77)
since the equation ( 76) has a polynomial solution in v, but λ can be periodic only if it is constant. Equation ( 75) then gives

Φ 20 = 0 on I -. (78)
The conformal Bianchi identities (B32), (B34), (B36) and (B38) on I -simplify to

Dψ 1 = 0, Dψ 2 -δψ 1 = -2 α ψ 1 , Dψ 3 -δψ 2 = -2 λ ψ 1 + 1 3 φD φ1 + φDϕ1 , Dψ 4 -δψ 3 = -3 λ ψ 2 + 2 α ψ 3 + φϕ1 + φ φ1 -4 ϕ1 φ1 + φ δϕ1 + φ δ φ1. ( 79
)
Applying D to these equations, we immediately see that

D 2 ψ n = 0 on I - (80)
for all n. By periodicity

Dψ n = 0 on I -, (81)
so all components of the Weyl spinor are v-independent on I -. Because ψ n = Ω -1 Ψ n , we have

D∆Ψ n = 0 on I -. (82)
Finally, we investigate the behaviour of the remaining components of the Ricci tensor, i.e., Φ 11 , Φ 12 , Φ 22 and Λ. The Ricci identity (A5h) immediately shows Λ = 0 on I -.

(83)

Since µ is identically zero not only on I -, but also in its neighbourhood, the Ricci identity (A5k) on I -reduces to

Φ 22 = -λ λ on I -, (84)
and therefore

DΦ 22 = 0 on I -. ( 85
)
Applying D on the Ricci identity (A5r) leads to

DΦ 21 = 0 on I -. (86)
The spin coefficients α and β on I -are given by (69). Inserting these into the Ricci identity (A5q) we find

Φ 11 = 1 2 on I -, (87)
so Φ 11 is obviously v-independent on I -. We have already shown that ϕ 1 and ϕ1 are v-independent on I -. Equation (B11) implies

Dϕ 2 -δϕ1 = (β -ᾱ)ϕ1 on I -. (88)
Applying D and assuming periodicity of the scalar field we conclude

Dϕ 2 = 0 on I -. (89)
Projections (B12) and (B13) of the wave equation and commutator (A3) applied to φ reveals, after differentiating with D, that D∆Q = 0 on I -, with

Q ∈ {ϕ 0 , ϕ1, ϕ 1 }.
To show the same for ∆ϕ 2 we apply D∆ to (B11) and obtain

D 2 ∆ϕ 2 + 2φD∆Λ = ϕ 1 D∆π -ϕ 2 D∆(ε + ε -ρ). (90)
From Ricci identities (A5f), (A5i), (A5l)-(A5o) we find that D∆Q = 0 on I -for Q ∈ {ρ, π, α, σ, ε, β}.

Applying D∆ to (A5h) shows (92)

D∆Λ = 0 on I -, and
Now we set up an induction, with inductive hypothesis:

Suppose that D∆ j Q = 0 on I -for 0 ≤ j ≤ k with Q ∈ H, C I , P I , ε, ρ, σ, λ, π, τ, κ, α, β, F, ψ n , Φ mn , Λ ,
and for 0

≤ j ≤ k + 1 with Q ∈ {S m , ϕ m }.
This holds for k = 0 by Lemma 5.1, so we need to deduce it for j = k + 1 from its validity for j ≤ k. Here we closely follow the procedure we used in [START_REF] Bičák | On asymptotically flat solutions of Einstein's equations periodic in time I. Vacuum and electrovacuum solutions[END_REF]. Applying D∆ k on (61) we find

D∆ k+1 H = -D∆ k (ε + ε),
where the r.h.s. vanishes on I -by the inductive hypothesis. By a similar argument we can deduce D∆ k+1 Q = 0 on I -

• for H, C I and P I from ( 61), ( 63) and ( 65);

• for κ, ε, π, τ, λ, β, σ, ρ and α from (A5c), (A5f), (A5i)-(A5o);

• for F from (B27), taking Φmn from ( 8) and Λ from ( 9) and ( 24);

• for Φ 00 , Φ 20 , Φ 01 and Φ 21 from (A6b), (A6d), ( A7b) and (A7d);

• for Λ, Φ 22 and Φ 11 from (A5h), ( A5k) and (A5q);

• for ψ 0 , ψ 1 , ψ 2 and ψ 3 from (B33), (B35), (B37) and (B39). Now, all quantities except for ψ 4 are proved to satisfy D∆ k+1 Q = 0 on I -. Applying D∆ k+1 on (B15), (B18), (B22), (B12), (B13) and (A3) shows D∆ k+2 Q = 0 on I -for Q ∈ {S 0 , S 1 , S 2 , ϕ 0 , ϕ1, ϕ 1 }.

Ricci identities (A5f), (A5i), (A5l)-( A5o) and (A5h) in this order imply

D∆ k+2 Q = 0 on I -for Q ∈ {ε, π, β, σ, ρ, α, Λ}.
Applying D∆ k+2 on (B11) implies D 2 ∆ k+2 ϕ 2 = 0 on I -and using periodicity we obtain D∆ k+2 ϕ 2 = 0 on I -.

Finally, acting by D∆ k+1 on (B38) we find D 2 ∆ k+1 ψ 4 = 0 on I -, and therefore, by periodicity,

D∆ k+1 ψ 4 = 0 on I -.
This completes the induction.

We have thus proved that all variables are v-independent on I -and, assuming analyticity in r, in a finite neighbourhood. Since our set of variables includes also the functions H, C I and P I constituting the components of metric tensor, we can conclude that K = ∂ v is a Killing vector of the unphysical metric. However, the conformal factor is v-independent as well, so the Lie derivative of the physical metric is

L K gab = -2 Ω -3 g ab L K Ω = 0,
i.e. K is also a Killing vector of the physical metric.

The norm of K is given by the component g vv in the coordinates x µ . The full form of the metric tensor g µν can be found in I, eq. ( 34). The norm of K is then

g(K, K) = g vv = 2H -2ω ω,
where ω = -C I R I and R I are O(1) functions (see (33) in I). Frame equations ( 61) and ( 63) imply H, C I = O(r 2 ), the Ricci identity (A5f) with (87) shows ε = -1/2 on I -. From these relations we find the norm of the Killing vector to be

g(K, K) = 2 r 2 + O(r 3 ).
We can see that K is null on I -and time-like in its neighbourhood. Our results are summarized in the following theorem. Theorem 5.2. A weakly-asymptotically-simple time-periodic solution of the Einsteinmassless-KG field equations which is analytic in a neighbourhood of I -in the coordinates introduced above necessarily has a Killing vector which is time-like in the interior and extends to a translation on I -.

The conformal-scalar field

The proof for the conformal-scalar field is essentially the same as in the case of massless-KG field, the only difference lying in the Bianchi identities. These are not so complicated as in the previous case and we present them in their full form in Appendix B, eqs. (B59)-(B66). Our results are summarized in the following theorem.

Theorem 5.3. A weakly-asymptotically-simple time-periodic solution of the Einsteinconformal-scalar equations which is analytic in a neighbourhood of I -in the coordinates introduced above necessarily has a Killing vector which is time-like in the interior and extends to a translation on I -.

We briefly outline the main steps of the proof. We use the same coordinate system and tetrad, defined by (60) and ( 61)-(66), so all consequences of the choice of the gauge remain unchanged. Projections of the wave equation (B10)-(B13) and equations for the conformal factor, (B14)-(B27) differ only in the presence of the physical scalar curvature Λ, which in this case is zero. The form of the Ricci identities does not depend on the type of the matter field. Summa summarum, eqs. (60)-(78) hold without change. Now it is straightforward to see from the conformal Bianchi identities (B59)-(B62) that the Weyl scalars ψ n are v-independent on I -. Next we return to eqs. (83)-(91), which are again valid. So, lemma (5.1) holds.

To finalize the proof we need to repeat the induction. In the previous case, in the inductive hypothesis we assumed that each quantity Q satisfies D∆ j Q = 0 on I -, where 0 ≤ j ≤ k, and in addition, D∆ k+1 Q = 0 on I -for Q ∈ {S a , ϕ a }. This was neccesary, since the Bianchi identities contained derivatives of these fields. In the inductive step we were able to prove D∆ k+1 Q = 0 on I -for ψ n 's and for all other quantities. Moreover, we proved

D∆ k+2 Q = 0 on I -for Q ∈ {S a , ϕ a }.
In this case, the Bianchi identities actually contain the second derivatives of the fields S a and ϕ a , i.e. the third derivatives of Ω and φ. This is not a problem, however, as all third derivatives are multiplied by Ω. Therefore terms with problematic D∆ k+2derivatives vanish on I -, and the induction can be repeated without change.

Inheritance

In the previous section we proved that if both gravitational and scalar fields are periodic near infinity, the spacetime is stationary there and the scalar field does not depend on time. However, there are examples known in which the gravitational field and its matter source do not share the same symmetries (see Paper I for a longer discussion). The question therefore is, whether a stationary gravitational field can be produced by a time-dependent source. In I we showed that this is not the case with an electromagnetic field -once the spacetime is stationary, the electromagnetic field must be too. Let us briefly recall the idea of the proof.

In the electromagnetic case the components of the physical Ricci spinor have the simple form Φmn = φm φn . It is clear, that if the Ricci spinor is to be stationary, the electromagnetic field can depend on time only through the phase of φm , i.e. φm = φm e iχ , where χ = χ(v, r, x I ), but the modulus φ is time-independent. Now, by the Bondi mass-loss formula φ 0 = 0 on I -. Using Maxwell's equations we deduced that φ m are time-independent on I -, and by induction also in its neighbourhood. Therefore, if an asymptotically flat electro-vacuum spacetime is stationary, the electromagnetic field has to inherit stationarity.

The situation is more complicated in the case of the massless-KG field, for now it is not obvious what kind of time-dependence of the scalar field φ is compatible with the stationarity of the spacetime. We first find this time-dependence and then the result follows, using the Bondi mass-loss formula and induction. Theorem 6.1. In a stationary, analytic, weakly-asymptotically-simple solution of the Einstein-massless-KG equations with stationarity Killing vector K = ∂/∂v, the physical massless-KG field must take the form φ = e iωv φ0 , where ∂ v φ0 = 0. If the metric is analytic in a neighbourhood of I -in the coordinates introduced above then φ is in fact time-independent.

For the first part, we use the coordinate system introduced above and assume the stationarity of the spacetime. Hence, K = ∂ v is a Killing vector of the metric and the Lie derivative L K reduces to simple partial derivative with respect to v, which will be also denoted by a dot. Since Λ is stationary, Einstein's equations [START_REF] Callan | A new improved energy-momentum tensor Ann[END_REF] imply

∂ v ( φc φc ) = 0.
The Lie derivative of the energy-momentum tensor (2) is then 4πL K T ab = ψa φb + φa ψb + φb ψa + ψb φa , where ψ = φ and ψa = ∇ a ψ. Let us decompose the fields φ and ψ into real and imaginary parts,

φ = X + iY, ψ = U + iW, (93)
with X, Y, U and W being real functions. In this notation we have

2π Ṫ ab = X a U b + Y a W b + X b U a + Y b W a = 0, ( 94
)
where X a = ∇ a X etc. We first consider the case when the gradient fields X a and Y a are proportional in some finite region so that, by analyticity, they are proportional everywhere. Thus X and Y are functionally dependent. If either were constant then that constant would be zero, since φ = 0 at infinity. Thus we may suppose that Y is a function of X and then φa = (1 + iY )X a , ˜ φ = (1 + iY ) ˜ X + iY gab X a X b , where the prime indicates derivative w.r.t X.

If Y = 0 then Y = aX + b for constant a, b, but once again b must vanish by asymptotic flatness so φ = (1 + ia)X and, after rescaling φ by a constant, we may assume Y = 0 whence also W = 0. Now (94) becomes X (a U b) = 0 from which necessarily U a = 0 and φ = constant, but then asymptotic flatness forces φ = 0.

If Y = 0 then ˜ X = 0 = gab X a X b . Now 4π Tab = 2(1 + Y 2 )X a X b , (95)
and one may impose on this expression the vanishing of L K Tab . Introduce

h := L K X,
then this is

L K (4π Tab ) = 4Y Y hX a X b + 2(1 + Y 2 )(h a X b + X a h b ) = 0.
For nonzero h, this is only possible if h a is proportional to X a , so that h is a function of X and this condition becomes

4Y Y h + 4h (1 + Y 2 ) = 0,
which can be integrated to give

h 2 (1 + Y 2 ) = C, a constant. Now from (95) 4π Tab K a K b = 2h 2 (1 + Y 2 ) = 2C,
but this expression must vanish at infinity for asymptotic flatness, so C = 0 so h = 0, and the scalar field inherits the symmetry of the metric. When X a and Y a are not proportional (except possibly on a set of measure zero), return to (94) and choose a vector field Z a with Z a X a = 0 but Z a Y a = 0. Contracting (94) with such Z a we find that W b is a linear combination of X b and Y b , from which we deduce

W = f (X, Y ).
Similarly, contracting (94) with a different Z a satisfying Z a X a = 0 and Z a Y a = 0 we arrive at

U = g(X, Y ).
Inserting this back into (94) we obtain

g X X a X b + 2(f X + g Y ) X (a Y b) + f Y Y a Y b = 0,
where the subscript on f or g indicates the corresponding partial derivative. Since X a and Y a are assumed to be linearly independent, all terms in the last equation must vanish separately. We thus have three differential equations for f and g. The general solution is

W = f = ωX + β, U = g = -ωY + γ,
with constant ω, β and γ. Regarding (93), for the field ψ we have ψ = iω(X + iY ) + (γ + iβ).

Since ψ = φ, we can solve the last equation to find φ = φ0 e iωv + const.

(96)

However, the constant second term must be set to zero, as the field itself must vanish at infinity. We have shown that the most general non-stationary scalar field compatible with stationarity of the spacetime is of the form φ(v, r, x I ) = φ0 (r, x I ) e iωv .

(

) 97
(It is not difficult to show the same result is obtained with a potential term V ( φ, φ) added as in subsection 2.3, with the extra condition that necessarily also V must have the form V = F ( φφ ).) We shall next show that nonzero ω leads to the vanishing of φ. The stationarity of the spacetime implies the constancy of the Bondi mass, so ϕ 0 is again zero on I -. Now, if ω = 0, the field φ is v-independent everywhere and is therefore stationary. On the other hand, if ω = 0, then expanding φ 0 in the variable r and using (96), we find i ω φ = 0 for 0 ≤ j ≤ k. Acting with ∆ k on (B11) leads to (recall that ρ and ε vanish on I -) i ω ∆ (k+1) φ = 0 on I -. and since the constant ω is assumed to be non-zero, it follows immediately that φ (k+1) = 0. Hence, by induction and analyticity, the field φ vanishes in a neighbourhood of I -. This completes the proof of Theorem 6.1 and of inheritance for the massless-KG field.

Let us now turn to the conformal-scalar field. Again, we demand the stationarity of the metric, and therefore also the stationarity of the energy-momentum tensor, but not the stationarity of the scalar field. Unfortunately, the complicated form of the energy-momentum tensor (36) does not allow us to find the most general timedependence of φ compatible with the stationarity of the metric, and thus we cannot proceed as before. In addition, we cannot deduce any concrete condition on I -, for we do not have a negative semi-definite mass-loss formula. Because of these complications we will only show that scalar field inherits the symmetry in a simpler case. Let us consider a complex conformal-scalar field with energy-momentum tensor

T C ab = 1 4π 2 φ(a φb) - 1 2 gab φc φc - 1 2 φ ∇a φb - 1 2 φ ∇a φb + φ φ Φab . (98)
The Bondi mass-loss formula (59) now takes the form

ṀB = - 1 2 √ π dS σ(0) σ(0) + 2 φ(0) φ(0) - 1 2 φ (0) φ(0) - 1 2 φ(0) φ(0) . (99)
Although we cannot exclude the existence of some more general time-dependence of φ, for the field of the form (97) the energy-momentum tensor (98) is stationary. In this case we can integrate by parts in (99) to find as in ( 59)

∆M B = - 1 2 √ π v+2π/ω v dv dS σ(0) σ(0) + 3 φ(0) φ(0) . (100)
Since we assume the stationarity of the spacetime, σ(0) = 0. The constancy of the Bondi mass then implies φ(0) = 0, i.e.

Dφ ≡ ϕ 0 = 0 on I -. (101)
Now we can proceed as in the case of massless scalar field. By (101) we have ω = 0 or φ (0) = 0. If ω = 0, the field is time-independent everywhere. If φ (0) = 0, or equivalently, φ = 0 on I -, we prove by induction that φ = 0 everywhere.

A special spherically symmetric solution -after choosing a suitable radial coordinate-reads

ds 2 = -1 - m r 2 dt 2 + 1 - m r -2 dr 2 + r2 dθ 2 + sin 2 θdφ 2 , φ = 3 4π m r -m . ( D75
)
The geometry is identical to that of an extreme Reissner-Nordström black hole, so it can be analytically continuated to r < m. However, φ and (∇ a φ)(∇ a φ) diverge at the "horizon" r = m. Nevertheless, this infinite scalar field does not imply an infinite barrier for test scalar charges and the solutions are often regarded as "black holes with scalar charge" [START_REF] Bekenstein | Black holes with scalar charge[END_REF]. In any case, both geometry and scalar field are analytic at r → ∞ satisfying our requirements.

Bekenstein's work inspired a number of more recent papers: for example, Einstein-conformal-scalar-field solutions were analyzed in arbitrary dimensions [START_REF] Xanthopoulos | Einstein gravity coupled to a massless conformal scalar field in arbitrary space-time dimensions[END_REF], self-interacting scalar fields were considered [START_REF] Fonarev | Exact Einstein scalar field solutions for formation of black holes in a cosmological setting[END_REF], and transversable wormholes from massless conformally coupled and other scalar fields non-minimally coupled to gravity were constructed [START_REF] Barceló | Scalar field, energy conditions and transversable wormholes[END_REF].

D2. FLRW metric

In this section we present simple homogenous isotropic solutions of the Einsteinconformal-scalar equations. We shall take the metric in the standard form ds 2 = dt 2 -a 2 (t) dr 2 1 -k r 2 + r 2 dθ 2 + sin 2 θ dφ 2 , (D76)

where k ∈ {-1, 0, 1}. The energy-momentum tensor is given by (38). Since this tensor is traceless, the scalar curvature must vanish: The conformally-invariant scalar field in the physical spacetime satisfies d'Alembert's equation φ = 0. Because the space-time is assumed to be homogeneous and isotropic, we suppose that the field does not depend on the spatial coordinates. D'Alembert's equation then reduces to

φ(t) = φ + 3 ȧ φ a = 0. (D78)
For given a(t), the solution can be found explicitly:

φ(t) = C 3 + C 4 dt a 3 (t)
.

In the following short discussion we consider three cases for the three possible values of k. We solve the wave equation, find the components of the energy-momentum tensor and see that these components do not exhibit the singularity formally present in (38). a) k = 0. Imposing the initial condition a(0) = 0 leads to

a(t) = √ 2 C t, ( D79
)
where C is an arbitrary positive constant. The general solution of (D76) is

φ(t) = α + β √ t .
Einstein's equations then imply Obviously, T ab is regular unless t = 0. This is the expected initial curvature singularity (for example, the Kretschmann invariant R abcd R abcd = 3/(2t 4 ) diverges for t = 0). As noted, above the term (1 -φ 2 ) may or may not vanish depending on the constants of integration but even when it does there is no singularity in T ab despite the form of (38). b) k = -1. Again, we demand a(0) = 0, so a(t) is of the form a(t) = t (t + C).

α = ±
The general solution of the wave equation is where χ is an arbitrary constant. Now (1 -φ 2 ) will vanish at some t > 0 for χ < 0 but not for χ > 0. The components of the energy-momentum tensor are

φ(t) = α + β 2 t + C 2 a(t) , (
T ab = C 2 32 π a 2 (t) diag 3 a 2 (t) , 1 1 + r 2 , r 2 , r 2 sin 2 θ , (D83)
and they are again singular only for t = 0, and not at φ 2 = 1. c) k = 1. Now we impose conditions a(0) = 0 and ȧ(T ) = 0, so that

a(t) = t (2 T -t).
The solution of the wave equation is

K 00 = H xx + H yy + 2 1 -f 2 f f uu -2f 2 u . (D93)
Solving the equation K 00 = 0 with respect to H we arrive at H = C(u, x + iy) + C(u, x -iy) +

x 2 + y 2 2 2 f 2 u -f f uu 1 -f 2 . (D94)
Here, C and f are arbitrary real functions. As we can now see, the metric function H is singular if ever f ≡ φ = ±1 and this, if it occurs, will be a curvature singularity.

In [START_REF] Klimčík | Interacting Einstein-conformal scalar waves[END_REF], a large class of solutions of the Einstein-conformal-scalar equations for colliding plane waves was found by employing the Bekenstein transformation [START_REF] Bekenstein | Exact solutions of Einstein-conformal scalar equations[END_REF].

Lemma 5 . 1 .

 51 thus (90) implies D 2 ∆ϕ 2 = 0 on I -, so by periodicityD∆ϕ 2 = 0 on I -.(91)Thus, we have proved the lemma The following quantities vanish on I -: H, C I , ρ, σ, π, τ, κ, ε, S 0 , S 1 , F, ψ 0 , Φ 00 , Φ 01 , Φ 02 , Λ, ϕ 0 , DP I , Dα, Dβ, Dλ, DS 2 , ∆S 0 , ∆S 1 , ∆S 2 , Dϕ 1 , Dϕ1, Dϕ 2 , D∆ϕ 0 , D∆ϕ 1 , D∆ϕ1, D∆ϕ 2 , Dψ 1 , Dψ 2 , Dψ 3 , Dψ 4 , DΦ 11 , DΦ 12 , DΦ 22 .

R = 6 a 2 k

 2 + ȧ2 + ä a = 0.Solutions to this equation area(t) = c 1 -k (t + c 2 ) 2 for k = 0, a(t) = c 1 √ 2 t + c 2 for k = 0.(D77)

  D81) and Einstein's equations give α = cosh χ, β = sinh χ, (D82)

  1, and β is nonzero but arbitrary. Note that φ 2 = 1 at t = β 2 /4 if αβ/|β| = -1, but φ 2 is never one if αβ/|β| = +1. The components of the energy momentum-tensor are

	T ab =	1 16 π	diag	3 2 t 2 ,	C t	,	C r 2 t	,	C r 2 sin 2 θ t	.	(D80)

Acknowledgments P. T. gratefully acknowledges hospitality and financial support from the Mittag-Leffler Institute, Djursholm, Sweden and the Charles University, Prague, and useful discussions with Gary Gibbons and John Stewart. M.S. is grateful to the Mathematical Institute of University of Oxford, and both M.S. and J.B. also thank to Albert Einstein Institute, Golm, for hospitality and support. J.B. acknowledges the discussions with Gary Gibbons and the partial support from the Grant GACR 202/09/00772 of the Czech Republic, of Grant No LC06014 and MSM0021620860 of the Ministry of Education. The work of M. S. was supported by the Grant GAUK no. 22708 of the Charles University and by the Grant GACR-205/09/H033, Czech Republic.

A2. Ricci identities

Dρ -δκ = ρ 2 + ( + ¯ ) ρ -κ 3α + β -π -τ κ + σσ + Φ 00 , (A5a)

∆λ

Appendix B: The conformal field equations

B1. The conformally-rescaled wave equation

The wave equation φ = 0 in the physical spacetime is not conformally invariant. If φ is the solution of the physical wave equation, then the unphysical scalar field φ satisfies equation (32),

The projections of this equation are the following:

Appropriate equations for the conformal-scalar field can be obtained from (B10)-(B13) by setting Λ = 0.

B2. Equations for the conformal factor

The projections of equation ( 23),

are the following :

has the following projections:

There is a misprint in Paper I: in eq. (B.2a) the sign at (ε + ε)S 0 should be "minus" as in (B14). This does not affect the results of Paper I.

(B27)

B3. Conformal Bianchi identities for the scalar field

Let us write the conformal Bianchi identities [START_REF] Xanthopoulos | Einstein gravity coupled to a massless conformal scalar field in arbitrary space-time dimensions[END_REF] for the scalar field in the form

where (for notation see ( 14))

Since both sides are totally symmetric in ABC, we denote their contractions with spinors o and ι by the number of ι's in the first index and number of ῑ's in the second, e.g.

The components of X ABCA read as follows:

We do not present the projections of Y ABCA in full detail since they are too long. However, the structure of all the terms entering this spinor allows one to reconstruct its components from a knowledge of the components of spinors (sϕ φ) and (∇sϕ), if appropriate interchanges of s, ϕ and φ are made. For expressions of the type (sϕ φ) we get

The expressions of type (∇sϕ) (and their projections) can be slightly simplified by observing that both s a and ϕ a are gradients of scalar functions, namely Ω and φ. Since the commutator ∇ A (A ∇ A B) annihilates any scalar quantity, we have

and thus

The components of (s∇ϕ) are

B4. Conformal Bianchi identities for the conformal-scalar field

The projections of the Bianchi identities (44)

are as follows:

(B66)

Appendix C

The asympotic solution of the Einstein-massless-scalar-field equations

Although we want the results at I -, we follow the usual convention and find the asymptotic solution of the field equations in the physical spacetime first in the neighbourhood of I + . The results can be easily translated to I -. For the solution, we closely follow the procedure presented in [START_REF] Stewart | Advanced general relativity[END_REF] for the vacuum spacetimes. The coordinates, tetrad and conformal transformations of the spin basis are identical to those used therein, and in this Appendix, since we don't consider unphysical quantities, we omit the tildes from physical quantities. We define

The components of the Ricci spinor and the scalar curvature are given by ( 8) and ( 9). The asymptotic behaviour of these quantities is as follows:

Assuming analyticity we can expand any quantity X = O(Ω k ) in a series of the form

Using the field equations, i.e. the Ricci and Bianchi identities and the frame equations, we arrive at following asymptotic solution for the spin coefficients (setting Φ mn = 0 and Λ = 0 we recover expansions valid for the vacuum case which can be found, e.g. in [START_REF] Stewart | Advanced general relativity[END_REF], section 3.10):

where

α (1) = ðσ (0) + a σ(0) , γ (2) = aðσ (0) -a ðσ (0) -

For the relevant Weyl scalars and Ricci tensor components we have

Appendix D: Selected solutions to the Einstein-conformal-scalar equations

We first briefly survey some explicit stationary solutions to the Einstein-conformalscalar equations which satisfy the requirements of our theorem. To explore the field equation ( 37) further, we also present two families of time-dependent solutions. Some are singular when φ 2 = 1, some are not and in some φ 2 never takes the value one.

D1. Stationary solutions

Over fifty years ago Buchdahl [START_REF] Buchdahl | Reciprocal Static Metrics and Scalar Fields in the General Theory of Relativity[END_REF] demonstrated how from any given static vacuum solution a one-parameter family of pairs of solutions of Einstein's equations with massless scalar field can be constructed. Later Bekenstein [START_REF] Bekenstein | Exact solutions of Einstein-conformal scalar equations[END_REF] showed how from any Einstein-scalar field solution the corresponding Einstein-conformal -scalar field solution can be found. In particular, considering any static vacuum solution in the form

two Einstein-conformal-scalar solutions are

where α = (1 -3β 2 ) 1/2 and β ∈ -

3 is a free parameter. Upper and lower signs, respectively, in (D74) correspond to two types of solutions A and B. If the solution (D73) is asymptotically flat, so is the type A solution. Hence, many solutions satisfying our assumptions are available. In this case, φ 2 always takes the value one for some time, but the components of the energy-momentum tensor are

, 1 1 -r 2 , r 2 , r 2 sin 2 θ , (D86)

and are nonsingular at φ 2 = 1.

In [START_REF] Bekenstein | Exact solutions of Einstein-conformal scalar equations[END_REF], cosmological solutions were also considered (both conformal scalar field and incoherent radiation), however, singularities in T ab for scalar field were not discussed.

D3. PP-waves

We can find pp-wave solutions with this source: consider the pp-wave with metric given by ds 2 = 2 H(u, x, y) du 2 + 2 du dvdx 2dy 2 .

(D87)

For simplicity, we assume that the scalar field φ = φ(u, x, y) does not depend on v. The wave equation is then φ = -φ xx -φ yy = 0, (D88) with subscripts denoting corresponding derivatives. We can take the general real solution to be φ(u, x, y) = f (u, x + iy)/2 + f (u, x -iy)/2, (D89)

where f is an arbitrary real function of two variables. Let us denote

so that Einstein's equation are K ab = 0. One of these equations is


Guillaume Montassier

Thibault Cholez

Guillaume Doyen

Rida Khatoun

Isabelle Chrisment

Olivier Festor

Content Pollution Quantification in Large P2P networks : a Measurement Study on KAD

Keywords: KAD, pollution of contents, pollution detection

Content pollution is one of the major issues affecting P2P file sharing networks. However, since early studies on FastTrack and Overnet, no recent investigation has reported its impact on current P2P networks. In this paper, we present a method and the supporting architecture to quantify the pollution of contents in the KAD network. We first collect information on many popular files shared in this network. Then, we propose a new way to detect content pollution by analyzing all filenames linked to a content with a metric based on the Tversky index and which gives very low error rates. By analyzing a large number of popular files, we show that 2/3 of the contents are polluted, one part by index poisoning but the majority by a new, more dangerous, form of pollution that we call index falsification.

I. INTRODUCTION

P2P networks, and particularly Distributed Hash Tables (DHT), are today a major usage of Internet involving several millions of users. Their scalability, robustness and small infrastructure costs make them a perfect architecture to support file sharing applications, among others. However, the lack of central authority and malicious peer's behaviors can disturb the services offered by the P2P system. One of the major issues affecting large open P2P file sharing networks is the pollution of the shared contents. The pollution is defined as an irregular usage of the network to avoid a peer to access a desired content. Different forms of pollution have been identified affecting P2P networks by corrupting either the indexation of files [START_REF] Liang | The Index Poisoning Attack in P2P File Sharing Systems[END_REF] or files' content itself [START_REF] Liang | Pollution in p2p file sharing systems[END_REF]. However, those studies were conducted several years ago on the FastTrack and Overnet P2P networks that are not used anymore today. Since 2005, no major investigation on P2P systems was done to quantify or describe the forms of pollution affecting current P2P networks.

We propose in this paper a measurement study of the pollution affecting the widely deployed KAD P2P network and we show that the pollution is widely diffused. In particular, we highlight and quantify a new form of pollution we call index falsification that aims at making a user access to a content totally unrelated to the desired one and potentially harmful.

This paper is structured as follows. Section II presents the related work on KAD and the different forms of pollution. Section III describes the index falsification pollution method and our strategy to detect it. Section IV presents the assessment of our pollution metric and our results on the quantification and the characterization of the pollution we obtained by investigating a large number of popular files in KAD. Finally, Section V concludes the paper and outlines our future work.

II. RELATED WORK

A. The KAD P2P network KAD is a widely deployed P2P file sharing network implemented in the eMule client. As a file sharing application, KAD's Kademlia based DHT uses a double-indexation mechanism to index the shared files. The first level associates keywords with files while the second associates files with sources (peers sharing the file). Each KAD node has a random 128 bits "KADID" determining its position in the DHT and the references of which it is in charge of. When sharing a file, the raw data and all the keywords associated with its name are hashed separately with a MD4 function generating an ID which is then published into the DHT. Firstly, the file's information (fileID, filename, etc) are published towards the hash of each keyword (keywordID). Secondly, the peer publishes its own information (IP address, port, etc) towards the hash of the file (fileID) to be indexed as a potential source.

Several research papers have been written about KAD, covering many aspects of this P2P network. Some studies like [START_REF] Steiner | A global view of kad[END_REF] focused on large scale monitoring of peers participating in the network showing that KAD is mainly used in European countries and in China. [START_REF] Hun | Why kad lookup fails[END_REF] highlighted some flaws decreasing the routing efficiency of KAD and proposed some fixes. The security of KAD has also been widely investigated. The Sybil attack, which consists in the insertion of malicious nodes in the DHT, is particularly critical on this network [START_REF] Steiner | Exploiting kad: possible uses and misuses[END_REF] and some countermeasures have been proposed against it [START_REF] Cholez | Efficient DHT attack mitigation through peers' ID distribution[END_REF]. However, despite the great interest of the academic community for KAD, no study actually monitored the pollution of contents, which is one of the major issues of the network.

B. Pollution in P2P file sharing networks

Several forms of pollution exist and were studied in real P2P networks. The first form, called data pollution, consists in the sharing of files (decoys) whose content is deliberately damaged. The authors of [START_REF] Christin | Content availability, pollution and poisoning in file sharing peer-to-peer networks[END_REF] showed that the best strategy to spread such a pollution is to limit the number of polluted files advertised and to make them very popular, that is to say, shared by a lot of sources. The second form of pollution, called index poisoning or meta-data pollution, consists in corrupting the indexation mechanism of the P2P system by advertising many fake files which are actually shared by no peer or by advertising unexisting sources. In 2004, Liang et al. were the first to study pollution [START_REF] Liang | Pollution in p2p file sharing systems[END_REF] by analyzing different files related to 7 popular songs shared in the FastTrack network and showed that, for some popular songs, more than 50% of the versions were polluted. In 2005 [START_REF] Liang | The Index Poisoning Attack in P2P File Sharing Systems[END_REF], they considered 10 popular songs and investigated the pollution in both the FastTrack and the Overnet P2P networks showing that both were polluted and that Overnet is particularly affected by the index poisoning pollution. More recently, a study [START_REF] Locher | Poisoning the Kad Network[END_REF] proved that the index poisoning attack can also affect the KAD network by corrupting DHT entries, either by publishing fake records on the responsible peers or by inserting malicious nodes which are close to them.

Several solutions have been proposed to limit the pollution. Liang et al. [START_REF] Liang | The Index Poisoning Attack in P2P File Sharing Systems[END_REF] proposed a blacklisting scheme to avoid polluters but the list is provided and updated in a central way opposed to the P2P paradigm. Reputation systems [START_REF] Costa | Reputation systems for fighting pollution in peer-to-peer file sharing systems[END_REF] can help fighting the pollution but they introduce a significant overhead and are vulnerable to malicious votes among other attacks. Winnowing [START_REF] Shin | WINNOWING : Protecting P2P Systems Against Pollution By Cooperative Index Filtering[END_REF] proposed a collaborative filtering done by the indexing peers to limit the index poisoning but it is vulnerable to malicious nodes insertion. None of the proposed solutions is actually deployed so far because of the limitations.

III. DETECTING KAD'S DHT POLLUTION

A. The index falsification pollution

We highlight and quantify a new form of pollution widely spread in the KAD network. While index poisoning advertises many unexisting files which can not be downloaded, index falsification pollution consists in advertising a single file under many different filenames, and consequently many different keywords, which are totally unrelated to the real content. Each wrong filename is artificially made popular by the polluters by exploiting a KAD weakness that allows a peer to manually publish the number of estimated sources for a given filename and which helps to make the pollution highly visible.

This form of pollution is the most harmful possible because it leads the user to download undesirable content, wasting the network resources in the process and because the downloaded file can be dangerous for the user safety. The real content can be a malware or a video hurting users' feelings (e.g. pornographic or paedophile contents). Besides, this pollution creates many false positives when monitoring illegal files within the KAD network. In fact, some users can be monitored accessing a file they did not even look for. While KAD's users can suffer from this pollution on a daily basis, no study so far investigated this problem.

B. Strategy for detecting the index falsification

The heart of index falsification is to attach many different names to a file. To detect this pollution, we must gather all the different filenames attached to a file and evaluate their consistency. However, KAD's double-indexation scheme makes this information hard to retrieve. From a keyword search, one can obtain the different files linked to the keyword and their details (filename, size, etc.). But all the different files collected from a keyword search include the desired keyword in their filename, for instance "avatar", and will appear consistent even if some files are indexed through other unrelated keywords in the DHT. From a source search, one can obtain all the sources of a file, no matter which filename is used by the sources. However, the filename is not an information published on the DHT at this level. Only the sources (IP address, port, etc.) are linked to the fileID. So, the diversity of filenames can not be obtained by regular DHT lookups, neither by a keyword search lookup nor by a source search lookup. However, we can obtain the different filenames linked to a fileID at the beginning of the download process. When a file is selected for download after a keyword search, the potential sources are first retrieved thanks to the DHT. Then, a TCP connection is initiated toward each source to request the download. A function implemented in KAD clients, but external to the KAD protocol, allows a peer to get detailed information about the file directly from the responding sources and can show conflicting filenames in case of pollution. Concerning a clean file (table I), the majority (30) of the responding sources share the same filename and the others are clearly related to the desired content. On the opposite, concerning a polluted file (table II), the sources show a lot of totally different filenames, conflicting with each other and with the desired content. We used a modified aMule client in order to collect, for a given file, all the filenames advertised by the responding sources. Given a fileID, we want to determine if the file is trustworthy or polluted by index falsification. Our detection is based on the overall consistency of the different filenames given by the sources. To compute the similarity of two filenames, we use a metric to evaluate the similarity between their set of keywords. Let X and Y be two sets of keywords, X being the keywords associated with the desired filename and Y being the keywords associated with a filename retrieved from a source. The Tversky index [START_REF] Tversky | Features of similarity[END_REF] is a generic similarity metric (when α = β) used in data mining and defined by:

S(X, Y ) = |X ∩ Y | |X ∩ Y | + α * |X -Y | + β * |Y -X| (1)
S(X, Y ) ∈ [0, 1] and more precisely returns 1 if both filenames are the same and 0 if they have no common keyword. We define our pollution coefficient P for each file X as a function of the average of similarity coefficients computed for all filenames Y i retrieved from the n sources.

P (X) = 1 - n i=1 S(X, Y i ) n (2)

IV. CONTENT POLLUTION QUANTIFICATION IN KAD

A. Investigating the shared contents

Collecting all the files shared within a P2P network is an impossible task. To quantify the pollution from a significative sample of the users' interest, we based our experiment on the top 100 of the most downloaded contents in 2010, according to one of the major BitTorrent indexation website 1 which receives more than one hundred million searchs a year. To have a most significative sample, for each content from top 100 we collected the twenty related files that show the highest number of estimated sources, resulting in 2000 files investigated. In fact, previous studies [START_REF] Cholez | Monitoring and Controlling Content Access in KAD[END_REF] and [START_REF] Lee | Understanding pollution dynamics in P2P file sharing[END_REF] showed that a higher number of sources estimated for a file increases its downloads.

Before collecting the different filenames for 2000 files, we wanted to know how fast the real sources of a file are found in order to define the duration of the experiment. Based on a Fig. 2: Distribution of files according to the pollution index sample of 150 files, we show that the number of responding sources quickly increases during the first 200s after the start of the download and is then stable (figure 1). We chose to wait up to 300s to collect the filenames, afterwards the number of missed sources is negligible. This result also indicates how fast the detection could be performed in real time.

B. Metric assessment

Every classification is inevitably prone to errors. In our case, some unpolluted files can be tagged as polluted (false positive) and inversely, polluted files can be tagged as unpolluted (false negative). To provide a reliable pollution detection, we need to define suitable parameters for the similarity metric and for the detection thresholds. To define these values and estimate the metric's error rates, we used a methodology close to [START_REF] Latapy | Quantifying paedophile queries in a large P2P system[END_REF] by asking a set of 10 experts to manually evaluate the pollution for a sample (20%) of the files we collected. By analyzing the different filenames, they tag files as polluted, clean or unclassified through a web interface showing the associated filenames as presented in tables I and II.

We tried some particular forms of the Tversky similarity metric and found that the best detection results according to the expert votes were given for α = β = 0.5 which is known as the Dice coefficient [START_REF] Manning | Foundations of statistical natural language processing[END_REF] and can also be written like:

S(X, Y ) = 2 * |X ∩ Y | |X| + |Y | (3)
Then, we set the detection thresholds to best match the expert votes: files for which P (X) is under 0.3 are tagged as clean while those above 0.7 are polluted. Concerning the few files between 0.3 and 0.7, we consider that the metric can't reliably determine their state. Figure 2 displays the distribution of sources according to our metric. It appears that we have a bimodal distribution with peaks at 0.1 and 0.9 showing that our metric naturally creates two major classes for our data. As a result, our metric based on the Dice coefficient with these detection thresholds gives low error rates: 3.5% of false positive and 0.8% of false negative.

We investigated the false positives of our metric and found that they are due to movies presenting both an original and a localized title. For example, we downloaded a file called "el Cigno Nero Sub Ita.avi" and the name for the majority of sources was "Black.Swan.2010.DVDSCR.XviD-TiMKY.avi". From the metric point of view, these two names do not appear related, only a semantic knowledge of the translated words or of the different localized names set for a movie can lead to the right classification given by the experts.

C. Quantification and characterization of KAD's pollution

To quantify the index falsification, we applied our final metric on the popular files we investigated. Table III shows the final classification of these files. First we can see that despite a large number of announced sources, no responding sources can be found for 20.5% of the files which clearly indicates an index poisoning pollution. However, the major form of pollution is the index falsification affecting 41.1% of the considered KAD's files. These two forms of pollution represent 61.5% of the 2000 popular files, leaving less than a third (28.6%) clean. According to the expert votes 78% of the unclassified files are clean and 12% polluted. Finally, we analyzed the number of corrupted files for each entry of the top 100. It appears that all the entries of the top 100 were concerned by this pollution, from the less infected: The big bang theory with 25% of polluted files out of the 20 considered, to the most: Avatar with all its top 20 files polluted. Besides, it appears that the index falsification pollution does not only target copyrighted content since the entry Ubuntu had fifteen polluted files out of twenty.

Moreover, we investigated more precisely the different filenames of polluted files to determine by which type of content they might be polluted with. We used two lists of keywords, one related to paedophile contents [START_REF] Latapy | Quantifying paedophile queries in a large P2P system[END_REF] and the other to pornographic contents. We then searched for those keywords among the filenames advertised for our 41.1% of files affected by index falsification. Table IV We presented in this paper a new type of pollution widely spread in the KAD network and called index falsification. We proposed a very efficient metric to detect it, assessed by the evaluation of experts. We applied our metric to a large number of popular contents to quantify the pollution and found that popular files are highly infected by pollution, with more than 41% of files infected by index falsification and more than 20% by index poisoning. Moreover, our results show that index falsification is an intentional and harmful pollution, linking undesirable contents (potentially paedophile and pornographic) through regular keywords and sometimes even related to cartoons.

In our future works, we will investigate the polluting behaviors in order to understand precisely how this pollution is achieved. Then, we will design a detection mechanism which can operate earlier in the download process to avoid the initialization of many connections towards the responding sources. Our solution will also need to be suitable for real implementations (by keeping backward compatibility and minimizing the overhead) in order to protect current P2P networks.

Fig. 1 :

 1 Fig. 1: % of the responding sources discovered in time

TABLE I :

 I Filename: The.Big.Bang.Theory.4x09.The.Boyfriend.Complexity.ENG... The.Big.Bang.Theory.4x09.The.Boyfriend.Complexity.ENG.-.sub.FR... 30 The.Big.Bang.Theory.4x09.The.Boyfriend.Complexity.VOSTFR.HD... 12 The Big Bang Theory 4x09 The Boyfriendplexity Vostfr Hdtv Xvid... 3 The.Big.Bang.Theory.S04E09.VOSTFR.HDTV.XviD.avi 2 409 The.Big.Bang.Theory.4x09.The.Boyfriend.Complexity.VOSTFR... 1 The Big Bang Theory -4x09 -VostFr.avi 1 The.Big.Bang.Theory.S04E09.VOSTFR.HDTV.XviD-TheOdusseus.avi 1 Example of consistent filenames retrieved from the responding sources for a clean file

	FileID: C0F8BFA37E0DD0A4585CD3B90B9F4D26
	Number of responding sources: 50
	Found filenames	#
	FileID: 7B9F403468CD821C38885E7777153C1C
	Number of responding sources: 175
	Found filenames	#
	Xxx Marc Dorcel -Russian Institute Lesson 1 (Sex, Porno, Lesbian...	4
	The Best Of The Doors.rar	2
	[DIVX-ITA]-Disney Pixar-Wall-E-2008-Italian Ld Dvdrip Xvid...	1
	[DIVX-ITA] The Twilight Saga New Moon.avi	1
	Dexter Fr Saison 3.rar	1
	Shrek.2.(Fr.DvdRipp).Teste.by.www.FreeDivx.org.avi	1
	Smallville 6x10 Hidro [DVD+DVB][Spanish-English][by jesuscas]...	1
	THE SOCIAL NETWORK [par emule island.com tp].avi	1
	Windows 2003 Server.iso	1
	...	...

Filename: Indiana Jones Et Les Aventuriers De L'Arche Perdue-Fr-Dvdrip...

TABLE II :

 II Part of conflicting filenames retrieved from the responding sources for a polluted file

TABLE III :

 III Global pollution quantification

  shows the resulting characterization: 8.8% of them are referenced by at least a paedophile filename and more than 55% by a pornographic name.

	Contents	quantification (%)
	Child pornography	8,8%
	Pornography	55,7%
	Other	35,3%

TABLE IV :

 IV Type of contents found in index falsification V. CONCLUSION

http://torrentfreak.com/bittorrent-zeitgeist-what-people-searched-for-in-2010-101227/