Robinson-Trautman solution with scalar hair
aa r X i v : . [ g r- q c ] M a y Robinson–Trautman solution with scalar hair
T. Tahamtan ∗ and O. Svítek † Institute of Theoretical Physics, Faculty of Mathematics and Physics,Charles University in Prague, V Holešovičkách 2, 180 00 Prague 8, Czech Republic (Dated: September 22, 2018)Explicit Robinson–Trautman solution with minimally coupled free scalar field is derived andanalyzed. It is shown that this solution contains curvature singularity which is initially naked butlater the horizon envelopes it. We use quasilocal horizon definition and prove its existence in laterretarded times using sub- and supersolution method combined with growth estimates. We showthat the solution is generally of algebraic type II but reduces to type D in spherical symmetry.
PACS numbers: 04.20.Jb, 04.70.Bw
I. INTRODUCTION
Solutions to Einstein equations with scalar field sourceprovide very useful tool for understanding relativity dueto the simplicity of the source. Recently, it becomesprogressively plausible that such fields might really exist(LHC) and potentially play a fundamental role in physics.In classical General Relativity they were used to studycounterexamples to black hole no-hair theorems and inmany other areas. These results were mostly based onhighly symmetric solutions and it is therefore importantto provide solutions with less or no symmetries to sub-sequently analyze if those results hold in more genericsituations and are not tied to a specific symmetry.Robinson–Trautman spacetimes represent an impor-tant class of expanding nontwisting and nonshearing so-lutions [1–3] describing non-spherical generalizations ofblack holes. In general, they do not posses any Killingvectors thus providing important solutions devoid of sym-metry. Many properties of this family in four dimensionshave been studied, especially in the last 25 years. In par-ticular, the existence, asymptotic behaviour and globalstructure of vacuum Robinson–Trautman spacetimes oftype II with spherical topology were investigated byChruściel and Singleton [4–6]. Robinson–Trautman so-lutions were shown to exist for generic, arbitrarily strongsmooth initial data for all positive retarded times, and toconverge asymptotically to corresponding Schwarzschildmetric. Extensions across the “Schwarzschild-like” fu-ture event horizon can only be made with a finite orderof smoothness. These results were generalized in [7, 8] toRobinson–Trautman vacuum spacetimes with cosmologi-cal constant. These cosmological solutions settle down toa Schwarzschild–(anti-)de Sitter solution at large times u . Finally, the Chruściel–Singleton analysis was ex-tended to Robinson-Trautman spacetimes including mat-ter, namely pure radiation [9, 10], showing that they ap-proach the spherically symmetric Vaidya–(anti-)de Sittermetric. Generally, the solutions of this family settle down ∗ Electronic address: [email protected]ff.cuni.cz † Electronic address: [email protected] to physically important solutions. The location of thehorizon together with its general existence and unique-ness for the vacuum Robinson–Trautman solutions hasbeen studied by Tod [12]. Later, Chow and Lun [13]analyzed some other useful properties of this horizonand made numerical study of both the horizon equa-tion and Robinson–Trautman equation. These resultswere later extended to nonvanishing cosmological con-stant [14]. The anisotropy of Robinson-Trautman hori-zon and its associated asymptotic momentum was alsoused in the analytic explanation of an "antikick" appear-ing in numerical studies of binary black hole mergers [15].Robinson–Trautman spacetimes (containing alignedpure radiation and a cosmological constant) were alsogeneralized to any dimension [11]. Existence of horizonswas subsequently analyzed in [16]. Finally, Robinson–Trautman solutions with p-form fields in arbitrary di-mension were derived recently [17]. One of the re-sults mentioned therein rules out the existence of alignedscalar field (where alignment refers to the gradient of thefield) for generic Robinson–Trautman case.The solutions for "stringy" Robinson–Trautman space-time corresponding to Einstein–Maxwell–dilaton systemwere obtained in [18]. Recently, scalar field solutions forEinstein–Maxwell–Lambda system with a conformallycoupled scalar field belonging to Plebański–Demiańskifamily (containing type D solutions of Robinson–Trautman class) were derived in [19].
II. VACUUM ROBINSON–TRAUTMANMETRIC AND FIELD EQUATIONS
The general form of a vacuum Robinson–Trautmanspacetime can be given by the following line element [1–3, 20]d s = − H d u − u d r + r ˜ P (d y + d x ) , (2.1)where 2 H = ∆( ln ˜ P ) − r ( ln ˜ P ) ,u − m/r − (Λ / r ,∆ ≡ ˜ P ( ∂ xx + ∂ yy ) , (2.2)and Λ is the cosmological constant. The metric dependson two functions, ˜ P ( u, x, y ) and m ( u ) , which satisfy thenonlinear Robinson–Trautman equation∆∆( ln ˜ P ) + 12 m ( ln ˜ P ) ,u − m ,u = 0 . (2.3)The function m ( u ) might be set to a constant by suitablecoordinate transformation for vacuum solution.The spacetime admits a geodesic, shearfree, twistfreeand expanding null congruence generated by k = ∂ r . Thecoordinate r is an affine parameter along this congruence, u is a retarded time coordinate, and x, y are spatial coor-dinates spanning transversal 2-space with their Gaussiancurvature (for r = 1) being given by K ( x, y, u ) ≡ ∆( ln ˜ P ) . (2.4)For general fixed values of r and u , the Gaussian cur-vature is K /r so that, as r → ∞ , they become locallyflat. III. SOLUTION COUPLED TO A SCALARFIELD
We consider the following action, describing a scalarfield minimally coupled to gravity, S = Z d x √− g [ R + ∇ µ ϕ ∇ µ ϕ ] (3.1)where R is the Ricci scalar for the metric g µν . The mass-less scalar field ϕ is supposed to be real and we use unitsin which c = ~ = 8 πG = 1. By applying the variationwith respect to the metric for the action (3.1), we getEinstein equations R µν − g µν R = T µν . (3.2)The energy momentum tensor generated by the scalarfield is given by T µν = ∇ µ ϕ ∇ ν ϕ − g µν g αβ ∇ α ϕ ∇ β ϕ (3.3)and the scalar field must satisfy corresponding field equa-tion (cid:3) ϕ ( u, r ) = 0 (3.4)where (cid:3) is a standard d’Alembert operator for our metric(3.5).For the matter of convenience we will be looking forthe metric in the following formd s = − H ( u, r ) + K ( u, x, y )) d u − u d r + R ( u, r ) P ( x, y ) (d x + d y ) (3.5)The scalar field is assumed to be function of u and r only( ϕ ( u, r )). The dependence on r means that the scalarfield is not aligned and thus is not ruled out by the results of [17]. The nontrivial components of the Ricci tensorcorresponding to the metric (3.5) are R uu = 2 (cid:18) R ,r R H ,r + H ,rr (cid:19) ( H + K ) + 2 R ,r R ( H + K ) ,u − R ( R ,u H ,r + R ,uu ) + P R ( K ,xx + K ,yy ) R rr = − R ,rr R (3.6) R ru = R ur = 2 R ,r H ,r − R ,ru R + H ,rr R xx = R yy = − P { k ( x, y ) + 2( H + K )( RR ,r ) ,r ++2 RR ,r H ,r − RR ,u ) ,r } where as usual () ,x i = ∂∂x i () and k ( x, y ) = ∆( ln P ( x, y )) (3.7)where ∆ is still given by expression (2.2) with ˜ P replacedby P .We will use the following form of equations equivalentto Einstein equations (3.2) coupled to energy momentumtensor (3.3) R µν = ϕ ,µ ϕ ,ν = ϕ ,u ϕ ,u ϕ ,r ϕ ,u ϕ ,r ϕ ,r (3.8)From the above equations describing gravitational fieldand field equation for the scalar field (3.4) we obtain thefollowing expressions for unknown metric functions andscalar field H ( u, r ) = r U ( u ) ∂U ( u ) ∂uR ( u, r ) = s U ( u ) r − C U ( u ) K ( u, x, y ) = k ( x, y )2 U ( u ) (3.9) ϕ ( u, r ) = 1 √ (cid:26) U ( u ) r − C U ( u ) r + C (cid:27) ∆ k ( x, y ) = α U ( u ) = γe ω u + ηu , in which C = 0 , α, η, γ, ω are constants and ω = α C . Inthe following we will assume C > , α > , η > , γ > IV. PROPERTIES OF THE SOLUTION
First, we should ensure that our solution really be-longs to the Robinson-Trautman family. This is simplyconfirmed by studying the properties of a null congruencegenerated by vector l = ∂ r . Such congruence is geodesic,nontwisting, nonshearing and its expansion is given byΘ l = 2 R ,r R = 2 U ( u ) rU ( u ) r − C . (4.1)Evidently, the above expression is positive only for r > C U ( u ) which may seem not satisfactory. However by in-specting the Kretschmann scalar κ ∼ R ( u, r ) (4.2)and using (3.9) we immediately see that the geometry hassingularities for r = ± r = ± C U ( u ) . Naturally, we are ledto constrain the range of coordinates to r ∈ (cid:16) C U ( u ) , ∞ (cid:17) .In this range the expansion (4.1) is everywhere positive,diverges at the singularity and approaches zero at infinity(as r → ∞ ). Also, one can check from the line elementthat the singularity is a standard pointlike one. Due tothe asymptotic behaviour of function U ( u ) (see (3.9)) thesingularity tends to r = 0 as u → ∞ . The singularityappears due to the divergence of the scalar field and itsenergy momentum tensor.Asymptotically ( u → ∞ ), the scalar field itself is van-ishing everywhere outside the singularity (see (3.9)) whileit diverges at r = 0. So there would be no scalar hairleft outside when the spacetime settles down to the fi-nal state. Indeed, our geometry approaches the originalRobinson–Trautman form (2.1) for u → ∞ when we de-fine ˜ P ( u, x, y ) = P ( x, y ) /U ( u ). In this case one can ap-ply the Chruściel–Singleton analysis [4–6] of asymptoticbehaviour to recover the spherical symmetry of the finalstate which neccesarrily points to Schwarzschild solution.When the singularity is present in our solution we willinvestigate if it is covered by a horizon. Due to dynamicalnature of the spacetime it is preferable to use the quasilo-cal definitions of horizon — apparent horizon [21], trap-ping horizon [22] or dynamical horizon [23]. The basic local condition is shared by all the standard horizon def-initions: these horizons are sliced by marginally trappedsurfaces with vanishing expansion of outgoing (ingoing)null congruence orthogonal to the surface. We will belooking for the horizon hypersurface in the following form r = M ( u, x, y ) (4.3)and study the expansion of compact slices of such hyper-surface given by u = u = const. (with M ( u , x, y ) = M ( x, y )). The requirement of compactness necessarilymeans that the two-spaces spanned by x and y are com-pact as well. We construct null vector fields orthogonalto surface r = M ( x, y ) l = ∂ r (4.4) k = ∂ u + (cid:20) P R ( M x + M y ) − ( H + K ) (cid:21) ∂ r ++ P R ( M x ∂ x + M y ∂ y ) (4.5) that satisfy normalization condition l · k = −
1. Fromthe geometry of the situation one can deduce that con-gruence l is outgoing while k is ingoing. The expansionof the congruence generated by l is always positive (see(4.1) and the discussion beneath) so we are looking forvanishing of expansion related to the other congruence k . These two conditions (Θ l > l = 0) meanthat we are looking for the past horizon according to thedefinition given by [22]. The second expansion is givenby Θ k = 1 R [∆ M − (ln R ) ,r ( ∇ M · ∇ M ) −− ( K + H )( R ) ,r + ( R ) ,u (cid:3) , (4.6)where Laplace operator and scalar product denoted bydot correspond to metric h ij dx i dx j = P ( x,y ) ( dx + dy )on the space Σ spanned by x, y . So the horizon is givenby the solution of the following quasilinear elliptic partialdifferential equation (cid:8) ∆ M − (ln R ) ,r ( ∇ M · ∇ M ) − ( K + H )( R ) ,r +( R ) ,u (cid:9) | r = M ( x,y )& u = u = 0 (4.7)where all dependence on r is replaced by the function M ( x, y ) and u is evaluated to arbitrary constant value u .It is impossible to solve this equation generally butfortunately we can get some useful information aboutthe existence of solution using the technique developedfor the case of Robinson–Trautman spacetime in higherdimensions [16]. The proof of existence of the solution tothe same type of quasilinear equation (∆ u = F ( x, u, ∇ u ))was given there by combining several steps motivated by[24] and using results from [25–27]. The main issues wereto provide an estimate for the function F of the form | F | ≤ B ( u )(1 + |∇ u | ) (where B ( u ) is increasing functionon R + ), and to show the existence of a sub- and a super-solution [28] u − ≤ u + , u ± ∈ C ,β (Σ) ∩ L ∞ (Σ) (here C ,β (Σ) are Hölder continuous functions of some suitableindex β ). Then we know there is a solution u ∈ C ,ι (Σ)(for some ι ) satisfying u − ≤ u ≤ u + .In our case, to provide an estimate of the form | M | ≤ B ( M )(1 + |∇ M | ) (the norm is taken with re-spect to the two-dimensional metric h ij ) for the hori-zon equation (4.7) when considered in the form ∆ M = F ( x, y, M, ∇ M ) where F = (ln R ( u , M )) ,r |∇ M | + k ( x, y ) M − C U ,u ( u ) U ( u )(4.8)one has to deal with the singular behaviour of (ln R ) ,r at r = C U ( u ) . We can do this either by removingthe vicinity of singularity from our domain r ∈ R + \ (cid:16) C U ( u ) (1 − δ ) , C U ( u ) (1 + δ ) (cid:17) or by continuing (with someappropriate smoothing) the divergent function on theproblematic interval (cid:16) C U ( u ) (1 − δ ) , C U ( u ) (1 + δ ) (cid:17) with aconstant value it attains on the boundary of the inter-val. Now, with all the coefficients of the equation finiteone can construct the bounding function B ( u ) easily andthus we can proceed to the construction of sub- and su-persolutions M ± .First, we note that due to the selection of sign for thefree constants made at the end of previous section ( C > , α > , η > , γ >
0) we obtain U ,u > u ∈ ( − η ω , ∞ ). Wecan then understand our solution as being given by ini-tial conditions specified at u in = − η ω which correspondsto usual understanding of Robinson–Trautman solution.As usual, we are looking for constant sub- and superso-lutions but we are unable to provide them independentlyof the value of u . Generally, we can find the sub- andsupersolutions in the following cases: • u < u = min( k ( x,y )) − max( k ( x,y )) δ − C η C ω M − = 0 (4.9) M + = C U ( u ) (1 − δ ) (4.10) • u > u = max( k ( x,y ))(1+ δ ) − C η C ω M − = C U ( u ) (1 + δ ) (4.11) M + = C U ,u ( u )min( k ( x, y )) U ( u ) (4.12)Both bounds u and u are in the restricted range of co-ordinate u . Evidently, the first case would provide exis-tence of solution only beneath the position of singularity(or, in other words, inside the singularity) which is irrele-vant and moreover we have already restricted the range of r ∈ (cid:16) C U ( u ) , ∞ (cid:17) . In the second case, one can easily checkthat the necessary condition M − ≤ M + is indeed satis-fied for u > u and we certainly have a horizon given by r = M ( x, y ) where M − ≤ M ( x, y ) ≤ M + . Note that wesuppose that min( k ( x, y )) > M + to be valid.If we allow k min ≡ min( k ( x, y )) ≤ k max ≡ max( k ( x, y ))) we are unable to provideconstant supersolution in the case u > u . Instead wecan use the knowledge of how Laplace operator acts on k ( x, y ) (3.9) and the observation that the first term ofthe definition of function F (4.8) is always positive toprovide the following non-constant supersolution M + = c [ k max − k ( x, y )] + C U ( u ) (1 + δ ) (4.13)where c = C [ C U ,u ( u ) − k min U ( u )(1 + δ )] U ( u )( α + k max k min − k min ) . (4.14)This estimate works if ( α + k max k min − k min ) > u this singularity appears to be initiallynaked and the horizon develops only in later time. V. ALGEBRAIC TYPE OF THE SOLUTION
Now, we would like to see if the geometry of our space-time is sufficiently general. Since vacuum Robinson–Trautman spacetime is generally of algebraic type II wewould like our solution to be at least of the same typeand not more special. Our preferred tetrad for determin-ing the Weyl scalars of our solution is given by differentnull vectors compared to (4.4) ˜l = ∂ r ˜k = ∂ u − ( H + K ) ∂ r (5.1) m = P √ R ( ∂ x + I∂y )where I is a complex unit. The Weyl spinor computedfrom this tetrad has only the following nonzero compo-nentsΨ = 14 U R (cid:20) P ( k ,yy − k ,xx + Ik ,xy ) −− ( k ,x − Ik ,y )( P ,x − IP ,y )]Ψ = √ P R ,r U R ( k ,x − Ik ,y ) (5.2)Ψ = 16 U R (cid:2) U k − ( U ,u r + k )( RR ,rr R ,r ) −− U RR ,ru + ( RU ,u + 2 U R ,r ) R ,r ]Now, we can easily determine the type irrespective ofpossible non-optimal choice of tetrad by using the reviewof explicit methods for determining the algebraic typein [29] that are based on [30]. Namely, when we useinvariants I = Ψ Ψ − Ψ + 3Ψ , J = det Ψ Ψ Ψ Ψ Ψ Ψ Ψ Ψ Ψ we can immediately confirm that I = 27 J is satisfied sothat we are dealing with type II or more special. At thesame time generally IJ = 0 so it cannot be just type III.Moreover, the spinor covariant R ABCDEF has nonzerocomponents R = Ψ (3Ψ Ψ − ) (5.3) R = 12 Ψ (3Ψ Ψ − ) (5.4)which means that generally the spacetime cannot be oftype D. So indeed our scalar field solution is of the mostgeneral type possible for the Robinson-Trautman vacuumclass. Which does not mean that there cannot be a scalarfield solution of type I. Moreover, inspecting the com-ponents of the Weyl spinor (5.2) one concludes that inthe special case of k ( x, y ) = const > x, y ) the algebraic type becomes D consistent with spher-ical symmetry. Finally, since Ψ = 0 implies Ψ = 0 wecannot have all components of spinor covariant Q ABCD (see [30]and [29]) vanishing while having nonvanishingWeyl spinor. This means that our family of solutionsdoes not contain type N geometries.
VI. CONCLUSION AND FINAL REMARKS
We have derived a Robinson–Trautman spacetime withminimally coupled free scalar field. We have shown thatit has a singularity for all retarded times created by thedivergence of the scalar field therein. This singularity isinitially (with respect to retarded time) naked and onlylater becomes covered by the quasilocal horizon. Notethat the energy momentum tensor of the free minimallycoupled scalar field trivially satisfies null energy condition(as well as weak and strong ones) and the naked singular-ity at the beginning of the evolution is probably causedby a slow buildup of effective energy density caused bythe scalar field at the singularity position which is enoughto form the singularity but not enough to envelop it inhorizon initially. This behaviour suggests similarity withthe appearance of a naked curvature singularity in Vaidyaspacetime with linear mass function. The naked singu-larity appears there initially depending on the speed ofgrowth of mass [31] and later becomes covered by hori- zon as well. From the properties of both null congru-ences orthogonal to the horizon we deduced that we aredealing with past horizon which is natural for standard(retarded) form of Robinson–Trautman spacetime.Our solution is asymptotically flat, contains a blackhole (at least in the later stage of development) and has ascalar field, so one is naturally interested in its connectionwith the no-hair theorems (see [32] for current review).As recently shown [33], for stationary black hole space-times there are no scalar hairs (even for time-dependentscalar field) which means that the dynamical nature ofRobinson–Trautman family is truly needed for our solu-tion to be feasible. Also, we have shown that the scalarfield vanishes outside the black hole in infinite retardedtime limit when the geometry settles down to the finalstate — Schwarzschild black hole.Finally, we have proved that our geometry is of al-gebraic type II (the most general type for vacuumRobinson–Trautman spacetimes) and if we restrict tospherically symmetric case it is of type D. However, thetype N subcase is not possible for our solution.
Acknowledgments
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