n-DBI gravity, maximal slicing and the Kerr geometry ∗ Fla´vio S. Coelho, Carlos Herdeiro, and Mengjie Wang Departamento de F´ısica da Universidade de Aveiro and I3N Campus de Santiago, 3810-183 Aveiro, Portugal (Dated: October2012) Recently [1], we have established that solutions of Einstein’s gravity admitting foliations with a certain geometric condition are also solutions of n-DBI gravity [2]. Here we observe that, in vacuum,therequired geometric condition isfulfilled bythewell known maximal slicing,often used in numerical relativity. As a corollary, we establish that the Kerr geometry is a solution of n-DBI gravity in thefoliation adapted to Boyer-Lindquist coordinates. PACSnumbers: 04.50.-h,04.50.Kd,04.20.Jb 3 1 There are at present important observational motiva- problems and absence of lapse dynamics [7]. It there- 0 tions to explore relativistic gravity beyond General Rel- fore merits further study as a learning ground for phe- 2 ativity (GR). Firstly, there is a large body of Cosmo- nomenology beyond GR. n logical evidence for an accelerating Universe [3], which, A way to learn about the properties of any model a J within GR, requiresanexotic form ofenergy,raisingthe of gravity is to consider its exact solutions. One sim- questionif suchexotic energycouldbe tradedbya mod- ple question concerning n-DBI gravity, raised in [1], is 6 ification in the laws of gravity. Secondly, the expected if the Kerr solution of vacuum GR, appropriately foli- ] opening of the gravitationalwave astronomy field in the ated, is also a solution of n-DBI. This question becomes c very near future, with the planned science runs for the even more interesting if we notice that only slowly ro- q second generation of gravitational wave observatories - tating black holes have been (recently) found in Hoˇrava- - r such as Advanced LIGO - will test GR and constrain Lifschitz gravity [8, 9]. The purpose of this note is to g alternative theories of gravity in new ways. Indeed, al- answer affirmatively to such question by noting that the [ ready the observation of the orbital period variation in geometric condition found in [1] for solutions of GR to 1 binary pulsar systems, in particular those with a large be solutions of n-DBI has a clear interpretation in the v mass ratio between the neutron stars, has set important theory of space-time foliations. 0 constraintsinscalar-tensorandTeVeStheoriesofgravity n-DBI gravity (without matter) is described by the 7 [4]. action [2] 0 1 Most alternative theories of gravity explored for phe- . nomenologicalpurposes keep the central diffeomorphism 3λ G 301 ifnravmareiasnhcaeveofalGsoRb;ebenutcomnosiddeelrsedw,itahnpotraefbelrereedxarmefpelreenbcee- S =−4πG2N Z d4x√−g"r1+ 6λNR−q# , (1) 1 ing the Einstein-aether theory [5]. In 2009, Hoˇrava pro- where λ,q are two constants, and : poseda nonfully covarianttheory of gravity,as a means v toaddressthewellknownnon-renormalizabilityproperty Xi of GR [6], which has drawn attention to gravity theories R=(4)R−2Dµ(nµDνnν) , (2) r where the symmetry group is reduced to the so-called a isthesumofthefourdimensionalRicciscalarwithatotal Foliation Preserving Diffeomorphisms (FPD). One such derivative term closely related to the Gibbons-Hawking- model that has been recently proposed is n-DBI gravity York boundary term [10, 11]. The unit vector n is time- [1, 2, 7]. This model, albeit not proposed as a funda- like anddefines a preferredfoliationof space-time; D is mental (i.e. quantum) theory of gravity, has appealing µ the four dimensional covariant derivative. properties, such as equations of motion that are higher Afundamentaldistinctionbetweenthesolutionsofthis orderinspatialderivatives,butonlysecondorderintime theory, as compared to GR or, say, f(R) gravity theo- derivatives, which would avoid loss of unitarity in the ries, is that they are defined by their intrinsic geometry quantum regime. Moreover,it revealedinteresting prop- - encoded in the four dimensional curvature - and the erties for phenomenological studies: it can explain early foliation chosen. The complexity of the field equations andlate time inflationin aunified way[2]; it containsas discourages an attempt at finding even the most generic solutions the standard spherically symmetric black hole spherically symmetric solution, which does not appear geometries of GR, appropriately foliated [1]; it is not to imply staticity in this model (i.e. Birkhoff’s theorem afflicted by the same pathologies as the initial Hoˇrava- does nothold), bydirectly tackling these equations. But Lifshitz proposal, namely, instabilities, strong coupling a considerable simplification occurs if we focus on the special case with constant , as we now describe. R First we observe that making a 3+1 ADM decompo- ∗ fl[email protected] sition adapted to an Eulerian observer tangent to n, we 2 have space-times - in a cosmological space-time volume ele- mentsnaturallychangewithtime. Thuswewillfocuson R=R+KijKij −K2−2N−1∆N , (3) the case with ΛC = 0. Then, (8) and (9), determine q and C to be q = C = 1, or, equivalently, = 0. For where N is the lapse, Kij the extrinsic curvature and vacuum solutions of the Einstein equationsRthis means R,∆ the 3-dimensional Ricci scalar and Laplacian. that D (nµD nν)=0 which is indeed verified as a con- µ ν Then, denoting sequence of K =D nν = 0. We thus arrive at the main ν message of this note: GN Any vacuum solution of Einstein’s equations in a foli- C 1+ , (4) ≡ 6λ R ation obeying the maximal slicing condition is a solution r of n-DBI gravity. the field equations reduce to There is a simple and nice way to double check this 6λ R N−1∆N + (1 qC)=0 , (5) result [13]. In [7] it was observed that the n-DBI action − GN − could be linearised, by introducing an auxiliary field e; then the action takes an Einstein-Hilbert form (albeit in a Jordan frame) j(K h K)=0 , (6) ij ij ∇ − 1 Se = d4x√ ge[ 2G Λ (e)] , (12) N C −16πG − R− N N Z £nKij = ∇i∇Nj −Rij −KKij +2KilKjl+hijGNΛC , where ΛC(e) is given by (8) with C → 1/e in the right handside. ThisreducestoGRcoupledtoacosmological (7) constant if the auxiliary field is constant. Since the aux- 1 iliary field equation of motion yields (4) with C 1/e, where£n = N(∂t−£N)istheLiederivativealongnand theconstancyofeisequivalenttotheconstancyo→f . In £ is the Lie derivative along the shift. The Hamilto- N R nianconstraint,momentumconstraintsandthe dynami- vacuum (4)R = 0 and thus, from (2), Dµ(nµDνnν) must calequationsarethose ofEinsteingravitywitha cosmo- beconstant,forwhichmaximalslicingisasufficientcon- logical constant dition. The above resultgives a concrete handle to obtain ex- 3λ 2 plicit solutions to n-DBI gravity, by invoking the liter- Λ = (2qC 1 C ) . (8) C G2 − − ature on maximal slicing. Let us first illustrate this by N reconsidering the spherically symmetric solutions. It follows, as observed in [1], that any solution of Ein- In [1] a family of spherically symmetric, time inde- stein’s gravity plus a cosmological constant admitting a pendent solutions to n-DBI was obtained, of which the foliation with constant is a solution of n-DBI gravity. relevant sub-set we wish to consider reads: R This slicing property canbe rephrasedin terms of(5). 2 2GNM1 C3 2 The foliation is such that ds = 1 + dT + − − r r4 (cid:18) (cid:19) 6λ 2 −1 R−N ∆N = GN(qC−1)=constant . (9)  1 2GdNrM1 + C3 +r2GNrM2 + Cr43dT +r2dΩ2. Let us also note that using the evolution equation for − r r4 K and the Hamiltonian constraint one arrives at the q  (13) ij following equation for the evolution of the trace of the As a four dimensional geometry, this is simply the extrinsic curvature: Schwarzschild manifold, with mass M = M1 + M2, and can be transformed into Schwarzschild coordinates 1 ∆N 2 (∂t £N)K = K + R+3GNΛC . (10) (t,r,θ,φ) by the non-FPD: N − − N − Educatedobservationofequations(9)and(10)unveils dt=dT 1 2GNrM2 + Cr43 dr . (14) thefollowingfact: choosingthemaximalslicingcondition − 1 2GNM s1 2GNM1 + C3 − r − r r4 for the foliation, defined as K = 0 = ∂ K [12], which t But taking as equivalent solutions only those related by clearly implies £nK =0, requires, from eq. (10), FPDs, these Schwarzschild geometries are, generically, −1 inequivalent. Moreover, scalar quantities constructed R N ∆N =3G Λ . (11) N C − from the extrinsic curvature are now curvature invari- Maximalslicingisinterpretedastherequirementthatthe ants, since they are preserved by FPDs. The ‘geometric volumeelementassociatedtoEulerianobserversremains invariant’ K reads: constant. This slicing is typically considered (mostly in 3GNM2 K = , (15) the field of numerical relativity) in asymptotically flat −√C3+2GNM2r3 3 and therefore taking M2 = 0 we obtain a family of lutionofGR ina maximalslicing is a solutionto n-DBI. Schwarzschildgeometries,parameterisedby C3, all max- The potential relevance of this observation depends, of imally sliced. These are described in the literature [12], course, on the physical interpretation of the solutions in where the constantC3 parameterisingthe differentmax- n-DBI gravity. In particular: is this Kerr geometry a imal slicings is the Estabrook-Wahlquist time [14]. black hole in some meaningful way? Indeed the concept WenowturnourattentiontotheKerrgeometry. Start ofablackholeasatrappedregion,causallydisconnected by noting that in the standard Boyer-Lindquist coordi- fromsomeappropriate‘exterior’requiresanunderstand- nates (t,r,θ,φ), the Kerr geometry is maximally sliced. ingofallpropagatingdegreesoffreedom. Inmodelswith To realize this it is enough to notice that, in these coor- abreakdownofLorentzinvariance,thisissuerequiresde- dinates,theADMformoftheKerrgeometryhasalapse, tailed analysis,since different modes of the gravitational shift and 3-metric of the form: field may obey different dispersion relations. In n-DBI gravity, however, the extra scalar graviton mode, which N =N(r,θ) , Ni∂i =Nφ(r,θ)∂φ , hij =hij(r,θ) . exists besides the two usual tensorial graviton modes of (16) GR, does not seem to propagate [7], and hence, we may Thenaturalfoliationintroducedbythesecoordinates,or- tentatively conclude that this Kerr geometry indeed de- thogonal to nµ =(1/N, Ni/N), has therefore extrinsic scribes a Kerr black hole in n-DBI gravity, as in GR. − curvature with trace Letusbrieflycommentonfoliationswithconstantbut 1 non-vanishing K. If we take q to have the value q = K =Dµnµ =−N√h∂i √hNi =0 . (17) 1+G2NΛC/(6λ), required by the Einstein gravity limit (cid:16) (cid:17) [1],wefind,from(8),thatC =1orC =1+G2 Λ /(3λ). N C Thus, this is a maximal slicing and therefore yields a Then,using(9),(10)andtheconstancyofKwegetK2 = solution of n-DBI gravity (since it is a solution of GR). 2G Λ or K2 = G3 Λ2/(3λ). Phenomenology of n- N C − N C In principle, by applying a non-FPD to the Kerr ge- DBI gravity requires a positive λ [2]; thus we conclude ometryinBoyer-Lindquistcoordinates,andimposingthe that an Einstein space foliated with constant K yields a preservation of maximal slicing yields a family of Kerr solution of n-DBI gravity (with the chosen q) if K2 = geometries, which solve the equations of n-DBI gravity. 2G Λ . It can be verified that the foliation of Kerr- N C Concretely,we should redefine the time coordinate t (A)dS naturallyinduced by Boyer-Lindquistcoordinates −→ T =t H(r,θ),whereH(r,θ)isknownastheheightfunc- does not obey this constraint. − tion,buildthenormalunitvectorn = ND T,withN determined by the normalization coµndi−tion oµf n, and fi- In closing, let us observe that Schwarzschild-dS in a McVittie slicing has K2 = 3G Λ [15]; this provides a nallyaddthe conditionofvanishingK, ∂ (√ gnµ)=0. N C µ − solution of n-DBI gravity with constant , but for q = One then finds an explicit PDE whose solutions yield R 1/C, rather than the aforementioned requirement. a set of Kerr geometries. It would be interesting - and indeedhasbeenattempted,mostlybythenumericalrela- Acknowledgements. We would like to thank S. Hi- tivitycommunity-tofindexplicitsolutionstothisPDE. rano for discussions. F.C. and M.W. are funded by But no explicit analytic solutions are known and finding FCT through the grants SFRH/BD/60272/2009 and them seems by no means a straightforwardtask. SFRH/BD/51648/2011. The work in this paper is also The analysis already made in [1], complemented by supported by the grants PTDC/FIS/116625/2010 and the observationmade herein,showsthat anyvacuumso- NRHEP–295189-FP7-PEOPLE-2011-IRSES. [1] C. Herdeiro, S. Hirano and Y. Sato, Phys.Rev. D84, [8] E. Barausse and T. P.Sotiriou, 1212.1334. 124048 (2011), [1110.0832]. [9] A. Wang, 1212.1876. [2] C. Herdeiro and S. Hirano, JCAP 1205, 031 (2012), [10] J. York,James W., Phys.Rev.Lett. 28, 1082 (1972). [1109.1468]. [11] G. Gibbons and S. Hawking, Phys.Rev. D15, 2752 [3] D.H. Weinberget al., 1201.2434. (1977). [4] P. C. Freire et al., Mon.Not.Roy.Astron.Soc. 423, 3328 [12] M. Alcubierre, Introduction to 3+1 numerical relativity, (2012), [1205.1450]. International series of monographs on physics (Oxford [5] T. Jacobson and D. Mattingly, Phys.Rev. D64, 024028 Univ.Press, Oxford, 2008). (2001), [gr-qc/0007031]. [13] S. Hirano, Private communication . [6] P.Horava, Phys.Rev.D79, 084008 (2009), [0901.3775]. [14] F. Estabrook et al., Phys.Rev.D7, 2814 (1973). [7] F. S. Coelho, C. Herdeiro, S. Hirano and Y. Sato, [15] M. Zilhao et al., Phys.Rev. D85, 104039 (2012), Phys.Rev.D86, 064009 (2012), [1205.6850]. [1204.2019].