n-DBI gravity, maximal slicing and the Kerr geometry

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by  Flavio S. Coelho

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
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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
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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
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[4]. action [2]
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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[e-mail] 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
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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.
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