Asymptotically Lifshitz spacetimes with universal horizons in $(1 + 2)$ dimensions

0.74 MB·

by  Sayandeb Basu

Asymptotically Lifshitz spacetimes with universal horizons in (1+2) dimensions
Sayandeb Basu∗
Physics Department, University of the Pacific, Stockton, CA 95211, USA
Jishnu Bhattacharyya†
School of Mathematical Sciences, University of Nottingham, University Park, Nottingham, NG7 2RD, UK
David Mattingly‡ and Matthew Roberson§
Department of Physics, University of New Hampshire, Durham, NH 03824, USA
Hoˇrava gravity theory possesses global Lifshitz space as a solution and has been conjectured to
provideanaturalframeworkforLifshitzholography. Wederivetheconditionsonthetwoderivative
Hoˇrava gravity Lagrangian that are necessary for static, asymptotically Lifshitz spacetimes with
flattransversedimensionstocontainauniversalhorizon,whichplaysasimilarthermodynamicrole
6 as the Killing horizon in general relativity. Specializing to z = 2 in 1+2 dimensions, we then
1 numerically construct such regular solutions over the whole spacetime. We calculate the mass for
0 thesesolutionsandshowthat,unliketheasymptoticallyanti-deSittercase,thefirstlawappliedto
2
the universal horizon is straightforwardly compatible with a thermodynamic interpretation.

n
a
J I. INTRODUCTION highercurvaturerelativisticgravity[4,5]1. Hoˇravagrav-

3 itythereforeservesasawell-behavedcandidatetheoryof
1 quantum gravity.
ConstructionofholographicdualsforLifshitzfieldthe-
Our interest is in using Hoˇrava gravity as a gravita-
oriesisanimportantandactivelineofresearch. Suchdu-
] tional dual to non-gravitational Lifshitz field theories.
h als release holographic approaches from the straitjacket
t ofrelativisticconformalfieldtheoryandtherebytremen- Typicallyaholographicconstructionfirstrequiresadual-
-
itybetweenazerotemperaturefieldtheoryonthebound-
p douslyexpandthetypesofsystemsholographicmethods
e can be applied to. Any gravitational dual to a Lifshitz ary and a bulk solution. Indeed, it has been argued that
h Hoˇrava gravity on a globally Lifshitz background pro-
field theory must possess solutions that exhibit Lifshitz
[ videsabettergravitationaldualforzerotemperatureLif-
symmetry somewhere in the spacetime. Lifshitz geome-
1 try is not a solution of the vacuum Einstein equations, shitz field theories, as certain quantities not reproduced
v however, and so gravitational duals of Lifshitz field the- in a relativistic gravitational dual naturally fall out from
4 Hoˇrava gravity when considering a global Lifshitz solu-
ories generally either possess extra tensor fields or oth-
7 tion [7].
erwise modify the Einstein-Hilbert action of general rel-
2 In the usual constructions, one extends a zero temper-
3 ativity. For example, spacetimes with Lifshitz geometry
ature field theory duality to finite temperature by con-
0 somewhereinthebulkcanbesolutionsofgeneralrelativ-
sidering gravitational solutions containing a black hole
. ity with extra gauge fields [1], Einstein-Maxwell-dilaton
1 in the bulk, with the Hawking temperature of the black
theory [2] and Einstein-Proca theory [3].
0
hole corresponding to the temperature of the dual the-
6 Lifshitzsymmetryeitherasymptoticallyorinthebulk
ory. In the Hoˇrava case, however, this identification be-
1 is not an inherent feature of any of the above theories,
comes immediately problematic as black hole thermody-
v: butmerelyaclassofsolutions. Thereisonegravitational namics in Hoˇrava gravity is poorly understood. Due to
i theory, however, where Lifshitz symmetry is in fact in-
X the non-relativistic Lifshitz symmetry in the UV, high
timately related to the structure of the theory: Hoˇrava-
energy excitations in Hoˇrava gravity can typically prop-
r Lifshitz theory, or Hoˇrava gravity for short [4]. Hoˇrava
a agate faster than light. Excitations propagate towards
gravity is a modified theory of gravity with a preferred
the future relative to the preferred foliation, and hence
foliation. The preferred foliation on the spacetime per-
thereisawell-definednotionofcausality[8], butUVex-
mits a splitting of spacetime into space and time in a
citations can escape from inside a Killing horizon of a
preferred manner, thereby allowing for the imposition of
static black hole solution in Hoˇrava gravity. Therefore
aLifshitzsymmetryonthetheoryathighenergies. This
the usual Killing horizons familiar from general relativ-
in turn renders the theory power counting renormaliz-
ity (and extensions such as apparent horizons appropri-
able without introducing ghosts, unlike what happens in
ate to more dynamic settings) no longer play the role of
causal boundaries. As a consequence, there is no generic
firstlawassociatedwithKillinghorizons[9]andhenceno
∗ sbasu@pacific.edu
† [email protected]
‡ [email protected] 1 Infact,projectable Hoˇravagravityisperturbativelyrenormaliz-
§ [email protected] able[6].
2
horizon thermodynamics. Hence it is unclear how to ex- inganddiscontinuitycalculationsusingeternaluniversal
tendanydualitybetweentheglobalLifshitzsolutionand horizon geometries indicate that they do radiate ther-
a zero temperature Lifshitz field theory on the boundary mally[17,18],althoughcalculationsincollapsinggeome-
to finite temperature. tries give a different picture [19]. Obviously for a com-
A possible prescription for establishing a finite tem- plete holographic construction a full thermodynamics of
perature duality is provided by analyzing the physics of universalhorizonsmustbebuilt. Inthispaperwetakea
universalhorizons. Universalhorizonsarethetruecausal moremodestgoal: ifaholographicconstructionforfinite
boundaries of bounded bulk regions in non-projectable temperature Lifshitz field theories using Hoˇrava gravity
Hoˇrava gravity [10, 11]. While the notion of the univer- is to be constructed, we need to, at the very least, find
salhorizoncanbeformalizedbeyondanysymmetries[8], staticsolutionsthatareasymptoticallyLifshitzandcon-
forourpresentpurposeitsufficestopresentthemwithin tain universal (and Killing) horizons in Hoˇrava gravity.
the context of spherically symmetric black hole space- These solutions, which we construct numerically, are the
timeswithflatorAdSasymptotics. Ifwelabeltheleaves focusofthispaper. Forearlierattemptsinthisdirection
of the preferred foliation by a scalar function T and de- see, e.g. [20, 21].
note one such leaf by Σ then each Σ can bend in such In order to minimize the algebraic complexity of the
T T
awayastostillcreateaneventhorizonevenforarbitrar- field equations we reduce to 1+2 dimensions, although
ily fast excitations, as shown in figure 1. Any excitation our approach is easily adaptable to higher dimensions
trapped inside the universal horizon (dotted region in as long as one assumes transverse planar, rather than
figure 1) has to move ‘backward in time’ with respect to spherical,symmetry. ThereductiontoD =1+2isnota
the preferred foliation in order to escape to infinity, and hindrance for eventual holographic uses, as for example
thereby violate causality. Universal horizons have been AdS /CFT dualityisoneofthebestunderstoodimple-
3 2
foundinD =1+2dimensioalHoˇravagravity,inanalogy mentations of holography.
toBTZblackholes[12],insphericalsymmetrywithAdS, The paper is organized as follows. In section II, we
and flat asymptotics in four dimensions [10, 11, 13], and introduce Hoˇrava gravity, the reduced action, and the
for the slowly rotating asymptotically flat case in four relevant equations of motion. In section III we review
dimensions [14]. the global Lifshitz solution in D =1+2 dimensions and
detailhowthechoiceofLifshitzasymptoticsrestrictsthe
coefficientsintheLagrangian. Wealsodiscusstheconse-
v
quence of the so called ‘spin-0 regularity’ in this section.
n
oziroh T = c o n sta nt IsmnoumsleaeceteiaxonandmIVpfilrewstesoldalwuestsciorainrbsee.poTruerhseennuctmoerderreiisncppsoernocdtciienodngurVSem.aFanridnragfloilvrye-,
las wesummarizeintheconclusionsVI.Throughoutthepa-
r
e per we use metric signature (−,+,+).
v
in
U r
Χa II. HORˇAVA GRAVITY AND EQUATIONS OF
MOTION
Killing horizon
A. The action and equations of motion
FIG. 1. Bending of the preferred (T = constant) hyper-
surfaces(thickbrownlines)iningoingEddington-Finkelstein
Hoˇrava gravity can be covariantly formulated as a
type coordinates in a static and spherically symmetric black
hole solution of Hoˇrava gravity. The Killing vector χa points scalar-tensor theory, where the dynamical scalar field T,
called the khronon, always admits a non-zero timelike
upward throughout everywhere. The vertical green line is a
constantrhypersurfaceanddenotestheusualKillinghorizon gradient everywhere on-shell. This allows one to con-
defined by g χaχb = 0. The universal horizon of a Hoˇrava struct a unit-timelike hypersurface orthogonal one-form
ab
gravityblackhole, denotedbytheverticalblueline, isalsoa u , called the æther, such that
a
constant r hypersurface (located at r = ruh) defined by the
conditionuaχa =0,whereua istheunittimelikenormalvec- ua =−N∇aT, gabuaub =−1 , (1)
tortotheconstantT hypersurfaces. Thedottedregioninside
where the function N is solved for via the unit norm
the universal horizon (i.e. for r < ruh) denotes a black hole
constraint as follows
region even for arbitrary fast excitations; the constant T hy-
persurfacesforthisregionarenotshowntokeepthediagram N−2 =−gab(∇ T)(∇ T) . (2)
clean. a b
Besides the usual diffeomorphism, Hoˇrava gravity is
Universalhorizonsdoobeyafirstlaw[15,16]. Tunnel- also invariant under arbitrary reparametrizations of the
3
khronon: T (cid:55)→ T˜ = T˜(T). Under such reparametriza- inboththeories). However,thecorrespondingbulkæther
tions N is required to transform as N (cid:55)→ N˜ = equations of motion in Einstein-æther theory is
(dT˜/dT)−1N, such that the æther remains manifestly
Æ(cid:126)a =0 , (6)
invariant under the reparametrizations of the khronon.
This allows one to express the (two-derivative trun-
where Æ(cid:126)a is the ‘component’ of the functional derivative
cated/IR limit) action of Hoˇrava gravity in D = (1+2)
of the action (3) with respect to the æther which is or-
dimensions in a manifestly covariant and reparametriza-
tion invariant manner as follows [22]2 thogonal to the æther itself (i.e. uaÆ(cid:126)a = 0), while the
khronon’s equations of motion in Hoˇrava gravity reads
1 (cid:90) √
S = 16πG d3x −g(−2Λcc+R+L)+Sghy+ST,b. ∇a[NÆ(cid:126)a]=0 . (7)
æ
(3) TheformalequivalenceoftheEinstein’sequations,taken
HereΛcc isthecosmologicalconstantwhichwillbetaken togetherwiththesimilaritiesof (6)and(7),makeitclear
tobenegativeinthiswork,Risthecurvaturescalar,and that any solution of the hypersurface orthogonal sector
L is the khronon’s Lagrangian given by Einstein-æther theory is also a solution of Hoˇrava grav-
ity [24], although the converse is generally not true.
L =−Zab (∇ uc)(∇ ud) . (4)
cd a b In this work, we will restrict ourselves to static solu-
tions of Hoˇrava theory in D =(1+2) with translational
The tensor Zac is given by
cd symmetry in the transverse space (see below). In a sim-
ilar setting, the æther in Einstein-æther theory is au-
Zab =c gabg +c δa δb +c δa δb −c uaubg , (5)
cd 1 cd 2 c d 3 d c 4 cd tomatically hypersurface orthogonal as dictated by the
symmetries. One may then argue along the lines of [13]
where c1, c2, c3, c4 are coupling constants. Sghy is to conclude that solutions of Hoˇrava theory with these
thestandardGibbons-Hawking-Yorkboundarytermand
symmetries,andadmitting a regular universal horizon in
S representsanyadditionalboundarytermsnecessary
T,b addition, are also the only solutions of Einstein-æther
due to the presence of the khronon field. We will return
theory with these properties (note that the asymptotic
to the boundary terms in section IIIC when we discuss
behaviourofthesolutionsisirrelevantinthisargument).
the total mass of solutions, but these boundary terms
Therefore, it suffices to solve the Einstein-æther equa-
are irrelevant for a derivation of the bulk equations of
tions of motion to obtain the desired solutions in Hoˇrava
motion. The bulk covariant equations of motion for the
gravity; this will be the approach taken in this paper.
metricandkhrononaregeneratedbyextremizingtheac-
Eventhoughtheindividualcouplingsc ,··· ,c appear
tion(3)undervariationsoftherespectivefields,withthe 1 4
directlyintheaction(3),onemayarguethatowingtothe
assumption that the æther is derived from the khronon
hypersurfaceorthogonalityoftheæther, onlythecombi-
via (1).
nations c = c +c , c = c +c and c = c +c
The khronon equation of motion and corresponding 13 1 3 123 2 13 14 1 4
show up explicitly in all subsequent expressions [10]. Fi-
solutions of Hoˇrava gravity are most efficiently analyzed
nally, it will be useful to note the following kinematical
by leveraging the relation between Hoˇrava gravity and
quantities: aa = ub∇ ua being the acceleration of the
Einstein-æthertheory,aswenowexplain. Theaction(3), b
æther congruence, K =∇ u +u a being the extrin-
with the æther only satisfying the unit norm constraint ab a b a b
siccurvatureoftheconstantkhrononhypersurfaces,and
(i.e. not hypersurface orthogonality) and being itself
K =gabK being the corresponding mean curvature.
treated as the fundamental field, leads to Einstein-æther ab
theory [23], a vector-tensor theory of gravity coupled to
a unit timelike vector field. One may subsequently re-
B. Equations of motion under staticity and
strict attention to the hypersurface orthogonal sector of
transverse space translation symmetry
Einstein-æther theory by imposing the hypersurface or-
thogonality condition on the æther (1) at the level of
It will be convenient to use ingoing Eddington-
the equations of motion. Neglecting all boundary terms,
Finkelstein type (EF) coordinates, in which the metric
theEinstein’sequationsgeneratedbyextremizingtheac-
on a static spacetime, with translational symmetry in
tion (3) under variations of the metric, leads to formally
the transverse space, becomes
identical Einstein’s equations for both Hoˇrava gravity
andthehypersurfaceorthogonalsectorofEinstein-æther ds2 =−e(r)dv2+2f(r)dvdr+r2dy2, (8)
theory [24] (see also [25] for a more recent discussion, es-
peciallyfromtheperspectiveoftheinitialvalueproblem where r is the canonical radial coordinate, and y is the
coordinate on the transverse space. Note that y is not a
bounded coordinate; rather −∞ < y < ∞. The Killing
vector associated with staticity, denoted by χa, is given
2 ThecompleteactionofHoˇravagravitycanalsobecovariantized by χa =∂v in these coordinates, while the Killing vector
via such ‘Stu¨ckelbering’ procedure [22]. In this work, however, associated with the translational symmetry in the trans-
weonlyworkwiththeIRlimitofthetheory. verse space (i.e. under y →y+ constant) is ∂ .
y
4
The æther one-form decomposes in these coordinates equations from (12) is a necessary but not sufficient con-
as dition on static solutions of the covariant equations. We
willthereforelookforsolutionsoftheequationsfrom(12)
f(r)dr
u =(u·χ)dv+ , (9) andthencheckiftheyarestaticsolutionsofthecovariant
a (s·χ)−(u·χ) equations of motion a posteriori.
The equations for e(r), f(r), and X(r) following
where (s · χ) = s χa, sa being the unique (‘outward
a from (12) are rather complicated coupled ODEs and are
pointing’) spacelike unit vector which is orthogonal to
not particularly illuminating, so we will not reproduce
both the æther and the transverse direction. As already
them if full here. However, there are some important
mentioned, the symmetries of the spacetime make the
structural aspects that need to be mentioned. First, one
æther hypersurface orthogonal as the above expression
maynotenotethatthetimeindependentaction(12)does
also manifestly reveals (the functions (u·χ) and (s·χ)
not contain any term that is quadratic in derivatives of
are functions of r only), while the unit-norm constraint
f(r), either via f(cid:48)(cid:48)(r) or f(cid:48)(r)2. As a result, the equa-
on the æther (1) is taken into account via
tion of motion for f(r) is an algebraic equation, which
simplifies the system considerably (and is the primary
e(r)=(u·χ)2−(s·χ)2 . (10)
motivationforchoosingX(r)asafundamentalfreecom-
ponent). In our subsequent numerical analysis, we sub-
Thefunctionse(r)andf(r)capturethefreemetriccom-
stitute this algebraic expression for f(r) back into the
ponents that one needs to solve for from the equations
equations of motion for e(r) and X(r) which yields two
of motion of Hoˇrava gravity. The (symmetry reduced)
second order differential equations for e(cid:48)(cid:48)(r) and X(cid:48)(cid:48)(r)
ætherhasoneadditionalfreecomponent. Itwillbealge-
in terms of e(r), e(cid:48)(r), X(r) and X(cid:48)(r).
braically beneficial to write this component via the vari-
Second,theresultingdifferentialequationsfore(r)and
able X(r) defined by
X(r)bothna¨ıvelyhaveasingularityataparticularvalue
X(r)=(s·χ)−(u·χ) . (11) of the pair e(r) and X(r). The source of this singularity
is a feature previously found in studies of black holes
In what follows, the equations of motion will be solved in Hoˇrava gravity known as the spin-0 horizon. In the
for the functions e(r), X(r) and f(r) for reasons to be present setting, unlike general relativity in D = (1+2),
explained below, and the functions (u·χ) and (s·χ) can Hoˇrava gravity is known to contain a propagating scalar
then be determined by inverting (10) and (11). or spin-0 mode with local (low energy) speed s relative
0
Instead of adapting the fully general covariant equa- totheætherframegivenbytheexpression[22](compare
tions of motion to the above symmetries, it is more con- with the corresponding expression in D =1+3 [26])
venient to substitute the above symmetry-adapted ex-
c
pressionsforthemetricandtheætherintotheaction(3) s2 = 123 . (13)
0 c (1−c )(1+c +2c )
directly, which yields the following time independent ac- 14 13 13 2
tion
The different local speed relative to the æther frame is
(cid:90) equivalently described by stating that the low energy
S =− dr(4rf2X4)−1 spin-0 mode propagates on the light cone of an effec-
tive spin-0 metric g(0) given by
[−2eX(f(r2(c −c )e(cid:48)X(cid:48) ab
123 14
−rX(r(c +c )X(cid:48)2+c e(cid:48))+c X3) g(0) =g −(s2−1)u u . (14)
123 14 2 123 ab ab 0 a b
+4rX3f(cid:48))+X2(f(−2r2X(c +c )e(cid:48)X(cid:48)
123 14 The low energy spin-0 mode has a corresponding causal
+r2(c123−c14)e(cid:48)2+rX2(r(c123−c14)X(cid:48)2 horizon,knownasthespin-0 horizon,anditsradialloca-
−2(c −4)e(cid:48)+4re(cid:48)(cid:48))+c X4+2c rX3X(cid:48)) tion is given by the largest root of |χ|2 ≡g(0)χaχb =0,
2 123 2 s0 ab
analogoustotheKillinghorizoningeneralrelativity. On
−4r2X2e(cid:48)f(cid:48)+8Λccr2f3X2)+e2f(r2(c123−c14)X(cid:48)2 this horizon the equations of motion break down (c.f.
+c123X2−2c2rXX(cid:48))] , (12) the discussion in [27]). In our case, this is reflected in
the equations of motion for e(r) and X(r) which take
where (cid:48) denotes differentiation with respect to r. The the form
equations of motion are then generated by extremizing
F (e,e(cid:48),X,X(cid:48),r,c )
the above time independent action with respect to vari- e(cid:48)(cid:48)(r)= e i (15)
ations of the three independent free components of the 8(1−c13c14(1+c13+2c2))r2X(r)6|χ|2s0
metric and the æther: e(r), f(r), and X(r). While so- F (e,e(cid:48),X,X(cid:48),r,c )
X(cid:48)(cid:48)(r)= X i (.16)
lutions to the equations thus obtained are not always
8(1−c c (1+c +2c ))r2X(r)5|χ|2
guaranteedtobestaticsolutionsoftheoriginalcovariant 13 14 13 2 s0
equationsofmotion, thesetofsolutionsoftheequations F andF arecomplicatedandunilluminatingfunctions
e X
from(12)is guaranteedtoincludesolutionsofthefullco- and hence their full form will be omitted. On the spin-
variantequations. Inotherwords,beingasolutionofthe 0 horizon this equation will generally be unstable unless
5
F and F also vanish. This regularity requirement will Themetric(19)ismanifestlyisometricunderconstant
e X
eventually reduce our black hole solutions down to a one translations of the time coordinate t, under t (cid:55)→ −t,
parameter family. We will return to this issue when we as well as under constant translations of the transverse
describe our numerical approach. space coordinate y. More interestingly, the metric (19)
The spin-0 horizon has a useful property in that it is also isometric under scale transformations of the form
can be ‘moved around’ relative to a Killing horizon via a
field redefinition. As noted in [28], under disformal field t(cid:55)→λzt, y (cid:55)→λy, r (cid:55)→λ−1r . (20)
redefinitions, i.e. redefinitions of the form
Clearly for z > 1, the scale invariance between t and y
g(cid:48) =g −(1−σ2)u u , ua =σ−1ua, σ >0, (17) is inhomogeneous.
ab ab a b
Weareeventuallyinterestedinconstructingblackhole
theaction(3) transformsintoitselfwith simplynewval-
solutions which are only asymptically Lifshitz, and for
ues of the c coefficients. In particular, the coefficients
i that purpose it will be useful to switch to ingoing EF
transform as
coordinates (8). In particular, the metric (19) of the
global Lifshitz spacetime takes the following form in EF
1−c(cid:48) =σ(1−c ), c(cid:48) =σc , c(cid:48) =c . (18)
13 13 123 123 14 14 coordinates (compare with (8)),
The speed of the spin-0 mode is not invariant under the
disformal redefinitions, and in fact, given an initial set ds2 =−(r/(cid:96)l)2zdv2+2(r/(cid:96)l)z−1dvdr+(r/(cid:96)l)2dy2 .
(21)
of coefficients one can always perform a field redefinition
The scale transformations analogous to (20) leaving the
suchthatthespin-0speedbecomesunity. Inotherwords,
metric (21) invariant are3
one can always set the Killing horizon and spin-0 hori-
zon to be co-located without loss of generality. This will
v (cid:55)→λzv , y (cid:55)→λy , r (cid:55)→λ−1r . (22)
simplify the numerical analysis.
In the present work, we wish to seek solutions of
As discovered in [7], the global Lifshitz metric (21) is
Hoˇrava gravity with Lifshitz asymptotics and a regular
a solution to the Hoˇrava gravity equations of motion,
universal horizon in the bulk. In particular, we need to
along with the following profile for the æther (compare
solve (15) with asymptotically Lifshitz boundary condi-
with (9))
tions on the metric and æther components. To that end,
we need to derive the appropriate asymptotic behaviour ua =−(r/(cid:96)l)zdv+(r/(cid:96)l)−1dr , (23)
ofthefunctionse(r),X(r)andf(r)asr →∞,aswellas
the conditions under which the solutions are also regular In particular, the æther satisfies (as per requirement) all
in the bulk of the spacetime, especially on their respec- the above symmetries including that under (22) and is
tive spin-0horizons. These issueswill be taken upinthe aligned with the Killing vector χa everywhere4. The so-
following section, which will also pave the way towards lution parameters z and (cid:96)l are fully determined by the
the numerical construction of the sought after solutions. parameters Λcc and c14 by the following relations
z(z+1) z−1
III. ASYMPTOTICALLY LIFSHITZ Λcc =− 2(cid:96)2 , c14 = z . (24)
l
SPACETIMES
The second relation means, in particular, that the Lif-
A. The global Lifshitz solution shitz exponent is uniquely determined by the coupling
c . Noticethattheglobalsolutionisindependentofthe
14
Before we can properly discuss asymptotically Lif- values of the couplings c13 and c2.
shitz spacetimes we first must discuss the global back-
groundLifshitzsolutionwhichplaysthesameroleglobal
AdS space does for asymptotically AdS spacetimes. In B. Asymptotic expansion
D = 1+2 dimensions in the canonical (Schwarzschild-
type) t, r and y coordinates (y being the transverse co- Moving on to static, asymptotically Lifshitz space-
ordinate), the global Lifshitz spacetime introduced in [1] times, it is not immediately clear under what conditions
isanobviousgeneralizationofAdS spacetime, butwith the various metric and æther coefficients admit a well-
3
inhomogeneous scale invariance between space and time. defined power series in r−1; this is a concern especially
In its standard/canonical form, the metric of the global
Lifshitz spacetime in D =1+2 is
ds2 =−(r/(cid:96)l)2zdt2+(r/(cid:96)l)−2dr2+(r/(cid:96)l)2dy2 , (19) 3 This follows from the definition of the v coordinate: dv =dt+
(r/(cid:96)l)−(z+1)dr.
where the constant z (cid:61) 1 is the (Lifshitz) scaling expo- 4 It can be easily proved that in a globally Lifshitz solution, the
nent and the fixed length scale (cid:96)l is the Lifshitz scale. equations of motion of Hoˇrava gravity forces the æther to be
For z =1, the metric (19) describes AdS . globallyalignedwiththeKillingvectorχa.
3
6
when z is non-integer. We must therefore construct a r−ν(cid:63) as follows:
useful parametrization of the asymptotic forms of the
various metric and æther coefficients around r =∞. E (r)=1+ e1(z+1)(cid:96)νl(cid:63) + e2(z+1)2(cid:96)2lν(cid:63) +O(r−3ν(cid:63)) ,
While in the global Lifshitz solution the æther is glob- 0 rν(cid:63) r2ν(cid:63)
aiclalyllyalLiginfsehditwzicthasethtehKisilwliinllgnvoetctboer tχhae, cinasaenevaesyrymwphteorte- U0(r)=1+ u1(zr+ν(cid:63)1)(cid:96)νl(cid:63) + u2(z+r2ν1(cid:63))2(cid:96)2lν(cid:63) +O(r−3ν(cid:63)) ,
in the spacetime. Rather, we merely require an asymp-
totic alignment between the æther and χa. The addi- F (r)=1+ f1(z+1)(cid:96)νl(cid:63) + f2(z+1)2(cid:96)2lν(cid:63) +O(r−3ν(cid:63)) .
tional measure for the ‘misalignment’ between the æther 0 rν(cid:63) r2ν(cid:63)
(28)
and χa is conveniently captured through the quantity
(s·χ)≡s χa, where sa, as introduced previously is the
a In particular, the O(r−nν(cid:63)) coefficient for some integer
unique outwards pointing unit spacelike vector orthog- n(cid:61)1hasbeendefinedwithanexplicitfactorof(z+1)n
onal to the æther and the transverse directions every-
forconveniencewiththeasymptoticanalysis,aswellasa
where. Intuitively, we wish to define an asymptotically factor of (cid:96)nlν(cid:63) has been included to make the coefficients
Lifshitz spacetime in the present context as a spacetime
dimensionless.
where the æther becomes aligned with the Killing vector
The above expansions yield analogous expansion for
andthemetricapproachestheglobalLifshitzsolutionas
(s·χ) and X(r) via (10) and (11). For the expansion
r → ∞. These conditions can be properly implemented
of (s·χ) in particular, we may start with the following
in the present coordinates by requiring
expression
lim √(u·χ) =−1 , (s·χ)2 =(r/(cid:96)l)2z(cid:2)U0(r)2−E0(r)(cid:3) ,
r→∞ −χ·χ
(s·χ) which follows from (10). If we plug in the ansatz (28)
lim √ =0 , (25) above, we end up with the following asymptotic be-
r→∞ −χ·χ
haviour for (s·χ)
f(r)
lim =1.
r→∞rz−1 (s·χ)2 =(r/(cid:96)l)2zO(r−nsν(cid:63)) ,
Since we wish our solutions to smoothly approach the which encompasses the possibility that, for some integer
global solution upon tuning some parameters (e.g. the n (cid:61)1, the first (n −1) terms in the series for U (r)2−
s s 0
mass) we can factor out the appropriate global Lifshitz E (r) are zero. In other words, (s · χ) may have the
0
behaviours from (u·χ), e(r) ≡ −(χ·χ), and f(r) and following asymptotic behaviour
write
e(r)=(r/(cid:96)l)2zE0(r) , (s·χ)=(r/(cid:96)l)z−ns2ν(cid:63)S0(r) , (29)
(u·χ)=−(r/(cid:96)l)zU0(r) , (26) along with
f(r)=(r/(cid:96)l)z−1F0(r) , S (r)=c + s1(z+1)(cid:96)νl(cid:63) + s2(z+1)2(cid:96)2lν(cid:63) +O(r−3ν(cid:63)) ,
0 æ rν(cid:63) r2ν(cid:63)
such that the conditions (25) becomes equivalent to
(30)
such that for a given n , c (cid:54)= 0 is a constant which
s æ
lim E (r)=1 , lim U (r)=1 , lim F (r)=1 .
0 0 0 captures the leading order behaviour of S (r). In partic-
r→∞ r→∞ r→∞ 0
(27) ular, the case n = 0 is not allowed on account of the
s
Asymptotically Lifshitz spacetimes are not, of presumed asymptotically Lifshitz behaviour; indeed, for
course, necessarily solutions of the Hoˇrava gravity all n (cid:61) 1 one finds that the second condition in the
s
equations of motion. Rather, for asymptotically Lifshitz definition of asymptotic Lifshitz-ness proposed in (25) is
solutions, the functions U (r), E (r) and F (r) not only also met. This expansion for (s · χ) also generates an
0 0 0
must have well-defined limits to r = ∞ but also must asymptotic expansion for X(r) via (11).
satisfy an asymptotic expansion of the equations of
motion. As we shall see, the asymptotic equations of
motions yield a significant restriction on the choice of C. Boundaries, mass and determination of ν(cid:63)
the c coefficients.
i
In order to compute the asymptotic equations of mo- To proceed in the expansion we need to determine ν ,
(cid:63)
tionweneedsomeconvenientparametrizationofthefall- whichcanbedonebyrequiringanon-zerobutfinitemass
offsofthesefunctionsasr →∞. Tothatend,wewillas- for the black hole solutions. Unlike the local equations
sume that given some z, there exists a number ν > 0 of motion, the total mass does depend on the boundary
(cid:63)
such that the functions U (r), E (r), and F (r) are ana- terms present in the action (3). Therefore, the first step
0 0 0
lytic at r = ∞ in r−ν(cid:63), i.e. , they all admit well defined is to deal with the additional possible boundary term
power series (albeit asymptotic) expansions in powers of S .
T,b
7
Boundarytermsareintroducedintoactionssothatthe ofthecorrespondingglobalsolution,hereLifshitz,which
variational principle is well defined. The variation of the satisfies the Einstein-æther theory equations of motion.
GHY term, for example, explicitly cancels the boundary Hence the asymptotic (Lifshitz) boundary condition im-
term generated when varying the Einstein-Hilbert term plies Æ(cid:126)a = 0 rather than ∇ (NÆ(cid:126)a) = 0. Consequently,
a
and imposing Dirichlet boundary conditions on the met- there is never a boundary contribution from the second
ric. Since Hoˇrava gravity has the Einstein-Hilbert term term in (31) for the present choice of ∂V.
in the bulk action (3), the GHY term is necessary if we
maintain Dirichlet boundary conditions for the metric. The lack of a boundary term proportional to δT
We must check, however, if a) the other terms in the matches the intuition gained by considering the funda-
HoˇravagravityactionarecompatiblewithDirichletmet- mental reparametrization invariance of Hoˇrava gravity.
ric conditions, b) what type of boundary conditions are Ontheboundary,suchvariationscanalwaysbeabsorbed
appropriate for the khronon, and c) if additional bound- by leveraging the reparametrization invariance. Con-
ary terms are generated from the khronon variation. sequently, setting Dirichlet boundary conditions on the
The variation of the bulk Hoˇrava gravity action (3) khronon is inappropriate. Rather, Neumann boundary
yields the following additional boundary variations condition is the appropriate type of boundary condition
for the khronon. We are requiring our spacetimes to be
δS = (cid:90) d2x√h(cid:104)ncB δgab+ncB δT +ncB a(∇(cid:126) δT)(cid:105), asymptotically Lifshitz, which in turn puts a condition
b abc c c a on ua at infinity; namely, it aligns with the asymptotic
∂V Killing vector that generates stationarity on I. Since
(31)
u is related to the gradient of T (1), such Dirichlet con-
wherena isthenormaltotheboundary∂V,h isthein- a
ab ditions on ua correspond to Neumann conditions on the
ducedmetricontheboundary,∇(cid:126)aistheprojectedspatial khronon. The appropriate condition to maintain ua ori-
covariant derivative on the preferred foliation, and Babc, entation at infinity (i.e. on I) is, in fact, precisely that
B b and B are tensors built out of g , h , u and
a a ab ab a thespatialgradientofthekhrononvariationvanishes,as
their derivatives. We immediately see that with Dirich-
non-zero spatial gradients are exactly what would ‘tilt’
letboundaryconditionsforthemetricthefirsttermvan- ua. Therefore, we impose Neumann conditions on the
ishesandhencetheboundaryanalysisforthemetricpro- khronon, in particular require ∇(cid:126) δT = 0 on I, which
ceeds exactly as it does in general relativity: addition a
kills the third boundary term in (31). In summary, at
of the GHY term and Dirichlet boundary conditions for
least for the kind of boundaries we have considered so
the metric makes the variational principle well defined
far, the only boundary term necessary in our construc-
for metric variations. Therefore the particular (compli-
tionistheusualGHYtermsincetheappropriatephysical
cated)expressionforB isirrelevantforoursubsequent
abc boundary conditions are Dirichlet for the metric varia-
discussions and we will omit it.
tions and Neumann for the khronon variations.
The khronon variation is more subtle, as we have
boundaryvariationsin(31)thatinvolvebothdirectvari- Things are more sutle if the spacetime admits a uni-
ations of the khronon and also derivatives of the varia- versalhorizon. Inthiscase,imposingDirichletboundary
tions. Insight can be gained by examining what consti- condition on the metric and Neumann boundary condi-
tutes the boundary ∂V, as well the structure of Ba and tion on the khronon on every boundary surface still suf-
B b, which are given by fices to kill the first and the last terms in (31), thereby
a
savingusfromintroducingadditionalboundarytermsin
Ba =−2NÆ(cid:126)a , Bca =2NZabcd∇bud , (32) theaction. However,theuniversalhorizonraisesthepos-
sibility of additional ‘inner boundaries’ in the spacetime
where Zab is defined in (5). In the simplest setting on which the contribution from the middle term in (31)
cd
with a spacetime without any horizons and/or singulari- isnotnecessarilyzeroatfirstsight. Toresolvethis,letus
ties, ∂V consists of the boundary at (spatial) infinity to takeacloserlookatuniversalhorizonsinasymptotically
be denoted by I in what follows5, as well as the bound- Lifshitz spacetimes.
aries at infinite past and future. Since we need to adapt
to the preferred foliation, the boundaries at infinite past In a stationary spacetime with flat asymptotics, the
andfuturearealsoslicesofthepreferredfoliation. There- universal horizon is a leaf of the preferred foliation that
fore, given the form of B (32), the contribution of the barely fails to reach the boundary at infinity [8]. Any
a
second term in (31) vanishes on the boundaries at infi- preferred slice that reaches spatial infinity never crosses
nite past and future, since na = ua on these surfaces. the universal horizon, but instead asymptotes to it. The
On the other hand, the field configuration on I is that causal structure is intuitively much more accessible in
spherically symmetric spacetimes, where the high degree
of symmetry allows one to appeal to e.g. figure 1 and
conclude that the universal horizon has to be a leaf of
5 DuetothemodifiedcausalstrutureofspacetimesinHoˇravagrav- thepreferredfoliationwhichissimultaneouslyaconstant
ity,theboundaryatspatialinfinityistheonlyrelevantboundary r hypersurface and therefore orthogonal to the Killing
atinfinity;see[8]forfurtherdetails. vector of stationarity χa. In other words, the universal
8
horizon is locally characterized by the condition Lifshitz black hole solution using the preferred foliation
is given by the familiar Hawking-Horowitz formula [29],
(u·χ)uh =0 , (33) 1 (cid:90) (cid:104) (cid:105)
whoseradiallocationwillbedenotedbyruh6. Theabove M =−8πGæ Nkˆ−Napabnb −Mgl , (34)
argument can be made more rigorous, and the condi- B
tion (33) still suffices as a local characterization of the where B is the suitable ‘one-boundary’ – cross-sections
universal horizon in the most general stationary space- of the boundary ∂V – on which the ‘surface terms’ in
times, as long as the quantity (a·χ) (cid:54)= 0 on the said the Hamiltonian (here generated purely from the GHY
surface [8]. term)contribute, Na istheshiftvector, p istheconju-
ab
In spacetimes with Lifshitz asymptotics, since the gate momentum of the induced metric on the preferred
Killing vector χa is timelike asymptotically, we have the foliation, kˆ is the trace of the extrinsic curvature of B,
desired asymptotic behaviour of the spacetime to utilize and Mgl is the mass of the background Lifshitz solution
the settings of [8]. Additionally, the quantity (a·χ)(cid:54)=0 (to be explained in more details below).
on the universal horizon (see figure 4). Hence, condi- To compute the integral (34) for the asymptotically
tion(33)alsoprovidesthesuitablelocalcharacterization Lifshitz solutions considered in this work, we may begin
of the universal horizon here, and the causal structure of by choosing the time translation vector along the Killing
the spacetimes we are dealing with is still qualitatively vector χa, for which the shift vector becomes the projec-
as captured in figure 1. tion of χa on the leaves of the preferred foliation. Fur-
In an asymptotically Lifshitz spacetime with a uni- thermore, by construction, there is no intersection of the
versal horizon (just as in the corresponding case of an ‘one-boundary’ B with any part of ∂V which itself is
asymptotically flat spacetime), one may divide up the a leaf of the preferred foliation (since na = ua on such
spacetime into two (causally) disjoint regions, namely surfaces). In particular, this kills any possible contri-
the ‘outside region’ which is the part of the spacetime bution from the boundaries at past and future infinity.
that is continuously connected to the boundary at infin- Forallpossible‘innerboundaries’includingtheuniversal
ity I and where (u·χ) < 0 holds everywhere, and the horizon(33),whichareallcharacterizedbythecondition
‘insideregion’whichisthecomplementofthe‘outsidere- (u·χ)=0aspreviouslydiscussed,thevanishingofthein-
gion’. Theboundaryofthe‘outsideregion’thenconsists tegrand in (34) can, in fact, be seen explicitly as follows:
of the boundary at infinity I, the boundaries at infinite for the present choice of the time translation vector, a
pastandfuture(forthe‘outsideregion’),andtheuniver- straightforward computation yields N =−(u·χ) [8] and
sal horizon denoting an inner boundary for the ‘outside kˆ =∇(cid:126) sa ∝(u·χ) and hence Nkˆ =0 on any (u·χ)=0
a
region’. One may invoke our previous logic to conclude hypersurface; in particular the vanishing of kˆ can be ap-
that (31) vanish on the boundary at infinity, as well as
preciated from the fact that sa ∝ χa on any (u·χ) = 0
the boundaries at infinite past and future, if Dirichlet
hypersurface,sothatbyKilling’sequation∇(cid:126) sa vanishes
and Neumann boundary conditions are imposed on the a
here. The second term in the integrand Nap nb van-
metric and the khronon, respectively. More importantly, ab
ishes simply because p is a linear combination of the
the (future) universal horizon coincides with the bound- ab
induced metric and the extrinsic curvature of the leaves
aryatinfinitefuture,ascanbeinferrede.g.fromfigure1,
of the preferred foliation while na = ua. Therefore, any
and hence (31) vanishes here as well. For the ‘inner re-
contributionto(34)comesonly fromthepartofBwhich
gion’, at least in the present setting, one may at most
‘resides within’ the boundary at infinity I, and this is
have a sequence of ‘inner horizons’ which are themselves
given by the line generated by the intersection of any
leavesofthepreferredfoliationcharacterizedbythecon-
preferred slice with I. Moreover, due to the asymptotic
dition(u·χ)=0(neitherofwhichareuniversalhorizons
alignment of ua with χa, the term containing the shift
however). Sincethesesurfacesareleavesofthepreferred
drops out, so that (34) for our solutions reduces to
foliation themselves, i.e. na = ua on each of them, the
boundary variation (31) vanishes on every possible inner ∞
1 (cid:90) r
boundaries as well. Therefore, even for the case of inter- M =− lim dy Nkˆ−Mgl .
est,theonlyboundarytermnecessaryinourconstruction 8πGæ r→∞ (cid:96)l
−∞
istheusualGHYtermwithDirichletboundarycondition
on the metric variations and Neumann boundary condi- As mentioned previously, Mgl is the mass of the back-
tion for the khronon variations. ground Lifshitz solution, and its relevance can be ex-
Now that the question of boundary terms has been plained as follows: the above expression without the
settled, we can proceed with calculating the mass. Since Mgl piece, when evaluated on the globally Lifshitz so-
the GHY term is the only term, the total mass M of a lutions, yields an infinity, whose origin is ultimately the
omnipresent vacuum energy. The quantity Mgl is pre-
cisely this ‘infinite mass’ ascribable to the globally Lif-
shitz background that needs to be subtracted to make
6 In a spacetime with multiple (disconnected) surfaces satisfying the above expression, applied to an asymptotically Lif-
(u·χ)=0,theoutermostoneistheuniversalhorizon. shitz solution, meaningful.
9
The total mass M as given above is still infinite, even expanding to first order yields a relationship on the c
i
with background subtraction, since B is a non-compact coefficients inadditiontothose establishedby theglobal
infinite line. As a remedy, we need to regulate the above Lifshitz solution (24), namely:
expression and work with a mass per unit length M of
4(1−c )(z−1)
theblackholesolutions. Tothatend,wemaymodifythe c = 13 . (37)
123 n (n −2)(z+1)2
above expression as s s
Therefore, physically acceptable solutions only exist for
L/2
1 1 (cid:90) r
M =− lim lim dy Nkˆ−Mgl . ns (cid:61)3 (38)
8πGæ r→∞L→∞L (cid:96)l
−L/2 and, quite remarkably, asymptotically Lifshitz solutions
existonlyfor‘discrete’choicesforc . Thesquaredspin-
where L is a ‘regulating length’ and all the M’s now 123
0 speed (13) then also becomes ‘quantized’ according to
stand for mass per unit length. The integral over the
transverse space is now trivial and allows us to cancel 4z
s2 = .
out the appearance of the regulating factor L. Using the 0 (1−c )[n (z+1)−2(z−1)](n (z+1)−4)
13 s s
appropriate asymptotic expressions from (28), we then (39)
end up with One can then easily show that for n (cid:61) 3 (38) and for
s
(r/(cid:96)l)(z+1) all z >1, the above expression for s20 is strictly positive.
M =− lim × Hence, all these backgrounds are physically acceptable.
r→∞ 8πGæ The restriction (38) implies that the analysis of the
[1 + (2u1−f1)(z+1)(cid:96)νl(cid:63)−1 +O(r−2ν(cid:63))]−Mgl . equations of motion for the next two subleading orders
(cid:96)l rν(cid:63) are completely universal. In particular, at O(r−(z+1)),
we find
Theleadingtermproportionalto(cid:96)−1 isthesourceofthe
divergent part in the mass due tolthe non-zero vacuum u =− rs , f =0 , s =0 . (40)
energy density as just noted, and Mgl is chosen to pre- 1 2(cid:96)l 1 1
cisely cancel this term. Once this is taken care of, the Note that the coefficient u is actually left undermined,
1
spacetime mass per unit length is simply given by allowing us to trade it for a length scale rs analogous to
the‘Schwarzschildradius’. Ifweplugthesevaluesin(36),
M =− lim (r/(cid:96)l)(z+1)(2u1−f1)(z+1)(cid:96)νl(cid:63)−1. the mass per unit length of the solutions take a cleaner
r→∞ 8πGæ rν(cid:63) form
This above quantity goes as r(z+1)−ν(cid:63). Therefore in the (z+1)rs
M = . (41)
limit r →∞, the expression for the mass per unit length 8πG (cid:96)2
æ l
diverges for ν < (z +1) while it goes to zero for ν >
(cid:63) (cid:63)
(z+1)butisfiniteandnon-zero(ingeneral)only forthe Next, at O(r−2(z+1)) we obtain
choice of
r2
s
u =− ,
ν(cid:63) =(z+1) . (35) 2 8(cid:96)2l
Wethereforefixν(cid:63)byrequiringthattheclassofsolutions f = (z2−1)rs2 ,
wearestudyingadmitsawell-defined,non-zeronotionof 2 8z2(cid:96)2
l
mass after the appropriate background subtraction. The c (n −2)(z+1)(n (z+1)−2(z−1))r2
expression (35) also is consistent with the standard re- s2 =− æ s 32(n +s 1)z2(cid:96)2 s .
sults for Lifshitz black branes (see e.g. [3]). Once this s l
(42)
valueofν isused,themassperunitlengthofanasymp-
(cid:63)
totically Lifshitz solution is given by The analysis, for a completely general n , can only be
s
carried out until this order, as already observed; to pro-
(z+1)(−2u +f )
M = 1 1 , (36) ceed further one needs to pick an n . However, the gen-
8πGæ(cid:96)l eral feature of all such solutions aressimilar: the solution
wherewehaveexpressedeverythingintermsofcanonical will initially depend on two free parameters, namely rs
quantitiesz and(cid:96)l. Wewillfurthermassagethisexpres- and cæ. We have already noted that rs is directly re-
sion in the next section after we solve the equations of lated to the mass of the solution (41). Although one
motionforlarger anddeterminethevaluesofu andf . cannot do this analytically, in principle c is fixed by
1 1 æ
demanding regularity on the spin-0 horizon, i.e. setting
F | =F | =0. Thisleavestheaoneparameterfam-
X s0 e s0
D. Restrictions on ci coefficients ilyofsolutionsspecifiedbyrs. Furthermore,thisanalysis
makesclearthatwemustchooseparticularvaluesforthe
Now that we have fixed ν , we may analyze the c coefficients in our numerical evolution or we will not
(cid:63) i
asymptoticequationsofmotion. Substitutingtheexpan- asymptote to a Lifshitz solution. We now turn to the
sions(28)and(30)intotheequationsofmotion(15)and numerical procedure.
10
IV. ASYMPTOTICALLY LIFSHITZ BLACK will eventually be related to the mass of the spacetime.
HOLES At r , the value of X(r) can also be chosen freely as
s0
it just changes the overall scale of the eventual solution,
The equations (15) do not yield exact solutions with but e(rs0) = 0 by definition of a Killing horizon (recall
Lifshitz asymptotics and universal horizons, so we have e(r)=−gabχaχb). Wethenanalyticallyexpande(r)and
toresorttonumerics. Thebasicapproachisdescribedin X(r) as a power series in (r−rs0) out to fourth order.
details below below and closely follows the route taken Solvingtheequationsofmotionanalyticallyorderbyor-
in [10]. der and imposing regularity by requiring that Fe and
Tobeginwith,weneedtomakechoicesforthevarious FX vanish on the spin-0 horizon relates the coefficients
couplings and paramaters. In this work, we have only for the near horizon expansion of e(r), e(cid:48)(r), X(r), and
focussed on the case of z = 2, although asymptotically X(cid:48)(r). All the coefficients are fixed other than a depen-
Lifshitz solutions with z >2 are expected to have quali- dence on a single additional, undetermined parameter µ,
tatively similar features. We will also set (cid:96)l =1 without whichexistsinadditiontors0 sincewehaveatthispoint
any loss in generality. Finally, we will apply a field re- onlyimposedregularityatthespin-0horizonbuthaven’t
definition and choose the values of c so that the spin-0 specified the asymptotic behaviour of the solutions. As
i
andKillinghorizonsarecolocated. SincetheLifshitzex- mentioned previously, only by requiring Lifshitz asymp-
ponent z is determined solely by c (24), the disformal toticsand spin-0horizonregularityareweabletoreduce
14
fieldredefinitions(18),whichdonotchangec ,preserve the solutions to a one parameter family. We start the
14
the Lifshitz exponent z as well, and hence also preserves evolution at r =rs0(1±O(10−5)) which yields an initial
(cid:96)l (24). However, as mentioned previously, the location accuracy in e(r), e(cid:48)(r), X(r) and X(cid:48)(r) vs. the exact
of the spin-0 horizon can be shifted. We use the field re- solution of O(10−20).
definitiontocolocatethespin-0andKillinghorizon, and
thenchoosecoefficientsthatsatisfythe‘discreteness’con-
dition on c (37). Our numerical results in this section B. Numerical evolution and normalization
123
and section V are for the coefficients choice
Given the initial values X(r ), r , and µ we evolve
1 9 161 s0 s0
c14 = 2 , c13 = 10 , c2 =−180 , (43) outwards from r = rs0(1±O(10−5)) respectively with
Mathematica. For generic values of µ the exterior solu-
andforn =4. Onemaycheckthatfortheabovechoice tioneventuallysignificantlydeviatesfromtheLifshitzge-
s
of coeffients one gets s =1 from (39). ometryand,infact,breaksdownatsomeradiusr . We
0 dev
With the above choices, our approach for finding search in the µ parameter space, which changes e(cid:48)(r )
s0
asymptotically Lifshitz solutions is as follows: and X(cid:48)(r ), to maximize r . In principle, by tuning
s0 dev
µ arbitrarily finely we can push r out to infinity and
1. Analyticallyexpandtheequationsofmotionabout dev
land on the ‘exact’ asymptotically Lifshitz solution. In
the spin-0 horizon and solve for e(r) and e(cid:48)(r) in
practice we tune µ until r is at least a factor of 104
terms of X(r) and X(cid:48)(r) there so that e(cid:48)(cid:48)(r) and dev
larger than r . This gives a very accurate asymptotic
X(cid:48)(cid:48)(r) remain regular. s0
Lifshitz region. It also, as promised, reduces the solu-
2. Evolve outwards and inwards from the spin-0 hori- tion space to a one parameter family controlled by rs0.
zon numerically. Evolving inwards with this µ then determines ruh.
TheinitialvalueofX(r)controlstheoverallscalingof
3. Iterate(`ala the‘shootingmethod’)e(cid:48)(r)andX(cid:48)(r) e(r), and X(r) in the solution. After each solution with
whilekeepinge(r)andX(r)fixedatthespin-0hori- some initial r has been found, we scale the asymptotic
s0
zon until the solution is asymptotically Lifshitz. solution such that the leading order term in e(r) goes
exactly as r4.
4. Perform an overall normalization on the solution,
which corresponds to choosing and initial value of
X(r) on the spin-0 horizon so that r−4e(r) → 1
C. Example solution
and r−2X(r)→1 as r →∞.
We now address each of these steps and then present Attheendofourprocedurewehaveafullsolutionover
some example numerical results. theentirespacetimeforthefunctionse(r)andX(r)(and
hence all other functions with can be expressed in terms
ofe(r),X(r)andtheirderivatives)thatisasymptotically
A. Analytic near spin-0/Killing horizon expansion Lifshitz and possesses a universal horizon. The Lifshitz
normalized coefficients (i.e. dividing the coefficients by
With the above choice of coefficients the singularity their approriate scaling in the globally Lifshitz case) for
in the equations of motion occur at the Killing horizon a typical solution is given in Fig. 2. Note that numer-
since the spin-0 horizon is colocated. The spin-0 hori- ical evolution inside the universal horizon is possible in
zon location r is a free parameter at this point but this construction and indeed Fig. 2 shows the behavior
s0