Published for SISSA by Springer
Received: October 13, 2016
Accepted: November 8, 2016 Published: November 21, 2016
P. Dempster,a D. Errington,a J. Gutowskib and T. Mohaupta
aDepartment of Mathematical Sciences, University of Liverpool
Peach Street, Liverpool L69 7ZL, U.K.
bDepartment of Mathematics, University of Surrey,
Guildford, GU2 7XH, U.K.
E-mail: mailto:[email protected]
Web End [email protected] , mailto:[email protected]
Web End [email protected] ,mailto:[email protected]
Web End [email protected] , mailto:[email protected]
Web End [email protected]
Abstract: We construct Nernst brane solutions, that is black branes with zero entropy density in the extremal limit, of FI-gauged minimal ve-dimensional supergravity coupled to an arbitrary number of vector multiplets. While the scalars take speci c constant values and dynamically determine the value of the cosmological constant in terms of the FI-parameters, the metric takes the form of a boosted AdS Schwarzschild black brane. This metric can be brought to the Carter-Novotn y-Horsk y form that has previously been observed to occur in certain limits of boosted D3-branes. By dimensional reduction to four dimensions we recover the four-dimensional Nernst branes of arXiv:1501.07863 and show how the ve-dimensional lift resolves all their UV singularities. The dynamics of the compacti cation circle, which expands both in the UV and in the IR, plays a crucial role. At asymptotic in nity, the curvature singularity of the four-dimensional metric and the run-away behaviour of the four-dimensional scalar combine in such a way that the lifted solution becomes asymptotic to AdS5. Moreover, the existence of a nite chemical potential in four dimensions is related to fact that the compacti cation circle has a nite minimal value. While it is not clear immediately how to embed our solutions into string theory, we argue that the same type of dictionary as proposed for boosted D3-branes should apply, although with a lower amount of supersymmetry.
Keywords: AdS-CFT Correspondence, Black Holes in String Theory, Gauge-gravity correspondence, Supergravity Models
ArXiv ePrint: 1609.05062
Open Access, c
[circlecopyrt] The Authors.
Article funded by SCOAP3. doi:http://dx.doi.org/10.1007/JHEP11(2016)114
Web End =10.1007/JHEP11(2016)114
Five-dimensional Nernst branes from special geometry
JHEP11(2016)114
Contents
1 Introduction 1
2 N = 2 gauged supergravity in ve dimensions 5
2.1 Lagrangian of ve-dimensional N = 2 gauged supergravity with vector mul
tiplets 52.2 Reduction to three dimensions 7
3 Five-dimensional Nernst branes 93.1 Solving the equations of motion 93.2 Properties of the solution 153.3 BPS solutions 203.4 Thermodynamics 213.5 Curvature properties of ve-dimensional Nernst branes 24
4 Four-dimensional Nernst branes from dimensional reduction 254.1 Review of four-dimensional Nernst branes 254.2 S1 bulk evolution 264.3 Dimensional reduction for A > 0 284.3.1 Four dimensional metrics and gauge elds 284.3.2 Momentum discretization, charge quantization and parameter counting 314.4 Dimensional reduction for A = 0 324.5 Curvature properties of four-dimensional Nernst branes 344.6 Curing singularities with decompacti cation 344.6.1 Curvature invariants 354.6.2 Tidal forces 36
5 Summary, discussion, and outlook 365.1 The ve- and four-dimensional perspective, and looking for a eld theory dual 365.2 The fate of the third law 375.3 Constructing solutions 38
A Rewriting the scalar potential 39
B Quasi-local computation of conserved charges 40B.1 The quasilocal stress tensor 40B.2 Mass, momentum and conserved charges 41
C Euclideanisation of the boosted black brane 42
D Five-dimensional tidal forces 44
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E Four-dimensional tidal forces 46E.1 A > 0 tidal forces 46E.2 A = 0 tidal forces 48E.3 Consistency with existing classi cation 48
F Normalization of the vector potential 49
1 Introduction
The Nernst law or third law of thermodynamics comes in two versions. The weak version, which states that zero temperature can only be reached asymptotically, is uncontroversial. In constrast, there is an ongoing discussion about the status of the strong version, originally formulated by Planck, which states that entropy goes to zero at zero temperature. See for example [1{4] for contrasting views on this. With regard to gauge/gravity duality, there is a natural tension between condensed matter systems, where the strong version is believed to apply generally or at least generically, and BPS and other extremal black hole solutions, where a regular, and hence normally nite, horizon is associated with a nite, and typically large entropy. This raises the question whether and how systems obyeing the strong version of the Nernst law can be modelled by gravitational counterparts.
Extremal black brane solutions obeying the strong version of the Nernst law have been found in a variety of theories [3, 5{9], including four-dimensional FI-gauged supergravity [10], where they were dubbed Nernst branes. More recently, a two-parameter family of Nernst branes parametrized by temperature T and a chemical potential was found in [11]. Asymptotically, these solutions approach hyperscaling violating Lifshitz (hvLif) geometries both at in nity and at the horizon, and therefore are interesting in the context of gauge/gravity duality with hyperscaling violation [12, 13], see also [14] and references therein. Four-dimensional Nernst branes share the typical problems of hvLif geometries, in that they exhibit curvature singularities [15, 16]. Moreover the scaling properties of the geometry at in nity suggested an entropy-temperature relation of the form S T 3, while the high-temperature asymptotics extracted from the equation of state was found to be S T .1 Since in addition the scalars showed runaway behaviour at in nity, and
the relation S T 3 is valid for AdS5, it was conjectured in [11] that the inconsistent UV
behaviour of four-dimensional Nernst branes signals a dynamical decompacti cation, and that the above problem would be cured by lifting the solutions to ve dimensions. This follows the general idea that scale covariant vacua can be obtained by dimensional reduction of scale invariant vacua [14].
In this paper we will verify this proposal and study the relation between ve-dimensional and four-dimensional Nernst branes in detail. The ve-dimensional Nernst branes will be constructed within FI-gauged minimal ve-dimensional supergravity with
1Since the brane world volume is in nite, extensive quantities such as entropy are supposed to be taken per unit volume.
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an arbitrary number of vector multiplets, by dimensional reduction to an e ective three-dimensional Euclidean theory and using the special geometry of the scalar sector. We will show that the singular asymptotic behaviour of four-dimensional Nernst branes is a compacti cation artefact, and that the ground state geometry is AdS5. A crucial role in understanding the relation between ve- and four-dimensional Nernst branes is played by the compacti cation circle, whose size changes dynamically along the direction transverse to the brane. The behaviour of this circle also solves another puzzle, namely the origin of the four-dimensional chemical potential. Five-dimensional Nernst branes turn out to be boosted AdS Schwarzschild branes, depending on two continuous parameters, the temperature T , and the linear momentum Pz. Since momentum turns into electric charge upon dimensional reduction, one might naively expect that four-dimensional Nernst branes depend on one continuous parameter, temperature T and on one discrete parameter, electric charge Q0. However, the solutions of [11] depend on an additional continuous parameter, the chemical potential . As it turns out, its origin can be traced to the fact that the compacti cation circle grows towards in nity and towards the horizon, and has a minimum in between. This minimum introduces a new scale, and since the minimal value of the radius can be varied continuously, this provides an additional continuous parameter.
The boosted AdS Schwarzschild metric we obtain by solving the ve-dimensional equations of motion is an Einstein metric and can be brought to Carter-Novotn y-Horsk y form. Such metrics describe the near horizon regions of dimensionally reduced D-branes and M-branes with superimposed pp-waves [17]. This does, however, not immediately provide us with a string theory embedding of our solutions, unless we switch o the vector multiplets. The solution we nd is valid for an arbitrary number of vector multiplets, and depends on the choice of the prepotential and on the choice of an FI-gauging through parameters cijk
and gi. While the scalars are constant, they still have to extremize the scalar potential, and therefore these parameters determine the e ective cosmological constant, and enter into the various integration constants of our solution. We will give explicit expressions in the paper. FI-gauged ve-dimensional N = 2 supergravity2 has so far only been obtained as a consistent truncation of a higher-dimensional supergravity in a very limited number of cases. The case without vector multiplets, that is pure gauged ve-dimensional N = 2 supergravity, can be obtained by reduction of IIB supergravity on Sasaki-Einstein manifolds Y p;q [18]. The STU-model and consistent truncations thereof can be obtained as consistent reductions of eleven-dimensional supergravity [19, 20]. Other consistent truncations involve hypermultiplets or massive vector multiplets and consequently have di erent types of gauging [21, 22]. The dimensionally reduced boosted D3-branes of [17] which lead to the same ve-dimensional metric should be considered as solutions of ve-dimensional gauged N = 8 supergravity, which can be obtained by reduction of IIB supergravity on S5. In this case the ve-dimensional cosmological constant is simply determined by the D3-charge, and we cannot account for the parameters cijk; gi of an FI-gauged supergravity
theory with vector multiplets. But while there is no obvious string theory embedding of our solutions, the ve-dimensional metric is still the same as for boosted D3-branes. Therefore
2We count supersymmetry in four-dimensional units.
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it is reasonable to assume that at least the same type of dictionary between geometry and eld theory will apply. We will come back to this in the conclusions. Throughout the paper we keep a strictly ve-dimensional perspective and work with relations between geometric and thermodynamical quantities without using any higher-dimensional or stringy input.
Our work includes a detailed study of the thermodynamical properties of ve-dimensional Nernst branes. Using the quasilocal energy momentum tensor we construct expressions for the mass M and momentum Pz. From the near horizon behaviour of the solution we obtain the entropy S and through the surface gravity of the Killing horizon, the temperature T . We verify the validity of the rst law, as well as the strong version of the third law. The solution is shown to be thermodynamically stable. We obtain an equation of state and show that the relation between entropy and temperature interpolates between S T 3 at high temperature and S T 1=3 at low temparture. This asymptotic be
haviour agrees with the literature on boosted D3-branes [17, 23] and veri es the prediction of [11]. One subtlety is that the metric admits a reparametrisation, which naively removes the integration constant corresponding to the temperature (as long as temperature is nonzero) from the solution.3 However, as the detailed analysis shows, when properly setting up thermodynamics using the quasi-local stress energy tensor, temperature is de ned by a reparametrisation invariant expression. The additional input that thermodynamics requires is the choice of the norm of the static Killing vector eld, which should be considered as part of the choice of the AdS5 groundstate.
Besides the trivial extremal limit, which is global AdS5, the solution admits a nontrivial double scaling limit, where temperature goes to zero, and the boost parameter goes to in nity while the momentum (density) is kept xed. This limit was studied (in di erent coordinates) in [17], where it was shown to result in a homogenoeus Einstein space of Kaigorodov type, which is 1/4 BPS. This analysis applies to our solution and implies that it supports 2 out of a maximum of 8 Killing spinors. In the extremal limit we also recover the ve-dimensional extremal Nernst branes of [24]. There are interesting parallels as well as di erences between boosted AdS Schwarzschild black branes and rotating black holes. Like for a Kerr black hole, boosted branes have an ergoregion that is a region before the event horizon where observers cannot stay static any more, but have to co-translate with the brane. Also, the Euclidean continuation of such a brane looks very similar to that of a Kerr black hole, and allows to derive the temperature by imposing the absence of a conical de cit. In other aspects the analogy breaks down, however. While supersym-metric rotating black holes cannot have an ergosphere [25], the ergoregion of an in nitely boosted black brane remains. We show that this is consistent with supersymmetry, because the Killing vector obtained as a Killing spinor bilinear is null rather than timelike. Since we are interested in how the lift to ve dimensions a ects the curvature singularities of four-dimensional Nernst branes, we work out explicit expressions for the ve- and four-dimensional curvature in our preferred coordinate systems. Part of these results have been obtained in the previous literature, and where results can be compared, we nd agree-
3A related observation was made in [17], where it was pointed out that one can locally remove the pp wave from the non-extremal solution by a coordinate transformation.
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ment. Our own contribution is to explicitly demonstrate how singular four-dimensional asymptotic hvLif metrics, when combined with the four-dimensional scalars encoding the dynamics of an additional compact direction, lift consistently to a geometry asymptotic to AdS5. This does not only show that the hvLif singularities are artefacts resulting from, as one might say, a bad slicing of AdS5, but the mechanism is dynamical in the sense that the run-away of the four-dimensional scalars encodes the decompacti cation of the fth dimension.
This removes all the sp curvature singularities and run-away behaviour of scalars that four-dimensional Nernst branes exhibit at asymptotic in nity.4 In addition, four-dimensional Nernst branes also have pp curvature singularities which lead to in nite tidal forces acting on freely falling observers. These occur at the horizon, and only at zero temperature. They are again accompanied by run-away behaviour of the scalar elds, which encodes the dynamical decompacti cation of a fth dimension. But in contrast to what happens at asymptotic in nity under decompacti cation, the pp singularity of the asymptotic hvLif space [16] is not removed but lifted to the pp singularity of a ve-dimensional Kaigorodov-type space-time [27]. This shows that pp singularities and in nite tidal forces are intricately related to the vanishing of the entropy (density), thus bringing us back full circle to the third law. We will continue this discussion in the nal section.
Outline of the paper. In section 2 we review ve-dimensional N = 2 FI gauged super-
gravity with vector multiplets and its dimensional reduction to three Euclidean dimensions. In section 3 we obtain ve-dimensional Nernst branes by solving the three-dimensional effective equations of motion and lifting the solution back to ve dimensions. We solve the full second order equations of motion but observe that imposing regularity conditions reduces the number of parameters by one half, so that the solution will satisfy a unique set of rst order equations, despite being non-extremal. By a coordinate transformation the solution can be brought to the form of a boosted AdS Schwarzschild black brane, and further to a metric of Carter-Novotn y-Horsk y type. We work out the thermodynamics in full detail, investigate the extremal limit, compare geometrical properties to those of rotating black holes, and analyse the behaviour of curvature. In section 4 we perform a reduction to four dimensions and show that we recover the four-dimensional Nernst branes of [11]. The relations between the geometrical and thermodynamical properties of ve-dimensional and four-dimensional Nernst branes is worked out in detail. In section 5 we interpret the results, obtain a consistent picture which ties together ve- and four-dimensional Nernst branes, and discuss its interpretation in the context of the gauge/gravity correspondence. We also come back to the question of a higher-dimensional string theory embedding, and use the fact that our solutions have the same metric as reduced boosted D3-branes to set up a gauge/gravity dictionary. We brie y explain how the method used in this paper to
4Following the terminology of [26] sp singularities correspond to a scalar invariant formed out of the Riemann tensor becoming in nite, while pp singularities are curvature singularities observed in a parallely propagated frame. These can occur even if all scalar curvature invariants are nite, and correspond to in nite tidal forces experienced by freely falling observers. This will be demonstrated in some detail later in the paper.
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generate solutions can be applied to nd more general solutions in the future. Finally we discuss open questions regarding the fate of pp curvature singularities and of the third law.
Various technical details have been relegated to appendices. Appendix A derives certain re-writings of the scalar potential, which are used in the main text. Appendix B contains the details of computing thermodynamic quantities using the quasi-local energy-momentum tensor. While in section 3 the Hawking temperature is obtained from the surface gravity of the Killing horizon, appendix C presents an alternative derivation using the Euclidean approach. This allows to again compare boosted branes to rotating black holes. Appendices D and E give the details for computing tidal forces in ve and in four dimensions, respectively. These details have been included to give, in combination with the main text, a full and self-contained account of curvature in ve and four dimensions. In appendix F we spell out the details of a well known fact about the normalization of vector potentials, for completeness, and because we are not aware of an easily digestible and su ciently detailed explanation in the literature.
2 N = 2 gauged supergravity in ve dimensions2.1 Lagrangian of ve-dimensional N = 2 gauged supergravity with vector
multiplets
We start with the ve-dimensional Lagrangian for N = 2 gauged supergravity coupled to n
vector multiplets [28]. Our conventions for the ungauged sector follow those of [29], albeit
with the opposite sign for the Einstein-Hilbert term:
e15L5 =
+
6p6e15cijk Fi FjAk + V5(h); (2.1)
; ^
; : : : are ve-dimensional Lorentz indices, while i; j; : : : = 1; : : : ; n + 1 label the ve-dimensional gauge elds. We use a formulation of the theory where the n-dimensional scalar manifold H is parametrised by n + 1 scalar elds hi which
are subject to real scale transformations. This formulation is natural in the context of the superconformal calculus and will turn out to be helpful for nding solutions. The construction of the theory of ve-dimensional vector multiplets coupled to supergravity using the superconformal calculus can be found in [30, 31], while the superconformal method in general is reviewed in [32]. We will in addition use the formulation of special real geometry developed in [33{36].
As explained in more detail in [35, 36], the scalars hi are special coordinates on an open domain U Rn+1, which is invariant under the action of the group R>0 by scale
transformations. The manifold U is the scalar manifold of an auxiliary theory of n + 1 superconformal vector multiplets, from which a theory of n vector multiplets coupled to Poincar e supergravity is obtained by gauge xing. U is a so-called conic a ne special real
(CASR) manifold. This means that it carries a Hessian metric which transforms with weight 3 under the R>0-action. When choosing special coordinates, which are a ne coordinates
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1
2 2 R(5)
3
4 2 aij(h)@ hi@ hj
1 4aij(h)Fi Fj[notdef]
with 2 = 8G5. Here ^
with respect to the Hessian structure and transform with weight 1 under scale transformations, the Hesse potential is a homogeneous cubic polynomial, H(h) = cijkhihjhk. In
this way one recovers the original de nition of [28]. The physical scalar manifold of the supergravity theory can be identi ed with the hypersurface H U de ned by
H(h) = cijkhihjhk = 1 : (2.2)
Note that the R>0-action is transverse to this hypersurface, so that we can identify H [similarequal] U=R>0. This is a real version of the superconformal quotients for four-dimensional vector multiplets and for hypermultiplets.
The manifold H will be referred to as a projective special real (PSR) manifold. In
the Lagrangian (2.1) we use the special coordinates hi, but it is understood that the constraint (2.2) has been imposed. Within the superconformal calculus this constraint is the D-gauge which gauge xes the local dilatations of the auxiliary superconformal theory in order to obtain the associated Poincar e supergravity theory in its conventional form. In (2.1) this is re ected by the Einstein-Hilbert term having its dimension-full prefactor
2, rather than being multiplied by a conformal compensator to make it scale invariant.
In (2.1) we have chosen to express both the scalar and the vector couplings using the symmetric, positive de nite tensor eld
aij(h) = @2
@hi@hj = 2
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; = 13 log H : (2.3)
Here we use a notation which suppresses indices which are summed over: chhh := cijkhihjhk, (chh)i := cijkhjhk, etc. The tensor @2 = aijdhidhj is a positive de nite
Hessian metric with Hesse potential on U. While it is di erent from the conical Hessian
metric gU = @2H, which has Lorentz signature, with the negative eigendirection along the
orbits of the R>0-action, the pullbacks of both metrics to the hypersurface H agree, so that
one can use either to obtain the positive de nite metric gH which encodes the self-couplings
of the n physical scalars. The couplings of the n + 1 physical vector elds are given by the restriction of the positive de nite metric aij to H.
The scalar potential V5(h) in (2.1) results from an FI-gauging parametrized by n + 1 gauging parameters gi. Using the expressions of [37] and [38] we nd
V5(h) = 2 [notdef] 61=3 h
(chhh)(ch)1[notdef]ij + 3hihj
(ch)ijchhh 32 (chh)i(chh)j (chhh)2
i
gigj: (2.4)
We have xed a convenient normalisation of the gauging parameters gi by comparing the dimensional reduction of (2.1) to the four-dimensional scalar potential of [11], evaluated for a \very special prepotential"
F (X) =
1 6
cijkXiXjXk
X0 ;
that is a prepotential which can arise by reduction from ve to four dimensions.5
5Speci cally, comparing to eq. (30) of [38] we have MI = 61/3hi, AI = 2 [notdef] 61/6Ai, and gPI =
1p2 gi.
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We remark that while on the physical scalar manifold we have to impose the constraint chhh = 1, we have kept factors of chhh explicit in (2.3) and (2.4). This is useful in keeping track of the scaling weights of elds, and thus checking that expressions are consistent with the scaling properties of the corresponding gauge-equivalent superconformal theory. We have chosen our conventions such that the scaling weights of the elds used in (2.1) match with [30, 31]. In particular, we take
w(hi) =
1
2; w(cijk) =
32; w(g ) = 2; w(A ) = 0; w( 2) = 3; w(gi) = 3 :
As a quick check, note that the Lagrangian (2.1) has scaling weight 5, so that the resulting action has scaling weight 0. Further, provided that we include the appropriate factors of chhh, the functions aij and V are homogeneous in hi, even in presence of the dimension-full factors and gi, which appear after imposing D-gauge.6 Note that throughout the remainder of this paper, we shall set 2 = 8G5 = 1.
2.2 Reduction to three dimensions
We now want to reduce the ve-dimensional theory to three (Euclidean) dimensions. We make the metric ansatz
ds2(5) = 62=32 dx0 + A04dx4
2 61=3
dx4 2
+ 61=3
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ds2(3); (2.5)
where all elds depend only on the coordinates of the three-dimensional space. In addition we choose to switch o all of the ve-dimensional gauge elds Ai = 0, i.e. we look only
for uncharged ve-dimensional solutions.7 The presence of the Kaluza-Klein one-form
A0 = A04dx4 p2 0dx4 indicates that we are looking for non-static ve-dimensional
solutions. Upon compacti cation of the x0 circle this will give rise to a non-trivial electric charge for the corresponding four-dimensional solution. Note that whilst the Killing vector @=@x0 is always space-like in ve dimensions, @=@x4 can be either time-like, space-like, or null, depending on the magnitude of A04. However, after performing the dimensional
reduction over x0, the x4 direction will always be time-like in four dimensions, and so we are able to use the same dimensional reduction technique as in [39], i.e. we reduce over both a space-like and time-like direction.
The resulting three-dimensional action is given by
e13L3 =
1
2R(3)
3
4aij(h)@ hi@ hj
142 (@)2
3
42 (@)2 +
312(@ 0)2 + V3(h); (2.6)
where the three-dimensional scalar potential is given by
V3(h) = 61=3
V5(h) =
2
h(chhh)(ch)1[notdef]ij + 3hihjigigj: (2.7)
6See [30, 31] and [32] for more details about the superconformal gauge xing.
7We remark that four-dimensional solutions which will lift to charged ve-dimensional solutions have been found in [11]. The detailed analysis of these solutions is left to future work.
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In order to solve the equations of motion resulting from (2.6) it turns out to be convenient to introduce the variables u, v and yi via
= u
1
2 v
1
2 ; = u
1
2 v
32 ; yi = vhi; ^
gij(y) =
3
4v2 aij(h); (2.8)
so that the three-dimensional Lagrangian (2.6) becomes
e13L3 =
1
2R(3) + ^
gij(y)@ yi@ yj
14u2 (@u)2 +
112u2 (@ 0)2 + V3(y): (2.9)
The scalar potential is given in terms of the new elds by
V3(y) = 2
h(cyyy)(cy)1[notdef]ij + 3yiyji gigj
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^gij(y) + 4yiyj gigj : (2.10)
The explicit steps used in getting to the second line are carried out in appendix A.
We note that the Lagrangian (2.9) has no explicit dependence on the eld v appearing in the metric ansatz. This re ects the fact that when taking the rescaled scalar elds yi as independent variables, the eld v becomes dependent, and can be recovered from the equation
v3 = cyyy;
= 3
which follows from the hypersurface constraint chhh = 1. In terms of the new elds u and v, the ve-dimensional metric ansatz (2.5) becomes
ds2(5) =
62=3
uv
dx0 p2 0dx4 2 61=3 uv (dx4)2 + 61=3v2ds2(3): (2.11)
The independent three-dimensional variables are: the metric ds2(3), the scalars yi which encode the n independent ve-dimensional scalars together with the Kaluza-Klein scalar v, the second Kaluza-Klein scalar u, and the scalar 0 which is dual to the Kaluza-Klein vector from the reduction over x4. The metric on the scalar submanifold parametrized by the yi,
^
gij(y) = 32
(cy)ij
cyyy
3 2
(cyy)i(cyy)j (cyyy)2
; (2.12)
is, up to a constant factor, isometric to the positive de nite Hessian metric (2.3) on the manifold U [similarequal] H [notdef] R>0 [similarequal] H [notdef] R. As shown in [35] this metric is isometric to
the product metric gH + dr2 on H [notdef] R. From (2.9) it is manifest that the scalar man
ifold Q of the three-dimensional Lagrangian carries a product metric, with the rst factor parametrized by yi and the second factor parametrized by u and 0. By inspection,8 the second factor is locally isometric to the metric of the pseudo-Riemannian symmetric space SU(1; 1)=SO(1; 1) [similarequal] AdS2, which can be thought of as the inde nite signature version
of the upper half plane (equivalently, of the unit disk) SL(2; R)=SO(2) [similarequal] SU(1; 1)=U(1).
To be precise u and 0 parametrise an open subset which can identi ed with the Iwasawa
8For a systematic analysis of the scalar manifolds occuring in reduction to three space-like dimensions, we refer the reader to [40, 41].
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subgroup of SU(1; 1), or, in physical terms, with the static patch of AdS2. The combined scalar manifold parametrized by yi; u; 0,
Q = H [notdef] R [notdef]
SU(1; 1)
SO(1; 1) ; (2.13)
has dimension n + 1 + 2 = n + 3.
If we perform the reduction of ve-dimensional supergravity with n vector multiplets to three Euclidean dimensions without any truncation, then the resulting scalar manifold is a para-quaternionic Kahler manifold
NPQK of dimension 2(2n + 2) + 4 = 4n + 8 [40, 41].
The submanifold Q is obtained by a consistent truncation and therefore it is a totally geodesic submanifold of
NPQK . We remark that Q is a (totally geodesic) submanifold of the (2n + 4)-dimensional totally geodesic para-Kahler manifold SPK described in [39, 42],
Q = H [notdef] R [notdef]
SU(1; 1)
SO(1; 1) SPK
NPQK :
It was shown there how to obtain explicit stationary non-extremal solutions of four- and ve-dimensional ungauged supergravity by dimensional reduction over time. As we will see in the following, it is still possible to obtain explicit solutions in the gauged case, where the eld equations of the three-dimensional scalars are modi ed by a scalar potential. While we will retrict ourselves to the submanifold Q in this paper, the higher dimensional para-Kahler submanifold SPK will be relevant when the present work is extended to more general, charged solutions, including the solutions found in [11].
3 Five-dimensional Nernst branes
3.1 Solving the equations of motion
We now turn to the three-dimensional equations of motion coming from (2.9). The equations of motion for yi; u and 0 read:
r2yi +
^ ijk(y)@ yj@ yk + 3^
ijk(y)^gjm(y)^gkn(y)gmgn 12(yjgj)^gik(y)gk = 0; (3.1)
r2u
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1 u(@u)2
13u(@ 0)2 = 0; (3.2)
r2 0
2u@ u @ 0 = 0; (3.3)
where we have introduced the Christo el symbols for the metric ^
gij(y):
^ ijk(y) = 12 ^
gil(y)@l^
gjk(y):
Meanwhile, the Einstein equations read
1
2R(3)[notdef] + ^
gij(y)@ yi@ yj
14u2 @ u @ u
+ 1
12u2 @ 0@ 0 + 3g
^gij(y) + 4yiyj gigj = 0: (3.4)
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We now proceed to solving the equations of motion (3.1){(3.4), and make the following brane-type ansatz for our three-dimensional line element:
ds2(3) = e4 d2 + e2 (dx2 + dy2); (3.5)
where = () is some function to be determined, and is a radial coordinate which parametrizes the direction orthogonal to the world-volume of the brane. This is the same brane-like ansatz for the three-dimensional line element as in [11]. Moreover we will impose that all of the elds yi, 0 and u depend only on . This coordinate has been chosen such that it is an a ne curve parameter for the curve C : [mapsto]!(yi(); u(); 0()) on the scalar manifold Q.
The Ricci tensor has components
R = 2 2
_ 2; Rxx = Ryy = e2
;
from which we nd that the Einstein equations (3.4) become
V3(y) = 12e4
; (3.6)
for = [negationslash]= , and
1
2
+ _ 2 = ^
gij(y) _
yi _
yj + _
u2 4u2
( _
0)2
12u2 ; (3.7) for = = , where we have used (3.6). We will now consider the equations of motion for each of 0, u and yi in turn.
0 equation of motion. The equation of motion (3.3) for 0 can be brought to the form
d d
1 u2
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_ = 0;
which is solved by
0 = p3Du2; (3.8)
for some integration constant D, where we have chosen the factor for later convenience. Once we solve the equation of motion for u we will further integrate (3.8) to obtain an expression for the Kaluza-Klein vector A0 = p2 0 appearing in the ve-dimensional metric.
u equation of motion. Substituting (3.8) in to the equation of motion (3.2) for u we nd
u
1 u _
_
u2 D2u3 = 0: (3.9)
Introducing the variable ~ = u1, this becomes
~
~2 D2
_ ~ = 0 : (3.10)
By di erentiation we obtain the necessary condition _
~
~ = ~...
~, which can be integrated to
~ = B20~, where B0 is a real constant.9 Parametrizing the general solution as
~() = A cosh(B0) + B
B0 sinh(B0); (3.11)
9Negative B20 would yield a solution periodic in , which we discard.
{ 10 {
with arbitrary constants A; B, and substituting back into the original equation (3.10) we nd the constraint
D2 = B2 B20A2 ;which imposes one relation between the four constants D; A; B; B0. It will turn out to be useful in what follows to consider A, B0 and := BB0A to be the independent quantities,
and to write everything in terms of these. In particular, we then have D2 = ( + 2B0A).We are also now in a position to further integrate (3.8), which we write as
_
0 = [notdef]p3 (
+ 2B0A)
~2 : (3.12)
For simplicity we will chose the negative sign in (3.12), and will not carry through the corresponding positive solution. Since 0 is dual to a Kaluza-Klein vector, this means that we have xed the sign of the charge that the solution carries.10 Substituting in (3.11) and integrating, we nd
0() = p3 B0 u()
p ( + 2B0A)
p3 B0
6 + 2B0A u()eB0dx4: (3.14)
yi equation of motion. The equation of motion (3.1) for the yi becomes
e4
yi + e4 ^ ijk(y) _
yj _
gjk(y);
10As we will see in the following, the solution carries electric charge from the four-dimensional point of view and linear momentum from the ve-dimensional point of view.
gjk(y) = ^
A sinh(B0) + B
B0 cosh(B0) 01; (3.13)
for some integration constant 01, which can be xed by imposing a suitable physicality
condition on the solution.At this point we anticipate that a horizon, if it exists, will be located at ! 1.
Moreover, as we will show in section 4, upon dimensional reduction we obtain a four-dimensional stationary (in fact static) solution with a Killing horizon. Such horizons admit, for nite temperature, an analytic continuation to a bifurcate horizon [43]. In order that the four-dimensional one-form A0() is well de ned, it must vanish at the horizon [2, 44],
see also appendix F.This xes
01 =
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p ( + 2B0A); and therefore the Kaluza-Klein one-form is given by
A0() = r
gij(y) gj(ykgk) = 0: (3.15)
To proceed, we rst contract (3.15) with the dual scalar elds yi := ^
gij(y)yj and make
yk + 3^ ijk(y)^gjm(y)^gkn(y) gmgn 12 ^
1
2yl@l^
use of the identity
^ ijk(y)yi = 12yi^
gil(y)@l^
gjk(y) =
{ 11 {
which follows from the fact that ^
gij(y) is homogeneous of degree 2 in the yi. We thus nd
e4
yiyi + e4 ^
gij(y) _
yi _
yj + V3(y) = 0;
which upon using (3.6) becomes
yiyi + ^
gij(y) _
yi _
yj =
1
2
: (3.16)
Given that ^
gij(y) _
yj = _
yi, we can integrate (3.16) to nd
_
yiyi =
1
2
_ + 14a0; (3.17)
for some integration constant a0, where the factor has been chosen for later convenience. Writing
_
yiyi = 34
(cyy)i _
JHEP11(2016)114
yi cyyy =
1 4
dd (log cyyy) ;
we can integrate (3.17) further to obtain
log cyyy = 2 + a0 + b0; (3.18)
for an integration constant b0. Again the prefactor has been chosen for later convenience. We now return to the Hamiltonian constraint (3.7). Using (3.11) and (3.8) this becomes:
1
2
+ _ 2 = 14B20 ^
gij(y) _
yi _
yj: (3.19)
We then have the following picture. The solutions yi() to (3.15) should satisfy the constraints (3.19) and constraint (3.17). One way to proceed, which is valid for generic ve-dimensional models, is to set all of the yi proportional to one another, i.e. we put yi = iy for some constants i, which satisfy
^
gij()ij =
3
4:
Note that since the (constrained) scalar elds hi can be recovered from the yi via hi = (cyyy)1=3yi, we see that this ansatz will result in constant ve-dimensional scalar elds.
Using (3.17) we obtain:
3
4
_
y y
2= 12 + _ 2 14B20; (3.20)
3
4
_
y y
=
1
2
_ + 14a0: (3.21)
Eliminating the quantity ( _
y=y) from (3.20){(3.21) we obtain an equation for the func-
tion ():
4 3
_ 2
2
3a0
_ + 12B20 +
16a20 = 0:
{ 12 {
This is precisely the same equation as was found in [11], and so can be solved in the same way by
e4 = 3ea0
sinh(! + ! ) !
3; (3.22)
for some integration constants and , where the quantity ! is given by
!2 := 23B20 +
13a20: (3.23)
From this, we can integrate (3.21) to nd
y() = e
1
2 a0
sinh(! + ! ) !
12;
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for some constant , and hence the yi are given by
yi() = ie
1
2 a0
sinh(! + ! ) !
12; (3.24)
where we have de ned i i= . We nally need to ensure that the solution (3.24) satis es
the original equations of motion (3.15). This xes i in terms of the gauging parameters gi and other integration constants as
i = [notdef]
3 3=2
8gi : (3.25)
Therefore the function v appearing in the line element (2.11) is given by
v() = (c )1=3e
1
2 a0
sinh(! + ! ) !
12: (3.26)
The signs in (3.25) should be chosen such that the function v() is real and positive for all > 0.
At this stage we have six independent integration constants ; ; a0; A; B0; which are a priori yet to be determined. However, following [11] we choose to set = 0 in what follows so that the asymptotic region is at = 0 and the near horizon region at ! 1.
We can then scale to set = 1.
In order for our solution to make sense as a black brane in ve dimensions, we need to impose some physicality constraints. In particular, we require that the ve-dimensional solution generically has a nite entropy density.11 Combining the ve-dimensional and three-dimensional metric ansatze (2.11) and (3.5) we see that nite entropy density corresponds to a nite value of v3=2u1=2e2 as ! 1 (i.e. at the horizon). To leading order
we nd
v3=2u1=2e2
!1
exp
14a0 34! +1 2B0
:
11Since the range of the coordinates (x, y, x0) is in nite, the entropy itself will diverge. By generic we mean that we allow that the solution has a limit, which hopefully will coincide with the zero temperature limit, where the entropy density becomes zero.
{ 13 {
In order that this be nite and non-zero we therefore require 3! = a0 + 2B0 which,
given (3.23), is equivalent to a0 = B0, further resulting in ! = B0. Hence, the physicality constraint further reduces the number of independent integration constants by one.
Before moving on to study properties of the solution, we summarise the story so far. The functions appearing in the ve-dimensional line element (2.11) are given by
v() = (c )1=3e
1
2 B0
sinh(B0) B0
12; (3.27)
u() = ~()1; ~() = A cosh(B0) + B
B0 sinh(B0); (3.28)
e4 = eB0
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sinh(B0) B0
3; (3.29)
A0() = r
6 + 2B0A u()eB0dx4; (3.30)
whilst the scalar elds hi parametrising the CASR manifold are constant and given by
hi = 1v yi = (c )1=3 i =
1gi clmng1lg1mg1n
1=3 : (3.31)
We have therefore found a family of solutions to the equations of motion (3.1){(3.4) depending on three non-negative parameters B0; ; A. Since the eld equations for the three-dimensional scalars yi(); v(); u() are of second order, and our ansatz amounts to three independent scalar elds (since the yi have been taken to be proportional), we should a priori have expected six independent integration constants. However, as we have seen, physical regularity conditions imposed on the lifted, higher-dimensional solution reduces the number of integration constants by one half. This is consistent with physical solutions being uniquely characterised by a system of rst order ow equations, despite that the equations of motion are of second order, as has been observed for other types of solutions before [36, 39, 42, 45].
We further remark that since the physical ve-dimensional scalar elds have turned out to be constant, their only contribution is to generate an e ective cosmological constant, whose value is determined by the value of the scalar potential at the corresponding stationary point. Since no ve-dimensional gauge elds have been turned on, our solution, which is valid for any ve-dimensional vector multiplet theory, can therefore be obtained from an e ective action, which only contains the Einstein-Hilbert term together with a cosmological constant, while the gauge elds and scalar elds have been integrated out.
A coordinate change. We introduce a new radial (more accurately: transversal) coordinate via
e2B0 = 1
2B0
W (); (3.32) so that the near horizon region is at = 2B0, and the asymptotic region is at ! 1.
Hence we can use to analytically continue the solution to the region 0 2B0 beyond
the horizon. In terms of we nd
u() = f()1W ()1=2; f() = A +
; (3.33)
{ 14 {
where we have de ned := B B0A. Moreover, we have
v() = (c )1=3(W )1=2; e4 = 3W 2; (3.34)
and
A0() = r
6 + 2B0A
W ()f() dx4: (3.35)
Introducing the notation
~
:=
1 6c
1=3;
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the ve-dimensional line element (2.11) becomes
ds2(5) =
1=2
6~
f()
dx0 r
6 + 2B0A
W ()f() dx4!2
1=2W ()~
f() (dx4)2
+ 6
~
2d2 2W () + 6
~
21=2(dx2 + dy2): (3.36)
3.2 Properties of the solution
Let us now turn to an investigation of the properties of the solutions constructed in section 3.1, which we recall depend on three independent parameters: A, B0 and . It is instructive to look at the cases A > 0 and A = 0 separately. Moreover, we focus rst on the situation B0 > 0, and will comment on the B0 = 0 case later.
Solutions with B0 > 0 and A > 0. In this situation it is convenient to introduce the notation:
~
:=
2B0A: (3.37)
After a suitable scaling of the boundary coordinates, and introducing the new radial coordinate r := 1=4, we can bring the ve-dimensional line element (3.36) to the form
ds2(5) =
r2l2 f(r)
0
@dx0 s
~ W (r)f(r) dx41
A
2 r2W (r)
l2f(r) (dx4)2
+ l2dr2
W (r)r2 +
r2l2 (dx2 + dy2): (3.38)
Here l is de ned by
l2 := 96~
2;
and, as we will see below, is the radius of an asymptotic AdS5 space, whilst
W (r) = 1
r4+r4 ; f(r) = A +
r4 ; r4+ := 2B0:
In order to interpret our solution, as well as to read o the various thermodynamic quantities associated with it, it is useful to introduce coordinates in terms of which the line
{ 15 {
element (3.38) becomes manifestly asymptotically AdS5. We observe that the solution is invariant under the combined rescalings
A ! A ; ! ; x0 !
x0p ; x4 !
p x4 where > 0 (3.39)
of parameters, with B0 invariant. Note that ~
is invariant, so that for A > 0 we obtain a two-parameter family of solutions parametrized by B0 and ~
. The coordinate
transformation
t = 1
pAx4; z =
pAx0 s
~ ) x4; (3.40)
absorbs A and brings the metric (3.38) to the form of a boosted AdS Schwarzschild black
brane:
ds2(5) =
l2dr2 r2W +
r2 l2
hW (ut dt + uz dz)2 + (uz dt + ut dz)2 + dx2 + dy2i: (3.41)
The constants
ut =
p1 + ~ ; uz =
p~ ; (3.42)
satisfy u2t u2z = 1 and parametrise a boost along the z-direction. By taking r ! 1 one
sees that (3.41) indeed asymptotes to AdS5 with radius l. The constant ~
parametrizes
the boost of the brane, while B0, as we will show below, is a non-extremality parameter and therefore related to temperature.
This metric can be rewritten by making the following co-ordinate transformation:
r = el1; x = ly1; y = ly2
t = l
r2+ (ut uz)X lr2+uzT; z =
lr2+ (ut uz)X + lr2+utT ; (3.43)
to obtain
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2dXdT + r4+dT 2 + (dy1)2 + (dy2)2
+(1 r4+e4l1)1d2 : (3.44)
This metric is the 5-dimensional generalized Carter-Novotn y-Horsk y metric constructed in [17].
We further remark that the line element (3.41) can be further simpli ed by setting R = r+r, ~T = t=r+, X = x=r+, Y = y=r+, Z = z=r+. This rescaling corresponds to formally setting r+ = 1 in the function W in (3.41), thus xing the coordinate of the horizon to r = 1. However, this reparametrization obscures the fact that r+ in (3.41) encodes the temperature, which, as we will show later, is de ned in a reparametrization invariant way.
{ 16 {
ds2(5) = e2l1dX2 + e21
Solutions with B0 > 0 and A = 0. Let us now look at the case where we take A = 0, so that f() = = in (3.36). In this case, after suitably rescaling the boundary coordinates and introducing the radial coordinate r as before, we nd that the ve-dimensional line element (3.36) becomes
ds2(5) =
l2r2
dx0 r4W (r) dx4 2 r6W (r) l2 (dx4)2 +l2dr2r2W (r) +r2l2 (dx2 + dy2): (3.45)
Making the coordinate rede nition
x4 = 12(t z); x0 +
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r4+2 x4 = t + z;
we can bring the metric (3.45) to the form (3.41) of a boosted AdS Schwarzschild black brane. The boost parameters are given by
ut = cosh ^
; uz = sinh ^
; (3.46)
where the quantity ^
is de ned via
e2 = 4 r4+ :
As in the case A > 0 we obtain a two-parameter family of black brane solutions. For A = 0 the parameters can be taken to be B0 (equivalently r+) and . We remark that while both the cases A > 0 and A = 0 can be mapped to two-parameter families of black branes, both families cannot be related smoothly by taking A ! 0.
Solutions with B0 = 0. If we take B0 ! 0 in (3.36) then the region 0 r (2B0)1=4
contracts to r = 0, which suggests that this limit is the extremal limit. We will show later that B0 = 0 does indeed correspond to vanishing surface gravity, and, hence vanishing
Hawking temperature, and, moreover, that the solution is a BPS solution.
For any value (zero or non-zero) of A we can then bring the metric to the form
ds2(5)[notdef]Ext =
l2dr2r2 +
dt2 + dx2 + dy2 + dz2 + r4 (dt + dz)2 : (3.47)
This solution agrees with the ve-dimensional extremal Nernst branes found in [24].12 We
can equivalently obtain this form of the metric from the boosted black brane (3.41) by taking the limitsr+ ! 0; ut ! 1; u2tr4+ ! = const: (3.48)
In the extremal limit can be interpreted as a boost parameter. The vacuum AdS5 solution is obtained by taking the zero boost limit ! 0. Thus in the extremal limit
determines the mass, or more precisely the mass per worldvolume or tension of the brane. The precise expressions for the mass and thermodynamic quantities will be calculated in section 3.4.
12However, the heated up branes of [46] appear to be di erent from our non-extremal solutions.
{ 17 {
r2 l2
The metric (3.47) displays an interesting scaling behaviour in the limit r ! 0. To
display it, we introduce coordinates x; x+ by13
t = x+ ; z = x x+ :
Then the metric becomes
ds2(5)[notdef]Ext =
l2dr2r2 +
(dx)2 2dxdx+ + dx2 + dy2 :
Dropping terms which are subleading in the near horizon limit r ! 0 we obtain
ds2(5)[notdef]Ext;NH =
l2dr2r2 +
r2 l2
1 +
r4
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r2 l2
r4 (dx)2 2dxdx+ + dx2 + dy2
: (3.49)
This metric is invariant under the scale transformations:
x [mapsto]! x ; y [mapsto]! y ; r [mapsto]! 1r ; x [mapsto]! 1x ; x+ [mapsto]! 3x+ :
Thus the asymptotic metric shows a scaling invariance similar to a Lifshitz metric with scaling exponent z = 3 (and no hyperscaling violation, = 0).14 The only di erence is that the coordinate x has scaling weight 1 rather than +1. This type of generalized scaling
behaviour was observed in [23, 47, 48], where the metric (3.49) was obtained by taking a particular limit of boosted D3-branes. We will come back to this in section 5, where we discuss the dual eld theory interpretation of our solutions.
The boosted black brane. The boosted black brane has similarities with Kerr-like black holes, with the linear momentum related to the boost playing a role analogous to the angular momentum. It is instructive to work this out in some detail, following the discussion of the Kerr solution in [49].
Let us rst look for the existence of static observers, who remain at constant (r; x; y; z) and as such have velocities parallel to the Killing vector eld @t. Therefore static observers exist in regions where @t is timelike, and the limit of staticity is at the value of r where
gtt = 0 , W (r)u2t + u2z = 0
, r4 = u2tr4+ r4+ :
This \ergosurface" is always located outside the event horizon, with the trivial exception of globally static (unboosted) spacetimes for which ut = 1 and the two surfaces overlap completely. This is di erent to the rotating case where ergosurface and event horizon always coincide at the north and south pole.
Beyond the limit of staticity there still exist stationary observers which are co-moving (more precisely, but less elegantly co-translating) with the brane. Observers which have
13For A = 1 these coordinates agree with x0 and x4 in the extremal limit. Moreover, the near-horizon limit preserves the symmetry that allows to set A = 1.
14Lifshitz metrics will be reviewed in section 4.1.
{ 18 {
xed (r; x; y) and a constant velocity in the z-direction have world lines tangent to Killing vector elds
(v) = @t + v @z ;
where the quantity v = const. will be referred to as the velocity. Such co-moving observers exist in regions where (v) is time-like. Killing vector elds of the form (v) become null for values of r where
gtt + 2vgtz + v2gzz = 0 ) v[notdef] =
gtz
gzz [notdef] s
gtz
gzz
2gtt gzz :
Thus there is a nite range of velocities v, given by v v v+, which co-moving observers
can attain. Note that at the limit of staticity, where gtt = 0, we nd that v+ = 0. Therefore v must be negative once the limit of staticity has been passed. The limit for co-moving observers is reached when v = v+ =: w, which happens at the point where
gttgzz g2tz = 0 :
It is straightforward to verify that this happens at the same value r+ of r where W (r+) = 0. The limiting velocity w is given by
w =
gtz
gzz
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r=r+
uzut ; (3.50)
and can be interpreted as the boost-velocity of the surface r = r+. Since W (r+) = 0 implies that grr(r+) = 0, it follows that on this surface outgoing null congruences have zero expansion, see [49] for the analogous case of a rotating black hole. Consequently r = r+ is an apparent horizon, and since the solution is stationary, an event horizon. Moreover this event horizon is a Killing horizon for the vector eld = @t + w@z = @t uzut @z and
we can interpret w as the boost-velocity of this horizon. Observe that the limit of staticity and the limit of stationarity are in general di erent, and only agree in the unboosted limit uz = 0 where we recover the AdS Schwarzschild black brane.
We note that there is frame dragging in our solutions, since the metric is non-static for uz [negationslash]= 0. Indeed, since the metric coe cients are independent of t and z, the covariant mo
mentum components pt and pz are conserved. But even when setting pz = 0, particles have a non-vanishing contravariant momentum component pz = gztpt [negationslash]= 0 in the z-direction.
The boost velocity of the metric varies between the horizon and in nity. It can be read o by writing the metric in the form
ds2(5) = N2(r)dt2 + M2(r)(dz v(r)dt)2 + [notdef] [notdef] [notdef]
where the omitted terms involve dx2; dy2 and dr2. An observer at xed r; x; y is co-moving with the space-time if their velocity is dz=dt = v. Bringing the metric (3.41) to the above form one nds
v =
=
(1 W )utuz u2t W u2z
{ 19 {
with limits
v !
r!r+
uzut = w 1 ;
and
v !
r!1
0 :
It is straightforward to check that for ut > 1 the boost speed [notdef]v(r)[notdef] is strictly monontonically
increasing from [notdef]v1[notdef] = 0 at in nity to [notdef]vhorizon[notdef] = [notdef]w[notdef] = uz=ut 1 at the horizon. Thus the boost speed is bounded by the speed of light and can only reach it at the horizon and in the extremal limit. Note that the asymptotic AdS space at in nity is not co-moving. This is di erent from Kerr-AdS, where the asymptotic AdS space is co-rotating, with implications for the black brane thermodynamics [50{52]. In particular, we will not need to subtract a background term, corresponding to the asymptotic AdS space, from our expressions for the boost-velocity in order to have quantities satisfying the rst law of thermodynamics. We will come back to this later when verifying the rst law.
3.3 BPS solutions
In this section, we consider the properties of the extremal solution in further detail. We begin by considering the solution (3.47). On making the co-ordinate transformation
r =
14 el1R; x = l
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14 y1; y = l
14 y2
t = 12l
14 (X 2T ); z =
1
2l
14 (X + 2T ) ; (3.51)
the metric (3.47) becomes
ds2 = e2l1RdX2 + e2l1R
2dXdT + (dy1)2 + (dy2)2 + dR2 : (3.52)
The metric (3.52) is a ve-dimensional generalized Kaigorodov metric, constructed in [17], which describes gravitational waves propagating in AdS5. The supersymmetry of this solution was investigated in [17], where it was shown that this solution preserves 1/4 of the supersymmetry. Furthermore, after making some appropriate co-ordinate transformations, this solution can be shown to correspond to a class of supersymmetric solutions which appears in the classi cation of supersymmetric solutions of minimal ve-dimensional gauged supergravity constructed in [53]. It is straightforward to show that the null Killing vector which is obtained as a spinor bilinear is given by @t @z, in the co-ordinates of (3.47).
The fact that this Killing vector is null rather than timelike is related to an interesting feature which distinguishes these BPS solutions from ve-dimensional rotating BPS black holes, namely the existence of an ergoregion, i.e. a region outside the horizon where it is not possible for observers to remain static. Note that in the BPS limit (3.48) the limit of staticity is at r = 1=4 0 which is outside the horizon at r = 0 (unless we switch
o momentum, = 0, and go to global AdS). Therefore the ergoregion persists in the BPS limit.
For stationary BPS black holes the Killing vector obtained as a spinor bilinear is the standard static Killing vector eld @t, which is timelike at in nity. For rotating black holes
{ 20 {
@t is di erent from the horizontal Killing vector eld @t + !@, which becomes null on the horizon. An ergoregion exists when @t becomes space-like outside the horizon. However if @t is a bilinear formed out of Killing spinors, supersymmetry implies that it must be either time-like or null. Hence, rotating BPS black holes cannot have an ergoregion. Moreover it can be shown that the event horizon of a rotating BPS black hole must be non-rotating [25]. As we have shown above, this is di erent for the extremal limit of an AdS-Schwarzschild black brane, which is a BPS wave solution in AdS5: the ergoregion persists in the BPS limit, and the (degenerate limit of the) horizon15 moves with the speed of light, since w = uz=ut ! 1. This is consistent with the solution being BPS, because the Killing
vector obtained as a Killing spinor bilinear is not @t, which is timelike at asymptotic in nity and becomes spacelike before the horizon, but @t @z, which is null everywhere for the BPS
solution. Moreover, the horizon turning into a purely left-moving wave is consistent with the familiar string theory description of a BPS state as a state with massless excitations moving in one direction only.
3.4 Thermodynamics
We turn to an investigation of the thermodynamics of the black brane solutions of section 3.2. The Hawking temperature is related to the surface gravity by T =
2 , where the
surface gravity of a Killing horizon is given by
2 =
1
2r r
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: (3.53)
Evaluating this for = @t + w@z, we nd the Hawking temperature T :
T = r+
l2 ut : (3.54)
We remark that the same result can be obtained by imposing that the Euclidean continuation of the solution does not have a conical singularity at the horizon, see appendix C.
In the zero boost limit ut = 1, uz = 0, we obtain the Hawking temperature of an AdS Schwarzschild black brane. In the extremal limit (3.48), where the boost parameters go to in nity ut; uz ! 1, while r+ ! 0, the Hawking temperature becomes zero, T ! 0,
irrespective of whether we keep nite or not.
Since our solutions are not asymptotically at, but rather asymptotic to AdS5, we
cannot apply the standard ADM prescription to compute the mass and linear momentum of our branes. Instead, we use the method based on the quasilocal stress tensor [54], see also [55] for a review in the context of the uid-gravity correspondence. Here we simply present the result, and relegate explicit calculational details to appendix B. To leading order in 1=r we nd that the quasilocal stress tensor takes the form
T = r4+
2l3r2 ( + 4u u ) + : : : ; (3.55)
15We will show later that in this limit the metric develops a singularity at the horizon, corresponding to freely falling observers experiencing in nite tidal forces.
{ 21 {
r=r+
where is the Minkowski metric on @Mr which denotes the hypersurface r = const: of
our space-time, with coordinates (t; x; y; z). As indicated we have omitted terms subleading in 1=r, since we are ultimately interested in expressions which are nite on the boundary @M = limr!1 @Mr of space-time. Note that T takes the form of the stress energy tensor
of a perfect ultra-relativistic uid (equation of state = 3p, where is the energy density and p is the pressure), with pressure proportional to r4+ T 4. The proportionality between
r+ and T is the same behaviour as for large AdS-Schwarzschild black holes. In the absence of a boost, it is known that AdS-Schwarzschild black branes behave thermodynamically like large (rather than small) AdS-Schwarzschild black holes [56].
Having obtained the quasilocal stress tensor, mass and linear momentum can be computed as conserved charges associated to the Killing vectors of our solution. Again, the details are relegated to the appendix B. The mass, which is the conserved charge associated with time translation invariance, is
M = (4u2t 1)r4+
2l5 V3; (3.56)
where V3 =
R d3x is the spatial volume of the brane, computed with the standard Euclidean metric dx2 + dy2 + dz2. Due to the in nite extention of the brane, the mass is in nite, and to obtain a nite quantity we must either compactify the world volume directions or de ne densities. We will do the latter by consistently splitting o a factor V3 from all extensive quantities.
Next we calculate the momentum in the z-direction, which is the conserved charge associated to z-translation invariance. The result is
Pz =
4r4+utuz
2l5 V3; (3.57) and vanishes as expected in the zero boost limit uz = 0, ut = 1. Notice that these charges satisfy Pz = M
4utuz 4u2t1
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, which resembles the motion of a non-relativistic body of mass M, moving at velocity vz = 4utuz4u2t
1 .
Finally, we calculate the Bekenstein-Hawking entropy of the solution by integrating the pull back of the metric over the horizon. Recalling that we are working in units where 8G5 = 1, we nd
S = 1
4G5
Z r=r+d3xp = 2
Z r=r+d3xp = 2r3+l3 utV3 ; (3.58)
where denotes the pullback of the metric to the surface r.
Using these, we can check that the thermodynamic variables satisfy the rst law:
M = T S + w Pz : (3.59)
We remark that obtaining (3.59) is a non-trivial consistency check for the correctness of the de nition of the thermodynamical quantities, which are initially ambiguous because they require background subtractions corresponding to renormalization of the boundary CFT [54], see also [52] for a discussion in the context of rotating black holes in higher than
{ 22 {
four dimensions. As noted before we do not need to apply a background subtraction for the translation velocity w, since the asymptotic AdS5 background is not co-translating. This is di erent for AdS-Kerr-type black holes, where the subtraction of the background rotation velocity is crucial for obtaining the correct thermodynamic relations [50{52]. We also note that T; M; Pz; S which are all de ned in a reparametrization invariant way, depend on the parameter r+, which is therefore a physical parameter, despite the fact that it could be absorbed into the coordinates in the line element (3.41). Moreover, without the ability of varying this parameter, one could not obtain the temperature/entropy term in the rst law. We refer to appendix B.2 for further details on this technical point.
The extremal limit of these quantities can be reached by taking r+ ! 0 and ut ! 1
with u2tr4+ ! xed. In this case we nd that the entropy density s := S=V3 vanishes in
the extremal limit, s ! 0 as T ! 0. Therefore our solutions satisfy the strong version of
the Nernst law, and will be referred to as Nernst branes.16 Moreover, since in this case w = 1, we nd M = [notdef]Pz[notdef], which is of course the saturation of the BPS bound, as it must
be given the results of section 3.3. As already remarked earlier, in the extremal limit the boost parameter controls the mass, and ! 0 is the limit where the solution becomes
globally AdS5.
We can eliminate the quantities r+ and ut in favour of the thermodynamical variables T and w via
ut = 1 p1 w2
; uz =
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w p1 w2
; r+ = l2(T ) p1 w2
:
In terms of T and w the mass of the solution is given by
M(T; w) = l3
2 V3
3 + w2 (1 w2)3
(T )4: (3.60)
Hence, we see that the heat capacity
CT
@M
@T
> 0; (3.61)
is positive, and the solution is thermodynamically stable. This is as expected, at least in the absence of a boost, since it is well known that AdS-Schwarzschild black branes behave thermodynamically like large AdS-Schwarzschild black holes [56]. As we see from (3.60), the introduction of a boost does not introduce thermodynamic instablility.
Expressing the entropy in terms of (T; w) we nd
S(T; w) = 2l3V3 (T )3
(1 w2)2
: (3.62)
Note that turning o the boost uz = 0, which corresponds to w = 0, we have S T 3,
which is the scaling behaviour expected for an AdS5 Schwarzschild black brane.
16Incidentially, this version of the Nernst law is due to Planck, but clearly Planck brane would be a bad choice of terminology.
{ 23 {
w
Indeed we can use (3.62) to investigate the behaviour of S as a function of T in both the high temperature and low temperature limits. The limit of high temperature (equivalently small boost velocity) is
uz ! 0; r+ ! 1; u2zr4+ ! = const:
This corresponds to [notdef]w[notdef] 1, and so we see from (3.62) that S T 3. The limit of low tem
perature (equivalently boost velocity approaching the speed of light) is the extremal limit
ut ! 1; r+ ! 0; u2tr4+ ! = const:
In this case, one can see that 1 w2 T 4=3, and so the entropy scales like S T 1=3. This
is the behaviour predicted for ve-dimensional lifts of four-dimensional Nernst branes [11]. We will comment further on the thermodynamic properties of our solutions in section 5.
3.5 Curvature properties of ve-dimensional Nernst branes
One motivation of the present work is the singular behaviour of the four-dimensional Nernst branes found in [11]. We will show in section 4 that the ve-dimensional Nernst branes found above are dimensional lifts of these four-dimensional Nernst branes. To investigate the e ect of dimensional lifting on such singularities, we now examine the behaviour of curvature invariants and tidal forces of the ve-dimensional solutions. From both the gravitational point of view, and with respect to applications to gauge-gravity dualities, one would like the solutions to have neither naked singularities, nor null singularities (singularities coinciding with a horizon), while the presence of singularities hidden behind horizons is acceptable. In practice, the presence of large curvature invariants or large tidal forces will also be problematic, given that the supergravity action we start with needs to be interpreted as an e ective action. Therefore large curvature invariants or tidal forces are indications that this e ective description breaks down due to quantum or, assuming an embedding into string theory, stringy corrections. This might also limit the applicability of gauge-gravity dualities to only part of the solution, where the corrections remain su ciently small.
Curvature invariants. For our ve-dimensional metric (3.41) we compute the Kretschmann scalar and Ricci scalar to be
K = 2 9r8+ 24r4+r4 + 20r8 r8l4 ; R =4
5r4 + 3r4+ r4l2 : (3.63)
Note that these only depend on the temperature T r+ and the curvature radius l of
the AdS5 ground state. For the extremal solution (r+ = 0) both curvature invariants take constant values which agree with those for global AdS5:
KAdS5 = 2d(d 1)
l4 =
40l4 ; RAdS5 =
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d(d 1)
l2 =
20 l2 :
For the non-extremal solution the curvature invariants tend to the AdS5 values asymptotically, but will blow up as r ! 0. Since this is behind the horizon, there are no naked or
null singularities related to the curvature invariants of ve-dimensional Nernst branes.
{ 24 {
Tidal forces. Even if all scalar curvature invariants are nite, there might still be curvature singularities related to in nite tidal forces. Such curvature singularities can be found by computing the components of the Riemann tensor in a parallely-propogatedorthonormal-frame (PPON) associated with the geodesic motion of a freely-falling ob-server. Following [26] they are called pp singularities, in contrast to sp singularities, where a scalar curvature invariant becomes singular. While such singularities are often considered milder than those associated to curvature invariants, they are nevertheless genuine singularities and have drastic physical e ects (spaghetti cation) on freely falling observers.
The details of this construction for the ve-dimensional extremal solution are relegated to appendix D. We only need to consider the extremal solution, since non-extremal solutions are manifestly analytic at the horizon r+ > 0. From table 3 in appendix D we observe that the non-zero components of the Riemann tensor in the PPON all have near horizon behaviour of the form
~Rabcd r with 0 ; (3.64)
with < 0 for all but one independent non-vanishing component. Hence, as the observer approaches the horizon of the extremal brane (r ! 0) these components will diverge,
resulting in infalling observers being subject to in nite tidal forces. This is the same behaviour as observed in four dimensions [11], and seems to be the price for having zero entropy. It is an interesting question whether stringy or other corrections could lift this singularity, and if so, whether it is possible to maintain zero entropy.
4 Four-dimensional Nernst branes from dimensional reduction
4.1 Review of four-dimensional Nernst branes
We now want to dimensionally reduce our ve-dimensional Nernst branes and compare the resulting four-dimensional spacetimes to those found in previous work [11]. Let us therefore review the relevant features, emphasising the problems that we want to solve. Four-dimensional Nernst branes depend on three parameters: the temperature T , the chemical potential and one electric charge Q0. Due to Dirac quantisation17 charge is discrete, and the solution depends on two continuous parameters. The asymptotic geometries, both at in nity and at the horizon, are of hyperscaling violating Lifshitz (hvLif) type:
ds2(d+2) = r2(d )=d r2(z1)dt2 + dr2 +
2
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: (4.1)
Here t is time, r parametrizes the direction transverse to the brane, and xi, i = 1; : : : ; d
are the directions parallel to the brane (with d = D 2 in D spacetime dimensions). For
= 0 the line element (4.1) is invariant under rescalings
(t; r; xi) [mapsto]!( zt; r; xi) :
17The four-dimensional theory admits both electric and magnetic charges, though for the solution in question only one electric charge has been turned on.
{ 25 {
Xi=1 dx2i
!
The parameter z, which measures deviations from relativistic symmetry (due to time scaling di erent from space) is called the Lifshitz exponent. For [negationslash]= 0 the metric is not scale
invariant but still scales uniformly, and is known as the hyperscaling violating exponent.
For four-dimensional Nernst branes the geometry at in nity is independent of the temperature. For nite chemical potential it takes the form of conformally rescaled AdS4,
which is of the above type, with z = 1, = 1. Moreover, the curvature scalar goes
to zero R(4) ! 0, while the scalar elds zA; A = 1 : : : n(4)V run o to in nity zA ! 1.
In contrast, for in nite chemical potential the geometry at in nity is asymptotic to hvLif with z = 3 and = 1. The behaviour of curvature and scalars is precisely the opposite as previously: the curvature scalar diverges R(4) ! 1, while the scalar elds go to zero
zA ! 0. The geometry at the horizon is independent of the chemical potential, but depends
on the temperature. For zero temperature the asymptotic geometry is again hvLif with z = 3, = 1, but approaching the opposite end of this geometry, so that the curvature scalar goes to zero R(4) ! 0. While there is no sp curvature singularity there remains a
pp curvature singularity at the horizon, that is, freely falling observers experience in nite tidal forces. Simultanously the scalar elds go to in nity zA ! 1. This type of behaviour
can be considered as a generalized form of attractor behaviour [6]. For nite temperature the geometry takes the expected form for a non-extremal black brane, the product of two-dimensional Rindler space with R2. The scalars and the curvature take nite values, so that the solutions are regular at the horizon for non-zero temperature.
The only element of hvLif holography that we will use is the entropy-temperature relation
S T (d )=z ;
valid for eld theories with hyperscaling violation [12].
Since the low-temperature asymptotics of the exact entropy-temperature relation of four-dimensional Nernst branes is S T 1=3, which matches the scaling properties of the
asymptotic zero temperature near horzion hvLif geometry with = 1, z = 3, gauge/gravity duality implies the existence of a corresponding three-dimensional non-relativistic eld theory with this scaling behaviour. As the solution is charged, the global solution should describe the RG ow starting from a UV theory corresponding to asymptotic in nity, and ending with this IR theory. Identifying this UV theory turned out to be problematic: the solution at in nity jumps discontinously between nite and in nite chemical potential, and is singular in either case. The more likely candidate (having no curvature singularity) is the conformally rescaled AdS4, which still does not look like a ground state, due to the runaway behaviour of the scalars. Moreover, while the geometric scaling properties indicate an entropy-temperature relation of the form S T 3, the high temperature asymptotic of
the four-dimensional Nernst brane solution is S T . As discussed before, this lead to the
conjecture that the solution decompacti es at in nity, and needs to be understood from a ve-dimensional perspective.
4.2 S1 bulk evolution
To relate ve-dimensional Nernst branes to four dimensions, we make the spacelike direction x0 compact, i.e. we identify x0 x0 + 2r0. Clearly then, to understand the
{ 26 {
JHEP11(2016)114
Figure 1. Plot showing the evolution of the compacti cation circle throughout the ve-dimensional bulk.
four-dimensional properties, it is crucial to rst understand the behaviour of the x0 circle. Writing (3.38) as18
ds2(5) = e2(dx0 + A04dx4)2 + eds2(4);
with
l2 ; (4.2) we nd the four-dimensional line element
ds2(4) =
Ar2l2 + r2l2 : (4.4)
Thus the geodesic length of the compacti cation circle S1 varies dynamically along the transverse direction, parametrized by r, of the four-dimensional spacetime, as shown in gure 1. Notice from (4.4) that for A > 0 there are two competing terms, resulting in decompacti cation both for r ! 1 and for r ! 0. The latter decompacti cation is only
reached in the extremal limit, since otherwise we encounter the horizon at r+ > 0. This
implies that in the non-extremal case the near horizon solution will still depend on the parameter A, while in the extremal case the near horizon solution becomes independent of A. The insensitivity of the extremal near horizon solution to changes of parameters which determine the asymptotic behaviour at in nity, in our case A, can be viewed as a version of the black hole attractor mechanism. Making the solution non-extremal results in the
18As it stands, (3.38) is specialized to the case A > 0 since it involves the variable ~
=
2B0A . However,
using (3.35), it is possible to write (3.38) in terms of a general Kaluza-Klein vector, valid for both A > 0 and A = 0. This then allows the reduction of both cases in parallel, leading to (4.3).
{ 27 {
JHEP11(2016)114
e2 = r2f(r)
r2W (r)l2f(r)1=2 dt2 + f(r)1=2l2 dr2r2W (r) +r2l2 f(r)1=2(dx2 + dy2); (4.3)
after identifying x4 t. From (4.2) we can read o the behaviour of the physical (geodesic)
length R0phys of the compacti cation circle:
(R0phys)2 = (2r0)2e2(r) = (2r0)2
r l
loss of attractor behaviour by making the near horizon solution sensitive to the asymptotic properties of the solution at in nity. Of course, the four-dimensional scalars run o to in nity instead of approaching nite x-point values, but they do so in a particular, ne-tuned way, which leads to a consistent lifting of the near horizon geometry ve dimensions. A remarkable feature of solutions with A > 0 is the existence of a critical point, Pcrit, where
the compacti cation circle reaches a minimal size at r4crit = =A. In contrast, for A = 0, this critical point does not exist and so, whilst the circle continues to decompactify as r ! 0
in the extremal case, it now shrinks monotonically with increasing r, ultimately becoming a null circle19 of zero size for r ! 1. This fundamentally di erent behaviour of the S1
means we must treat the dimensional reduction of the A > 0 and A = 0 cases separately in what follows. Additionally, we clearly see that A is the parameter responsible for the asymptotic behaviour at in nity from a ve-dimensional point of view. This resembles the role played by the parameter h0 in the four-dimensional solutions of [11]: this connection will be made manifest in the following subsections.
In the case A > 0, the compacti cation introduces a new continuous parameter, the parametric radius r0 of the circle. We now observe that the identi cation x0 [similarequal] x0 +
2r0 breaks the scaling symmetry (3.39), which made the parameter A irrelevant for ve-dimensional (uncompacti ed) solutions. For A > 0 there is a circle of minimal size at r4crit = =A, with geodesic size R0crit given by
(R0crit)2 = 8 r20
l2 p A :
The size of this minimal circle depends only on the combination r20pA and is therefore invariant under any increase in A that is compensated for by a reduction in r0 and vice-versa. This ability to trade r0 for A means that A can be used as the physical parameter controlling the minimal circle size, whilst r0 becomes redundant. It is natural to set r0 =
pA, as this is precisely what is needed such that the expression for the four-dimensional charge, Q0, calculated later in (4.17), is independent of the compacti cation radius, which is natural for a quantity which was de ned in [11] in a purely four-dimensional context.
In the case A = 0, there is no such invariant length and we can see this in a number of ways. Firstly, the A ! 0 limit pushes r4crit = A ! 1 and so no minimal circle exists.
Secondly, with A = 0, the geodesic size of the compacti cation circle is found from (4.4) to be (R0phys)2 = (2r
0)2 r2l2 and depends only on ; since this is already a parameter of the ve-dimensional solution, there is nothing else to be accounted for and no need for additional parameters. One can try to obtain an invariant length from the size of the circle on the horizon, R0phys(r+), which, assuming non-extremality, will at least be nite. However, it is clear from (4.4) that this will be a function of both and r+, which again are already existing parameters of the ve-dimensional A = 0 solution.
4.3 Dimensional reduction for A > 0
4.3.1 Four dimensional metrics and gauge elds
In [11] a family of four-dimensional Nernst branes was found, which depend on one electric charge Q0 and two continuous parameters B(4d)0 and h0, which can be expressed alterna-
19The norm-squared of the tangent vector @x0 goes to zero in this limit.
{ 28 {
JHEP11(2016)114
tively in terms of temperature T (4d) and chemical potential . It was also observed that the four-dimensional solutions with nite chemical potential exhibited a speci c singular behaviour in the asymptotic regime, which suggested to be interpreted as a decompacti -cation limit. Given the behaviour of the compacti cation circle, the natural candidate for a lift of four-dimensional Nernst branes with nite chemical potential is the A > 0 family of ve-dimensional Nernst branes.
We begin by comparing the four-dimensional Nernst brane solutions with nite chemical potential (h0 [negationslash]= 0) as found in [11] to the four-dimensional metric in (4.3) obtained by
dimensionally reducing our ve-dimensional solution with A > 0. Setting = r4 in (3.30) of [11] gives:
ds24 = H1=2W (4d)r3dt2 +
16H1=2 W (4d)
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dr2r + H1=2r3(dx2 + dy2) ; (4.5)
where W (4d) = W (4d)(r) = 1 2B
(4d) 0
r4 and
(4d)0 h0=Q0
r4
H(r) = C
"
Q0 B(4d)0
sinh B(4d)0h0
Q0 +
Q0eB
#
=: cH0(r) : (4.6)
Here Q0 parametrizes the four-dimensional electric charge, the continuous parameter h0 [negationslash]= 0
corresponds to a chemical potential , with [notdef] [notdef] < 1,20 and the continuous parameter
B(4d)0 0 corresponds the temperature T (4d) 0. The constant C is determined by the
choice of a prepotential and a gauging of the four-dimensional theory. More precisely, it is determined by the cubic coe cients cijk and gauging parameters gi, but since we are assuming that this solution can be lifted to ve-dimensions, these are the same parameters that enter into our ve-dimensional theory in (2.1). The precise form of C can be read o from the unnumbered equation between (3.30) and (3.31) in [11]. At this point we anticipate that the functions W (4d) and W in the four- and ve-dimensional solutions can be identi ed, which allows us to drop the superscrips 4d on B0 and T . Since we can no longer rescale the coordinate r, matching the coe cients of dr2 between the metrics (4.3) and (4.5) xes the relation between the functions f(r) and H(r) to bel2f = 162H = 162CH0 :
Then the remaining metric coe cients match if we rescale t; x; y by constant factors involving l.21 Writing out the functions f and H and comparing, we obtain:
162C Q0
B0 sinh
Q0 = l2A ; (4.7)
162CQ0eB0h0=Q0 = l2 :
20Due to the speci c choices made for certain signs, the chemical potential will turn out to be negative. This is correlated with a choice of sign for the electric charge. There is another branch of the solution, which we dont give explicitly, where these signs are reversed.
21Alternatively, we could absorb l into r, but then by comparing the functions W we will conclude that the respective parameters B0 di er by a factor l4. Given the relation of B0 to the position of the event horizon and to temperature, we prefer not to do this.
B0h0
{ 29 {
While the ve-dimensional line element is non-static, the four-dimensional one is static, but as an additional degree of freedom we have a Kaluza-Klein gauge eld, given by
A0t(r) = 0 =
1 p2A04 =
p6 p2
uz ut
W (r) f(r) =
p3wW (r)f(r) : (4.8)
Here we use the de nitions and conventions of section 2.2, and with regard to four-dimensional quantities, we use the conventions of [29], which were also used in [11].
By matching the expression for _
0 given by (3.12) with the -derivative of (3.38) of [11], we can identify the Kaluza-Klein vector with the four-dimensional gauge eld provided that
p3 ( + 2B0A)
A2 =
B202Q0 sinh2 B0h0Q0; (4.9)
1 +
B0A = coth
B0h0
Q0 :
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From this we can nd
Q0 =
1 6
p3 ( + 2B0A) ; (4.10)
h0 = Q0
B0 arcoth
1 +
B0A
; (4.11)
which expresses the four-dimensional parameters Q0; h0 in terms of the ve-dimensional parameters A; ; B0. Comparing (4.7) to (4.9) we nd that these relations are mutually consistent provided that
162C = 2p3l2 : (4.12)
This equations relates the overall normalizations of metrics (4.3) and (4.5) and of the underlying vector multiplet actions.
The four-dimensional chemical potential is given by the asymptotic value of the gauge eld At, which is chosen such that At(r+) = 0, as explained in appendix F. Having matched the ve-dimensional Kaluza-Klein vector with the four-dimensional gauge eld of [11], the corresponding expressions for the chemical potential must also match.22 For reference, we provide the following expression in terms of both four- and ve-dimensional parameters,
= 12
B0 Q0
coth B0h0Q0 2 = 2Q0A =
p3 A
r
+ 2B0A ; (4.13)
where we used (4.9). Notice from (4.10) that Q0 < 0 which then forces h0 < 0 by (4.11), which is consistent with the remark in [11] that sign(h0) = sign(Q0). Moreover we observe that Q0 < 0 implies < 0. This re ects the correlation in the signs of the charge Q0 and of the chemical potential . We have, for concreteness and simplicity, restricted ourselves to solutions where _
0 > 0, which have turned out to correspond to negative charge and negative chemical potential. Conversely, solutions with _
0 < 0 will carry positive charge
22This can be seen explicitly by applying (4.9) to (3.39) in [11] and comparing to the asymptotic value of (4.8).
{ 30 {
and positive chemical potential. This is consistent with the fact that in relativistic thermodynamics the chemical potentials of particles and antiparticles di er by a minus sign.
For completeness we note a few further signs which are implied by our decisision to focus on solutions with _
0 < 0 (and, hence, 0 > 0). From (4.7) we deduce that the four-dimensional constant C must be negative, C < 0, which explains the minus sign in (4.12). Furthermore, it is clear from (4.6) that H0(r) < 0 such that the harmonic
function H(r) > 0, which we need in order that the roots of H, which appear in our
expression for the solution, are real.
4.3.2 Momentum discretization, charge quantization and parameter counting
Since the reduction is carried out over the x0 direction, it is instructive to calculate the Killing charge associated to the Killing vector @0 = @=@x0. For A > 0, (3.40) tells us this is related to the Killing vector @z of the ve-dimensional spacetime via
@0 = pA@z:
Since the charge associated with @z is the brane momentum (3.57), the Killing charge corresponds to momentum in the x0 direction, and can be determined as follows
P 0 = pAPz [similarequal]
Nr0 =
p ( + 2B0A) (4.16)
and comparing to (4.10), we see explicitly how the quantization of the internal momentum implies the quantization
Q0 [similarequal] pAP
2 pA
p ( + 2B0A) ; (4.14)
where we have omitted V3 and l for simplicity. The periodicity of the x0 direction implies that momentum takes discrete values,
P 0 [similarequal]
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NpA; N 2 Z [ [notdef]0[notdef] ; (4.15)
where we have taken into account that P 0 0. Rearranging this asN [similarequal] pAP
0 [similarequal] N [similarequal] 2
0 [similarequal] N; N 2 Z [ [notdef]0[notdef] (4.17)
of the four-dimensional charge. Note that while the spectrum of P 0 changes with the radius r0 = pA of the compacti cation circle, the four-dimensional electric charge Q0 is independent of it. As already mentioned before, P 0 and Q0 being negative results from choosing 0 positive, and solutions with positive P 0 and Q0 can be obtained by ipping signs in (3.13). Our choice of signs is consistent with the choices made in [11], in particular the same anti-correlation between the signs of A0t = 0 and Q0 can be observed in the equation above (3.38) of [11].
Let us end this discussion by comparing the number of parameters describing the Nernst branes in di erent dimensions. Five-dimensional Nernst branes are parametrized by three continuous paramters (A; B0; ), but for A > 0 we have the scaling symmetry (3.39),
{ 31 {
which tells us that A is redundant, and that we can parametrize solutions by the two independent and continuous parameters (B0; ~
), which then correspond to temperature and boost momentum. Upon compacti cation a new length scale is introduced that breaks the scaling symmetry present in ve dimensions. Consequently, the four-dimensional solution picks up an extra parameter; we need to specify the three independent and continuous parameters (B0; ; A) in order to completely de ne the metric (4.3). In terms of physical parameters, the four-dimensional solution depends on temperature, charge and chemical potential (T; Q0; ). These are all independent but, as we have seen, since the momentum has a component in the direction we compactify over, it becomes discrete, which corresponds directly to the discretization of four-dimensional electric charge. As such, the ve-dimensional solution involves two independent and continuous thermodynamic parameters whilst the four-dimensional solution has three independent parameters, two of which are continuous and one of which is discrete.
4.4 Dimensional reduction for A = 0
The two parameter family of four-dimensional Nernst branes found in [11] exhibits discontinuities in the asymptotic behaviour of both the geometry and the scalar elds when taking the limit h0 ! 0, or equivalently, [notdef] [notdef] ! 1. This discontinuity can be accounted for
by the discontinuous asymptotic behaviour of the compacti cation circle in the limit A ! 0
as seen in gure 1. We should therefore expect that the in nite chemical potential four-dimensional solutions of [11] with h0 = 0 can be recovered from the A = 0 ve-dimensional solution with one dimension made compact.
To demonstrate this relationship we take the four-dimensional Nernst brane metric (4.5) obtained in [11] and set h0 = 0 in (4.6) which reduces the function H(r) to
H(r) = CQ0 r4 :
Substituting this back into (4.5) gives the following metric
ds24 = C1=2Q1=20W (4d)r5dt2 +
r5W
1=2l3 dt2 +
l 1=2
r3W dr2 +
r 1=2
l3 (dx2 + dy2); (4.19) where we have used (4.3) with A = 0. Again we identify the functions W (4d) and W
appearing in the above metrics, which means the parameters B0 and T will be the same in both cases. As before, this prevents rescaling of the coordinate r and then, by comparing dr2 terms in (4.18) and (4.19), we establish the following relationship between four- and ve-dimensional quantities
162CQ0 = l2 : (4.20)
Again, the remaining metric coe cients can be made to match by rescaling t; x; y by constant factors involving l. Following the same procedure as in section 4.3.1, we match
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16C1=2Q1=20dr2
W (4d)r3 + C1=2Q1=20r(dx2 + dy2): (4.18)
On the other hand, the dimensional reduction of the A = 0 class of ve-dimensional Nernst branes gives
ds24 =
the gauge eld and Kaluza-Klein vector by comparing expressions for _
0. Speci cally, we match (3.12) with the -derivative of (3.38) in [11]. The two are equivalent provided that
Q0 =
2p3; (4.21)
which expresses the four-dimensional electric charge in terms of the ve-dimensional boost parameter . This is a much simpler expression than in the A > 0 case and we observe that it matches the A ! 0 limit of (4.11). Considering the discontinuities we have encountered
previously when taking A ! 0 limits, this seems at rst surprising but just re ects that Q0
is a well de ned paramater for the four-dimensional solutions of [11], for any choice of and T . Having established Q0 < 0, we see from (4.11) that A ! 0 corresponds to h0 ! 0,
and thus from (4.13) that ! 1. Lastly, we can substitute (4.21) into (4.20) to nd
the relationship between the overall normalizations of the metrics (4.18) and (4.19),
162C = 2p3l2: (4.22)
Clearly this requires C < 0 as before and, in fact, is exactly the same relationship as for the A > 0 case in (4.12), which is expected since C and l are only sensitive to the four-and ve-dimensional multiplet actions respectively, and these are indpendent of A. Again, since we have matched the gauge elds by comparing _
0, the chemical potentials must match and this is indeed the case; using the asymptotic value of (4.8) with A = 0, we nd = 1 which agrees with the negatively charged, h0 = 0 solutions in [11].
The parameter counting becomes simpler in the A = 0 case. Five-dimensional Nernst branes are parameterized by two independent and continuous parameters (B0; ), or equivalently temperature and momentum. However, as we have seen in section 4.2, no new length scale is introduced by the reduction and consequently, the four-dimensional solution obtained via dimensional reduction also depends on exactly two independent parameters, (B0; ), which are su cient to completely determine (4.18) since A = 0 is xed. Using (4.21), these are equivalent to (T; Q0) with = 1. The di erence between the
ve-dimensional and four-dimensional parameters is that the S1 causes charge quantization. This means that whilst both B0 and are continuous in ve dimensions, reducing to four dimensions forces one parameter, namely Q0 , to become discrete.
One di erence between the A = 0 solution and the A > 0 solution is that for the A = 0 solution the compacti cation circle has no critical value. Therefore we cannot relate the momentum P 0 to the electric charge Q0 using r0 as a reference scale. This is not a problem since we could relate Q0 to ve-dimensional quantities through (4.21), and, moreover, we have seen that the relation between Q0 and ve-dimensional quanities has a well de ned limit for A ! 0. A related feature of the A = 0 solution is that compacti cation circle has
no minimal size, and contracts to zero for r ! 1. That means that there is a region in this
solution, where the circle has sub-Planckian, or sub-stringy size. While this is problematic for an interpretation as a four-dimensional solution, the lifted ve-dimensional solution is simply AdS5, and can be decribed consistently within ve-dimensional supergravity.
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JHEP11(2016)114
4.5 Curvature properties of four-dimensional Nernst branes
The four-dimensional solutions with A > 0 and A = 0, obtained in sections 4.3 and 4.4, exactly match the h0 < 0 and h0 = 0 solutions of [11] respectively. In [11] these four-dimensional solutions were observed to be hyperscaling-violating Lifshitz metrics. It is known from [16] that such solutions su er from various curvature singularities, and we shall now investigate this by computing the singular behaviour of the metrics (4.3) and (4.19).
Curvature invariants. As with the ve-dimensional spacetimes in section 3.5 we can determine the presence of curvature singularities of our four-dimensional solutions by looking at the Kretschmann scalar and Ricci scalar associated to the metrics (4.3) and (4.19). Indeed, since any singular behaviour in the curvature will already be present for the extremal solutions, we will concentrate only on the case r+ = 0. The curvature invariants are calculated (using Maple) to be
KA>04 =
r2 351A4r16 + 1476A3r12 + 2586A2r8 2 + 1284Ar4 3 + 959 4
4L2 (Ar4 + )5 ;
RA>04 =
3 15A2r8 + 34Ar4 + 15 2
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2
qAr4+ r4 (Ar4 + )2 rL;
45r2p L : (4.23)
For A > 0, or equivalently [notdef] [notdef] < 1, we nd that the Ricci scalar behaves as R r1 for large r, and R r for r ! 0, whilst the Kretschmann scalar scales as K r2 and
K r2 in these respective regions. Hence, the curvature invariants will remain nite
along the solution. However, for the A = 0 solution we will still have the same behaviour at r ! 0, but asymptotically we nd R r and K r2. We therefore have a naked
curvature singularity as we approach the boundary of the spacetime.
Tidal forces. In order to investigate whether the four-dimensional solutions of [11] admit in nite tidal forces in the near-horizon regime we will follow the analysis of [16], albeit considering a slightly simpler set-up in which the infalling observer is moving only in the radial direction i.e. has zero transverse momentum. The technical details of this procedure can be found in appendix E.
Our results in tables 4 and 5 show that, for both A > 0 and A = 0, there exist components of the Riemann tensor, as measured in the PPON, that diverge as r ! 0.
This indicates that the radially infalling observer will experience in nite tidal forces at the extremal horizon, r+ = 0. As before, tidal forces will remain nite on non-extremal horizons, r+ > 0.
4.6 Curing singularities with decompacti cation
A summary of the singular behaviour of our four- and ve-dimensional solutions can be found in tables 1 and 2. Notice that since B0 and A control the near horizon and asymptotic geometries respectively, we can use these to catalogue any singularities. We will now
{ 34 {
KA=04 = 959r2
4 L2 ; RA=04 =
B0; h0 Near Horizon Asymptotic
Curvature
Singularity 1
Tidal Forces
Curvature
Singularity 1
Tidal Forces
B0 = 0; A = 0 [notdef] [check] [check] [notdef] B0 = 0; A > 0 [notdef] [check] [notdef] [notdef]
B0 > 0; A = 0 [notdef] [notdef] [check] [notdef] B0 > 0; A > 0 [notdef] [notdef] [notdef] [notdef]
Table 1. Summary of singular behaviour of four-dimensional Nernst brane.
B0; A Near Horizon Asymptotic
Curvature
Singularity 1
Tidal Forces
Curvature
Singularity 1
Tidal Forces
B0 = 0; A = 0 [notdef] [check] [notdef] [notdef] B0 = 0; A > 0 [notdef] [check] [notdef] [notdef]
B0 > 0; A = 0 [notdef] [notdef] [notdef] [notdef] B0 > 0; A > 0 [notdef] [notdef] [notdef] [notdef]
Table 2. Summary of singular behaviour of ve-dimensional Nernst brane.
explain how the singularities present in the four-dimensional hyperscaling-violating Lifshitz solutions of section 4.5, except those related to in nite tidal forces at extremal horizons, can be removed by dimensional lifting to the asymptotically AdS solutions of section 3.5.
4.6.1 Curvature invariants
Dimensional reduction relates the ve-dimensional Ricci scalar to its four-dimensional counterpart by23
R5 eR4:
As can be seen from table 1 and table 2, the only situation where we encounter a curvature singularity is the asymptotic regime of the four-dimensional solution with h0 = 0, or
equivalently A = 0. In this instance we have R4 r from (4.23) whilst e 1=r from (4.2)
resulting in R5 being asymptotically constant and exactly equal to the value of global AdS5 as seen in section 3.5. Recalling that the dilaton e measures the geodesic length of the x0 circle, we can now account for the presence of an asymptotic curvature singularity in this class of four-dimensional Nernst branes. Speci cally, the four-dimensional, =
1, asymptotic curvature singularity emerges from a bad slicing, of the parent AdS5
hyperboloid by a circle that gets pinched at in nity. It was shown previously, that the independent four-dimensional scalars are all proportional to each other, see formula (3.29) in [11]. It was also observed that for in nite chemical potential, these scalars approach zero asymptotically. From the ve-dimensional point of view, the single pro le of the four-dimensional scalars determines the pro le of the Kaluza-Klein scalar. Therefore the four-dimensional scalars approaching zero corresponds to the shrinking of the compacti cation
23Similarly, the Kretschmann scalars are related by K5 e2K4. The appearence of the second power of
the dilaton re ects the fact that the Kretschmann scalar is quadratic in the curvature.
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circle. When combining this with the singular behaviour of the four-dimensional metric, we obtain AdS5.
In the A > 0, or equivalently [notdef] [notdef] < 1, case the four-dimensional solution of [11]
is asymptotically conformal to AdS4, or CAdS4 for short. We see from (4.23) that the curvature invariants of CAdS4 behave as R4 1=r and vanish asymptotically. At the
same time, this is compensated by e r from (4.2), meaning the circle now blows up
at large r such that R5 remains asymptotically constant and equal to RAdS5. Thus, in this case the asymptotic behaviour of the four-dimensional metric and scalars is reversed compared to the A = 0 case, but still leads to the same ve-dimensional asymptotic geometry after lifting.
4.6.2 Tidal forces
As can be seen from tables 1 and 2, tidal forces are asymptotically irrelevant24 and so we are only concerned with the situation near the horizon. It is clear that in nite tidal forces are present at the horizon of the extremal Nernst brane in four-dimensions, and are not removed by dimensional lifting. This seems to be the price for obtaining the strong version of Nernsts law.
5 Summary, discussion, and outlook
5.1 The ve- and four-dimensional perspective, and looking for a eld theory dual
Let us summarize and discuss our results. Starting from FI-gauged ve-dimensional super-gravity with an arbitrary number of vector multiplets, we have obtained a two-parameter family of Nernst branes, labelled by temperature and momentum. These solutions interpolate between AdS5 and an event horizon, and have an entropy-temperature relation interpolating between S T 3 at high temperature/low boost and S T 1=3 at low temper
ature/high boost. The relation S T 3 is consistent with the scaling properties of AdS5.
Given that we are working within ve-dimensional gauged N = 2 supergravity, the dual UV eld theory should be a conformally invariant four-dimensional N = 1 eld theory. Since the metric is the same as in the duality between gauged N = 8 supergravity and N = 4 Super Yang Mills, one might expect it to be a conformally invariant N = 1 Super Yang Mills theory or a deformation thereo , but without having a higher dimensional embedding which allows one to understand the role of the parameters cijk and gi of the gauge theory, we cant say much more.
We have seen how the ve-dimensional lift of four-dimensional Nernst branes removes all the singularities at asymptotic in nity as well as the mismatch between geometrical and thermodynamic scaling relations. To understand the variation of the compacti cation circle along the transverse direction, which from the four-dimensional point of view is encoded in the scalar elds, is crucial. The apparently singular behaviour of the four-dimensional geometry is exactly compensated for by the singular behaviour of the scalars,
24See appendices D and E for reasons why.
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JHEP11(2016)114
or, put di erently, by the behaviour of the circle one has to add to obtain asymptotically AdS5. Moreover, the compacti cation circle also accounts for the four-dimensional chemical potential, which has no counterpart in the un-compacti ed ve-dimensional solution. However, once we decide to make the boost direction compact the dynamics forces the circle to expand at both ends, and the resulting minimum introduces a new parameter which we can relate to the chemical potential. As proposed in [11], we can interpret the apparently singular UV behaviour of four-dimensional Nernst branes as a dynamical decompacti cation limit, which tells us that the description as a four-dimensional system breaks down and has to be replaced by a ve-dimensional one.
The ve-dimensional solution admits a non-trivial extremal limit, where the boost parameter is sent to in nity, while the momentum (density) is kept xed. The resulting extremal near horizon geometry should de ne a eld theory with entropy-temperature relation S T 1=3. In the context of boosted D-branes and M-branes, the proposed inter
pretation is a conformal eld theory in the in nite momentum frame, which carries a nite momentum density [17]. Moreover, it was proposed in [23, 47, 48, 57] that the compacti -cation of the direction along the boost corresponds to discrete light cone quantisation. In this respect it is interesting to look at the asymptotic scaling symmetries of the ve- and four-dimensional extremal solutions near the horizon. In ve dimensions the metric looks like a Lifshitz metric with z = 3 and = 0, except that the direction along the boost has weight 1 instead of +1. Upon reduction to four dimensions, the asymptotic geometry, and
if we go to in nite chemical potential even the global geometry, is a hyperscaling violating Lifshitz geometry with z = 3 and = 1 [11]. That is, by reduction over the boost direction one trades the non-trivial scaling of this direction for an overall scaling of the metric. Following [23, 47, 48, 57] we propose to associate a four- and a three-dimensional eld theory to the near-horizon ve- and four-dimensional geometries, respectively, with the three-dimensional theory encoding the zero mode sector of the discrete light cone quantisation of the four-dimensional theory. Both theories are non-relativistic with Lifshitz exponent z = 3, and supersymmtric with two supercharges.25 The four-dimensional theory is scale invariant and arises by deforming a four-dimensional relativistic N = 1 supersymmetric theory by a nite momentum density, while the three-dimensional theory is scale covariant.
5.2 The fate of the third law
From a strictly gravitational point of view, one should still worry about the pp curvature singularities which persist in the extremal limit irrespective of whether we consider four-dimensional or ve-dimensional Nernst branes. While sometimes considered to be mild, they are genuine curvature singularities which make the solution geodesically incomplete. Moreover, they are not cured by stringy [prime]-corrections [16], and strings probing pp singularities get in nitely excited [15]. While at nite temperature there is technically no singularity, near extremality objects falling towards the event horizon will still experience very large tidal forces [59]. This behaviour is, if not an inconsistency, at least a sign that the singularity has physical relevance. Moreover, the pp singularity is clearly caused by the
25According to the analysis of [58], extremal four-dimensional Nernst branes are BPS.
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JHEP11(2016)114
way the metric complies with the strong version of Nernsts law, namely through a warp factor which scales any nite piece of the world volume26 to zero volume. It is not obvious at all how pp singularities could be removed while keeping the strong version of Nernsts law. For small BPS black holes, R2-corrections remove null curvature singularities, by making the area nite [60]. But as these singularities are of the sp type, it is not clear what this implies for pp singularities. One example where a pp singularity is removed is the D6 brane of type IIA supergravity, using an M-theory embedding [61]. The e ect of higher curvature corrections on pp type singularities has been investigated in [62, 63]. One can also approach the problem from the eld theory side. For example, in [64] they study the in nite momentum frame CFT dual to a boosted brane and nd evidence that the CFT resolves the geometric singularity. In our case it would be interesting to understand the dual four-, or possibly, the three-dimensional IR eld theory, and to investigate whether it is non-singular, and whether its ground state is unique or degenerate. And if the ground state is unique, one would need to understand whether this means that (i) pp-singularities are acceptable, (ii) they are not, but the dual eld theory can be used to construct a quantum geometry of some sort, (iii) or if there is some kind of breakdown of gauge/gravity duality in the extremal limit. Points (i) and (iii) are not necessarily mutually exclusive, since one might invoke the process version of the third law to assure that the extremal limit cannot be reached by any physical process.
5.3 Constructing solutions
This paper is part of a series of papers where explicit, non-extremal solutions of ve- and four-dimensional ungauged and gauged supergravity have been constructed using time-like dimensional reduction in combination with special geometry [39, 42, 45]. As explained in section 2, solutions correspond to curves on a particular submanifold of the paraquaternionic Kahler manifold obtained by reduction to three dimensions, which satisfy the geodesic equation deformed by a potential. As part of the solution we have obtained an explicit expression for a stationary point of the ve-dimensional scalar potential, corresponding to an AdS5 vacuum, for an arbitrary number of vector multiplets and general
FI-gauging. While we initially obtain solutions to the full second order eld equations, with the corresponding number of integration constants, we have seen that once we impose regularity of the lifted ve-dimensional solution at the horizon27 the number of intergration constants is reduced by one half, so that the solution satis es a unique set of rst order equations. Such behaviour has been observed before, and been interpreted as a remnant of the attractor mechanism [45].28 For our ve-dimensional solutions the scalars are constant, so that the only sense in which we have attractor behaviour is that the scalars sit at a stationary point of the scalar potential. However, from the four- and three-dimensional perspective we have scalar elds which need to exhibit a particular, ne-tuned, asymptotic behaviour at the horizon in order to make the ve-dimensional solution regular. This is
26Here nite refers to the Euclidean metric de ned by the coordinates x, y, z, which we use to refer extensive quantities to unit world volume.
27This is done in the generic situation, that is in particular for nite temperature.
28A related idea seems to be that of hot attractors [65].
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very similar to attractor behaviour, and the e ect of reducing the number of integration constants by one half is the same. Such universal features of scalar dynamics deserve further study.
In the present paper we have made a very particular choice of the ansatz, which was tailored to obtaining the ve-dimensional lift of the four-dimensional Nernst branes of [11]. In the future we will study systematically other choices, which will lead to other and more general solutions. Already in [11] a four-dimensional magnetic solution was found, and we expect that it is possible to obtain dyonic solutions as well. It would also be interesting to revisit the issue of embeddings into ten- and eleven-dimensional supergravity.
Acknowledgments
The work of JG is supported by the STFC grant ST/1004874/1. The work of TM is partially supported by STFC consolidated grant ST/G00062X/1. The work of DE was supported by STFC studentship ST/K502145/1 and by the Scottish International Educational Trust. TM would like to thank Gabriel Cardoso and Suresh Nampuri for extensive and helpful discussions about their work on Nernst branes.
Data management. No additional research data beyond the data presented and cited in this work are needed to validate the research ndings in this work.
A Rewriting the scalar potential
Our goal in this appendix is to obtain a workable expression for the scalar potential V3 appearing in (2.9). Let us concentrate on the term (cyyy)(cy)1[notdef]ij. This is to be interpreted as the matrix inverse to (cyyy)1(cy)ij in the sense that
(cyyy)(cy)1[notdef]ij (cy)jkcyyy = ik: (A.1)
Now, using the expression (2.12) for ^
gij(y):
^
gij(y) = 32
(cy)ij
cyyy
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3 2
(cyy)i(cyy)j (cyyy)2
;
23 ^gjk(y) + 32(cyy)j(cyy)k(cyyy)2 : (A.2)
We now introduce the dual scalars yi via
@ yi := ^
gij(y)@ yj; yi = 34
(cyy)icyyy = ^
gij(y)yj:
we have
ik = (cyyy)(cy)1[notdef]ij
Hence, (A.2) becomes
23 ^gjk(y) + 83yjyk : (A.3)
{ 39 {
ik = (cyyy)(cy)1[notdef]ij
In other words, the quantity (cyyy)(cy)1[notdef]ij is just the inverse of the term in square brackets in (A.3). Thankfully, the latter is easily invertible. Indeed, we nd
3 2
^gij(y) + 2yiyj
[notdef]23 [^gjk(y) + 4yjyk] = ik :
Hence we can rewrite
(cyyy)(cy)1[notdef]ij = 32 ^
gij(y) + 3yiyj; (A.4)
so that the scalar potential term in (2.9) becomes
V3 = 3
^gij(y) + 4yiyj gigj: (A.5)
B Quasi-local computation of conserved charges
We use the form of our ve-dimensional line element given in (3.41), which can be rewritten as
ds2 = l2dr2 r2W +
r2 l2
+ r4+r4 u u
dx dx ; (B.1)
where u = (ut; 0; 0; uz). Note that u u = 1 so we can interpret this as a velocity
vector. Following the procedure of [54] we want to calculate the quasilocal stress tensor T associated with the metric (B.1).
B.1 The quasilocal stress tensor
Given a timelike surface @Mr at constant radial distance r we de ne the metric on
@Mr via the ADM-like decomposition
ds2 = N2dr2 + (dx + N dr)(dx + N dr): (B.2)
We de ne the extrinsic curvature via
:=
1
2 (r ^
n + r ^
n ) ; (B.3)
where ^
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n is the outward-pointing normal vector to the surface @Mr. For solutions asymp
toting to AdS5 the procedure of [54] tells us that the quasilocal stress tensor is then given by29
T = ( ) ( )
3l
l2G ( ); (B.4)
where = is the trace of the extrinsic curvature, and G is the Einstein tensor for .
For the case at hand we see that the metric (B.1) decomposes according to (B.2) with
N2 = l2r2W ; N = 0; (r) =
r2 l2
+ r4+r4 u u
: (B.5)
29We remind the reader that in this paper we work in units where 8G = 1.
{ 40 {
n to a surface of constant r is given by
^
n = rl W 1=2(r) ;r;
from which we nd the extrinsic curvature
=
r 2l
The unit normal vector ^
1 r4+ r4
1=2@r = r2 l3
1 r4+ r4
1=2
r4+r4 u u
: (B.6)
In order to calculate the trace of this we need an expression for the inverse metric ,
which is given by
= l2 r2
" r4+ r4
1 r4+ r4
1 u u
#
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; (B.7)
where u = u , etc. This can be used to compute the trace of the extrinsic curvature
= =
2 l
1
#
1 r4+ r4
1=2
"2 + r4+ r4
1 r4+ r4
: (B.8)
Putting all this together, and noting that G ( ) = 0, we can use (B.4) to nd the resulting
gravitational stress-energy tensor induced on the boundary @Mr,
T = r4+
2l3r2 ( + 4u u ) + : : : ; (B.9)
where the dots represent terms which are subleading in the limit r ! 1.B.2 Mass, momentum and conserved charges
The quasilocal stress tensor (B.9) can be used to compute well-de ned mass and other conserved charges for the spacetime (B.1). Let be a spacelike hypersurface in @M =
limr!1 @Mr and make the ADM decomposition
dx dx = N2 dt2 + ab(dxa + Na dt)(dxb + Nb dt); (B.10) where [notdef]xa[notdef] are coordinates spanning , which has metric ab. Let U be the timelike unit
normal to . Then for any isometry of , which we take to be generated by a Killing vector , we can de ne a conserved charge Q by
Q =
Z dd1x p (U T ) : (B.11)
In particular, the mass of the solution is given by taking = @t, whilst the momentum in the direction xa is given by taking = @a.
For the boosted black brane we can make the ADM decomposition (B.10) of the metric (B.5) with
xx = yy = r2l2 ; zz =
r2 l2
1 + r4+r4 u2z ;
Nz = r4+
r4 uzut
1 + r4+r4 u2z 1;
N2 = r8+
l2r6 u2zu2t
1 + r4+r4 u2z 1
+ r2
l2
1 r4+r4 u2t :
{ 41 {
The timelike unit normal to has components
Ut =
l r
1 + r4+r4 u2z 1=2 1 r4+ r4
1=2 ;
Uz = lr4+
r5 utuz
1 + r4+r4 u2z 1=2 1 r4+ r4
1=2 :
Using these expressions, as well as the components of the quasilocal stress tensor (B.9), we can calculate the mass and linear momentum associated with the boosted black brane (B.1). Taking = @t and = @z we obtain the expressions (3.56) and (3.57)
for the mass and linear momentum respectively.
Finally, let us add some further comments on the fact that r+, and hence temperature, is a physical parameter despite that it can be absorbed by rescaling coordinates in (3.41). From (B.11), (3.53), (3.58) it is manifest that all quantities entering into the rst law are geometric quantities (norms of vectors elds, and integrals of functions over submanifolds using the induced metric) which are independent of the choice of coordinates. Applying the coordinate transformation R = r+r; ~T = t=r+; X = x=r+; Y = y=r+; Z = z=r+ to these expressions, it is straightforward to see that the parameter r+ is not eleminated, but scaled out as an overall prefactor. In particular
@t = r+@T ; @z = r+@Z ;
while
V3 =
Z dxdydz = r3+ Z dXdY dZ
so that irrespective of our choice of coordinates T r+, S r3+, M r4+ and Pz r4+.
It is precisely this r+-dependence of the thermodynamic quantities that gives rise to the correct temperature/entropy term in the rst law. Put di erently, when working in the rescaled coordinates ( ~T; R; X; Y; Z) the parameter r+ is hidden in the choice of the vector eld and the volume V3.
C Euclideanisation of the boosted black brane
As is well known from the study of Kerr black holes, obtaining the Hawking temperature by Euclidean methods is much more subtle for non-static spacetimes. For this reason, we nd it useful to give an explicit demonstration of how this works in the case of boosted (non-static) black branes. The treatment of the linear case given below will be parallel to the analysis of the Kerr black hole in [49].
A Euclidean continuation of the boosted black brane solution (3.41) can be obtained by setting t = i and uz = i , and taking and to be real. Observe that following the standard treatment of the Kerr solution, we do not only continue time but also the boost parameter w = uz=ut, which is analogous to the angular momentum parameter of the
Kerr solution in Boyer-Lindquist coordinates.
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The Euclidean section of the boosted black brane in (3.41) is then
ds2(5)E =
l2 r2
dr2
W +
r2l2 W (ut d + dz)2 +
r2l2 ( d + ut dz)2 + dx2 + dy2
:
We now explore the near horizon geometry by adapting a similar calculation used to examine the Kerr-Newman solution in [66]. Introducing the new radial variable R by R2 = rr+, the function W has the expansion
W = 4
r+ R2 + [notdef] [notdef] [notdef] ;
around the horizon. Expanding up to order R2, the metric takes the form
ds2(5)E;NH =
l2 r+
dR2 + 4r+l2 R2d~2 +r2+ + 2r+R2l2 (d2 + dx2 + dy2) ;
where we have replaced the coordinates and z by the new coordinates
~ = ut + z ; = utz :We remark that, in contradistinction to the Kerr-Newman solution discussed in [66], (i) the coordinate is linear rather than angular, i.e. we do not need to impose an identi cation on it; and (ii) the coordinate ~ is well de ned, since ut and are constant, so that utd + dz is exact. The horizon is at R = 0. The coordinates x; y; parametrize a three-dimensional plane with a metric which is at up to corrections of order R2. This part of the metric is clearly regular for R ! 0. The variables R and ~ parametrize a surface with metric
ds2Cone = l2
r+
1
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1
R2 r+
R2 r+
dR2 + 4R2 r2+l4 d~2
;
which is, up to a subleading term of order R2, the metric of a cone with apex at R = 0. Thus ~ is an angular variable and the surface parametrized by R and ~ is topologically a disk. Imposing the absence of a conical singularity at R = 0 xes the periodicity of ~ to be
~ [similarequal] ~ + 2
l2 2r+ :
Since the coordinate is linear (has no identi cations) we can determine the periodicities of and z from
(~;) [similarequal]
~ + 2 l22r+ ; ,(; z) [similarequal] ( + A; z + B) ;
with
A = 2 ut l2
2r+ ; B = 2
l2 2r+ :
The Hawking temperature T is read o from the periodicity of by [similarequal] + T 1, so that
T = r+
l2ut ;
which agrees with the result found by computing the surface gravity (3.54).
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To interpret the periodicity of z, remember that the boost velocity at the horizon is
w =
uzut = i
ut :
Thus
B = iw 1
T ;
so that the identi cations take the form
(; z) [similarequal] + T 1; z + iwT 1
;
which is analogous to the identi cation for the Euclidean Kerr solution, see for example [66].
D Five-dimensional tidal forces
In this appendix we shall construct the frame elds describing the PPON associated to an observer freely falling towards the ve-dimensional extremal black brane in (3.47). The frame-dragging e ects associated to the branes boost in the z direction mean that an observer who starts falling radially inward from in nity will acquire a velocity in the z direction. We want to pick our rst frame eld to be the vector eld generating the geodesic motion of the observer. To do this, we follow the procedure of [15, 16, 67] and introduce the frame eld
(^e0) =
d d
JHEP11(2016)114
= _t (@t) + _z (@z) + _r (@r) ; (D.1)
where is the proper time of our observer or, equivalently, the a ne parameter for the geodesic motion, and a dot denotes di erentiation with respect to . Note that for simplicity we consider an observer who is not moving in the x and y directions.
It is clear that to obtain ^
e0, we must rst obtain _
t; _
z and _
r. To do this, we recall that associated to each of the Killing vector elds @t; @z; @x; @y of (3.47) there is an integral of motion. These conserved quantities are the energy and momenta,
E = gt _
x =
r2l2 r2l2 _t r2l2 _z ; (D.2)
pz = gz _
x =
r2l2 + r2l2 _z + r2l2 _t ; (D.3)
px = gx _
x = r2 l2 _
x = 0 ; (D.4)
py = gy _
x = r2 l2 _
y = 0 : (D.5)
De ning the quantities
:= r2
l2 +
r2l2 ; :=
r2l2
r2l2 ; :=
r2l2 ;
{ 44 {
we can simultaneously solve (D.2) and (D.3) to nd
t = l4
_ r4 ( E + pz) ; (D.6)
z = l4
_ r4 ( pz E) : (D.7)
Notice that both of these velocities diverge as we approach the horizon at r+ = 0. This
divergence tells us that this particular coordinate system is not valid beyond the horizon. However, for our current purposes, this is not a problem as we are only interested in tidal forces close to, but outside, the horizon. In order to write down ^
e0, we still need to obtain
_
r. For this we use that g _
_
r =
are the roots of E2 + 2 Epz p2z = 0. Notice that had we instead
picked the positive root in (D.8), describing an outgoing timelike geodesic, _
r will become
complex for su ciently large r; this indicates that geodesic cannot reach the boundary but in fact hits a turning point and returns to the bulk [68{70]. For this reason, we will only be interested in near horizon tidal forces.
We can now substitute the above expressions for _
t; _
z; _
r into (D.1) to obtain the following
expression for the rst frame eld
Whilst the frame eld ^
e0 correctly describes the parallel propagation, it is not correctly normalised. To form an orthonormal basis of frame elds we can apply Gram-Schmidt procedure to the set of linearly independent frame elds ^
ea = [notdef]^
e0; ^
e1 = @r; ^
e2 = @z; ^
ei = @i[notdef]
where i = x; y. This was done using Maple and returns a basis of frame elds that we shall denote [notdef]ea[notdef] without the hat. These still correctly characterise the parallel propagation but
at the same time are fully orthonormal in the sense that they satisfy g (ea) (eb) = ab.
The full expressions for the individual frame elds [notdef]ea[notdef] are quite complicated and not
especially illuminating so we omit them here. However, we can then use the frame elds as transformation matrices to obtain the components of the Riemann tensor as measured in the PPON via
~Rabcd = R (ea) (eb) (ec) (ed) : (D.10)
The non-zero components of the PPON Riemann tensor are again rather complicated and so rather than provide full expressions, we instead list their scaling behaviour in the near horizon regime in table 3.
{ 45 {
JHEP11(2016)114
x _
x = 1 for a timelike observer, which is equivalent to
rr2l2 +l2Ar2 (E V+) (E V) ; (D.8)
where weve taken the negative root to represent a radially infalling observer and V[notdef] =
1
pz [notdef] r
2pz l2
(^e0) = l4r4 ( E + pz) (@t) +
l4r4 ( pz E) (@z)
rr2l2 + l2r2 (E V+) (E V) (@r) : (D.9)
Component Near horizon behaviour~R0101 const
~R0102 r13
~R0112 r13
~R0202 r13
~R0212 r13
~R0i0j ijr6
~R0i1j ijr6
~R0i2j ijr6 ~R1212 r13 ~R1i1j ijr6
~R1i2j ijr6
~R2i2j ijr6
~Rijkl r3 ( il jk ik jl)
Table 3. Near horizon scaling behaviour of the non-zero components of the ve-dimensional Riemann tensor, ~Rabcd, as measured in the PPON.
E Four-dimensional tidal forces
To investigate the tidal forces present for the four-dimensional extremal Nernst brane solutions of [11] we must treat the cases with nite and in nite four-dimensional chemical potential separately as we have done throughout the paper. These have metrics given in (4.3) and (4.19) respectively. We shall proceed in a similar fashion to appendix D except for the assumption that the infalling observer is now moving only in the radial direction and has no transverse momentum in either the x or y directions. This is slightly di erent to the analysis of [16] and means the tangent vector for the timelike geodesic on which our radially infalling observer is travelling is given by
T =
_t; _r;~0 ;
where dot denotes di erentiation with respect to the observers proper time, .
E.1 A > 0 tidal forces
The extremal version of (4.3) is given by
ds2A>0; Ext = r
l
r2 l2 1 +
Ar4
1=2 dt2 +
l2 1 +
Ar4
JHEP11(2016)114
1=2r2 dr2
+r2
l2
1 + Ar4
1=2
dx2 + dy2
!
: (E.1)
The energy is again an integral of motion:
E = gtt _
t = r3
l3 1 +
Ar4
1=2 _
t ) _
t = l3E 1 +
Ar4
1=2 r3 :
{ 46 {
For a timelike geodesic we have
g T T = 1 ) _
r =
1 l1=2r
v
u
u
tl3E2
r3
1 +
Ar4
1=2 ;
where we pick the negative square root to represent an observer falling radially inwards. We could equally well pick the positive root and consider an outgoing geodesic but _
r
will become complex for large r, meaning the geodesic encounters a turning point and is re ected back into the bulk. This is reminiscent of the situation in appendix D and in fact, this inability of timelike geodesics to reach the boundary is an example of a property that hyperscaling-violating Lifshitz spacetimes can inherit from their parent Anti de-Sitter spacetimes. All of this means that we need only focus on the ingoing observer and near horizon tidal forces. Another similarity with appendix D is the divergence of _
t and _
r as
r ! 0; again this indicates the coordinates are only valid up the horizon which is absolutely
ne for the analysis of tidal forces.Next we align the frame eld30 e0 with the vector eld d
d responsible for generating the integral curve along which the observer is moving:
(e0) =
d d
= _t@ t + _ r@ r
= l3E 1 +
Ar4
JHEP11(2016)114
1=2r3 @ t 1 l1=2r
v
u
u
tl3E2
r3
1=2 @ r:
The observer is moving in the (t; r) directions and so there are two frame elds associated to this: e0 and e1. Since the observer isnt moving in any of the xi (i 2) directions, the
frames ei for i 2 are just given by the square roots of the inverse metric components i.e.
(ei) = lr 1 +
Ar4
1=4 @ i:
It remains to nd the frame e1 such that the [notdef]ea[notdef] form a PPON. We have picked e0 to
describe the parallel propagation and so we just need a second frame eld, e1, that is orthonormal to both e0 and ei; i 2. It follows from simple linear algebra that
(e1) =
l3=2 1 +
Ar4
1 +
Ar4
1=2 r3
v
u
u
tl3E2
r3
lEr @ r:
1 +
Ar4
1=2 @ t +
It is interesting to note that in the case of the static four-dimensional metric, the frame elds are already orthonormal whereas in appendix D, where the ve-dimensional metric is non-static, this is not the case and we had to perform an additional Gram-Schmidt procedure at this point.
30We use unhatted frame elds in four-dimensions to distinguish from their hatted cousins in ve-dimensions.
{ 47 {
Component Near horizon behaviour~R0101 r
~R0i0j ijr4
~R0i1j ijr4
~R1i1j ijr4
~Rijkl r ( il jk ik jl)
Table 4. Near horizon scaling behaviour of the non-zero components of the four-dimensional A > 0 Riemann tensor, ~Rabcd, as measured in the PPON.
Component Near horizon behaviour~R0101 r
~R0i0j ijr4
~R0i1j ijr4
~R1i1j ijr4
~Rijkl r3 ( il jk ik jl)
Table 5. Near horizon scaling behaviour of the non-zero components of the four-dimensional A = 0 Riemann tensor, ~Rabcd, as measured in the PPON.
We next use Maple to nd the components of the Riemann tensor in a coordinate basis with lowered indices, R , and then multiply by frame elds to obtain the local tidal forces felt by the observer as in (D.10). We again omit the full expressions and instead list in table 4 the scaling behaviour of the non-zero components in the near horizon regime.
E.2 A = 0 tidal forces
Here we repeat the same procedure as above for the A = 0 extremal metric. The extremal version of (4.19) is given by
ds2A=0; Ext =
r5 1=2l3 dt2 +
JHEP11(2016)114
1=2l
r3 dr2 +
1=2r
l3 dx2 + dy2
: (E.2)
The resulting nonzero components of the Riemann tensor as measured in the PPON are given in table 5.
E.3 Consistency with existing classi cation
The near horizon scaling behaviours of the PPON Riemann tensor components in tables 4 and 5 agree. This is consistent with the fact that the parameter A only a ects the asymptotic geometry, which is why the metrics (E.1) and (E.2) both take the same form in the small r limit: speci cally, a hyperscaling-violating Lifshitz metric with parameters (z; ) = (3; 1) as observed in [11].
It is worthwhile to check the consistency of the results of this appendix with the complete classi cation of hyperscaling-violating Lifshitz singularities obtained in [16]. It can be shown that our (z; ) = (3; 1) geometry is equivalent to a (n0; n1) = (10; 4) geometry
{ 48 {
in their notation. This would place our near horizon metric into Class IV of the analysis in [16], making it both consistent with the Null Energy Condition and indicative of a null curvature singularity (in nite tidal forces) at r = 0.
F Normalization of the vector potential
For the four-dimensional chemical potential At(r = 1), to be uniquely de ned, it is
crucial that the vector potential is normalized such that At(r+) = 0. While this is widely used and the reason well known, see for example [2, 44], we would like to review the full argument here for completeness.
Assume that we are given a static space-time which has a Killing horizon with Killing vector eld . If the norm of has a simple zero at the horizon, in other words, if the solution is non-extremal, then the space-time can be continued analytically to a space-time which contains a bifurcate horizon [43]. This means that the horizon has a spatial section 0 where the Killing vector eld vanishes. If A is a well-de ned one-form on this space-time, then A() = 0 on 0. Since the horizon is generated by the ow of the Killing vector eld , and if assuming that the one-form A is invariant under , LA = 0 (where
L denotes the Lie derivative), it follows that A() = 0 on the whole horizon. Outside the horizon we can de ne a time coordinate t, such that = @t. Then the horizon limit of the component At of the one-form is At ! A() = 0.
In our application, we have non-extremal solutions with Killing horizons, generated by , given by = @t outside the horizon. Moreover not only the metric but also the vector eld is assumed static (invariant under t), and therefore At has to vanish on the horizon. By continuity this continues to hold in the extremal limit.
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References
[1] R.M. Wald, The Nernst theorem and black hole thermodynamics, http://dx.doi.org/10.1103/PhysRevD.56.6467
Web End =Phys. Rev. D 56 (1997) http://dx.doi.org/10.1103/PhysRevD.56.6467
Web End =6467 [https://arxiv.org/abs/gr-qc/9704008
Web End =gr-qc/9704008 ] [http://inspirehep.net/search?p=find+EPRINT+gr-qc/9704008
Web End =INSPIRE ].
[2] S.A. Hartnoll, Lectures on holographic methods for condensed matter physics, http://dx.doi.org/10.1088/0264-9381/26/22/224002
Web End =Class. Quant. http://dx.doi.org/10.1088/0264-9381/26/22/224002
Web End =Grav. 26 (2009) 224002 [https://arxiv.org/abs/0903.3246
Web End =arXiv:0903.3246 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:0903.3246
Web End =INSPIRE ].
[3] E. DHoker and P. Kraus, Charged Magnetic Brane Solutions in AdS (5) and the fate of the third law of thermodynamics, http://dx.doi.org/10.1007/JHEP03(2010)095
Web End =JHEP 03 (2010) 095 [https://arxiv.org/abs/0911.4518
Web End =arXiv:0911.4518 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:0911.4518
Web End =INSPIRE ].
[4] S.A. Hartnoll, Horizons, holography and condensed matter, in Black holes in higher dimensions, G. Horowitz ed., Cambridge University Press, Cambridge U.K. (2012).
[5] M. Ammon, J. Leiber, and R.P. Macedo, Phase diagram of 4D eld theories with chiral anomaly from holography, http://dx.doi.org/10.1007/JHEP03(2016)164
Web End =JHEP 03 (2016) 164 [https://arxiv.org/abs/1601.02125
Web End =arXiv:1601.02125 ].
[6] K. Goldstein, S. Kachru, S. Prakash and S.P. Trivedi, Holography of charged dilaton black holes, http://dx.doi.org/10.1007/JHEP08(2010)078
Web End =JHEP 08 (2010) 078 [https://arxiv.org/abs/0911.3586
Web End =arXiv:0911.3586 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:0911.3586
Web End =INSPIRE ].
{ 49 {
JHEP11(2016)114
[7] K. Goldstein, N. Iizuka, S. Kachru, S. Prakash, S.P. Trivedi and A. Westphal, Holography of dyonic dilaton black branes, http://dx.doi.org/10.1007/JHEP10(2010)027
Web End =JHEP 10 (2010) 027 [https://arxiv.org/abs/1007.2490
Web End =arXiv:1007.2490 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1007.2490
Web End =INSPIRE ].
[8] J.P. Gauntlett, J. Sonner and T. Wiseman, Holographic superconductivity in M-theory, http://dx.doi.org/10.1103/PhysRevLett.103.151601
Web End =Phys. http://dx.doi.org/10.1103/PhysRevLett.103.151601
Web End =Rev. Lett. 103 (2009) 151601 [https://arxiv.org/abs/0907.3796
Web End =arXiv:0907.3796 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:0907.3796
Web End =INSPIRE ].
[9] G.T. Horowitz and M.M. Roberts, Zero temperature limit of holographic superconductors, http://dx.doi.org/10.1088/1126-6708/2009/11/015
Web End =JHEP 11 (2009) 015 [https://arxiv.org/abs/0908.3677
Web End =arXiv:0908.3677 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:0908.3677
Web End =INSPIRE ].
[10] S. Barisch, G. Lopes Cardoso, M. Haack, S. Nampuri and N.A. Obers, Nernst branes in gauged supergravity, http://dx.doi.org/10.1007/JHEP11(2011)090
Web End =JHEP 11 (2011) 090 [https://arxiv.org/abs/1108.0296
Web End =arXiv:1108.0296 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1108.0296
Web End =INSPIRE ].
[11] P. Dempster, D. Errington and T. Mohaupt, Nernst branes from special geometry, http://dx.doi.org/10.1007/JHEP05(2015)079
Web End =JHEP 05 http://dx.doi.org/10.1007/JHEP05(2015)079
Web End =(2015) 079 [https://arxiv.org/abs/1501.07863
Web End =arXiv:1501.07863 ].
[12] L. Huijse, S. Sachdev and B. Swingle, Hidden Fermi surfaces in compressible states of gauge-gravity duality, http://dx.doi.org/10.1103/PhysRevB.85.035121
Web End =Phys. Rev. B 85 (2012) 035121 [https://arxiv.org/abs/1112.0573
Web End =arXiv:1112.0573 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1112.0573
Web End =INSPIRE ].
[13] X. Dong, S. Harrison, S. Kachru, G. Torroba and H. Wang, Aspects of holography for theories with hyperscaling violation, http://dx.doi.org/10.1007/JHEP06(2012)041
Web End =JHEP 06 (2012) 041 [https://arxiv.org/abs/1201.1905
Web End =arXiv:1201.1905 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1201.1905
Web End =INSPIRE ].
[14] E. Perlmutter, Hyperscaling violation from supergravity, http://dx.doi.org/10.1007/JHEP06(2012)165
Web End =JHEP 06 (2012) 165 [https://arxiv.org/abs/1205.0242
Web End =arXiv:1205.0242 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1205.0242
Web End =INSPIRE ].
[15] G.T. Horowitz and B. Way, Lifshitz singularities, http://dx.doi.org/10.1103/PhysRevD.85.046008
Web End =Phys. Rev. D 85 (2012) 046008 [https://arxiv.org/abs/1111.1243
Web End =arXiv:1111.1243 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1111.1243
Web End =INSPIRE ].
[16] K. Copsey and R. Mann, Singularities in hyperscaling violating spacetimes, http://dx.doi.org/10.1007/JHEP04(2013)079
Web End =JHEP 04 (2013) http://dx.doi.org/10.1007/JHEP04(2013)079
Web End =079 [https://arxiv.org/abs/1210.1231
Web End =arXiv:1210.1231 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1210.1231
Web End =INSPIRE ].
[17] M. Cveti c, H. Lu and C.N. Pope, Space-times of boosted p-branes and CFT in in nite momentum frame, http://dx.doi.org/10.1016/S0550-3213(99)00002-4
Web End =Nucl. Phys. B 545 (1999) 309 [https://arxiv.org/abs/hep-th/9810123
Web End =hep-th/9810123 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/9810123
Web End =INSPIRE ].
[18] A. Buchel and J.T. Liu, Gauged supergravity from type IIB string theory on Y p;q manifolds, http://dx.doi.org/10.1016/j.nuclphysb.2007.03.001
Web End =Nucl. Phys. B 771 (2007) 93 [https://arxiv.org/abs/hep-th/0608002
Web End =hep-th/0608002 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/0608002
Web End =INSPIRE ].
[19] M. Cveti c, H. Lu and C.N. Pope, Geometry of the embedding of supergravity scalar manifolds in D = 11 and D = 10, http://dx.doi.org/10.1016/S0550-3213(00)00215-7
Web End =Nucl. Phys. B 584 (2000) 149 [https://arxiv.org/abs/hep-th/0002099
Web End =hep-th/0002099 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/0002099
Web End =INSPIRE ].
[20] A. Azizi, H. Godazgar, M. Godazgar and C.N. Pope, The embedding of gauged STU supergravity in eleven dimensions, http://dx.doi.org/10.1103/PhysRevD.94.066003
Web End =Phys. Rev. D 94 (2016) 066003 [https://arxiv.org/abs/1606.06954
Web End =arXiv:1606.06954 ] [http://inspirehep.net/search?p=find+J+PHRVA,D94,066003
Web End =INSPIRE ].
[21] D. Cassani, G. DallAgata and A.F. Faedo, Type IIB supergravity on squashed Sasaki-Einstein manifolds, http://dx.doi.org/10.1007/JHEP05(2010)094
Web End =JHEP 05 (2010) 094 [https://arxiv.org/abs/1003.4283
Web End =arXiv:1003.4283 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1003.4283
Web End =INSPIRE ].
[22] J.P. Gauntlett and O. Varela, Universal Kaluza-Klein reductions of type IIB to N = 4 supergravity in ve dimensions, http://dx.doi.org/10.1007/JHEP06(2010)081
Web End =JHEP 06 (2010) 081 [https://arxiv.org/abs/1003.5642
Web End =arXiv:1003.5642 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1003.5642
Web End =INSPIRE ].
[23] H. Singh, Special limits and non-relativistic solutions, http://dx.doi.org/10.1007/JHEP12(2010)061
Web End =JHEP 12 (2010) 061 [https://arxiv.org/abs/1009.0651
Web End =arXiv:1009.0651 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1009.0651
Web End =INSPIRE ].
[24] S. Barisch-Dick, G. Lopes Cardoso, M. Haack and S. Nampuri, Extremal black brane solutions in ve-dimensional gauged supergravity, http://dx.doi.org/10.1007/JHEP02(2013)103
Web End =JHEP 02 (2013) 103 [https://arxiv.org/abs/1211.0832
Web End =arXiv:1211.0832 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1211.0832
Web End =INSPIRE ].
[25] M. Cvetic, G. W. Gibbons, H. Lu and C.N. Pope, Rotating black holes in gauged supergravities: thermodynamics, supersymmetric limits, topological solitons and time machines, https://arxiv.org/abs/hep-th/0504080
Web End =hep-th/0504080 .
{ 50 {
JHEP11(2016)114
[26] S. Hawking and G. Ellis, The large scale structure of space-time, Cambridge University Press, Cambridge U.K. (1973).
[27] J. Podolsky, Interpretation of the Siklos solutions as exact gravitational waves in the anti-de Sitter universe, http://dx.doi.org/10.1088/0264-9381/15/3/019
Web End =Class. Quant. Grav. 15 (1998) 719 [https://arxiv.org/abs/gr-qc/9801052
Web End =gr-qc/9801052 ] [http://inspirehep.net/search?p=find+EPRINT+gr-qc/9801052
Web End =INSPIRE ].
[28] M. Gunaydin, G. Sierra and P.K. Townsend, The geometry of N = 2 Maxwell-Einstein supergravity and Jordan algebras, Nucl. Phys. B 242 (1984) 244.
[29] V. Cortes and T. Mohaupt, Special geometry of euclidean supersymmetry III: the local r-map, instantons and black holes, http://dx.doi.org/10.1088/1126-6708/2009/07/066
Web End =JHEP 07 (2009) 066 [https://arxiv.org/abs/0905.2844
Web End =arXiv:0905.2844 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:0905.2844
Web End =INSPIRE ].
[30] E. Bergshoe et al., Superconformal N = 2, D = 5 matter with and without actions, http://dx.doi.org/10.1088/1126-6708/2002/10/045
Web End =JHEP http://dx.doi.org/10.1088/1126-6708/2002/10/045
Web End =10 (2002) 045 [https://arxiv.org/abs/hep-th/0205230
Web End =hep-th/0205230 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/0205230
Web End =INSPIRE ].
[31] E. Bergshoe , S. Cucu, T. de Wit, J. Gheerardyn, S. Vandoren and A. Van Proeyen, N = 2 supergravity in ve-dimensions revisited, http://dx.doi.org/10.1088/0264-9381/23/23/C01
Web End =Class. Quant. Grav. 21 (2004) 3015 [https://arxiv.org/abs/hep-th/0403045
Web End =hep-th/0403045 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/0403045
Web End =INSPIRE ].
[32] D.Z. Freedman and A. Van Proeyen, Supergravity, Cambridge University Press, Cambridge U.K. (2012).
[33] D.V. Alekseevsky and V. Cort es, Geometric construction of the r-map: from a ne special real to special Kahler manifolds, http://dx.doi.org/10.1007/s00220-009-0803-7
Web End =Comm. Math. Phys. 291 (2009) 579 [https://arxiv.org/abs/0811.1658
Web End =arXiv:0811.1658 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:0811.1658
Web End =INSPIRE ].
[34] T. Mohaupt and K. Waite, Instantons, black holes and harmonic functions, http://dx.doi.org/10.1088/1126-6708/2009/10/058
Web End =JHEP 10 (2009) http://dx.doi.org/10.1088/1126-6708/2009/10/058
Web End =058 [https://arxiv.org/abs/0906.3451
Web End =arXiv:0906.3451 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:0906.3451
Web End =INSPIRE ].
[35] V. Cortes, T. Mohaupt and H. Xu, Completeness in supergravity constructions, http://dx.doi.org/10.1007/s00220-012-1443-x
Web End =Commun. http://dx.doi.org/10.1007/s00220-012-1443-x
Web End =Math. Phys. 311 (2012) 191 [https://arxiv.org/abs/1101.5103
Web End =arXiv:1101.5103 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1101.5103
Web End =INSPIRE ].
[36] T. Mohaupt and O. Vaughan, Non-extremal black holes from the generalised R-map, http://dx.doi.org/10.1007/978-3-319-00215-6_6
Web End =Springer http://dx.doi.org/10.1007/978-3-319-00215-6_6
Web End =Proc. Phys. 144 (2013) 233 [https://arxiv.org/abs/1208.4302
Web End =arXiv:1208.4302 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1208.4302
Web End =INSPIRE ].
[37] A. Ceresole and G. DallAgata, General matter coupled N = 2, D = 5 gauged supergravity, http://dx.doi.org/10.1016/S0550-3213(00)00339-4
Web End =Nucl. Phys. B 585 (2000) 143 [https://arxiv.org/abs/hep-th/0004111
Web End =hep-th/0004111 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/0004111
Web End =INSPIRE ].
[38] S. Cremonini, K. Hanaki, J.T. Liu and P. Szepietowski, Black holes in ve-dimensional gauged supergravity with higher derivatives, http://dx.doi.org/10.1088/1126-6708/2009/12/045
Web End =JHEP 12 (2009) 045 [https://arxiv.org/abs/0812.3572
Web End =arXiv:0812.3572 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:0812.3572
Web End =INSPIRE ].
[39] P. Dempster and T. Mohaupt, Non-extremal and non-BPS extremal ve-dimensional black strings from generalized special real geometry, http://dx.doi.org/10.1088/0264-9381/31/4/045019
Web End =Class. Quant. Grav. 31 (2014) 045019 [https://arxiv.org/abs/1310.5056
Web End =arXiv:1310.5056 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1310.5056
Web End =INSPIRE ].
[40] V. Cort es, P. Dempster and T. Mohaupt, Time-like reductions of ve-dimensional supergravity, http://dx.doi.org/10.1007/JHEP04(2014)190
Web End =JHEP 04 (2014) 190 [https://arxiv.org/abs/1401.5672
Web End =arXiv:1401.5672 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1401.5672
Web End =INSPIRE ].
[41] V. Cort es, P. Dempster, T. Mohaupt and O. Vaughan, Special geometry of euclidean supersymmetry IV: the local c-map, http://dx.doi.org/10.1007/JHEP10(2015)066
Web End =JHEP 10 (2015) 066 [https://arxiv.org/abs/1507.04620
Web End =arXiv:1507.04620 ].
[42] D. Errington, T. Mohaupt and O. Vaughan, Non-extremal black hole solutions from the c-map, http://dx.doi.org/10.1007/JHEP05(2015)052
Web End =JHEP 05 (2015) 052 [https://arxiv.org/abs/1408.0923
Web End =arXiv:1408.0923 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1408.0923
Web End =INSPIRE ].
[43] I. Racz and R.M. Wald, Global extensions of space-times describing asymptotic nal states of black holes, http://dx.doi.org/10.1088/0264-9381/13/3/017
Web End =Class. Quant. Grav. 13 (1996) 539 [https://arxiv.org/abs/gr-qc/9507055
Web End =gr-qc/9507055 ] [http://inspirehep.net/search?p=find+EPRINT+gr-qc/9507055
Web End =INSPIRE ].
[44] S. Kobayashi, D. Mateos, S. Matsuura, R.C. Myers and R.M. Thomson, Holographic phase transitions at nite baryon density, http://dx.doi.org/10.1088/1126-6708/2007/02/016
Web End =JHEP 02 (2007) 016 [https://arxiv.org/abs/hep-th/0611099
Web End =hep-th/0611099 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/0611099
Web End =INSPIRE ].
{ 51 {
JHEP11(2016)114
[45] T. Mohaupt and O. Vaughan, Non-extremal Black Holes, Harmonic Functions and Attractor Equations, http://dx.doi.org/10.1088/0264-9381/27/23/235008
Web End =Class. Quant. Grav. 27 (2010) 235008 [https://arxiv.org/abs/1006.3439
Web End =arXiv:1006.3439 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1006.3439
Web End =INSPIRE ].
[46] K. Goldstein, S. Nampuri and
A. V eliz-Osorio, Heating up branes in gauged supergravity,
http://dx.doi.org/10.1007/JHEP08(2014)151
Web End =JHEP 08 (2014) 151 [https://arxiv.org/abs/1406.2937
Web End =arXiv:1406.2937 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1406.2937
Web End =INSPIRE ].
[47] K. Narayan, On Lifshitz scaling and hyperscaling violation in string theory, http://dx.doi.org/10.1103/PhysRevD.85.106006
Web End =Phys. Rev. D 85 http://dx.doi.org/10.1103/PhysRevD.85.106006
Web End =(2012) 106006 [https://arxiv.org/abs/1202.5935
Web End =arXiv:1202.5935 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1202.5935
Web End =INSPIRE ].
[48] H. Singh, Lifshitz to AdS ow with interpolating p-brane solutions, http://dx.doi.org/10.1007/JHEP08(2013)097
Web End =JHEP 08 (2013) 097 [https://arxiv.org/abs/1305.3784
Web End =arXiv:1305.3784 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1305.3784
Web End =INSPIRE ].
[49] E. Poisson, A relativists toolkit: the mathematics of black-hole mechanics, Cambridge University Press, Cambridge U.K. (2007).
[50] S.W. Hawking, C.J. Hunter and M. Taylor, Rotation and the AdS/CFT correspondence, http://dx.doi.org/10.1103/PhysRevD.59.064005
Web End =Phys. Rev. D 59 (1999) 064005 [https://arxiv.org/abs/hep-th/9811056
Web End =hep-th/9811056 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/9811056
Web End =INSPIRE ].
[51] M.M. Caldarelli, G. Cognola and D. Klemm, Thermodynamics of Kerr-Newman-AdS black holes and conformal eld theories, http://dx.doi.org/10.1088/0264-9381/17/2/310
Web End =Class. Quant. Grav. 17 (2000) 399 [https://arxiv.org/abs/hep-th/9908022
Web End =hep-th/9908022 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/9908022
Web End =INSPIRE ].
[52] G.W. Gibbons, M.J. Perry and C.N. Pope, The rst law of thermodynamics for Kerr-Anti-de Sitter black holes, http://dx.doi.org/10.1088/0264-9381/22/9/002
Web End =Class. Quant. Grav. 22 (2005) 1503 [https://arxiv.org/abs/hep-th/0408217
Web End =hep-th/0408217 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/0408217
Web End =INSPIRE ].
[53] J.P. Gauntlett and J.B. Gutowski, All supersymmetric solutions of minimal gauged supergravity in ve-dimensions, http://dx.doi.org/10.1103/PhysRevD.70.089901
Web End =Phys. Rev. D 68 (2003) 105009 [Erratum ibid. D 70 (2004) 089901] [https://arxiv.org/abs/hep-th/0304064
Web End =hep-th/0304064 ] [http://inspirehep.net/search?p=find+J+%22Phys.Rev.,D68,105009%22
Web End =INSPIRE ].
[54] V. Balasubramanian and P. Kraus, A stress tensor for Anti-de Sitter gravity, http://dx.doi.org/10.1007/s002200050764
Web End =Commun. http://dx.doi.org/10.1007/s002200050764
Web End =Math. Phys. 208 (1999) 413 [https://arxiv.org/abs/hep-th/9902121
Web End =hep-th/9902121 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/9902121
Web End =INSPIRE ].
[55] N. Ambrosetti, J. Charbonneau and S. Weinfurtner, The uid/gravity correspondence: lectures notes from the 2008 summer school on particles, elds and strings, https://arxiv.org/abs/0810.2631
Web End =arXiv:0810.2631 [http://inspirehep.net/search?p=find+EPRINT+arXiv:0810.2631
Web End =INSPIRE ].
[56] M. Ammon and J. Erdmenger, Gauge/gravity duality, Cambridge University Press, Cambridge U.K. (2015).
[57] J. Maldacena, D. Martelli and Y. Tachikawa, Comments on string theory backgrounds with non-relativistic conformal symmetry, http://dx.doi.org/10.1088/1126-6708/2008/10/072
Web End =JHEP 10 (2008) 072 [https://arxiv.org/abs/0807.1100
Web End =arXiv:0807.1100 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:0807.1100
Web End =INSPIRE ].
[58] G. DallAgata and A. Gnecchi, Flow equations and attractors for black holes in N = 2 U(1) gauged supergravity, http://dx.doi.org/10.1007/JHEP03(2011)037
Web End =JHEP 03 (2011) 037 [https://arxiv.org/abs/1012.3756
Web End =arXiv:1012.3756 ] [http://inspirehep.net/search?p=find+J+%22JHEP,1103,037%22
Web End =INSPIRE ].
[59] G.T. Horowitz and S.F. Ross, Naked black holes, http://dx.doi.org/10.1103/PhysRevD.56.2180
Web End =Phys. Rev. D 56 (1997) 2180 [https://arxiv.org/abs/hep-th/9704058
Web End =hep-th/9704058 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/9704058
Web End =INSPIRE ].
[60] A. Dabholkar, R. Kallosh and A. Maloney, A stringy cloak for a classical singularity, http://dx.doi.org/10.1088/1126-6708/2004/12/059
Web End =JHEP http://dx.doi.org/10.1088/1126-6708/2004/12/059
Web End =12 (2004) 059 [https://arxiv.org/abs/hep-th/0410076
Web End =hep-th/0410076 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/0410076
Web End =INSPIRE ].
[61] R. Guven, The conformal Penrose limit and the resolution of the pp-curvature singularities, http://dx.doi.org/10.1088/0264-9381/23/2/001
Web End =Class. Quant. Grav. 23 (2006) 295 [https://arxiv.org/abs/hep-th/0508160
Web End =hep-th/0508160 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/0508160
Web End =INSPIRE ].
[62] S. Barisch-Dick, G. Lopes Cardoso, M. Haack and
A. V eliz-Osorio, Quantum corrections to extremal black brane solutions, http://dx.doi.org/10.1007/JHEP02(2014)105
Web End =JHEP 02 (2014) 105 [https://arxiv.org/abs/1311.3136
Web End =arXiv:1311.3136 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1311.3136
Web End =INSPIRE ].
[63] K. Hristov, S. Katmadas and I. Lodato, Higher derivative corrections to BPS black hole attractors in 4d gauged supergravity, http://dx.doi.org/10.1007/JHEP05(2016)173
Web End =JHEP 05 (2016) 173 [https://arxiv.org/abs/1603.0003
Web End =arXiv:1603.0003 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1603.0003
Web End =INSPIRE ].
{ 52 {
JHEP11(2016)114
[64] D. Brecher, A. Chamblin and H.S. Reall, AdS/CFT in the in nite momentum frame, http://dx.doi.org/10.1016/S0550-3213(01)00170-5
Web End =Nucl. http://dx.doi.org/10.1016/S0550-3213(01)00170-5
Web End =Phys. B 607 (2001) 155 [https://arxiv.org/abs/hep-th/0012076
Web End =hep-th/0012076 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/0012076
Web End =INSPIRE ].
[65] K. Goldstein, V. Jejjala and S. Nampuri, Hot attractors, http://dx.doi.org/10.1007/JHEP01(2015)075
Web End =JHEP 01 (2015) 075 [https://arxiv.org/abs/1410.3478
Web End =arXiv:1410.3478 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1410.3478
Web End =INSPIRE ].
[66] R.B. Mann and S.N. Solodukhin, Conical geometry and quantum entropy of a charged Kerr black hole, http://dx.doi.org/10.1103/PhysRevD.54.3932
Web End =Phys. Rev. D 54 (1996) 3932 [https://arxiv.org/abs/hep-th/9604118
Web End =hep-th/9604118 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/9604118
Web End =INSPIRE ].
[67] Y. Lei and S.F. Ross, Extending the non-singular hyperscaling violating spacetimes, http://dx.doi.org/10.1088/0264-9381/31/3/035007
Web End =Class. http://dx.doi.org/10.1088/0264-9381/31/3/035007
Web End =Quant. Grav. 31 (2014) 035007 [https://arxiv.org/abs/1310.5878
Web End =arXiv:1310.5878 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1310.5878
Web End =INSPIRE ].
[68] J. Maldacena, The Gauge/gravity duality, https://arxiv.org/abs/1106.6073
Web End =arXiv:1106.6073 [http://inspirehep.net/search?p=find+EPRINT+arXiv:1106.6073
Web End =INSPIRE ].
[69] U. Moschella, The de Sitter and Anti-de Sitter sightseeing tour, lecture notes, available at http://www.bourbaphy.fr/moschella.pdf
Web End =http://www.bourbaphy.fr/moschella.pdf .
[70] G. Gibbons, Part III: applications of di erential geometry to physics, lecture notes, available at http://www.damtp.cam.ac.uk/research/gr/members/gibbons/dgnotes3.pdf
Web End =http://www.damtp.cam.ac.uk/research/gr/members/gibbons/dgnotes3.pdf .
JHEP11(2016)114
{ 53 {
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Abstract
Abstract
We construct Nernst brane solutions, that is black branes with zero entropy density in the extremal limit, of FI-gauged minimal five-dimensional supergravity coupled to an arbitrary number of vector multiplets. While the scalars take specific constant values and dynamically determine the value of the cosmological constant in terms of the FI-parameters, the metric takes the form of a boosted AdS Schwarzschild black brane. This metric can be brought to the Carter-Novotný-Horský form that has previously been observed to occur in certain limits of boosted D3-branes. By dimensional reduction to four dimensions we recover the four-dimensional Nernst branes of arXiv:1501.07863 and show how the five-dimensional lift resolves all their UV singularities. The dynamics of the compactification circle, which expands both in the UV and in the IR, plays a crucial role. At asymptotic infinity, the curvature singularity of the four-dimensional metric and the run-away behaviour of the four-dimensional scalar combine in such a way that the lifted solution becomes asymptotic to AdS 5. Moreover, the existence of a finite chemical potential in four dimensions is related to fact that the compactification circle has a finite minimal value. While it is not clear immediately how to embed our solutions into string theory, we argue that the same type of dictionary as proposed for boosted D3-branes should apply, although with a lower amount of supersymmetry.
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