Lifshitz Spacetimes Research Articles

High-dimensional Lifshitz-type spacetimes, universal horizons, and black holes in Hořava-Lifshitz gravity

In this paper, we present all $[(d+1)+1]$-dimensional static diagonal vacuum solutions of the non-projectable Ho\v{r}ava-Lifshitz gravity in the IR limit, and show that they give rise to very rich Lifshitz-type structures, depending on the choice of the free parameters of the solutions. These include the Lifshitz spacetimes with or without hyperscaling violation, Lifshitz solitons, and black holes. Remarkably, even the theory breaks explicitly the Lorentz symmetry and allows generically instantaneous propagations, universal horizons still exist, which serve as one-way membranes for signals with any large velocities. In particular, particles even with infinitely large velocities would just move around on these boundaries and cannot escape to infinity. Another remarkable feature appearing in the Lifshitz-type spacetimes is that the dynamical exponent $z$ can take its values only in the ranges $1 \le z < 2$ for $d \ge 3$ and $1 \le z <\infty$ for $d = 2$, due to the stability and ghost-free conditions of the theory.

Interpolating solution fromAdS5to hyperscaling violating Lifshitz spacetime

We construct two interpolating solutions in type II string theory which interpolate between an AdS$_5$ in the UV and a hyperscaling violating three (spatial) dimensional Lifshitz space-time in the IR. The first solution is non-supersymmetric and is obtained from a known intersecting non-supersymmetric D3-brane with chargeless D0-brane solution of type IIB string theory, by restricting some parameters characterizing the solution and going to a new coordinate. In the IR the dilaton is non-constant in general and the metric is three (spatial) dimensional hyperscaling violating Lifshitz with dynamical critical exponent $z=(3+3\gamma)/(3-\gamma)$ and hyperscaling violation exponent $\theta = 12/(3-\gamma)$, where $\gamma$ is a real parameter and can take continuous values from $-1$ to $+1$. At the two extreme values, i.e., for $\gamma = \pm 1$, the dilaton is constant. The second solution is supersymmetric and is obtained from the known F-D2 bound state solution of type IIA string theory by zooming into a particular region of space. In the UV, the proper description is given in a T-dual frame which is AdS$_5$, whereas in the IR, it gives a three dimensional hyperscaling violating Lifshitz with $z=3$ and $\theta=2=d-1$.

Lifshitz effects on holographic p-wave superfluid

In the probe limit, we numerically build a holographic p-wave superfluid model in the four-dimensional Lifshitz black hole coupled to a Maxwell-complex vector field. We observe the rich phase structure and find that the Lifshitz dynamical exponent z contributes evidently to the effective mass of the matter field and dimension of the gravitational background. Concretely, we obtain that the Cave of Winds appeared only in the five-dimensional anti-de Sitter (AdS) spacetime, and the increasing z hinders not only the condensate but also the appearance of the first-order phase transition. Furthermore, our results agree with the Ginzburg–Landau results near the critical temperature. In addition, the previous AdS superfluid model is generalized to the Lifshitz spacetime.

Thermodynamics of Lifshitz black holes

We specialize the Wald formalism to derive the thermodynamical first law for static black holes with spherical/torus/hyperbolic symmetries in a variety of supergravities or supergravity-inspired theories involving multiple scalars and vectors. We apply the formula to study the first law of a general class of Lifshitz black holes. We analyse the first law of three exact Lifshitz black holes and the results fit the general pattern. In one example, the first law is $TdS + \Phi dQ=0$ where $(\Phi,Q)$ are the electric potential and charge of the Maxwell field. The unusual vanishing of mass in this specific solution demonstrates that super-extremal charged black holes can exist in asymptotic Lifshitz spacetimes.

Lifshitz flows in IIB and dual field theories

We construct solutions describing flows between AdS and Lifshitz spacetimes in IIB supergravity. We find that flows from AdS$_5$ can approach either AdS$_3$ or Lifshitz$_3$ in the IR depending on the values of the deformation from AdS$_5$. Surprisingly, the choice between AdS and Lifshitz IR depends only on the value of the deformation, not on its character; the breaking of the Lorentz symmetry in the flows with Lifshitz IR is spontaneous. We find that the values of the deformation which lead to flows to Lifshitz make the UV field theory dual to the AdS$_5$ geometry unstable, so that these flows do not offer an approach to defining the field theory dual to the Lifshitz spacetime.

Universal horizons and black holes in gravitational theories with broken Lorentz symmetry

In this paper, we first show that the definition of the universal horizons studied recently in the khronometric theory of gravity can be straightforwardly generalized to other theories that violate the Lorentz symmetry, by simply considering the khronon as a probe field and playing the same role as a Killing vector field. As an application, we study static charged (D + 1)-dimensional spacetimes in the framework of the healthy (nonprojectable) Horava–Lifshitz (HL) gravity in the infrared (IR) limit, and find various solutions. Some of them represent Lifshitz spacetimes with hyperscaling violations, and some have black hole structures. In the latter, universal horizons always exist inside the Killing horizons. The surface gravity on them can be either larger or smaller than the surface gravity on the Killing horizons, depending on the spacetimes considered. Although such black holes are found only in the IR, we argue that black holes with universal horizons also exist in the full theory of the HL gravity. A simple example is the Schwarzschild solution written in the Painleve–Gullstrand coordinates, which is also a solution of the full theory of the HL gravity and has a universal horizon located inside the Schwarzschild Killing horizon.

Scattering amplitudes in Lifshitz spacetime

We consider the calculation of scattering amplitudes in field theories dual to Lifshitz spacetimes. These amplitudes provide an interesting probe of the IR structure of the field theory; our aim is to use them to explore the observable consequences of the singularity in the spacetime. We assume the amplitudes can be related by T-duality to a Wilson loop, as in the AdS case, and determine the bulk minimal surfaces for the simplest cusp Wilson loop. We use this to determine the leading IR singularity in the amplitude. We find there is a stronger IR singularity for than for z = 1, with a coefficient that vanishes as .

Towards the uniqueness of Lifshitz black holes and solitons in new massive gravity

We prove that the z=1 and z=3 Lifshitz black hole solutions of New Massive Gravity in three dimensions are the only static axisymmetric solutions that can be cast in a Kerr-Schild form with a seed metric given by the Lifshitz spacetime. Correspondingly, we study the issue of uniqueness of Lifshitz solitons when considering an ansatz involving a single metric function. We show this problem can be mapped to the previous one and that the z=1 and z=1/3 Lifshitz soliton solutions are the only ones within this class. Finally, our approach suggests for the first time an explanation as to what is special about those particular values of the dynamical critical exponent z at finite temperature.

A Note on Black Holes in Asymptotically Lifshitz Spacetime

We investigate several aspects of exact black hole solutions in asymptotically Lifshitz spacetime. Firstly, we calculate the tidal forces and find that in the near horizon region of such black hole backgrounds, the tidal forces diverge in the near extremal limit. Secondly, we evaluate the Wilson loops in both extremal and finite temperature cases. Finally, we obtain the corresponding shear viscosity and square of the sound speed and find that the ratio of shear viscosity to entropy density takes the universal value 1/4π in arbitrary dimensions while the square of the speed of sound saturates the conjectured bound 1/3 in five dimensions.

Holographic thermalization with Lifshitz scaling and hyperscaling violation

A Vaidya type geometry describing gravitation collapse in asymptotically Lifshitz spacetime with hyperscaling violation provides a simple holographic model for thermalization near a quantum critical point with non-trivial dynamic and hyperscaling violation exponents. The allowed parameter regions are constrained by requiring that the matter energy momentum tensor satisfies the null energy condition. We present a combination of analytic and numerical results on the time evolution of holographic entanglement entropy in such backgrounds for different shaped boundary regions and study various scaling regimes, generalizing previous work by Liu and Suh.

Lifshitz black holes in Einstein-Yang-Mills theory

We find that the four dimensional cosmological Einstein-Yang-Mills theory with $SU(2)$ gauge group admits Lifshitz spacetime as a base solution for the dynamical exponent $z>1$. Motivated by this, we next demonstrate numerically that the field equations admit black hole solutions which behave regularly on the horizon and at spatial infinity for different horizon topologies. The solutions depend on one parameter, the strength of the gauge field at the horizon, which is fine-tuned to capture the Lifshitz asymptotics at infinity. We also discuss the behavior of solutions and the change in Hawking temperature for black holes that are large or small with respect to the length scale $L$, which is itself fixed by the value of the cosmological constant.

Electromagnetic quasinormal modes of an asymptotically Lifshitz black hole

Motivated by the recent interest in the study of the spacetimes that are asymptotically Lifshitz and in order to extend some previous results, we calculate exactly the quasinormal frequencies of the electromagnetic field in a D-dimensional asymptotically Lifshitz black hole. Based on the values obtained for the quasinormal frequencies we discuss the classical stability of the quasinormal modes. We also study whether the electromagnetic field possesses unstable modes in the D-dimensional Lifshitz spacetime.

Non-integrability in non-relativistic theories

Generic non-relativistic theories giving rise to non-integrable string solutions are classified. Our analysis boils down to a simple algebraic condition for the scaling parameters of the metric. Particular cases are the Lifshitz and the anisotropic Lifshitz spacetimes, for which we find that for trivial dilaton dependence the only integrable physical theory is that for z = 1. For the hyperscaling violation theories we conclude that the vast majority of theories are non-integrable, while only for a small class of physical theories, where the Fermi surfaces belong to, integrability is not excluded. Schrödinger theories are also analyzed and a necessary condition for non-integrability is found. Our analysis is also applied to cases where the exponential of the dilaton is a monomial of the holographic coordinate.

Lifshitz spacetimes, solitons, and generalized BTZ black holes in quantum gravity at a Lifshitz point

In this paper, we study static vacuum solutions of quantum gravity at a fixed Lifshitz point in (2+1) dimensions, and present all the diagonal solutions in closed forms in the infrared limit. The exact solutions represent spacetimes with very rich structures: they can represent generalized BTZ black holes, Lifshitz space-times or Lifshitz solitons, in which the spacetimes are free of any kind of space-time singularities, depending on the choices of the free parameters of the solutions. We also find several classes of exact static non-diagonal solutions, which represent similar space-time structures as those given in the diagonal case. The relevance of these solutions to the non-relativistic Lifshitz-type gauge/gravity duality is discussed.

Interpolating from Bianchi attractors to Lifshitz and AdS spacetimes

We construct classes of smooth metrics which interpolate from Bianchi attractor geometries of Types II, III, VI and IX in the IR to Lifshitz or $AdS_2 \times S^3$ geometries in the UV. While we do not obtain these metrics as solutions of Einstein gravity coupled to a simple matter field theory, we show that the matter sector stress-energy required to support these geometries (via the Einstein equations) does satisfy the weak, and therefore also the null, energy condition. Since Lifshitz or $AdS_2 \times S^3$ geometries can in turn be connected to $AdS_5$ spacetime, our results show that there is no barrier, at least at the level of the energy conditions, for solutions to arise connecting these Bianchi attractor geometries to $AdS_5$ spacetime. The asymptotic $AdS_5$ spacetime has no non-normalizable metric deformation turned on, which suggests that furthermore, the Bianchi attractor geometries can be the IR geometries dual to field theories living in flat space, with the breaking of symmetries being either spontaneous or due to sources for other fields. Finally, we show that for a large class of flows which connect two Bianchi attractors, a C-function can be defined which is monotonically decreasing from the UV to the IR as long as the null energy condition is satisfied. However, except for special examples of Bianchi attractors (including AdS space), this function does not attain a finite and non-vanishing constant value at the end points.

Mimic the optical conductivity in disordered solids via gauge/gravity duality

We study the optical conductivity in a (2+1)-dimensional non-relativistic field theory holographically dual to a (3+1)-dimensional charged Lifshitz black brane within the Einstein–Maxwell-dilaton theory. Surprisingly, we find that the optical AC conductivity satisfies the nontrivial (non-)power law scaling in the high frequency regime rather than approaching to a constant when the dynamical critical exponent z>1, which is qualitatively similar to those in various disordered solids in condensed matter systems. Besides, this (non-)power law scaling behavior shows some universality, which is robust against the temperatures. We argue that the peculiar scaling behavior of AC conductivity may stem from the couplings of the dilaton field with the gauge fields and also the logarithmic behavior near the boundary in the Lifshitz spacetime.

What do non-relativistic CFTs tell us about Lifshitz spacetimes?

Abstract We study the reconstructability of (d + 2)-dimensional bulk spacetime from (d + 1)-dimensional boundary data, particularly concentrating on backgrounds which break (d + 1)-dimensional Lorentz invariance. For a large class of such spacetimes, there exist null geodesics which do not reach the boundary. Therefore classically one might guess some information is trapped in the bulk and thus invisible at the boundary. We show that this classical intuition correctly predicts the quantum situation: whenever there are null geodesics which do not reach the boundary, there are also “trapped scalar modes” whose boundary imprint is exponentially suppressed. We use these modes to show that no smearing function exists for pure Lifshitz spacetime, nor for any flow which includes a Lifshitz region. Indeed, for any (planar) spacetime which breaks (d + 1)-dimensional Lorentz invariance at any radius, we show that local boundary data cannot reconstruct complete local bulk data.

Scalar boundary conditions in Lifshitz spacetimes

We investigate the conditions imposable on a scalar field at the boundary of the so-called Lifshitz spacetime which has been proposed as the dual to Lifshitz field theories. For effective mass squared between −(d + z − 1)2 /4 and z 2 − (d + z − 1)2 /4, we find a one-parameter choice of boundary condition type. The bottom end of this range corresponds to a Breitenlohner-Freedman bound; below it, the Klein-Gordon operator need not be positive, so we cannot make sense of the dynamics. Above the top end of the range, only one boundary condition type is available; here we expect compact initial data will remain compact in the future.

Logarithmic two-point correlation functions from a z =2 Lifshitz model

Abstract The Einstein-Proca action is known to have asymptotically locally Lifshitz spacetimes as classical solutions. For dynamical exponent z = 2, two-point correlation functions for fluctuations around such a geometry are derived analytically. It is found that the retarded correlators are stable in the sense that all quasinormal modes are situated in the lower half-plane of complex frequencies. Correlators in the longitudinal channel exhibit features that are reminiscent of a structure usually obtained in field theories that are logarithmic, i.e. contain an indecomposable but non-diagonalizable highest weight representation. This provides further evidence for conjecturing the model at hand as a candidate for a gravity dual of a logarithmic field theory with anisotropic scaling symmetry.

From AdS to Schrödinger/Lifshitz dual space-times without or with hyperscaling violation

It is observed that the (intersecting) branes of M/string theory, which are known to give AdS geometry (directly or upto a conformal transformation) in the near horizon limit, do also lead to Schr\" odinger/Lifshitz dual space-times (without or with hyperscaling violation) upon using appropriate solution generating transformation and dimensional reduction. We show that the dynamical exponents of the Schr\" odinger and the Lifshitz space-times obtained in this way always add upto 2. We illustrate this by several examples, including M2-, M5-branes of M-theory and D$(p+1)$-branes ($p\neq 4$, since in this case the near horizon limit does not give AdS geometry) of string theory as well as many of their intersecting solutions. The Schr\" odinger space-time can be obtained by the standard wave generating technique along one of the brane directions (for single brane) or one of the common brane directions (for intersecting branes) and then interchanging the light-cone coordinates by double Wick rotations, whereas, the Lifshitz space-time can be obtained by dimensionally reducing (for M-theory) along the wave direction or taking T-duality (for string theory) along the same direction. We thus obtain Schr\" odinger/Lifshitz dual space-times without or with hyperscaling violation from the same M/string theory solutions and they preserve some fraction of the supersymmetry.