Horava-Lifshitz Research Articles

Cosmological Friedmann equation in infrared modified Hořava–Lifshitz gravity via generalized Misner–Sharp mass

In various gravity theories, Friedmann equations can be cast to a form of the first law of thermodynamics in a Friedmann–Robertson–Walker (FRW) cosmological setup. However, this result failed in recent infrared (IR) modified Hořava–Lifshitz (HL) gravity. The difficulty stems from the fact that HL gravity is Lorentz-violating. Motivated by this problem, we use the Misner–Sharp mass to investigate the thermodynamics near the apparent horizon in HL cosmology. We find that the Friedmann equations can be derived from the first law of thermodynamics. The Misner–Sharp mass used here inherits the specific properties of HL gravity since it is directly from the gravitational action of HL theory. We also prove that the first law of thermodynamics with logarithmic entropy still holds at the apparent horizon in FRW. The results suggest that the general prescription of deriving the field equation from thermodynamics still works in the HL cosmology.

Friedmann inflation in Horava–Lifshitz gravity with a scalar field

We study Friedmann inflation in general Horava–Lifshitz (HL) gravity with detailed and nondetailed and also without the projectability conditions. Accordingly, we derive the modifications in the Friedmann equations due to single scalar field potentials describing power-law and minimal-supersymmetrically extended inflation. By implementing four types of the equations-of-state characterizing the cosmic background geometry, the dependence of the tensorial and spectral density fluctuations and their ratio on the inflation field is determined. The latter characterizes the time evolution of the inflation field relative to the Hubble parameter. Furthermore, the ratio of tensorial-to-spectral density fluctuations is calculated in dependence on the spectral index. The resulting slow-roll parameters apparently differ from the ones deduced from the standard General Relativity (Friedmann gravity). We also observe that the tensorial-to-spectral density fluctuations continuously decrease when moving from nondetailed HL gravity, to Friedmann gravity, to HL gravity without the projectability, and to detailed HL gravity. This regular pattern is valid for three types of cosmic equations-of-state and different inflation potential models. The results fit well with the recent Planck observations.

Infrared Horava–Lifshitz gravity coupled to Lorentz violating matter: a spectral action approach

We continue our study of Horava–Lifshitz (HL) type theories using the methods of spectral geometry. In this work we construct the infrared action of gravity and matter coupled to gravity in the most general way respecting the foliation preserving diffeomorphisms. This is done with the help of the spectral action principle based on some generalized Dirac operators. The gravity part reproduces the infrared limit of the HL gravity, while the matter part gives the generalization of the earlier suggested models. Due to the fact that the same Dirac operator is used in the construction of both sectors, the parameters of the gravity and matter parts are related. We expect that this potentially could naturally exclude fine tunings needed to get some desired properties as well as open new possibilities for the experimental tests of the model.

Hořava-Lifshitz gravity from dynamical Newton-Cartan geometry

Abstract Recently it has been established that torsional Newton-Cartan (TNC) geometry is the appropriate geometrical framework to which non-relativistic field theories couple. We show that when these geometries are made dynamical they give rise to Hořava-Lifshitz (HL) gravity. Projectable HL gravity corresponds to dynamical Newton-Cartan (NC) geometry without torsion and non-projectable HL gravity corresponds to dynamical NC geometry with twistless torsion (hypersurface orthogonal foliation). We build a precise dictionary relating all fields (including the scalar khronon), their transformations and other properties in both HL gravity and dynamical TNC geometry. We use TNC invariance to construct the effective action for dynamical twistless torsional Newton-Cartan geometries in 2+1 dimensions for dynamical exponent 1 < z ≤ 2 and demonstrate that this exactly agrees with the most general forms of the HL actions constructed in the literature. Further, we identify the origin of the U(1) symmetry observed by Hořava and Melby-Thompson as coming from the Bargmann extension of the local Galilean algebra that acts on the tangent space to TNC geometries. We argue that TNC geometry, which is manifestly diffeomorphism covariant, is a natural geometrical framework underlying HL gravity and discuss some of its implications.

Observational Data Fitting to Constrain Variable Modified Chaplygin Gas in the Background of Horava-Lifshitz Gravity

FRW universe in Horava-Lifshitz (HL) gravity model filled with a combination of dark matter and dark energy in the form of variable modified Chaplygin gas (VMCG) is considered. The permitted values of the VMCG parameters are determined by the recent astrophysical and cosmological observational data. Here we present the Hubble parameter in terms of the observable parameters Ω d m0, Ω v m c g0, H 0, redshift z and other parameters like α, A, γ and n. From Stern data set (12 points), we have obtained the bounds of the arbitrary parameters by minimizing the χ 2 test. The best-fit values of the parameters are obtained by 66 %, 90 % and 99 % confidence levels. Next due to joint analysis with BAO and CMB observations, we have also obtained the bounds of the parameters (A, γ) by fixing some other parameters α and n. The best fit value of distance modulus μ(z) is obtained for the VMCG model in HL gravity, and it is concluded that our model is perfectly consistent with the union2 sample data.

Casimir effect in the Hořava–Lifshitz gravity with a cosmological constant

We calculate the Casimir energy of a massless scalar field confined between two nearby parallel plates formed by ideal uncharged conductors, placed tangentially to the surface of a sphere with mass M and radius R. To this end, we take into account a static and spherically symmetric solution of Hořava–Lifshitz (HL) gravity, with a cosmological constant term, in lower orders of approximation, considering both weak-field and infrared limits. We show that the Casimir energy, just in the second order weak-field approximation, is modified due to the parameter of the HL gravity as well as to the cosmological constant.

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.

On geodesics in space-times with a foliation structure: a spectral geometry approach

Motivated by the Horava–Lifshitz (HL)-type theories, we study the physical motion of matter coupled to a foliated geometry in a non-diffeomorphism invariant way. We use the concept of a spectral action as a guiding principle in writing down the matter action. Based on the deformed Dirac operator compatible with the reduced symmetry-foliation preserving diffeomorphisms, this approach provides a natural generalization of the minimal coupling. Focusing on the infrared version of the Dirac operator, we derive the physical motion of a test particle and discuss in what sense it can still be considered as a geodesic motion for some modified geometry. We show that the apparatus of non-commutative geometry could be very efficient in the study of matter coupled to the HL gravity.

Energy of a slowly-rotating black hole in Horava-Lifshitz gravity by using approximate Lie symmetry methods

The energy content of the Lee-Kim-Myung slowly-rotating black hole (Lee et al., Eur. Phys. J. C 70, 361 (2010)), in the Horava-Lifshitz (HL) theory of gravity is investigated by using approximate Lie symmetry methods for differential equations. The energy of this slowly-rotating black hole in the HL gravity is found to be is rescaled by a r-dependent factor. From the rescaling of energy, the rotation is observed to be added to the mass of the black hole, and this effect decreases with increasing coordinate r. This expression of energy rescaling reduces to the Schwarzschild mass for the limits in which the Lee-Kim-Myung slowly rotating black hole solution in the HL gravity reduces to the Schwarzschild solution in general relativity.

Vaidya solutions in general covariant Hořava–Lifshitz gravity without projectability: Infrared limit

In this paper, we have studied nonstationary radiative spherically symmetric spacetime, in general covariant theory (U(1) extension) of Hořava–Lifshitz (HL) gravity without the projectability condition and in the infrared (IR) limit. The Newtonian prepotential φ was assumed null. We have shown that there is not the analogue of the Vaidya's solution in the Hořava–Lifshitz Theory (HLT), as we know in the General Relativity Theory (GRT). Therefore, we conclude that the gauge field A should interact with the null radiation field of the Vaidya's spacetime in the HLT.

Covariant renormalizable modified and massive gravity theories on (non)commutative tangent Lorentz bundles

The fundamental field equations in modified gravity (including general relativity; massive and bimetric theories; Hořava–Lifshitz (HL); Einstein–Finsler gravity extensions etc.) possess an important decoupling property with respect to nonholonomic frames with 2 (or 3) + 2 + 2 + ⋯ spacetime decompositions. This allows us to construct exact solutions with generic off-diagonal metrics depending on all spacetime coordinates via generating and integration functions containing (un-)broken symmetry parameters. Such nonholonomic configurations/models have a nice ultraviolet behavior and seem to be ghost-free and (super-)renormalizable in a sense of covariant and/or massive modifications of HL gravity. The apparent noncommutativity and breaking of Lorentz invariance by quantum effects can be encoded into fibers of noncommutative tangent Lorentz bundles for corresponding "partner" anisotropically induced theories. We show how the constructions can be extended to include conjectured covariant renormalizable models with massive graviton fields and effective Einstein fields with (non)commutative variables.

Observational constraints on modified Chaplygin gas in Horava–Lifshitz gravity with dark radiation

We present Cosmological models with modified Chaplygin gas (MCG) in the framework of Horava-Lifshitz (HL) theory of gravity both with and without detailed balance. The equation of state (EOS) for a MCG contains three unknown parameters namely, $A$, $\alpha$, $B$. The allowed values of some of these parameters of the EOS are determined using the recent astrophysical and cosmological observational data. Using observational data from $H(z)-z$, BAO peak parameter, CMB shift parameter we study cosmologies in detailed-balance and beyond detailed-balance scenario. In this paper we take up the beyond detailed-balance scenario in totality and contribution of dark radiation in the case of detailed-balance scenario on the parameters of the EOS. We explore the effect of dark radiation on the whole range the of effective neutrino parameter to constrain matter contributing parameter $B$ in both the detailed-balance and the beyond-detailed balance scenario. It has been observed that greater the dark radiation less the matter contribution in the MCG in both the scenario considered here. In order to check the validity of beyond detailed balance scenario we plot supernovae magnitudes ($\mu$) with redshift of Union2 data and then the variation of state parameter with redshift is studied. It has been observed that beyond detailed balance scenario is equally suitable in HL gravity with MCG.

Emergence of spaces and the dynamic equations of FRW universes inthe f(R) theory and deformed Hořava-Lifshitztheory

It has been shown that Friedmann equation of FRW universe can be derivedfrom the idea which says cosmic space is emergent as cosmic time progressesand our universe is expanding towards the state with the holographicequipartition by Padmanabhan. In this note, we give a general relationshipbetween the horizon entropy and the number of the degrees of freedomon the surface, which can be applied to quantum gravity.we also obtain the corresponding dynamic equationsby using the idea of emergence of spaces in the f(R) theory anddeformed Hořava-Lifshitz(HL)theory.

Lifshitz Gravity for Lifshitz Holography

We argue that Hořava-Lifshitz (HL) gravity provides the minimal holographic dual for Lifshitz-type field theories with anisotropic scaling and a dynamical exponent z. First we show that Lifshitz spacetimes are vacuum solutions of HL gravity, without need for additional matter. Then we perform holographic renormalization of HL gravity, and show how it reproduces the full structure of the z=2 anisotropic Weyl anomaly in dual field theories in 2+1 dimensions, while its minimal relativistic gravity counterpart yields only one of two independent central charges in the anomaly.

Quantum singularities in Hořava-Lifshitz cosmology

The recently proposed Horava-Lifshitz (HL) theory of gravity is analyzed from the quantum cosmology point of view. By employing usual quantum cosmology techniques, we study the quantum Friedmann-Lemaitre-Robertson-Walker (FLRW) universe filled with radiation in the context of HL gravity. We find that this universe is quantum mechanically nonsingular in two different ways: the expectation value of the scale factor $<a>(t)$ never vanishes and, if we abandon the detailed balance condition suggested by Horava, the quantum dynamics of the universe is uniquely determined by the initial wave packet and no boundary condition at $a=0$ is indeed necessary.

Trans-Planckian problem in the healthy extension of Horava-Lifshitz gravity

Planck scale physics may influence the evolution of cosmological fluctuations in the early stages of cosmological evolution. Because of the quasi-exponential redshifting, which occurs during an inflationary period, the physical wavelengths of comoving scales that correspond to the present large-scale structure of the Universe were smaller than the Planck length in the early stages of the inflationary period. This trans-Planckian effect was studied before using toy models. The Horava-Lifshitz (HL) theory offers the chance to study this problem in a candidate UV complete theory of gravity. In this paper we study the evolution of cosmological perturbations according to HL gravity assuming that matter gives rise to an inflationary background. As is usually done in inflationary cosmology, we assume that the fluctuations originate in their minimum energy state. In the trans-Planckian region the fluctuations obey a non-linear dispersion relation of Corley-Jacobson type. In the "healthy extension" of HL gravity there is an extra degree of freedom which plays an important role in the UV region but decouples in the IR, and which influences the cosmological perturbations. We find that in spite of these important changes compared to the usual description, the overall scale-invariance of the power spectrum of cosmological perturbations is recovered. However, we obtain oscillations in the spectrum as a function of wavenumber with a relative amplitude of order unity and with an effective frequency which scales nonlinearly with wavenumber. Taking the usual inflationary parameters we find that the frequency of the oscillations is so large as to render the effect difficult to observe.

Static electromagnetic fields and charged black holes in general covariant theory of Hořava-Lifshitz gravity

In this paper, we study electromeganetic static spacetimes in the nonrelativisitc general covariant theory of the Hořava-Lifshitz (HL) gravity, proposed recently by Hořava and Melby-Thompson, and present all the electric static solutions, which represent the generalization of the Reissner-Nordström solution found in Einstein's general relativity (GR). The global/local structures of spacetimes in the HL theory in general are different from those given in GR, because the dispersion relations of test particles now contain high-order momentum terms, so the speeds of these particles are unbounded in the ultraviolet (UV). As a result, the conception of light-cones defined in GR becomes invalid and test particles do not follow geodesics. To study black holes in the HL theory, we adopt the geometrical optical approximations, and define a horizon as a (two-closed) surface that is free of spacetime singularities and on which massless test particles are infinitely redshifted. With such a definition, we show that some of our solutions give rise to (charged) black holes, although the radii of their horizons in general depend on the energies of the test particles.

Modified Chaplygin gas in Horava-Lifshitz gravity and constraints on itsBparameter

We present cosmological models with modified Chaplygin gas (MCG) in the framework of Horava-Lifshitz (HL) theory of gravity. The equation of state (EOS) for a MCG is a three-variable equation determined by $A$, $\ensuremath{\alpha}$, $B$. The allowed values of these parameters of the EOS are determined using the recent astrophysical and cosmological observational data. For this we consider $H(z)\ensuremath{-}z$, BAO peak, CMB shift parameter data to obtain a suitable range of values of the parameters for a viable cosmological model. Considering the suitable range of effective neutrino parameter ($\ensuremath{\Delta}{N}_{\ensuremath{\nu}}$), the ranges of values of the B-parameter along with ${A}_{S}$ (defined by $A$, $\ensuremath{\alpha}$, and $B$) and $\ensuremath{\alpha}$ are determined here by chi-square minimization.

Black holes and global structures of spherical spacetimes in Horava-Lifshitz theory

We systematically study black holes in the Horava-Lifshitz (HL) theory by following the kinematic approach, in which a horizon is defined as the surface at which massless test particles are infinitely redshifted. Because of the nonrelativistic dispersion relations, the speed of light is unlimited, and test particles do not follow geodesics. As a result, there are significant differences in causal structures and black holes between general relativity (GR) and the HL theory. In particular, the horizon radii generically depend on the energies of test particles. Applying them to the spherical static vacuum solutions found recently in the nonrelativistic general covariant theory of gravity, we find that, for test particles with sufficiently high energy, the radius of the horizon can be made as small as desired, although the singularities can be seen in principle only by observers with infinitely high energy. In these studies, we pay particular attention to the global structure of the solutions, and find that, because of the foliation-preserving-diffeomorphism symmetry, ${Diff}(M,{\cal{F}})$, they are quite different from the corresponding ones given in GR, even though the solutions are the same. In particular, the ${Diff}(M,{\cal{F}})$ does not allow Penrose diagrams. Among the vacuum solutions, some give rise to the structure of the Einstein-Rosen bridge, in which two asymptotically flat regions are connected by a throat with a finite non-zero radius. We also study slowly rotating solutions in such a setup, and obtain all the solutions characterized by an arbitrary function $A_{0}(r)$. The case $A_{0} = 0$ reduces to the slowly rotating Kerr solution obtained in GR.

EVOLUTION OF ELECTROMAGNETIC AND DIRAC PERTURBATIONS AROUND A BLACK HOLE IN HOŘAVA GRAVITY

The evolution of electromagnetic and massless Dirac perturbations in the spacetime geometry of Kehagias–Sfetsos (KS) black hole in the deformed Hořava–Lifshitz (HL) gravity is investigated and the associated quasinormal modes (QNMs) are evaluated from time domain integration data. We find a considerable deviation in the nature of field evolution in HL theory from that in the Schwarzschild spacetime and QNM region extends over a longer time in HL theory before the power-law tail decay begins. The dependence of the field evolution on the HL parameter α is also studied. In the time domain picture we find that the length of QNM region increases with α. But the late-time decay of field follows the same power-law tail behavior as in the case of Schwarzschild black hole.