Entropy of black holes in the deformed Ho\v{r}ava-Lifshitz gravity

We find the entropy of Kehagias-Sfetsos black hole in the deformed Ho\v{r}ava-Lifshitz gravity by using the first law of thermodynamics. When applying generalized uncertainty principle (GUP) to Schwarzschild black hole, the entropy $S=A/4+(\pi/\omega)\ln(A/4)$ may be interpreted as the GUP-inspired black hole entropy. Hence, it implies that the duality in the entropy between the Kehagias-Sfetsos black hole and GUP-inspired Schwarzschild black hole is present.

It seems that the GUP-inspired Schwarzschild black hole was related to black holes in the deformed HL gravity [11]. We could not confirm a solid connection between the GUP [21,22] and the black hole of deformed HL gravity, although partial connections were established between them. However, it was known that the GUP provides naturally a UV cutoff to the local quantum field theory as quantum gravity effects [23,24]. Also, the GUP density function may be replaced by a cutoff function for the renormalization group study of deformed HL gravity [25]. We have found GUP-inspired graviton propagators and compared these with UV-tensor propagators in the deformed HL gravity. Two were similar, but the p 5 -term arisen from Cotton tensor was missed in the GUP-inspired graviton propagator.
This shows that a power-counting renormalizable theory of the HL gravity is closely related to the GUP.
In this Letter, we will make a further progress on exploring the connection between the GUP and black hole of the deformed HL gravity. We obtain the entropy of KS black hole in the deformed HL gravity. This entropy may be interpreted as the GUP-inspired black hole entropy when applying the GUP to Schwarzschild black hole.

HL gravity
Introducing the ADM formalism where the metric is parameterized the Einstein-Hilbert action can be expressed as where G is Newton's constant and extrinsic curvature K ij takes the form Here, a dot denotes a derivative with respect to t. An action of the non-relativistic renormalizable gravitational theory is given by [1] where the kinetic terms are given by with the DeWitt metric and its inverse metric The potential terms is determined by the detailed balance condition as Here the E tensor is defined by with the Cotton tensor C ij Explicitly, E ij could be derived from the Euclidean topologically massive gravity where ǫ ikl is a tensor density with ǫ 123 = 1.
In the IR limit, comparing L 0 with Eq.(2) of general relativity, the speed of light, Newton's constant and the cosmological constant are given by The equations of motion were derived in [26] and [3]. We would like to mention that the IR vacuum of this theory is Lifshitz spacetimes [7]. Hence, it is interesting to take a limit of the theory, which may lead to a Minkowski vacuum in the IR sector. To this end, one may deform the theory by introducing a soft violation term of " then, take the Λ W → 0 limit [10]. We call this as the "deformed HL gravity". This theory does not alter the UV properties of the HL gravity, while it changes the IR properties. That is, there exists a Minkowski vacuum, instead of Lifshitz vacuum. In the IR limit, the speed of light and Newton's constant are given by

Entropy of KS black hole
A spherically symmetric solution to the deformed HL gravity was obtained by considering the line element In this case, we have K ij = 0 and C ij = 0. Hence, it is emphasized that we have relaxed both the projectability restriction and detailed balance condition [1,27] since the lapse function N depends on the spatial coordinate r as well as a soft violation term of µ 4 R is included. Substituting the metric ansatz (15) intoL V with L K = 0, one has the reduced Lagrangiañ where a parameter ω λ = 8µ 2 (3λ − 1)/κ 2 specifies the deformed HL gravity.
For λ = 1, the KS solution is given by [10] f KS = N 2 with ω(= ω λ=1 ) = 16µ 2 /κ 2 . In the limit of ω → ∞ (equivalently, κ 2 → 0), it reduces to the Schwarzschild metric function From the condition of f KS (r ± ) = 0, the outer (inner) horizons are given by In order to have a black hole solution, it requires that Furthermore, the extremal black hole is obtained from the condition of degenerate horizon with f ′′ KS (r e ) = 4ω/3. Thermodynamic quantities of mass, temperature, and heat capacity for the KS black hole are defined as [11] M (r ± ) = 1 + 2ωr 2 ± 4ωr ± , T = 2ωr 2 In the limit of ω → ∞, these reduce to corresponding quantities of Schwarzschild black hole as Now we wish to derive the entropy by considering that the first law of thermodynamics holds for black hole in the deformed HL gravity: Then, the entropy is calculated as which leads to [18] S = π r 2 + + 1 ω ln r 2 If one chooses then we have a compact expression of the entropy with A/4 = πr 2 + and G = 1. We note that in the limit of ω → ∞, Eq. (28) reduces to the Bekenstein-Hawking entropy of Schwarzschild black hole as It is clear that the logarithmic term represents the feature of KS black hole in the deformed HL gravity. Accordingly, we have to interpret this logarithmic term to understand why the entropy of KS black hole takes the form (28).

GUP-inspired Schwarzschild black hole
A meaningful prediction of various theories of quantum gravity (string theory and loop quantum gravity) and black holes is the presence of a minimum measurable length or a maximum observable momentum. This has provided the GUP which modifies commutation relations between position coordinates and momenta. Also the black hole solution of deformed HL gravity reminds us the Schwarzschild black hole modified with the GUP [11].
Hence, it is very interesting to develop a close connection between GUP and HL gravity. A generalized commutation relation 2 of 2 We note that the GUP is in the heart of the quantum gravity phenomenology. Certain effects of quantum gravity are universal and thus, influence almost any system with a well-defined Hamiltonian [28].
In general, the GUP satisfies the modified Heisenberg algebra [23,24] [xi, pj] = ih δij + βp 2 δij + β ′ pipj , [pi, pj] = 0, where pi is considered as the momentum at high energies and thus, (30) can be interpreted to be the UVcommutation relations. In order to achieve the commutativity, we have to choose β ′ = 2β. In this case, the minimal length which follows from the modified Heisenberg algebra is given by (∆x)min =h √ 5β. We emphasize that the presence of the minimal length represents a feature of the GUP. In order to see the leads to the generalized uncertainty relation with l p = Gh/c 3 the Planck length. Here a parameter α =h √ β/l p is introduced to take into account the GUP effect. The Planck mass is given by m p = hc/G. The above implies a lower bound on the length scale Furthermore, the GUP may be used to derive temperature for the modified Schwarzschild black hole by identifying ∆p with the energy (temperature) of radiated photons [29]. The momentum uncertainty for radiated photons can be found to be ∆x 2α 2 l 2 The left inequality implies small corrections to the Heisenberg's uncertainty principle for ∆x ≫ αl p as ∆p ≥h/∆x +hα 2 l 2 p /(∆x) 3 + · · · [30]. On the other hand, the right inequality means that ∆p cannot be arbitrarily large in order that the GUP in (32) makes sense. For simplicity, we use the Planck units of c =h = G = k B = 1 which imply that l p = m p = 1 and β = α 2 . Considering the GUP effect on the near-horizon and ∆x = 2r + = 4M , the relation (35) reduces to Replacing β with 2/ω, the above leads to a relation Here we wish to mention that a replacement of β → 2/ω was performed because both sides of Eq. (36) have mathematically the same form as Eq. (19). It seems that Eq. (37) GUP-inspired black holes, it is sufficient to consider βp 2 because this term determines the minimal size of the black hole.
indicates a connection between quantum and classical properties of KS black holes in the deformed HL gravity.

Discussions
We have found the entropy of Kehagias-Sfetsos black hole in the deformed HL gravity by using the first law of thermodynamics. The presence of logarithmic term ln[A/4] seems to be universal for the HL gravity because it appeared in topological black hole solutions [4] and LMP solution [8].
We would like to mention that the GUP seems to be a powerful tool to study quantum gravity effects. The GUP with the relation ∆x = 2r + = 4M of the Schwarzschild radius r + and its ADM mass M make sense because quantum gravity effects of GUP is universal.
Thus, the Schwarzschild black hole was modified if one assumes the GUP. It seems that the GUP explains a part of quantum gravity effects but not whole of these. A relevant relationship of the black hole entropy-area based on string theory and loop quantum gravity is given by [34] where ρ is a model-dependent parameter. Therefore, the GUP was widely used to obtain quantum correction (41) to the Bekenstein-Hawking entropy of Schwarzschild black hole [31,32,33].
In this work, we have attempted to explain the logarithmic term (π/ω) ln[A/4] for the entropy of black hole in the deformed HL gravity by considering the GUP-inspired entropy to Schwarzschild black hole. The corresponding quantities may be β in the generalized commutation relation (31) and 1/ω of parameter in the deformed HL gravity (16). In the limit of β → 0, we recover the Heisenberg uncertainty relation without quantum gravity effects, while in the limit of ω → ∞, the entropy of Schwarzschild black hole is recovered without the logarithmic term. This may imply a close connection between GUP and HL gravity.
However, we recognize that the entropy of KS black holes (standard black hole thermodynamics) was being compared with that of the GUP-inspired Schwarzschild black hole (non-standard black hole thermodynamics). This implies that there exists a sort of correspondence (duality) between two systems, but not that the GUP is a fundamental property of Hořava-Lifshitz gravity. Especially, we realize that logarithmic terms in black hole entropy appeared in many different models. At this stage, thus, the logarithmic term in the KS black hole entropy does not represent a definite signal that the GUP is a underlying principle of Hořava construction.
Consequently, we have shown that the duality in the entropy between the KS black hole from the HL gravity and GUP-inspired Schwarzschild black hole is present.