*Summary:*

*◉ Edwin Son and colleagues found that neutron star (NS) in Horava-Lifshitz (HL) gravity with a larger radius and heavier mass than a NS in GR remains stable without collapsing into a black hole.*

◉ *They observed that HL gravity has a relatively weaker gravitational acceleration than that of GR near the central region of the NS. This observation explains why the heavier NS rather than the NS in GR, remains stable in HL gravity without collapsing into a black hole.*

General relativity (GR) has been a successful theory of gravity for explaining the motion of planets and stars at the macroscopic scale up to now, because it is strongly valid under the weak gravitational field approximation. However, the theory is believed to be theoretically incomplete in terms of a unified theory with a quantum nature, for example, in describing the physics at strong gravitational fields near compact astrophysical objects such as black holes or neutron stars (NSs), mysterious darkness such as dark energy/matter, the origin of the Universe such as inflationary blowup, and the hierarchical phase transition at the very early stage of the Universe. Although there have been many attempts to address these problems by introducing new particles and/or alternative theories of gravitation, a breakthrough is still required from the theoretical as well as experimental/observational point of view.

One of the discrepancies between gauge theories and GR is that the latter is not renormalizable due to ultraviolet (UV) divergence. The UV divergence appears in extreme circumstances and/or at high energies, where the quantum effect has to be taken into account, for example, at the early stage of the Universe or in the vicinity of a black hole. A renormalizable gravity theory is naturally supposed to have improved properties at the UV scale. One of the theories embodying such a philosophy has been proposed by Horava with the aim of formulating a UV-complete theory by sacrificing local Lorentz symmetry, which is called Horava-Lifshitz (HL) gravity. HL gravity approximates to GR at the infrared (IR) scale, whereas it becomes a different type of gravity at the UV scale by the introduction of anisotropic scaling between time and space. This anisotropic scaling is key to renormalizability in HL gravity although it breaks the local Lorentz symmetry. Many intensive studies in diverse fields of applications have been conducted in search of a possible candidate for quantum gravity.

Tolman and Oppenheimer and Volkoff formulated the so-called TOV equation in GR to describe the equilibrium state of compact stars such as white dwarfs and (Neutron stars) NSs in a spherically symmetric and static configuration. By solving the TOV equation, we can obtain the mass-radius relation and, consequently, can estimate the maximum mass of a compact star and its radius. In particular, for a NS, the maximum mass and radius are sensitive to the stiffness of the selected equation-of-state (EOS) model. Hence, by investigating the TOV equation with the realistic EOS parameters, we can precisely understand the UV aspect of a NS in HL gravity in comparison with the case in GR.

That’s what Edwin Son and colleagues have done. In their recent paper, they investigated the structure of a NS in HL gravity, in comparison with that in GR. They derived an explicit form of the TOV equations in HL gravity. By solving them with realistic EOS models, they obtained the mass-radius relation of a NS, which estimates how HL gravity deviates from GR in terms of the mass and the radius of a NS. They found that a NS in HL gravity with a larger radius and heavier mass than a NS in GR remains stable without collapsing into a black hole. But why?

Well, from mass profiles of the neutron stars, they observed that HL gravity has a relatively weaker gravitational acceleration than that in GR near the central region of the NS. Even though the NS in HL gravity is much heavier than the NS in GR, its gravity is weaker at r ≤ 6 km. This observation explains why the heavier NS rather than the NS in GR, remains stable in HL gravity without collapsing into a black hole.

The weaker gravity of HL is also observed from the curve of the same-mass NS (when NS of same mass considered in HL and GR) at the bottom panel of Fig. 1 (a). Note that the maximum accelerations inside the heaviest NSs are almost the same in both GR and HL gravity; however, the accelerations may be slightly different depending on the EOS models.

The central density of the NS is the same as that of the NS in GR gravity, and the density and pressure decrease slowly as ‘r’ increases and vanish at larger radii, with the result that the NS (RL gravity) becomes naturally heavier than the NS in GR. (Refer fig. 1b)

The weaker gravitational attraction in HL gravity can be easily understood by computing the Kretschmann curvature scalar. The overall behavior of the Kretschmann curvature scalar for the KS black hole (KSBH) solution in HL is slowly growing, unlike the SBH solution in GR, which implies that the gravitational force induced by the curvature scalar in HL gravity is weaker than that in GR in the limit of r → 0. Then the mass obtained by integrating Eq. m’ = 4πr²ρ, from r = 0 to r = R also becomes heavier, eventually.— said Chan Park, Co-author of the study

They concluded that to validate HL gravity, theoretical investigations and future observations of GWs from compact binary mergers containing at least one NS will yield more constraints on the physical observables of a NS and, eventually, will determine the fate of HL gravity.

**Reference***: Kyungmin Kim, John J. Oh, Chan Park, Edwin J. Son, “Neutron Star Structure in Hořava-Lifshitz Gravity”, ArXiv, pp. 1-6, 2021. https://arxiv.org/abs/1810.07497v2*

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