Limits on the mass of compact objects in Hořava-Lifshitz gravity

TL;DR

The paper investigates how HL gravity modifies theoretical mass limits for compact objects by deriving HL TOV equations and analyzing mass-radius bounds. It shows that a uniform-density (udl) bound exists as a HL analogue of Buchdahl's limit, and that for arbitrary EOS this bound remains universal, with many heavy configurations residing inside horizons but still bounded above. A HL causal limit (ssl) is computed using a fiducial-density–dependent framework, revealing that HL can permit larger maximum masses than GR for certain parameter choices, with SSL curves converging toward horizon features near the minimal black hole where $M_c=q$. Together, these results explain why HL gravity can accommodate heavier neutron stars and more compact objects than GR and highlight the sensitivity of the bounds to HL-specific scales like $q$. The findings offer a pathway to constrain HL parameters through astrophysical observations of compact objects, while acknowledging that a fully Lorentz-violating theory may require a deeper understanding of the relation between light-speed and sound-speed limits.

Abstract

It is known that there exist theoretical limits on the mass of compact objects in general relativity. One is the Buchdahl limit for an object with an arbitrary equation-of-state that turns out to be the limit for an object with uniform density. Another one is the causal limit that is stronger than the Buchdahl limit and is related to the speed of sound inside an object. Similar theoretical limits on the mass of compact objects in deformed Hořava-Lifshitz (HL) gravity are found in this \paper. Interestingly, the both curves of the uniform density limit and the sound speed limit meet the horizon curve at the minimum of the horizon, where a black hole becomes extremal, i.e., $M=q$, considering the Kehagias-Sfetsos vacuum that is an asymptotic flat solution in the HL gravity.

Figures (4)

• Figure 1: The udl, the upper bound of the mass $M$ of a compact object with uniform density is depicted with respect to its radius $R$, compared with the Buchdahl limit in GR for Schwarzschild and Reissner-Nordström vacua. The horizon radii of ks, Reissner-Nordström ($r_+^\text{\acs{ks}} = r_+^\text{RN}$) and Schwarzschild ($r_h^\text{Schwarzschild}$) black holes are also plotted.
• Figure 2: Scatter plots of simulated compact objects \ref{['fig:uniform']} for uniform density and \ref{['fig:random']} for arbitrarily monotonic eos with positive pressure are presented. The panels \ref{['fig:uniform:zoom']} and \ref{['fig:random:zoom']} are the detailed views near the minimal black hole of the panels \ref{['fig:uniform']} and \ref{['fig:random']}, respectively. We see that the compact objects does not exceeds the udl in both cases, which means that the udl is indeed the universal limit of compact objects made of an arbitrary matter, provided that the eos is monotonic and the pressure inside the object is everywhere positive. This is consistent with the Buchdahl limit in GR, which is actually the uniform density limit. The objects inside the horizon are new solutions in hl gravity, which have not been seen in gr. Indeed, in gr limit $q \to 0$, the line $M = (R/R_c)^3 M_c$ becomes the vertical axis and the only solutions below the udl curve remain. The new solutions are well confined inside the horizon so that they are de facto black holes to an observer outside the horizon.
• Figure 3: \ref{['fig:causal:factor']} The numerical factor in the causal limit in gr becomes a function of the fiducial density $\rho_u$ in the ssl in hl. \ref{['fig:causal:mr']} The ssl in hl is compared to the causal limit in gr, considering non-rotating, asymptotically flat vacuum solutions. The black hole horizons and the udl of ks vacuum also depicted. All three curves, the horizon, the udl and the ssl, meet at the minimum of the horizon (star), which represents the minimal black hole.
• Figure 4: Mass-radius relations of the eos selected in the previous work Kim:2018dbs are shown to be below the ssl curve for \ref{['fig:mr:q4k']}$q=4\text{ km}$ and \ref{['fig:mr:q400']}$q=400\text{ m}$.