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

This paper establishes theoretical mass limits for compact objects in deformed Hořava-Lifshitz gravity, demonstrating that both the uniform density and causal sound speed bounds converge at the extremal black hole horizon ($M=q$) within the Kehagias-Sfetsos vacuum solution.

The Gist

Imagine the universe as a giant, cosmic construction site. For decades, physicists have been building models of the heaviest, densest objects in the universe: neutron stars. These are the cosmic equivalent of a mountain squeezed into the size of a city.

For a long time, we used a set of blueprints called General Relativity (Einstein's theory) to understand how heavy these stars can get before they collapse into black holes. But recently, astronomers found some neutron stars that are so heavy they seem to break the old blueprints. It's like finding a skyscraper that defies the laws of physics as we knew them.

This paper explores a new set of blueprints called Hořava-Lifshitz (HL) gravity. Think of this as a "remastered" version of Einstein's theory, designed to work better at the tiniest scales (like the quantum level) while still looking like Einstein's theory at the large scales we see every day.

Here is the breakdown of what the paper discovers, using some everyday analogies:

1. The Old Rules: The "Weight Limit" Signs

In Einstein's universe, there are two main "weight limit" signs for these cosmic stars:

• The Uniform Density Limit (Buchdahl Limit): Imagine a balloon. If you blow it up too much, it pops. In physics, if a star gets too dense and heavy for its size, it collapses. There's a mathematical "ceiling" on how heavy it can be.
• The Causal Limit (Sound Speed Limit): Imagine sound traveling through a material. In our universe, nothing can travel faster than light. Inside a star, if the "sound" (pressure waves) tries to move faster than light to support the star's weight, the star collapses. This sets a second, stricter weight limit.

2. The New Rules: The "Hořava-Lifshitz" Playground

The author, Edwin J. Son, asks: What happens if we use the new HL gravity blueprints instead of Einstein's?

He finds that in this new universe, the rules change in a fascinating way:

• Heavier Stars are Possible: Just like a stronger building material allows for taller skyscrapers, HL gravity allows neutron stars to be much heavier than Einstein's theory predicted, without collapsing.
• The "Speed Limit" Changes: The limit on how fast sound can travel inside the star changes. In the new theory, stars can support more weight before the "sound" hits the cosmic speed limit.

3. The Big Discovery: The "Three Roads" Meet

The most exciting part of the paper is a visual discovery. The author draws three lines on a graph:

1. The Horizon Line: The point where a star becomes a black hole (the event horizon).
2. The Weight Limit Line (Uniform Density): The heaviest a star can be before collapsing.
3. The Sound Speed Line: The limit based on how fast sound travels inside.

In Einstein's universe, these lines are far apart. There is a big gap between the heaviest possible star and the smallest possible black hole.

In Hořava-Lifshitz gravity, something magical happens: As the stars get smaller and denser (approaching the size of a "minimal" black hole), all three lines converge and meet at a single point.

The Analogy: Imagine three roads running parallel to each other. In Einstein's world, they stay separate forever. In this new theory, as you drive toward a specific destination (the smallest possible black hole), the roads curve inward and merge into one. This means that in this new theory, the heaviest possible star and the lightest possible black hole are almost the same thing. The "gap" between them disappears.

4. Why Does This Matter?

• Solving the Mystery: We have observed neutron stars that are too heavy for Einstein's rules. This paper suggests that if HL gravity is correct, these heavy stars aren't breaking the rules; they are just following a different, more flexible set of rules.
• The "Gap" Filler: There is a mysterious "mass gap" in the universe where we don't see many objects (between the heaviest neutron stars and the lightest black holes). This paper suggests that in HL gravity, objects can exist right in that gap, or even cross over from being a star to a black hole more smoothly.
• The Minimal Black Hole: The theory predicts a "minimal" black hole (a tiny black hole) that acts as a boundary. Once a star gets heavy enough, it doesn't just collapse; it transforms into this minimal black hole, and the math shows that the star and the black hole become indistinguishable at that point.

Summary

Think of General Relativity as a rigid, old fence that says, "You cannot build a star heavier than X." This paper suggests that the universe might actually have a more flexible fence (Hořava-Lifshitz gravity). With this new fence, we can build heavier stars, and the line between a "super-heavy star" and a "tiny black hole" blurs until they meet. It's like realizing that the wall between two rooms isn't a solid barrier, but a door that opens when you get heavy enough.

This doesn't mean Einstein was "wrong," but rather that his rules might be an approximation of a deeper, more complex reality that allows for these massive, exotic objects we are now starting to see.

Technical Summary

Here is a detailed technical summary of the paper "Limits on the mass of compact objects in Hořava-Lifshitz gravity" by Edwin J. Son.

1. Problem Statement

In General Relativity (GR), the mass of compact objects (like neutron stars) is constrained by theoretical upper limits:

• The Buchdahl Limit (Uniform Density Limit - UDL): For an object with uniform density, the compactness $C \equiv G_N M / c^2 R$ is bounded by $4/9$.
• The Causal Limit (Sound Speed Limit - SSL): A stricter bound derived from the requirement that the speed of sound inside the object must not exceed the speed of light ($c_s \leq c$), typically limiting neutron star masses to $\approx 3 M_\odot$.

Recent observations of neutron stars with masses $\gtrsim 2 M_\odot$ challenge standard GR models, even with stiff equations of state (EOS). Modified gravity theories, such as Hořava-Lifshitz (HL) gravity, offer potential explanations. HL gravity is a UV-complete theory of gravity that introduces anisotropic scaling between time and space. Previous work suggested HL gravity allows for much heavier compact objects than GR. However, the specific theoretical mass limits (UDL and SSL) within the deformed HL framework had not been explicitly derived. This paper aims to derive these limits for the Kehagias-Sfetsos (KS) vacuum solution in HL gravity.

2. Methodology

The author employs a combination of analytical derivation and numerical simulation:

• Theoretical Framework: The study utilizes the deformed HL gravity action with $z=3$ scaling and a softly broken detailed balance condition. The KS vacuum solution (asymptotically flat) is used as the background metric.
• Derivation of TOV Equations: The author derives the explicit form of the Tolman-Oppenheimer-Volkoff (TOV) equations for HL gravity. These equations govern the hydrostatic equilibrium of a spherically symmetric star in this modified gravity.
  • The equations involve a deformation parameter $q$ (related to the coupling constant $\omega$), which vanishes in the GR limit ($q \to 0$).
  • The metric function $f(r)$ is redefined in terms of a mass function $m(r)$ to facilitate solving the system.
• Analytical Derivation of UDL:
  • Assuming a uniform density $\rho_0$ and positive pressure, the TOV equations are solved analytically.
  • The condition for finite pressure everywhere inside the star is used to derive an upper bound on the mass $M$ as a function of radius $R$ and parameter $q$.
• Numerical Simulation for Arbitrary EOS:
  • To test the universality of the UDL, the TOV equations are solved numerically using the 4th-order Runge-Kutta method (SciPy).
  • Simulations cover a wide range of central densities and arbitrary monotonic EOSs with positive pressure.
• Derivation of SSL:
  • To find the causal limit, the author assumes a "luminal" EOS (speed of sound equals the speed of light) for densities above a fiducial value $\rho_u$.
  • The TOV equations are solved numerically using an 8th-order Runge-Kutta method to maximize the mass of the compact object.
  • The results are fitted to series expansions for two regimes: near the minimal black hole ($R \sim q$) and far from it ($R \gg q$).

3. Key Contributions and Results

A. The Uniform Density Limit (UDL) in HL Gravity

• Analytical Bound: The paper derives a new upper bound for the mass of a uniform density star in KS vacuum. Unlike the constant $4/9$in GR, the bound in HL depends on the ratio$R/q$.
• Convergence to Extremal Black Hole: A critical finding is that as the radius $R$ approaches the minimal black hole radius $R_c = q$ (where the mass $M = q$), the UDL curve converges with the event horizon curve.
  • In the limit $R \to q$, the compactness $M/R$ approaches 1 (specifically, the object becomes an extremal black hole).
  • This contrasts with GR, where the Buchdahl limit ($M/R = 4/9$) is strictly below the Schwarzschild horizon ($M/R = 1/2$).
• Universal Limit: Numerical simulations with arbitrary EOSs confirm that the UDL derived for uniform density acts as a universal upper bound for any compact object with positive pressure and monotonic density in HL gravity. Objects exceeding this limit collapse into black holes (hidden behind the horizon).

B. The Sound Speed Limit (SSL) in HL Gravity

• Mass Enhancement: The SSL in HL gravity allows for significantly higher maximum masses compared to GR, particularly when the deformation parameter $q$ is large (e.g., $q \sim 4$ km).
• Dependence on Fiducial Density: The maximum mass $\mu_{HL}$ is a function of the fiducial density $\rho_u$.
  • For low $\rho_u$, $\mu_{HL} \approx \mu_{GR} \approx 3 M_\odot$.
  • For high $\rho_u$ (or large $q$), $\mu_{HL}$ grows exponentially. For instance, with $q \sim 5$ km and $\rho_u \gtrsim 10^{15}$ g/cm³, the limit exceeds $5 M_\odot$.
• Convergence: Similar to the UDL, the SSL curve converges with the horizon curve at the minimal black hole point ($M=q, R=q$).
• Observational Relevance: The results suggest that HL gravity can accommodate neutron stars with masses in the "lower mass gap" (between the heaviest neutron stars and lightest black holes, e.g., $\sim 4 M_\odot$), potentially explaining candidates like GRO J0422+32.

C. Series Expansions

The paper provides series expansions for the limits near the minimal black hole ($R \sim q$) and far away ($R \gg q$):

• Near $R \sim q$: Both UDL and SSL approach the horizon line $M=R$ (in units where $q=1$), allowing compactness $M/R < 1$.
• Far from $R \gg q$: The limits reduce to the standard GR Buchdahl limit ($4/9$) and the causal limit, with corrections proportional to$(q/R)^2$.

4. Significance and Discussion

• Resolution of Mass Limits: The study demonstrates that the strict mass limits of GR are relaxed in HL gravity. The convergence of the UDL and SSL to the extremal black hole horizon implies that compact objects in HL can be significantly more compact ($M/R \to 1$) than in GR ($M/R \leq 4/9$).
• Implications for Neutron Stars: The theory provides a natural mechanism for the existence of massive neutron stars ($> 3 M_\odot$) without requiring exotic, unphysical equations of state.
• Caveats: The author notes a limitation regarding the SSL: HL gravity violates Lorentz symmetry, meaning the speed of light is not a universal constant in the same way as in GR. Therefore, the definition of "causal" (sound speed $\leq$ light speed) requires further investigation within the specific context of HL's preferred foliation. However, the geometric convergence of the limits to the horizon remains a robust feature of the theory.
• Conclusion: The paper establishes that in Hořava-Lifshitz gravity, the theoretical upper bounds on compact object masses are not fixed constants but depend on the deformation parameter $q$. Crucially, these bounds allow for objects to approach the properties of extremal black holes, effectively closing the "compactness gap" between neutron stars and black holes observed in GR.