The black hole and FRW geometries of non-relativistic gravity (0905.0477v1)

Abstract: We consider the recently proposed non-relativistic Ho\v{r}ava-Lifshitz four-dimensional theory of gravity. We study a particular limit of the theory which admits flat Minkowski vacuum and we discuss thoroughly the quadratic fluctuations around it. We find that there are two propagating polarizations of the metric. We then explicitly construct a spherically symmetric, asymptotically flat, black hole solution that represents the analog of the Schwarzschild solution of GR. We show that this theory has the same Newtonian and post-Newtonian limits as GR and thus, it passes the classical tests. We also consider homogeneous and isotropic cosmological solutions and we show that although the equations are identical with GR cosmology, the couplings are constrained by the observed primordial abundance of ${}^4{\rm He}$.

• The paper demonstrates a non-relativistic gravity model with Lifshitz scaling where quadratic fluctuations reveal modified graviton propagation and stability conditions.
• It constructs a static, spherically symmetric black hole solution that reduces to the Schwarzschild metric at large distances, validating classical gravitational tests.
• The study derives FRW cosmological solutions with constraints on the coupling parameter λ, aligning its predictions with Big Bang nucleosynthesis observations.

Non-Relativistic Gravity: Black Hole and FRW Geometries

The exploration of gravity through non-relativistic frameworks such as Hořava-Lifshitz gravity presents a compelling alternative to the standard relativistic formulations established by General Relativity (GR). This paper by Kehagias and Sfetsos addresses a four-dimensional model of gravity that eschews the full diffeomorphism invariance of GR, instead embracing Lifshitz scaling characterized by spatial and temporal anisotropy, specifically with the scaling exponent $z=3$.

Core Contributions

The authors examine a specific limit of this theory, which is designed to accommodate a Minkowski vacuum in the infrared (IR) domain. By modifying the theory with an operator related to the Ricci scalar of a three-dimensional geometry, they investigate the consequences on both fundamental and derived gravitational phenomena.

Quadratic Fluctuations and Gravity Propagation

The paper explores quadratic fluctuations around the Minkowski vacuum, elucidating that the theory supports two distinct propagating polarizations of the graviton, akin to those in GR but manifesting different behaviors due to the modifications in the theory. Notably, while GR gravitons propagate at the speed of light, this non-relativistic model predicts a gravitational wave speed contingent on the running of the coupling parameter $\lambda$. Specifically, for $\frac{1}{3} < \lambda < 1$, a ghost instability arises unless $\lambda$ appropriately converges to the fixed point $\lambda = 1$ in the IR regime.

Black Hole Solutions

A significant result is the construction of a static, spherically symmetric, black hole solution, which is comparable to the Schwarzschild solution in GR. The paper confirms that for large distances, this solution reduces to the Schwarzschild metric, thereby validating the classical tests of gravity through the Newtonian and post-Newtonian approximations. The presence of two event horizons and confined ranges for singularity avoidance highlights the complex dynamics introduced by the altered gravitational equations.

Cosmological Implications

For cosmological interpretations, the paper tackles homogeneous and isotropic solutions modeled by the Friedmann-Robertson-Walker (FRW) geometry. While the resulting Friedmann equation appears similar to that in GR, deviations emerge from the coupling $\lambda$, constraining its range through Big Bang nucleosynthesis constraints to $(0.926, 1.095)$. This constriction has crucial implications for understanding early universe dynamics and potential observational validations or refutations of the theory.

Concluding Remarks and Implications

This research not only reinforces the parallels between certain GR results and the non-relativistic gravity framework but also underscores deviations that could be observable under specific conditions. While further empirical scrutiny is necessary to ascertain the pertinence of such deviations in describing physical reality, the theoretical groundwork constitutes a vital component in the ongoing dialogue between quantum field theories and classical gravity.

As the paper articulates, future investigations could focus on the broader implications of such theories in cosmology and astrophysics, particularly regarding their predictions of dark matter and dark energy, which might elucidate unexplained phenomena within the current standard model of cosmology.

Future work could also focus on the specific experimental setups that can effectively constrain or potentially falsify these theoretical predictions by leveraging gravitational wave observations and alternative astrophysical data. The exploration continues to contribute to the understanding of gravity beyond the confines of relativistic invariance.