Consistent Extension (BPS)
Blas, Pujolàs, and Sibiryakov 2009: a non-projectable extension of Hořava gravity that addresses scalar graviton pathologies. The 'healthy extension' colloquially named in the 2011 follow-up.
Placeholder for a 3D visualisation of Hořava-Lifshitz Gravity. The interactive scene will land in Phase 3. Hořava proposed in 2009 that gravity becomes power-counting renormalizable in the ultraviolet if the symmetry between space and time is broken at high energies. The theory has anisotropic scaling: time scales as one power and space scales as three powers under a renormalization group transformation. The cost is that full Lorentz invariance, foundational to general relativity and the Standard Model, becomes an emergent low-energy property rather than a fundamental symmetry. The 2009 proposal generated significant initial enthusiasm followed by sustained technical challenges; modern variants (consistent extension, foundational analyses) address known pathologies with mixed success. The program is in a frontier-niche standing today.
§1 · The claim, in one sentence
Blas, Pujolàs, and Sibiryakov proposed in 2009 a non-projectable extension of Hořava-Lifshitz gravity that addresses the scalar-graviton pathologies of the original formulation. By allowing the lapse function to depend on space as well as time, the theory gains an additional symmetry-breaking pattern that decouples the problematic scalar mode at low energies. The 2009 paper, *Consistent Extension of Horava Gravity*, is the most-cited Hořava extension at roughly 628 INSPIRE citations; the colloquial 'healthy extension' name originates in their 2011 follow-up *Models of non-relativistic quantum gravity: The Good, the bad and the healthy*.
§2 · Why it might be true
The original Hořava 2009 framework imposed projectability, the requirement that the lapse function (the time component of the metric) depend only on time, not on space. This simplified the action and the symmetry analysis but produced the strong-coupling problem and the scalar-graviton problem in the IR. BPS 2009 dropped projectability and introduced additional terms involving the spatial gradient of the lapse, making the theory non-projectable.
The technical effect is that the scalar graviton mode acquires a mass term in the low-energy limit, decoupling from the standard tensor modes. The theory recovers something close to general relativity in the appropriate limit without the IR pathologies of the original. The Lorentz-violating structure remains at high energies; the theory remains power-counting renormalizable.
Subsequent BPS work (the 2011 *Good, the bad and the healthy* paper at arXiv:1007.3503, 384 citations) systematically classified the possible non-projectable extensions and identified the 'healthy' subset that is phenomenologically viable. The 2009-2011 BPS framework is the modern technical foundation for Hořava-Lifshitz phenomenology and is the basis for most contemporary cosmological work in the theory.
The family stance
Spacetime is fundamentally not Lorentz-invariant. At ultra-high energies, time and space scale differently. Full Lorentz symmetry emerges at low energies but is not part of the deep description. The theory becomes power-counting renormalizable as a consequence.
§2.5 · Evidence
- The BPS construction explicitly addresses the original Hořava formulation's most severe pathology (the scalar graviton problem); this is a substantive technical achievement
- Subsequent independent work (Mukohyama, Sotiriou, Visser, Weinfurtner) has verified and extended the BPS framework, indicating community engagement
- The healthy-extension version remains compatible with current solar-system tests of gravity and with constraints from gravitational-wave observations
- The 2011 *Good, the bad and the healthy* paper provides a systematic classification that helps distinguish viable from pathological extensions
§3 · What you'd need to test it
- Non-projectable Hořava gravity has a scalar graviton mode with a mass that depends on the parameters of the extension; the mode decouples from low-energy physics
- The theory recovers something close to general relativity in the infrared limit, modulo Lorentz-violating corrections that vanish at sufficiently low energies
- Modified gravitational-wave dispersion relations arise from the Lorentz-violating structure and are in principle testable by binary-inspiral gravitational-wave observations
- Specific cosmological signatures arise in early-universe physics (modifications to the primordial perturbation spectrum that depend on the parameters of the extension)
§4 · Where it breaks
- The non-projectable extension introduces additional free parameters; the theory's predictive power is correspondingly reduced
- Lorentz violation remains the foundational feature, and the empirical constraints on Lorentz violation from precision experiments continue to push the breaking scale higher
- The 2011 'healthy' classification is technical and the boundary between healthy and unhealthy extensions depends on parameter choices that may not have natural priors
- The framework still does not produce a complete UV completion of gravity; it addresses the IR pathologies but the high-energy physics remains the same as the original Hořava theory
Go deeper
The technical structure of the BPS extension involves additional terms in the action of the form a_i a^i, where a_i is the spatial gradient of the logarithm of the lapse. These terms break the projectability assumption and introduce a new spatial-derivative structure that interacts with the scalar mode.
The 'Good, the bad and the healthy' nomenclature in the 2011 paper refers to three classes of non-projectable extensions, depending on the signs and magnitudes of the new coupling constants. The 'healthy' class is the phenomenologically viable subset; the 'bad' class has IR instabilities; the 'good' class is intermediate. The classification is a useful organizing principle for the modern Hořava-Lifshitz literature.
Cross-references: the sibling Original Hořava Formulation variant covers the 2009 framework that BPS extends. The Foundational Analysis variant by Sotiriou, Visser, and Weinfurtner covers the systematic theoretical analyses that established the framework's constraints and bounds. The Mukohyama cosmological work (cited in goDeeper of the Foundational Analysis variant, not given its own variant slot in this PR) explored the implications for inflation and primordial perturbations.
Variants in this family
▸§5 · Who built it, and when(2 sources, 2 established)
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