Quantum Vacuum at Extreme Curvature — α LGQV
Volume II of the Research Programme: Local Gravity of Quantum Vacuum
Boris Kriger, Institute of Integrative and Interdisciplinary Research Department of Cosmology and Theoretical Physics boriskriger@interdisciplinary-institute.org ORCID: 0009-0001-0034-2903
Overview
This is Volume II of the research programme Local Gravity of Quantum Vacuum (α LGQV). Volume I (A Dark-Sector-Free Cosmology) established that the cosmological constant Λ and the quantum vacuum energy ρ_vac are physically distinct, derived the vacuum–matter coupling α = 0.005 from QCD sigma terms with zero free parameters, and showed that the gravitating vacuum reproduces the principal observations attributed to dark matter and dark energy — from flat rotation curves to the cosmic web — without new particles or new forces.
Volume II extends the programme to extreme densities and curvatures. The central question: what happens to the gravitating vacuum where general relativity predicts singularities — inside collapsed objects and at the origin of the universe?
The central result: singularities are artifacts of extrapolating classical general relativity beyond its domain of validity. The same elastic limit of spacetime that produces inflation at cosmological scales produces finite-density vacuum cores at astrophysical scales. Collapsed objects are not holes in spacetime but complex physical systems with rich internal structure, accessible through gravitational-wave spectroscopy.
Building on Volume I
Volume II presupposes the results of Volume I (Papers #1–#17). The key results carried forward are:
R1. Λ and ρ_vac are physically distinct. Five independent formalisms support the separation: trace-free Einstein gravity, Kaloper–Padilla sequestering, Volovik’s condensed-matter analogy, Solà Peracaula’s running vacuum, and Brown’s QKDE framework. [Papers #1, #1a]
R2. The vacuum energy depends on local matter density: ρ_vac(ρ_m) = Λ₀ − αρ_m. Derived from one-loop QFT in curved spacetime via the Schwinger–DeWitt expansion. [Papers #2, #3]
R3. The coupling α is derived from the QCD sigma terms of the nucleon: α_bare = (σ_π + σ_s)/m_N ≈ 0.096, reduced by the baryon fraction (×0.156) and nonlinear screening (÷3), giving α = 0.005. The observed value from fσ₈ data is 0.003. Agreement to a factor of 1.7 with zero free parameters. [Paper #3a]
R4. Nonlinear self-screening: N-body simulations show that G_eff = G(1 + 2α) confines itself to the shrinking overdense volume fraction as structure grows. The σ₈ enhancement is ~3× smaller than linear theory predicts. This simultaneously resolves the JWST early galaxy problem (enhanced early growth) and the S₈ tension (moderate late-time σ₈) — same α, same physics, different epochs. [Papers #8, #9]
R5. The Starobinsky potential V(ψ) = V₀(1 − e^{−√(2/3)ψ/M_Pl})² is derived — not postulated — from the R + R² gravitational action as the elastic potential of spacetime. Inflation is the elastic recoil of maximally deformed spacetime, not a separate mechanism. [Paper #6]
R6. Inside gravitationally bound systems, vacuum energy is trapped and its gravity is uncompensated. The isothermal vacuum profile reproduces flat rotation curves, the baryonic Tully–Fisher relation (M_b ∝ v⁴), the radial acceleration relation with a₀ ~ √(Λ₀G) ~ 10⁻¹⁰ m/s² (derived, not fitted), the Bullet Cluster morphology, and galaxies without dark mass (NGC 1052-DF2, DF4). [Paper #12]
R7. The cosmic web is the unique fixed point of the matter–vacuum–metric cycle, proved via the Brouwer and Banach theorems. Its topology is determined by α and cosmological parameters alone, independent of primordial initial conditions. [Paper #13]
R8. The entire programme uses only established physics — QCD, general relativity, E = mc², and the Casimir effect. No new particles. No new forces. No fine-tuning. One parameter derived from nuclear physics. [Papers #1–#17]
The Questions of Volume II
Twelve questions, arising directly from the results of Volume I, define the territory of Volume II:
Q1. What replaces singularities? If the R² correction limits curvature and the chiral phase transition creates pressure at supranuclear densities, what is the concrete internal structure of collapsed objects?
Q2. How does the ansatz ρ_vac(ρ_m) behave at extreme densities? The linear ansatz loses validity at ρ_m > Λ₀/α. What is the nonlinear generalization?
Q3. Does a third step of the Pauli ladder exist? Degenerate electrons stabilize white dwarfs; degenerate neutrons stabilize neutron stars; does deconfined vacuum pressure stabilize the next step?
Q4. What is the role of the 2000-fold vacuum enhancement at z ~ 12 in the formation of supermassive black hole seeds? What are the characteristic masses of the vacuum-assisted direct collapse channel?
Q5. Why is the mass range 300–10⁴ M☉ between stellar and supermassive black holes nearly empty?
Q6. Is the event horizon a physical surface or an observational artifact? What are the consequences for defining collapsed objects?
Q7. How do gravitational waves transmit information about internal structure? What are the concrete predictions from quasi-normal modes, echoes, and tidal Love numbers?
Q8. What is the fate of magnetic fields during collapse? Superconductivity, colour confinement, toroidal configuration — what is the magnetic architecture of the vacuum core?
Q9. Does time flow inside a collapsed object? What are the two timescales, and what are their consequences for thermodynamics, entropy, and information?
Q10. Why are wormholes impossible in the framework of the gravitating vacuum?
Q11. What is the nature of the “pole” of spacetime? Not a singularity, not a bounce, not a “before” — what?
Q12. The mass hierarchy of 10⁵⁵ between inflationary and late-time scales — can it be narrowed?
The Papers
Method
Paper #18 — On Method, the Unity of the Programme, and the Questions of Volume II
The conventional publication system requires each paper to be a self-contained unit validated by journal peer review. This creates a structural problem for a programme of thirty papers where each builds on all previous. A parallel system exists: preprint servers, direct communication, and peer review for improvement rather than gatekeeping. This programme implements transparent peer review: every paper documents all objections, responses, and revisions.
The two volumes constitute a single unified work. Numbering is continuous: Volume I (Papers #1–#17 plus companions), Volume II (Papers #18–#32). Each paper is self-contained in exposition — not in proofs — and includes a cumulative results table, a brief method statement, counterarguments and responses, limitations, and a full review history.
This paper presents the complete results of Volume I, the twelve questions of Volume II, and the structure of the fifteen papers that address them.
The Nonlinear Regime
Paper #19 — The Nonlinear Vacuum: Beyond the Linear Ansatz at Extreme Densities
The linear ansatz ρ_vac(ρ_m) = Λ₀ − αρ_m is the leading-order Taylor expansion, valid when ρ_m ≪ ρ_Planck. At extreme densities it must be extended. Higher-order terms in the Schwinger–DeWitt expansion (the a₂ coefficient: R², R_μν R^μν) become significant. The chiral phase transition at ρ ~ 3–5 ρ_nuclear marks the boundary where the QCD vacuum restructures and the linear approximation fails qualitatively.
At z ~ 12, the critical density ρ_crit(z) = Λ₀(z)/α is 2000× higher than today, extending the domain of validity of vacuum physics deep into the collapse process — a consequence of the (1+z)³ scaling established in Volume I.
The Third Step
Paper #20 — The Third Step: Vacuum Pressure as the Stabilizer of Gravitational Collapse
A self-similar pattern operates at successive density scales. Degenerate electron gas creates pressure that stabilizes white dwarfs up to the Chandrasekhar limit (~1.4 M☉). Degenerate neutron gas creates pressure that stabilizes neutron stars up to the TOV limit (~2.2 M☉). At supranuclear densities, quark deconfinement releases the chiral condensate energy ~(250 MeV)⁴, creating repulsive vacuum pressure with equation of state w = −1. The colour-flavour-locked (CFL) superconducting phase provides additional stabilization.
The absence of a third step in standard general relativity is not a physical result but a deficit of theory. Every previous infinity in physics — the ultraviolet catastrophe, the electron self-energy, weak-interaction divergences — signalled new physics, not real infinity. The singularity signals the same thing.
Internal Structure
Paper #21 — The Internal Structure of Collapsed Rotating Objects
All astrophysical collapsed objects rotate (conservation of angular momentum makes non-rotating collapse impossible). In the Kerr solution, the singularity is a ring, not a point — already in classical general relativity, internal structure is unavoidable. The inner (Cauchy) horizon is unstable (Poisson–Israel mass inflation). The region beyond the ring (negative mass, closed timelike curves) is causally pathological.
In the gravitating vacuum framework, the ring singularity is replaced by a toroidal vacuum core of finite density and curvature. The inner horizon instability is regularized by absorption in the core. The pathological region beyond the ring does not exist — spacetime terminates on the physical core. The layered structure from inside out: vacuum toroid → chiral transition shell → neutron zone → ergosphere → vacuum atmosphere.
Magnetic Fields
Paper #22 — Magnetic Architecture of the Vacuum Core
Magnetic flux is conserved during collapse (Alfvén’s theorem) as long as matter remains conducting. It remains conducting — and more: at nuclear densities, protonic superconductivity (type II) quantizes the field into vortices. At deconfinement, CFL colour superconductivity expels the field via the Meissner effect, concentrating it on the toroid surface at B ~ 10¹⁴–10¹⁶ G.
The magnetic field is not a relic but a structural element. It stabilizes the toroid against MHD instabilities (sausage and kink modes) — the same physics that confines plasma in tokamaks. The collapsed object is a natural tokamak. The Grad–Shafranov equation in general relativity governs the equilibrium. Magnetic catalysis (Gusynin, Miransky, Shovkovy, 1994) shifts the chiral boundary in strong fields, creating a self-consistent feedback loop between magnetic structure and vacuum phase transition.
Time
Paper #23 — Time Inside: The Two Clocks of a Collapsed Object
Time does not stop at the horizon — that is a coordinate artifact. Time does not end at the singularity — there is no singularity. Proper time flows continuously inside the vacuum core, slowed by gravitational redshift but never halted: dτ/dt_∞ = √(1 − r_S/R) ≪ 1, but > 0.
Two timescales govern the physics: external time t_∞ (the observer’s clock, on which accretion, orbital dynamics, and mergers occur) and internal time τ (the core’s clock, on which thermodynamic relaxation, MHD evolution, and quantum fluctuations occur). Thermodynamic equilibrium is achieved in internal milliseconds. The magnetic configuration is frozen by superconductivity. The object grows layer by layer under accretion, each shell “freezing” at the surface as seen from outside.
The Soviet term “frozen star” (zastyvshaya zvezda) is physically more accurate than “black hole”: the object is not a hole but a star whose internal evolution is frozen relative to the external observer — but not stopped.
The Horizon
Paper #24 — The Event Horizon Reconsidered: An Observer’s Property, Not an Object’s
The event horizon is defined through null geodesics reaching a distant observer. It is a statement about observation, not about the object. A local observer crossing the horizon notices nothing — no surface, no barrier, no signal. The metric is smooth. For a supermassive black hole, tidal forces at the horizon are negligible.
A collapsed object should be defined by its internal physics (equation of state, density, pressure), not by whether light can escape from its surface. Three possibilities — vacuum star without horizon, regular black hole with horizon but no singularity, classical black hole with singularity — are distinguished by internal physics. The no-hair theorem applies only to vacuum solutions; objects with vacuum cores have “hair” (additional multipole moments determined by internal structure).
This is the Copernican shift of Volume II: from observer-centric definitions (what we cannot see) to object-centric definitions (what the object is).
Gravitational Waves
Paper #25 — Gravitational Waves from the Interior: Three Channels of Information
Gravitational waves are perturbations of spacetime itself, not fields propagating within spacetime. This fundamental asymmetry changes their relationship with the horizon.
Channel 1: Quasi-normal modes. The QNM spectrum encodes the full internal structure through boundary conditions, as a bell’s sound encodes its shape and material. For classical Kerr black holes, QNMs depend only on M and J (no-hair theorem). For objects with vacuum cores, additional overtones appear.
Channel 2: Gravitational echoes. Reflections from the internal structure (chiral phase transition surface, CFL boundary) produce delayed signals after the primary merger waveform. Time delay Δt ~ (r_S/c) × ln(R/r_S − 1) ~ milliseconds for stellar masses — within LIGO sensitivity.
Channel 3: Tidal Love numbers. For classical black holes in GR, tidal Love numbers are identically zero — the object is perfectly rigid. For vacuum stars, Love numbers are finite, determined by the core equation of state. Already measurable: LIGO measured neutron star Love numbers in GW170817. The same technique applied to objects above the TOV limit tests for internal structure.
Concrete predictions for LIGO/Virgo/KAGRA, LISA, Einstein Telescope, and Cosmic Explorer.
Supermassive Seeds
Paper #26 — Vacuum-Assisted Direct Collapse and the Formation of Supermassive Seeds at z > 10
At z ~ 12, vacuum energy ρ_vac(z) ≈ 2000Λ₀ creates external pressure on protogalactic clouds. This pressure suppresses fragmentation (which would produce small stars) and enables monolithic collapse into massive seeds — the vacuum-enhanced DCBH pathway (Scenario C of Paper #11, Volume I).
The minimum mass for vacuum-assisted collapse M_vac(z) scales steeply: M_vac ∝ (1+z)^{−9/2}. At z = 12: M_vac ~ 10⁴–10⁵ M☉. At z = 5: M_vac ~ 10⁷ M☉ — no suitable clouds remain. The window for seed formation is z ~ 10–15, after which the mechanism self-terminates as vacuum dilutes with expansion.
All SMBH seeds formed in this narrow window. Subsequent growth is by accretion only. Early SMBHs carry the imprint of the 2000× vacuum in their internal structure — potentially different from late-forming stellar-mass black holes.
The Mass Desert
Paper #27 — The Mass Desert: Why Intermediate-Mass Black Holes Are Rare
The observed mass function of black holes shows a striking gap: stellar black holes (3–100 M☉) are detected in dozens by LIGO/Virgo; supermassive black holes (10⁶–10¹⁰ M☉) inhabit the centres of virtually all massive galaxies; between them — four orders of magnitude nearly empty.
The gravitating vacuum framework explains this naturally. Two formation channels exist with non-overlapping mass ranges. Channel 1 (stellar collapse) has an upper ceiling ~100–300 M☉, set by pair-instability physics. Channel 2 (vacuum-assisted direct collapse) has a lower threshold ~10⁴ M☉, set by the minimum cloud mass for which vacuum pressure can suppress fragmentation. Between 300 and 10⁴ M☉ — no efficient formation mechanism.
Sequential mergers cannot bridge the gap: gravitational recoil (100–4000 km/s) ejects merger products from host clusters (escape velocity ~50 km/s). The mass desert is a prediction, not a puzzle.
Against Wormholes
Paper #28 — Against Wormholes: Eight Arguments from the Gravitating Vacuum
Eight independent arguments, each sufficient alone:
(1) No singularity → no analytic continuation → no Einstein–Rosen bridge. (2) The toroidal vacuum core physically fills the would-be throat. (3) The null energy condition is satisfied (ρ + p = Kφ̇² ≥ 0 in QKDE) → exotic matter is forbidden → stabilization is impossible. (4) The elastic vacuum actively closes topological deformations (flatness feedback). (5) The unique fixed point of Φ_cycle (Paper #13) fixes the topology → alternative topologies are not solutions. (6) Baryon number conservation → no “other side.” (7) Causality is a structural property of the fixed point → closed timelike curves are excluded. (8) Vacuum energy fills the interior → the throat is not empty.
Wormholes are artifacts of idealization: vacuum solutions, point/ring singularities, analytic continuation through regions where the metric is undefined. In a universe with gravitating vacuum, none of these idealizations holds.
The Pole
Paper #29 — The Pole of Spacetime: No Singularity, No “Before,” No Bounce
Hawking’s analogy: “What is north of the North Pole?” is meaningless — not because we lack knowledge, but because the coordinate “north” is exhausted. The gravitating vacuum gives this analogy concrete physical content.
At the state of maximum deformation (R ~ M², the plateau of the Starobinsky potential), spacetime approaches de Sitter geometry asymptotically. The density is finite: ρ ~ V₀ ~ M²M²_Pl ~ 10⁷³ kg/m³ (large, but 23 orders below Planckian). The curvature is finite. The temperature is finite. Time does not begin — the time coordinate reaches its pole, as latitude reaches 90° at the North Pole.
“Before” has no meaning — not because time was created, but because the coordinate is exhausted. There is no bounce (which would require a “before” phase of contraction). There is no singularity (the elastic limit prevents infinite curvature). Inflation is not an epoch that “begins” and “ends” — it is the property of the pole: maximally deformed spacetime recoiling. The further from the pole, the weaker the recoil, the slower the expansion, the more structure emerges.
Synthesis
Paper #30 — Three Myths Dissolved: Time Does Not Stop, Singularities Do Not Exist, “Before” Has No Meaning
Three persistent misconceptions of standard theory are dissolved by a single mechanism — the elastic limit of spacetime, realized through the R² correction to the gravitational action:
Myth 1: Time stops at the event horizon. → Time slows but does not stop. The horizon is a coordinate artifact. Inside — physics continues.
Myth 2: Singularities are real features of spacetime where physics ends. → Singularities are artifacts of extrapolation. The Pauli ladder continues: vacuum pressure at the third step halts collapse at finite density, finite curvature, finite temperature.
Myth 3: The question “what was before the Big Bang” is meaningful (or meaningfully unanswerable). → “Before” does not exist. The universe is a four-dimensional manifold with a pole, not an edge. The pole is the state of maximum elastic deformation, smooth and finite.
One principle — three consequences. The same physics at astrophysical scales (vacuum cores) and cosmological scales (the pole).
The Open Problem
Paper #31 — The Mass Hierarchy Problem: From 10¹³ GeV to 10⁻³³ eV
The single open problem of the programme. The inflationary scalaron mass M ~ 10¹³ GeV (fixed by the CMB power spectrum amplitude) and the late-time screening field mass m_ψ ~ H₀ ~ 10⁻³³ eV differ by a factor of 10⁵⁵. Interpretive unification is achieved: both are manifestations of the same elastic principle at different deformation scales. Dynamical unification — a single mechanism operating continuously across 55 orders of magnitude — is not achieved.
This paper examines renormalization group running, condensed-matter analogies (acoustic vs. seismic modes in a solid differ by orders of magnitude, not 10⁵⁵), and possible pathways including asymptotic safety and cosmological evolution of the effective mass. An honest statement of what remains unsolved.
Observational Programme
Paper #32 — Observational Signatures: A Complete Falsification Programme for Volume II
All predictions of Volume II, collected with specific numbers, thresholds, and timelines:
LIGO/Virgo/KAGRA (current and future runs): Gravitational echoes with Δt ~ milliseconds after stellar-mass mergers. Tidal Love numbers for objects with M > 3 M☉ (zero for classical black holes, finite for vacuum stars). QNM overtones from vacuum core structure.
LISA (launch ~2035): EMRI spectroscopy — multipole moments Q₂ ≠ −Ma² indicating internal structure. Mergers of SMBH seeds at z > 10 with masses ~10⁴–10⁵ M☉. Absence of mergers in the 300–10⁴ M☉ range at any redshift.
Einstein Telescope and Cosmic Explorer: MHD overtones from magnetic vacuum cores in high-magnetization mergers. Post-merger signal structure distinguishing vacuum stars from classical black holes.
EHT next generation (space baseline): Fine structure of the shadow from internal geometry — additional photon rings beyond the Kerr prediction.
JWST: SMBH seeds with masses ~10⁵ M☉ at z > 12. Absence of seeds with masses ~10³ M☉ at any redshift (would contradict the vacuum-assisted formation channel).
CMB-S4 and LiteBIRD: Spatial curvature Ω_K > 0 (small but positive, consistent with the pole geometry). Low-ℓ power deficit from finite inflation duration. B-mode spectrum characteristics.
eROSITA and Athena: IMBH number density ≤ 10⁻³ Mpc⁻³ (orders of magnitude below hierarchical merger predictions).
What would falsify the programme: (1) Detection of a dark matter particle. (2) Demonstration that Λ = 8πGρ_vac from a fundamental principle. (3) α excluded in the range 0.001–0.01 at >5σ by joint Planck+DESI+LSST analysis. (4) Rotation curves measured smoothly beyond the predicted capture radius r_c with no decline. (5) No correlation of M_dark/M_baryon with distance from group centre for satellite galaxies. (6) Tidal Love numbers measured to be exactly zero for objects above the TOV limit.
How to Read Volume II
For the core physical argument: Paper #20 (the third step), Paper #21 (internal structure), Paper #25 (gravitational wave channels).
For the formation and demographics of black holes: Papers #26 and #27.
For the nature of time and spacetime: Papers #23 and #29.
For the complete falsification programme: Paper #32.
Volume II builds directly on Volume I. Readers unfamiliar with Volume I should begin with Paper #18, which provides a complete summary of all prior results.
Links
Volume I: A Dark-Sector-Free Cosmology
Volume II: This page.
Complete programme page: Local Gravity of Quantum Vacuum — α LGQV
All papers are available as preprints with documented peer review history.
Correspondence: boriskriger@interdisciplinary-institute.org ORCID: 0009-0001-0034-2903
Institute of Integrative and Interdisciplinary Research 