“Alternative Cosmologies: Finite-Sphere Kinematic Dynamics and Big-Bang-as-White-Hole Hypotheses”

AbstractI formulated two hypotheses and placed them in a testable, quantitative framework:(A) Finite-Sphere Kinematic Cosmology (FSKC) — the cosmic redshift–distance relation and other “expansion” phenomena arise from large-scale ordered motions of galaxies inside a finite 3-space (e.g., a 3-sphere ) rather than from metric expansion. I construct a covariant model in which a static spatial metric is endowed with a divergence-dominated velocity field on . I derive redshift, distance dualities, surface-brightness scaling, and time-dilation relations, and compare them—term by term—with empirical tests (SNe Ia time dilation, Tolman surface–brightness dimming, BAO standard ruler, and CMB blackbody spectrum). These comparisons isolate the necessary—and highly constraining—conditions under which FSKC could be viable.(B) Big-Bang-as-White-Hole (BBWH) — the observable Universe is the time-reverse of gravitational collapse: a (possibly quantum-stabilized) white-hole–like origin ejects matter/radiation, with the Big Bang identified with the formation/bounce of that white hole. I lay out two concrete realizations: (i) classical white-hole interiors matched to cosmology via Israel junction conditions; and (ii) quantum-gravity “black-to-white” transitions (Loop Quantum Cosmology/Planck-star scenarios). I discuss stability, horizon structure, thermodynamics (Weyl–curvature/low-entropy past), and confront the same four empirical pillars (CMB spectrum, SNe Ia time dilation, BAO/CMB acoustic physics, and Tolman test).For both hypotheses, I provide falsifiable predictions: distinctive signals in redshift drift, cosmic parallax/proper-motion vector harmonics, CMB spectral-distortion amplitudes, BAO phase coherence, and stochastic GW backgrounds. I close with a prioritized observational program.Key observational anchors used below include: COBE/FIRAS CMB spectrum (blackbody to parts in ) and future spectral-distortion sensitivities; Planck 2018 cosmological parameters; SNe Ia time dilation; BAO measurements; and classical white-hole instability results alongside quantum-gravity bounce models. —1. Motivation and Observational StatusThe standard CDM model—an FLRW spacetime with nearly flat spatial sections—jointly fits CMB anisotropies, BAO, SNe distances, and large-scale structure with six base parameters. Planck 2018 measures, e.g., km s^{-1} Mpc^{-1} and . The CMB spectrum is exquisitely Planckian (COBE/FIRAS), strongly constraining any non-thermal photon processing since ; future missions (PIXIE/PRISM) target -type distortions at . SNe Ia exhibit time dilation consistent with the scaling expected in expanding FLRW spacetimes; recent DES analyses extend precision to . BAO provide a comoving standard ruler linking late-time clustering to early-time plasma acoustics. Finally, the Tolman test detects surface-brightness dimming consistent with expansion once luminosity evolution is modeled; static “tired-light” scalings are strongly disfavored. Any alternative must reproduce these pillars—or make crisp, testable departures.—2. Hypothesis A: Finite-Sphere Kinematic Cosmology (FSKC)2.1 Geometry and kinematicsAssume a static spatial metric with positive curvature (topology ), e.g.ds^2 = c^2\,dt^2 – a_0^2\left[d\chi^2+\sin^2\chi\,(d\theta^2+\sin^2\theta\,d\phi^2)\right],Let galaxies be “test particles” moving on this manifold with a global velocity field that is (on large scales) irrotational and radially outward from some distinguished point . To recover isotropy for observers not at , one must postulate either (i) a randomized cellular structure in which local divergence mimics isotropy statistically, or (ii) a symmetric vector-harmonic decomposition on whose dipole/quadrupole moments average to below observed limits. Either way, the field must satisfy a continuity equation and a Poisson-like equation for a gravitational potential such that\nabla\cdot \mathbf{v} \approx 3H_{\rm eff}\qquad\text{(locally),}2.2 Cosmological redshift without expansionSuppose light from a source at comoving geodesic length experiences a cumulative kinematic Doppler plus gravitational redshift1+z=\gamma(1+\mathbf{v}\cdot\hat{\mathbf{n}}/c)\,\exp\!\left[\frac{\Delta\Phi}{c^2}\right]\approx 1 + \frac{\mathbf{v}\cdot\hat{\mathbf{n}}}{c} + \frac{\Delta\Phi}{c^2}+\dotsThe non-trivial content lies in higher-order relations that connect area distances, flux, and time dilation.2.3 Surface brightness and time dilationIn an expanding FLRW universe,S_{\rm obs} = S_{\rm em}\,(1+z)^{-4},In FSKC with a static metric, Liouville’s theorem forces surface brightness to be invariant under pure special-relativistic Doppler unless photon rate or area distance laws are modified. To reproduce , the model must embed either:a gravitational-time-dilation factor that stretches all temporal processes by , anda non-Euclidean angular-diameter relation on that yields the extra beyond flux dilution.Concretely, the model must enforce\frac{dt_{\rm obs}}{dt_{\rm em}}=(1+z),\qquad d_A(z)\propto \frac{1}{1+z}\,f_{S^3}(z),2.4 BAO and CMB spectrumBAO encode an early-plasma sound horizon that leaves a standard ruler in galaxy clustering and imprints the CMB acoustic pattern. Any static-metric/redshift mechanism must still generate a co-moving ruler with the right scale and phase coherence across the CMB/BAO transfer functions. Current BAO measurements and Planck’s acoustic-peak structure strongly prefer an expanding early universe with tightly coupled photon–baryon fluid. The CMB spectrum’s near-perfect blackbody requires early thermalization and severely limits late photon-processing (Compton/IC, photon injection). A non-expanding mechanism must reproduce FIRAS limits , , or predict detectable distortions at PIXIE-class sensitivities—both are highly constraining. 2.5 Summary for FSKCFSKC is not a priori impossible, but it is highly fine-tuned. To be viable it must:1. enforce universal time dilation,2. reproduce Tolman dimming,3. explain the BAO/CMB acoustic ruler and phase, and4. respect COBE/FIRAS spectral limits.Each requirement is met naturally by metric expansion but must be engineered in FSKC.—3. Hypothesis B: Big-Bang-as-White-Hole (BBWH)3.1 Classical white holes and matchingIn maximally extended Schwarzschild (Kruskal) geometries, a white-hole region is the time-reverse of a black hole. One can attempt to model the Big Bang as a white-hole–like past singularity and match a cosmological interior to an exterior via Israel junction conditions, generalizing the Oppenheimer-Snyder construction (FRW interior ↔ Schwarzschild exterior). However, classical white holes are generically unstable: arbitrarily small accretion across the past horizon triggers collapse into a black hole (“death of white holes”). This is a robust result since Eardley (1974) and subsequent analyses. As cosmology, a persistent classical white hole exposed to ambient radiation (e.g., the CMB!) is therefore extremely problematic.3.2 Quantum bounces and “Planck stars”Quantum-gravity models can evade classical instabilities. In Loop Quantum Cosmology (LQC), effective dynamics replace the big-bang singularity with a bounce when energy density reaches a critical value; the spacetime extends to a pre-bounce branch. Related “Planck-star” scenarios for black holes posit a quantum-pressure halt of collapse and a black-to-white transition with enormous external time dilation—suggesting a cosmological analogue: a white-hole-like birth whose instability is mitigated by quantum effects. More speculative approaches consider a brane emerging from the interior of a higher-dimensional black/white hole, or anisotropic white-hole interiors with cosmological perturbations. These yield specific predictions for primordial spectra and horizon structures. 3.3 Thermodynamics and the arrow of timeAny BBWH model must supply the low-entropy initial state required by the arrow of time—often expressed via Penrose’s Weyl curvature hypothesis (suppression of free gravitational degrees of freedom near the initial surface). Bounce/white-hole pictures must either impose or derive low initial Weyl curvature while recovering standard thermodynamics thereafter. 3.4 Observational confrontationsCMB spectrum and anisotropies: BBWH must produce a thermalized plasma and acoustic oscillations with the observed peak sequence; Planck’s precise constraints apply. SNe Ia time dilation & Tolman: These follow from FLRW-like expansion after the white-hole birth/bounce. Thus, BBWH models that reduce to effective FLRW at can satisfy these tests. Instability: Purely classical white holes are disfavored by Eardley-type arguments; quantum-stabilized transitions (LQC/Planck stars) remain viable, but their durations, echoes, or spectral signatures are model-dependent and testable. 3.5 Relation to past-incompletenessThe Borde–Guth–Vilenkin (BGV) theorem shows that spacetimes with positive average expansion are past-incomplete; a boundary (e.g., a bounce/white-hole-like surface) is required. BBWH provides a candidate completion, but says nothing by itself about what physics resolves that boundary—quantum gravity must do the work. —4. Mathematical Sketches4.1 FSKC redshift–distance and reciprocityLet photons follow null geodesics of the static metric while their frequency measured by comoving detectors satisfies\frac{d\ln\nu}{d\lambda}= -\frac{1}{c^2}\,\frac{d\Phi}{d\lambda} – \frac{1}{c}\,\hat{k}\!\cdot\!\frac{d\mathbf{v}}{d\lambda},d_L=(1+z)^2 d_A,4.2 BBWH matching (thin shell)Let be a cosmological interior with metric and an exterior (e.g., Schwarzschild-de Sitter). On the junction hypersurface , Israel conditions impose continuity of the induced 3-metric and relate jumps in extrinsic curvature to a surface stress-energy :\left[K_{ab}-h_{ab}K\right]= -8\pi G\, S_{ab}.—5. Discriminating Predictions and TestsT1. Redshift drift ()FLRW predicts with sign changes tied to . ELT-class spectroscopy can detect cm s^{-1} yr^{-1} drifts over decade baselines. FSKC generically predicts different because the redshift is sourced by velocity/potential fields rather than ; BBWH that reduce to CDM match the standard curve. (Sandage–Loeb test.) T2. Cosmic parallax / proper-motion vector harmonicsLarge-scale, non-Hubble flows on produce E/B-mode patterns in extragalactic proper motions (vector spherical harmonics). Gaia-successor astrometry can bound or detect such patterns; standard FLRW predicts only tiny signals (secular aberration drift and local motions). T3. CMB spectral distortionsAny non-expansion energy exchange (FSKC) or non-adiabatic BBWH reheating can drive distortions above CDM expectations. PIXIE-like missions provide decisive constraints. T4. BAO phase and ruler stabilityFSKC must re-derive the Mpc co-moving ruler and phase coherence across matter/radiation eras; 21-cm intensity mapping at will stress this. BBWH that become standard radiation-dominated FLRW recover it by construction. T5. SNe Ia time dilation beyond Next-generation NIR SNe at test whether time stretching persists universally. Deviations would strongly favor (or rule out) FSKC variants. T6. Gravitational-wave backgrounds / echoesBBWH/Planck-star transitions could leave non-standard stochastic GW backgrounds or late-time “echo” phenomenology from black-to-white conversions; null results constrain parameter space. —6. What Would Constitute “Success”?For FSKCA self-consistent solution of the Einstein (or modified-gravity) equations with a static spatial metric and a stress-energy content that generates the required and , and reproduces:(i) time dilation, (ii) Tolman dimming, (iii) BAO ruler and phase, and (iv) COBE/FIRAS limits.Distinctive redshift-drift and cosmic-parallax predictions subsequently confirmed.For BBWHA quantum-consistent resolution of the white-hole instability that yields an effective FLRW expansion with correct thermal history and acoustic physics, and a predictive new signature (e.g., a calculable spectral-distortion floor, or a GW background feature).—7. Practical Program (near-term)1. Model building (theory):FSKC: specify and solve for on ; enforce reciprocity and compute , , , and the Tolman scaling from first principles.BBWH: construct junction models with consistent with energy conditions (or controlled violations), then add perturbations to assess stability and primordial spectra. 2. Data confrontations:Fit redshift drift forecasts to ELT/HIRES and SKA timelines. Use Gaia-NIR successor forecasts to constrain vector-harmonic amplitudes of extragalactic proper motions. Quantify allowed CMB spectral-distortion budgets in each model vs. PIXIE-class sensitivity. Ensure BAO phase-coherence reproduction in FSKC (challenging). —8. DiscussionFSKC offers an appealingly direct kinematic intuition (“apparent expansion = organized motion in a finite space”), but as soon as one demands all first-principles cosmological tests—notably time dilation, Tolman, BAO phase/ruler, and CMB spectral purity—the model becomes tightly constrained. These requirements are automatically satisfied in FLRW but must be intricately encoded in FSKC.BBWH aligns with the idea that expansion begins at a boundary (consistent with BGV past-incompleteness) and invites quantum-gravity completion. Classical white-hole realizations are not viable (instability), but bounce-type models (LQC/Planck stars) remain open and potentially predictive if they deliver distinct late- or early-time signatures. —9. ConclusionBoth hypotheses can be cast in rigorous form and are falsifiable against the same four pillars that buttress CDM. At present, the empirical weight—CMB spectrum, SNe Ia time dilation, BAO, Tolman test—strongly favors genuine metric expansion with early-universe thermal physics. That said, FSKC and BBWH make distinctive, measurable predictions (especially in redshift drift, cosmic parallax, and CMB spectral distortions). Pursuing these tests is scientifically valuable: either the alternatives are cleanly excluded, or we uncover cracks that point to new physics.—Selected references (illustrative)CMB spectrum & distortions: Fixsen (1996, COBE/FIRAS); NASA/LAMBDA FIRAS results; PIXIE concept and spectral-distortion review. Planck parameters: Planck 2018 (Aghanim et al.). SNe Ia time dilation: Blondin et al. (2008); DES 2024 measurement. BAO reviews/results: White (review); Alam et al./BOSS DR12 (representative); general BAO overview. Tolman test: Lubin & Sandage series. White-hole instability: Eardley (1974) and follow-ups. Quantum bounces / Planck stars: Ashtekar & Singh (LQC review); Rovelli & Vidotto (Planck stars). Junction conditions: Israel (1966) and modern treatments. Redshift-drift test: Sandage–Loeb concept. Cosmic proper motions: real-time cosmology/astrometry outlooks.