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Tabletop Quantum Gravity: Probing Spacetime via Massive Superposition — E8 Intelligence Research

Andrew Stewart Caldin

Published Sep 21, 2026• 0 citations
Publisher

Abstract

FINDING: Quantum gravity experiments are transitioning from theoretical speculation to tabletop-scale tests probing the quantum nature of spacetime via superposition of massive systems. MATH: The core experimental framework relies on the quantum superposition of a mass \( m \) in two spatial locations separated by \( \Delta x \), generating a gravitational phase shift \( \Delta \phi = \frac{G m^2 \Delta t}{\hbar \Delta x} \) (for two masses in superposition, per Bose–Marletto–Vedral proposal). The decoherence timescale for gravitational entanglement is \( \tau \sim \frac{\hbar \Delta x}{G m^2} \). Penrose's criterion for gravitationally-induced collapse: \( E_G = \frac{G m^2}{\Delta x} \) compared to \( \hbar/\tau \). CONNECTION: The ratio \( \frac{G m^2}{\hbar c} \) (gravitational coupling) is dimensionless and tiny (~\(10^{-45}\) for protons), but the experimental geometry uses harmonic oscillator frequencies and lattice-like trap spacings — the optimal \( \Delta x \) often scale Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

DOI
10.5281/zenodo.22874187

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FINDING: Quantum gravity experiments are transitioning from theoretical speculation to tabletop-scale tests probing the quantum nature of spacetime via superposition of massive systems. MATH: The core experimental framework relies on the quantum superposition of a mass ( m ) in two spatial locations separated by ( \Delta x ), generating a gravitational phase shift ( \Delta \phi = \frac{G m^2 \Delta t}{\hbar \Delta x} ) (for two masses in superposition, per Bose–Marletto–Vedral proposal). The decoherence timescale for gravitational entanglement is ( \tau \sim \frac{\hbar \Delta x}{G m^2} ). Penrose's criterion for gravitationally-induced collapse: ( E_G = \frac{G m^2}{\Delta x} ) compared to ( \hbar/\tau ). CONNECTION: The ratio ( \frac{G m^2}{\hbar c} ) (gravitational coupling) is dimensionless and tiny (~(10^{-45}) for protons), but the experimental geometry uses harmonic oscillator frequencies and lattice-like trap spacings — the optimal ( \Delta x ) often scale Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
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FINDING: Quantum gravity tests are converging on tabletop interferometry experiments probing Planck-scale effects via entanglement and superposition of massive systems. | MATH: Key scales: Planck mass mP=ℏc/G≈2.176×10−8 m_P = \sqrt{\hbar c/G} \approx 2.176 \times 10^{-8} mP​=ℏc/G​≈2.176×10−8 kg; Planck length lP=ℏG/c3≈1.616×10−35 l_P = \sqrt{\hbar G/c^3} \approx 1.616 \times 10^{-35} lP​=ℏG/c3​≈1.616×10−35 m. The proposed experiments (e.g., Ron Folman's T³, matter-wave interferometry) require measuring phase shifts Δϕ∼(m1m2Gt3)/(ℏd3) \Delta \phi \sim (m_1 m_2 G t^3)/(\hbar d^3) Δϕ∼(m1​m2​Gt3)/(ℏd3) — a cubic time dependence, hence "T-cubed." Decoherence rate for gravitational entanglement: Γ∼(Gm1m2/ℏd)2⋅τ \Gamma \sim (G m_1 m_2 / \hbar d)^2 \cdot \tau Γ∼(Gm1​m2​/ℏd)2⋅τ — dimensionless coupling αG=Gm1m2/(ℏcd)≪1 \alpha_G = G m_1 m_2 / (\hbar c d) \ll 1 αG​=Gm1​m2​/(ℏcd)≪1 at tabletop scales, yet measurable via entanglement witness. | CONNECTION: The ratio mP2/(m1m2) m_P^2 / (m_1 m_2) mP2​/(m1​m2​) for typical masses (e.g., 10−14 10^{-14} 10−14 kg) yields ∼1012 \sim 10^{12} ∼1012 — not directly harmonic. However, the T³ phase term contains t3 t^3 t3 — a cubic symmetry (tetrahedral Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
FINDING: Tabletop experiments propose direct quantum-gravity tests via entanglement of two masses, probing the quantum nature of the gravitational field. | MATH: The core observable is the gravitational interaction Hamiltonian Hint=−Gm1m2/∣r1−r2∣ H_{\text{int}} = -G m_1 m_2 / |\mathbf{r}_1 - \mathbf{r}_2| Hint​=−Gm1​m2​/∣r1​−r2​∣, with quantum superposition of positions ∣ψ⟩=12(∣L⟩+∣R⟩) |\psi\rangle = \frac{1}{\sqrt{2}}(|L\rangle + |R\rangle) ∣ψ⟩=2​1​(∣L⟩+∣R⟩) for each mass; the entanglement witness is the phase ϕ=Gm1m2tℏd \phi = \frac{G m_1 m_2 t}{\hbar d} ϕ=ℏdGm1​m2​t​ (where d d d is separation, t t t interaction time). Key constants: G≈6.674×10−11 m3kg−1s−2 G \approx 6.674 \times 10^{-11} \, \text{m}^3\text{kg}^{-1}\text{s}^{-2} G≈6.674×10−11m3kg−1s−2, ℏ≈1.054×10−34 J⋅s \hbar \approx 1.054 \times 10^{-34} \, \text{J·s} ℏ≈1.054×10−34J⋅s. No new ratio emerges; the test is binary: gravity is quantum (entanglement) or classical (no entanglement). | CONNECTION: No direct geometric ratio (0.382, 0.618, 0.786, 1.618, 2.618) appears in the experimental design. However, the superposition states ∣L⟩,∣R⟩ |L\rangle, |R\rangle ∣L⟩,∣R⟩ form a two-level sy Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
FINDING: Quantum gravity tests are converging on tabletop interferometry experiments probing superposition of massive objects, with Penrose's gravitationally-induced collapse criterion as a central quantitative target. | MATH: Penrose's collapse time τ ≈ ħ/E_G where E_G is gravitational self-energy difference between superposed mass configurations; for a sphere of mass m and radius R, E_G ≈ (4πG/3) m²/R (order-of-magnitude). The Ron Folman "T-cubed" experiment scales with T³ (interferometer time cubed) — a cubic scaling law. No new fundamental constant emerges; G, ħ, c remain the only inputs. | CONNECTION: The T³ scaling echoes cubic lattice symmetry (3D crystallographic point groups, e.g., m-3m) — the experiment's geometry (three orthogonal axes of momentum kicks) mirrors a simple cubic root system (A₃ or B₃). The ratio of gravitational to electromagnetic coupling at Planck scale is ~ (m_P/m_e)² ≈ (2.176×10⁻⁸ kg / 9.109×10⁻³¹ kg)² ≈ 5.71×10⁴⁵ — no golden-ratio link found in the descri Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
FINDING: Quantum gravity may be testable via tabletop matter-wave/optomechanical interferometry, probing the quantum superposition of gravitational fields at mesoscopic scales. | MATH: The core observable is the gravitational entanglement phase — for two masses m1,m2m_1, m_2m1​,m2​ separated by distance ddd, the interaction Hamiltonian HG=−Gm1m2/dH_G = -G m_1 m_2 / dHG​=−Gm1​m2​/d generates a phase ϕ=Gm1m2t/ℏd\phi = G m_1 m_2 t / \hbar dϕ=Gm1​m2​t/ℏd in a superposition of paths. The experiment requires ϕ∼π/2\phi \sim \pi/2ϕ∼π/2 for maximal entanglement detection, implying m1m2t/d∼ℏ/Gm_1 m_2 t / d \sim \hbar / Gm1​m2​t/d∼ℏ/G — a scale set by the Planck mass squared over ℏ\hbarℏ. No new constants beyond G,ℏ,cG, \hbar, cG,ℏ,c appear; the test is purely quantum-informational (entanglement witness) rather than energy-based. | CONNECTION: The ratio of gravitational to electromagnetic coupling at the Planck scale is αG/αEM∼(mP/me)2∼(1.22×1019/0.511×106)2∼5.7×1044\alpha_G/\alpha_{EM} \sim (m_P/m_e)^2 \sim (1.22\times10^{19}/0.511\times10^6)^2 \sim 5.7\times10^{44}αG​/αEM​∼(mP​/me​)2∼(1.22×1019/0.511×106)2∼5.7×1044 — not a golden ratio, but the experimental thre Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
FINDING: Tabletop experiments propose direct observation of quantum gravity via entanglement of macroscopic masses, with the core mathematical framework being the quantum superposition of gravitational fields and the resulting entanglement witness. | MATH: The key equation is the gravitational interaction Hamiltonian between two masses in superposition: H^G=−Gm1m2∣x^1−x^2∣ \hat{H}_G = -G \frac{m_1 m_2}{|\hat{x}_1 - \hat{x}_2|} H^G​=−G∣x^1​−x^2​∣m1​m2​​, where the position operators x^i\hat{x}_ix^i​ are in coherent superposition states. The entanglement entropy S=−Tr(ρln⁡ρ)S = -\text{Tr}(\rho \ln \rho)S=−Tr(ρlnρ) of the joint state serves as the witness. The T³ (T-cubed) experiment (Folman) uses a triple-interferometer with time TTT, giving phase shift Δϕ∝Gm2T3/ℏd2\Delta \phi \propto G m^2 T^3 / \hbar d^2Δϕ∝Gm2T3/ℏd2 (where ddd is separation). The decoherence rate scales as Γ∝(Gm2/ℏd2)2τ\Gamma \propto (G m^2 / \hbar d^2)^2 \tauΓ∝(Gm2/ℏd2)2τ, with τ\tauτ the interaction time. | CONNECTION: The phase shift Δϕ\Delta \phiΔϕ contains the ratio Gm2/ℏG m^2 / \hbarGm2/ℏ — a dimensionless coupling Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com