Published on January 12, 2025 5:11 PM GMT
Epistemic Status: Speculative.
This post describes a purely mathematical rearrangement within well-known theories (quantum mechanics + general relativity). No new physics is introduced; we unify established equations in an unconventional way. The goal is to spark curiosity and invite feedback about potential semi-classical or pilot-wave approaches to quantum gravity.
Summary in Rationalist Terms:
• Claim: There is a lesser-known global integral identity that connects the “quantum potential” from the Bohm–Madelung representation of quantum mechanics to spatial curvature in a 3+1 slice of general relativity.
• Why Care? If correct, it is a neat “bridge equation” that ties quantum amplitude gradients (wavefunction shapes) to 3D curvature. This might matter for attempts to unify quantum theory and gravity without adding new constants or fields.
• Caveat: This is not a complete theory of quantum gravity. It is a demonstration that hooking two known frameworks together can yield novel relationships that standard treatments do not emphasize.
1. Background: Two Pillars, Rarely Combined
(a) Bohm–Madelung Quantum Mechanics:
• We write the wavefunction psi(x,t) as psi = sqrt(rho)exp(iS/hbar).
• This yields a fluid-like set of equations with a quantum potential Q = - (hbar^2 / 2m) (nabla^2 sqrt(rho) / sqrt(rho)).
• Interpretation: Q often remains in the background of standard QM, but it can provide insights in pilot-wave or hidden-variable interpretations.
(b) 3+1 Decomposition of General Relativity:
• Spacetime is sliced into spatial hypersurfaces (Sigma_t) evolving in time.
• The 3D metric h_ij on each slice and the extrinsic curvature K_ij satisfy constraint equations, including the Hamiltonian constraint:
(3)R + K^2 - K_ijK^ij = 16 pi G rhoeff.
• Normally, one lumps “quantum matter fields” into T{mu nu} without focusing on the distinct form of the Bohmian quantum potential.
2. The New-Looking Equation
By carefully combining the pilot-wave stress-energy with the 3+1 Hamiltonian constraint, one obtains a global integral relationship of the schematic form:
Integral over Sigma_t of (3)R sqrt(h) d^3x = 16 pi G Integral over Sigma_t of (rho_m + rho_quantum) sqrt(h) d^3x + [terms involving extrinsic curvature].
Here:
• (3)R is the scalar curvature of the 3D slice.
• rho_m ~ m rho is the usual matter density if each “particle” has mass m.
• rho_quantum encodes the quantum potential’s effect (it depends on gradients of sqrt(rho)).
• The integral of rho Q over space is nonzero if the wavefunction amplitude has significant spatial gradients.
3. Why Might This Matter to Rationalists?
3.1 Incremental Knowledge and Hypothesis-Generation:
• We only use established equations from QM (Bohm–Madelung) and GR (ADM). There are no new parameters or couplings.
• It highlights how amplitude gradients in a quantum fluid may directly source or shape spatial curvature.
• Potential relevance to:
– Early-universe cosmology.
– Black hole interiors or horizon structure.
– Speculative “dark sector” effects.
3.2 Epistemic Humility: Obvious Next Questions:
• We used a single, nonrelativistic psi. Full consistency might require a relativistic field or QFT approach.
• Pilot-wave mechanics is an alternative formulation of QM but is still consistent with standard predictions; some may question how “real” Q is.
• Actual measurability is unclear; these effects might be tiny or overshadowed by classical curvature sources.
• The boundary conditions, smoothness assumptions, or topological constraints need rigorous checking.
3.3 Could It Solve the Big Puzzle?
• No single identity can unify quantum theory and gravity outright. But bridging standard frameworks can lead to new conceptual strategies.
• Seeing a direct link between wavefunction amplitude gradients and curvature might stimulate fresh approaches.
4. Why Is This Post Potentially Relevant?
• LessWrong often values “Open-Problem Literacy.” Even a small rearrangement of known equations can unlock bigger conceptual leaps or new research directions.
• This example shows how rewriting standard formulae can reveal relationships that mainstream texts rarely emphasize.
• Feedback is welcomed. If there are mathematical or conceptual oversights, the rationalist community is well-placed to spot them. If there is a path to experiment, that is even more exciting.
5. Conclusion & Next Steps
Bottom Line: We identified a global constraint linking the integral of rhoQ to the integral of (3)R over a spatial slice. While it is not an all-in-one quantum gravity solution, it is an intriguing synergy of the Bohm–Madelung “quantum fluid” and Einstein’s 3+1 constraints. Further development could involve:
• Extending to multi-particle or quantum field scenarios.
• Looking for small gravitational anomalies in carefully designed quantum experiments.
• Checking boundary conditions and topological assumptions for cosmic or black hole spacetimes.
References :
1. Madelung, E. (1927), “Quantum Theory in Hydrodynamic Form,” Zeitschrift für Physik, 40.
2. Bohm, D. (1952), “A Suggested Interpretation of the Quantum Theory in Terms of ‘Hidden’ Variables,” Physical Review, 85, 166–193.
3. Arnowitt, R., Deser, S., Misner, C.W. (1962), “The Dynamics of General Relativity,” in Gravitation: An Introduction to Current Research, ed. L. Witten, Wiley.
4. Parker, L., Toms, D. (2009), “Quantum Field Theory in Curved Spacetime.”
TL;DR for LessWrong:
• We used standard but rarely combined pieces of physics: Bohm–Madelung quantum mechanics + ADM 3+1 GR.
• Derived a global constraint that directly relates the integral of the quantum potential term to the integral of spatial curvature.
• Possibly relevant for cosmic or black-hole scale quantum effects.
• Feedback/critique is welcome, especially regarding boundary conditions, topological constraints, and experimental testing.
Thanks for reading, and I look forward to any critiques or suggestions on next steps.
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