General Relativity and Newton's Constant with Scretching–Planck Matter: Four Routes and a Geometric Identity
Year: 2026
Type: preprint
Abstract: We formulate superconductivity in Scretching–Planck (SP) density–phase variables by writing the Cooper-pair order parameter as (\psi=\sqrt{P},e^{iS/\hbar}), with charge magnitude (2e) and effective pair mass (m^*). In these variables, the Ginzburg–Landau free energy gives the stationary balance (Q_{\mathrm{SP}}+\tfrac12m^*v_s^2+\alpha+\beta P=0) and the corresponding supercurrent equation. The phase dynamics recover the London equations and penetration depth. Fluxoid quantization gives the flux quantum (h/(2e)=2.0678\times10^{-15}) Wb and accounts for Little–Parks periodicity. Coupling two condensates gives the Josephson relations, with (K_J=2e/h=483597.848) GHz/V. The SP amplitude-gradient term sets the Ginzburg–Landau coherence length. The calculated normal–superconducting interface energy changes sign at (\kappa=0.707107), and the vortex calculation gives (H_{c1}/H_c=1.000000) at (\kappa=1/\sqrt2) and recovers the large-(\kappa) limit. A disorder model for niobium with (\kappa=1.61) gives (\mu_0H_{c1}=112) mT. In (d)-wave rings with a net (\pi) phase shift, the same phase condition permits half-integer fluxoid quantization. A momentum-space density–phase representation of BCS pairing recovers the weak-coupling ratios (2\Delta_0/(k_BT_c)=3.5279) and (\Delta C/(\gamma T_c)=1.426). Across eight elemental superconductors, the condensation-energy relation reproduces the measured zero-temperature thermodynamic critical fields of Al, In, Sn, and Ta within (2.4%). Deviations of (7%) to (15%) for Hg, Pb, and Nb illustrate the limits of the weak-coupling estimate. Free-electron estimates of penetration depth and coherence length place even Nb on the type-I side in the clean-limit calculation, illustrating the sensitivity of that classification to material parameters. Nine solved problems are included.
Author: Scretching, Daniel
Language: English
Subfield: Condensed Matter Physics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225755670
Cites:
Cited by:
Related to: