Spacetime Curvature and Time Dilation Inside a Stationary Capsule Enclosed by a Near-Light-Speed Rotating Energy Shell: The Börekci Energy Field Apparatus, the Börekci Metric and Bethe–Heitler Antimatter Production

This work presents a pre-experimental design intended to place a laboratory bound on the gravitational coupling of confined electromagnetic energy. The apparatus is built in a sealed chamber beneath a mountain. Outermost stand two 15 T superconducting C-electromagnets facing one another with their mouths opposed; each rests on its own independent platform 12 m above the floor and is mounted, through a bore in the middle of its outer belly, on a thick pin driven into the side wall along the X axis, about which it turns at 120 rpm. Because that spin axis passes through the capsule centre, no point of a magnet can come within 9 m of that centre, so a 1.0 m clearance to the stationary 8 m radius outer capsule is preserved at every angle and the field seen by the capsule is a continuously rotating one with unit duty cycle. Within the outer capsule an elliptical inner capsule holds terrestrial air at 1 atm. Between the two, a 50 MV electron arc and a co-directional 1 PW laser flow enclose the inner capsule from pole to pole, with lead vapour injected at 0.1 atm. Every component contribution to the stress-energy tensor is computed individually in the weak-field approximation. The total metric deviation is Φ/c² = 2.23×10^-23, a time difference of 1.93×10^-18 s over 24 h. The Börekci metric is formulated with a shell-coupling parameter ξ that reduces to standard linearised general relativity at ξ = 1; a pronounced time dilation would require ξ of order 2×10^22, and the purpose of the proposed experiment is to bound ξ rather than to assert it. A three-clock protocol with a calibration channel on the rotating rim, where the ordinary kinematic dilation is 8.29×10^-14, is proposed together with a 1/100 scale test cell. Bethe-Heitler production of 3.06×10^17 positrons in 24 h and a full energy and cost analysis are also given.