Curvature regularization and dynamical vacuum structure in pseudo-complex general relativity

Abstract Whether classical spacetime remains well-defined at arbitrarily high curvature is a central question. In general relativity, singularities signal the breakdown of the classical description, motivating modifications of spacetime’s short-distance structure. Pseudo-complex general relativity (pcGR) extends spacetime geometry to pseudo-complex coordinates, naturally introducing two metric sectors, the physical metric $$g_{\\mu \\nu }$$ g μ ν and an auxiliary field $$f_{\\mu \\nu }$$ f μ ν . The magnitude of the pseudo-imaginary component defines the invariant acceleration scale $$a_0$$ a 0 , a direct consequence of the pseudo-complex structure. The simultaneity condition, requiring both idempotent sectors to satisfy the pseudo-complex Einstein equations independently, fixes the auxiliary field algebraically, with no new propagating degrees of freedom, and sets a lower bound on the lapse that regularizes curvature. The regularization is achieved through the combined effect of the lapse-gap condition $$ e^{\\nu (r)} > a_0 $$ e ν ( r ) > a 0 , which ensures the radial metric component remains nondegenerate, together with the regularity conditions at the areal-radius origin, $$ B(0)=1 $$ B ( 0 ) = 1 and $$ B'(0)=0 $$ B ′ ( 0 ) = 0 , which emerge naturally from the pseudo-complex geometry. These conditions jointly eliminate the Schwarzschild-type curvature divergence, rather than the gap condition acting alone. In cosmology, we show that the pseudo-complex field equations admit a formal reduction to an effective evolution equation for the vacuum. component $$\\rho _\\textrm{pc}(t)$$ ρ pc ( t ) : the auxiliary functions $$f_0$$ f 0 and $$f_2$$ f 2 , together with $$\\ddot{a}$$ a ¨ , are formally eliminated within the restricted parametrization adopted in the reduction, yielding a second-order evolution equation in which no auxiliary functions remain explicitly. The reduced evolution equation admits asymptotic solutions whose behavior is determined by the pseudo-complex geometry. In particular, it admits late-time solutions approaching an effective constant vacuum energy, suggesting a geometric origin for dark energy with scale set by $$a_0$$ a 0 . pcGR thus realizes, through purely geometric conditions, bounded curvature, intrinsic cutoff scales, dynamical vacuum, and absence of global symmetries, while features depending on quantum spectra have no classical analog.