Abstract
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.