Schwarzschild black-hole interiors border on spacelike singularities representing classical information leaks. We show that local quantum physics is decoupled from these leaks due to dynamically generated boundaries, called Zeno borders. Beyond Zeno borders black-hole interiors become asymptotically silent, and quantum fields evolve freely toward the geodesic singularity with vanishing probability measure for populating the geodesic boundary. Thus Zeno borders represent a probabilistic completion of Schwarzschild black holes within the semiclassical framework.
A modified de Broglie–Bohm approach is generalized to the Schwarzschild black hole. By using this method, the quantum potential and the quantum trajectories of the black hole are investigated. And we find that the linear combination of two particular solutions of the black hole wavefunction is not physical although each of them is physical, if we think that the quantum gravity should reduce into its corresponding classical counterpart in which the gravity vanishes. It seems to confirm the argument, given by Alwis and MacIntire, that a possible resolution on the quantum gravity is to give up the superposition principle.
Bohmian mechanics offers a deterministic alternative to conventional quantum theory through well-defined particle trajectories. While successful in nonrelativistic contexts, its extension to curved spacetime-and hence quantum gravity-remains unresolved. Here, we develop a covariant extension of Bohmian mechanics in curved spacetime that removes the need for metric quantization. From a Lagrangian formulation, we derive a generalized guidance equation in which Bohmian trajectories generate hidden curvature, replacing metric superposition with a statistical ensemble constrained by Heisenberg uncertainty, offering a novel perspective on quantum gravity. Consequently, in our approach, measuring the gravitational potential at a point unveils a pre-existing trajectory and its associated curvature-a departure from the observer-centric paradigm of standard quantum mechanics-providing an alternative in which gravitational effects emerge from deterministic quantum trajectories rather than wavefunction collapse. Numerical simulations in Robertson-Walker and cigar soliton spacetimes reveal that while quantum interference is curvature-sensitive, Zitterbewegung remains invariant, distinguishing fundamental quantum effects. Moreover, deviations from the Born rule in inhomogeneous spacetimes are observed and suggest gravity-induced quantum non-equilibrium. This new approach has far-reaching implications for the role of determinism and potential observational signatures of quantum non-equilibrium in cosmology.
The effects of the de Broglie-Bohm quantum potential on a test particle of mass
m are investigated in a conformally-flat geometry. A real, nonlinear, scalar field
Ψ is introduced and related directly to the conformal factor and to the effective mass
M(r,t) of the particle. The radial acceleration of a static observer in the conformally-flat metric is negative (the field is repulsive), and the corresponding proper acceleration resembles that one of the "peak radius" where the radial probability density has a global maximum during the wave packet spreading phenomenon. The timelike radial geodesics are computed in the conformally-flat spacetime in double-null coordinates and proves to be hyperbolae in the region
r>t.
We take the non-relativistic limit
c→∞ of the free Klein-Gordon field on static spacetimes at three levels: one-particle resolvents, quasi-free states and the time-zero net. After subtraction of the rest energy, the one-particle Hamiltonian is an explicit function of a Schrödinger-type operator whose potential on the Schwarzschild exterior is exactly Newtonian. On regular static stars it converges to the Newtonian Schrödinger operator in norm-resolvent sense at rate
c−2, with convergence of eigenvalues and an explicit first post-Newtonian correction. On the Schwarzschild exterior the Boulware one-particle spectrum has no eigenvalues for any
c; the convergence is strong but not in norm, the limit selects the Friedrichs realisation of the gravitational hydrogen atom, and its bound states emerge by spectral concentration. The same holds on the Reissner-Nordström exterior. In all cases the rescaled two-point functions converge to those of the Schrödinger-Fock vacuum. The hydrogenic states are limits of resonances: in each partial wave there is exactly one near each hydrogenic level, its real part carries the first post-Newtonian correction, and its width, proportional to
c−(4l+3), is the one found for gravitational atoms by matched asymptotics. At the level of nets the time-zero Weyl algebra is the same for every
c, and the vacuum and the dynamics converge on it, yet for smooth metrics the algebra of each bounded region with smooth boundary is a type III
1 factor at finite
c and a type I factor with a pure product vacuum in the limit. What collapses is the entanglement of the vacuum; thermal states in flat space and on stars survive the limit. With positivity of the energy imposed modulo the central charge, the axioms of Galilean nets hold in the limit for every static star and for Schwarzschild, and the vacuum is not separating.
One of the important and famous topics in general theory of relativity and gravitation is the problem of geodesic deviation and its related singularity theorems. An interesting subject is the investigation of these concepts when quantum effects are considered. Since the definition of trajectory is not possible in the framework of standard quantum mechanics (SQM), we investigate the problem of geodesic equation and its related topics in the framework of Bohmian quantum mechanics in which the definition of trajectory is possible. We do this in a fixed background and we do not consider the backreaction effects of matter on the space–time metric.