Who’s afraid of naked singularities? Probing timelike singularities with finite energy waves

Abstract

To probe naked spacetime singularities with waves rather than with particles we study the well posedness of initial value problems for test scalar fields with finite energy so that the natural function space of initial data is the Sobolev space. In the case of static and conformally static spacetimes we examine the essential selfadjointness of the time translation operator in the wave equation defined in the Hilbert space. For some spacetimes the classical singularity becomes regular if probed with waves while stronger classical singularities remain singular. If the spacetime is regular when probed with waves we may say that the spacetime is ''globally hyperbolic.'' ͓S0556-2821͑99͒00620-7͔

Quantum singularity of Levi-Civita spacetimes

“…The volume element used to define H is V −1 times the natural volume element on Σ. We do not choose the first Sobolev norm H 1 proposed by Ishibashi and Hosoya [8]; that choice is related to Dirichlet boundary conditions at the singularity [4]. If we initially define the domain of A to be C ∞ 0 , A is a real positive symmetric operator and self-adjoint extensions always exist [5].…”

Are Classically Singular Spacetimes Quantum Mechanically Singular as Well?

“…The second way to remove the quantum singularity is to change the Hilbert space for the wavefunction from L 2 to the first Sobolev space. Ishibashi and Hosoya [11] show that requiring both the wave function and the derivative of the wave function to be square integrable will remove the singularity in the KleinGordon case for a cosmic string. Of course, this is not the usual quantum mechanical Hilbert space and one may not be comfortable adding this extra condition.…”

Bäcklund Transformation and N-Soliton Solutions for the Cylindrical Nonlinear Schrodinger Equation from the Diverging Quasi-Plane Envelope Waves

“…From the figures, we can also observe that the finite regions with the envelope waves existing centralize almost all of the amplitudes and energies, and the values of the soliton amplitude tend to zero when x → ±∞, while the values of t are arbitrary real numbers. In general, to deal with the classical solutions of differential equations, the notion of weak derivatives is introduced in the Sobolev spaces [43]. Here, to search for the analytic soliton solutions and integrability, we have taken advantage of the Wronskian technique and Hirota method, and do not need to introduce the notion of weak derivatives in the Sobolev spaces.…”