5 research outputs found
Von-Neumann Stability and Singularity Resolution in Loop Quantized Schwarzschild Black Hole
Though loop quantization of several spacetimes has exhibited existence of a
bounce via an explicit evolution of states using numerical simulations, the
question about the way central singularity is resolved in the black hole
interior has remained open. The quantum Hamiltonian constraint in loop
quantization turns out to be a finite difference equation whose stability is
important to understand to gain insights on the viability of the underlying
quantization and resulting physical implications. We take first steps towards
addressing these issues for a loop quantization of the Schwarzschild interior
recently given by Corichi and Singh. Von-Neumann stability analysis is
performed using separability of solutions as well as a full two dimensional
quantum difference equation. This results in a stability condition for black
holes which have a very large mass compared to the Planck mass. For black holes
of smaller masses evidence of numerical instability is found. In addition,
stability analysis for macroscopic black holes leads to a constraint on the
choice of the allowed states in numerical evolution. With the caveat of using
kinematical norm, sharply peaked Gaussian states are evolved using the quantum
difference equation and singularity resolution is obtained. A bounce is found
for one of the triad variables, but for the other triad variable singularity
resolution amounts to a non-singular passage through the zero volume. States
are found to be peaked at the classical trajectory for a long time before and
after the singularity resolution, and retain their semi-classical character
across the zero volume. Our main result is that quantum bounce occurs in loop
quantized Schwarzschild interior at least for macroscopic black holes.Comment: 10 pages, 10 figures; Discussion of results expanded. To appear in
CQ