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Chinese Physics B

Modified de Broglie–Bohm approach to the Schwarzschild black hole

Bobo Wang

Published Aug 1, 2008• 1 citation
Publisher

Abstract

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.

DOI
10.1088/1674-1056/17/8/012

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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.
In this work we apply the de Broglie-Bohm interpretation of quantum mechanics to the quantized spherically symmetric black-hole coupled to a massless scalar field. The wave-functional used was first obtained by Tomimatsu using the standard ADM quantization and a gauge that places the observer close to the black-hole horizon. Using the causal interpretation, we compute quantum trajectories determined by the initial conditions. We show that the quantum trajectories for the black-hole mass can either increase or decrease with time. The quantum trajectories that show increasing mass represent the usual black-hole behavior of continuous energy absorption. The mass-decreasing quantum trajectories are a purely quantum mechanical phenomena. They can be physically interpreted as describing a black-hole that evaporates.
We revisit the Schwarzschild singularity in a semiclassical setting where the background geometry is classical and quantum effects enter through Bohmian (quantal) trajectories associated with a Klein Gordon wave packet. Using the Madelung-Bohm decomposition of the Klein Gordon wavefunction, we show that the quantum-modified motion is equivalent to geodesic motion in an effective metric conformally related to Schwarzschild, with a conformal factor fixed by the wavefunction amplitude. Solving the wavefunction equation near r→0r\to 0r→0 determines this factor and yields finite curvature invariants, in suitable coordinates the interior extends smoothly and the effective spacetime is geodesically complete. This suggests that quantum dynamics on a fixed classical background can regularize the Schwarzschild singularity without a full theory of quantum gravity.
The quantum trajectories in the de Broglie-Bohm formulation of quantum mechanics depend on an additional quantum potential derived from the full wave solution of Schrödinger's equation. The task of supplying collectively all the correct quantum results strongly alters the characteristics of the corresponding classical trajectories, which underlie semiclassical approximations to the evolving wave function. Both classical and quantum trajectories are here considered to be conservative with no influence of an external environment, even though this is the source of eventual classicality in quantum systems, that is, decoherence. The concept of integrability, closely correspondent in classical and quantum mechanics, is not preserved by the quantum trajectories. General systems, in which classical chaotic motion participates, are much harder to treat semiclassically, but quantum trajectories can be chaotic even for integrable systems. This discrepancy between the character of classical and quantum trajectories in the de Broglie-Bohm interpretation does not clarify the singular classical-quantum transition.
Abstract We generalize the de Broglie-Bohm (dBB) formulation of quantum mechanics to the case of quantum gravity (QG) by using the effective action for a QG theory. This is done by replacing the dBB equations of motion with the effective action equations of motion, which is beneficial even in the non-gravitational case, since in this way one avoids the violations of the Heisenberg uncertainity relations and the absence of the classical trajectories for stationary bound states. Another advantage of the effective action formalism is that one can obtain the field configurations in the case of a quantum field theory (QFT). The proposed QG generalization is natural for Bohmiam mechanics because a dBB wavefunction is really a wavefunction of the Universe and in order to define the effective action for an arbitrary initial state one needs a QG path integral. The QG effective action can be constructed by using the piecewise flat quantum gravity (PFQG) theory and the PFQG effective action can be approximated by the QFT effective action for General Relativity coupled to matter, with a cutoff determined by the average edge length of the spacetime triangulation. One can then calculate the corresponding field configurations and from these field configurations one can obtain the trajectories for the corresponding elementary particles.