Abstract
A localised particle in Quantum Mechanics is described by a wave packet in position space, regardless of its energy. However, from the point of view of General Relativity, if the particle's energy density exceeds a certain threshold, it should be a black hole. In order to combine these two pictures, we introduce a horizon wave-function determined by the particle wave-function in position space, which eventually yields the probability that the particle is a black hole. The existence of a minimum mass for black holes naturally follows, albeit not in the form of a sharp value around the Planck scale, but rather like a vanishing probability that a particle much lighter than the Planck mass be a black hole. We also show that our construction entails an effective Generalised Uncertainty Principle (GUP), simply obtained by adding the uncertainties coming from the two wave-functions associated to a particle. Finally, the decay of microscopic (quantum) black holes is also described in agreement with what the GUP predicts.
Highlights
Introduction and motivationA general property of the Einstein theory is that the gravitational interaction is always attractive and we are not allowed to neglect its effect on the causal structure of spacetime if we pack enough energy in a sufficiently small volume
From the point of view of General Relativity, if the particle’s energy density exceeds a certain threshold, it should be a black hole. To combine these two pictures, we introduce a horizon wave function determined by the particle wave function in position space, which eventually yields the probability that the particle is a black hole
The paper is organized as follows: we introduce the main ideas that define the horizon wave function associated with any localized Quantum Mechanical particle; in Sect. 3, we apply the general construction to the simple case of a particle described by a Gaussian wave function at rest in flat space-time, for which we explicitly obtain the probability that the particle is a black hole, we recover the generalized uncertainty principle (GUP) and a minimum measurable length, and estimate the decay rate of a black hole with mass around the Planck scale; in Sect. 4, we comment on our findings and outline future applications
Summary
A general property of the Einstein theory is that the gravitational interaction is always attractive and we are not allowed to neglect its effect on the causal structure of spacetime if we pack enough energy in a sufficiently small volume This can occur, for example, if two particles (for simplicity of negligible spatial extension and total angular momentum) collide with an impact parameter b shorter than the Schwarzschild radius corresponding to the total center-mass energy E of the system, that is, b p. The aim of this paper is precisely to introduce the definition of a wave function for the horizon that can be associated with any localized Quantum Mechanical particle [14] This tool will allow us to put on quantitative ground the condition that distinguishes a black hole from a regular particle. The paper is organized as follows: we introduce the main ideas that define the horizon wave function associated with any localized Quantum Mechanical particle; in Sect. 3, we apply the general construction to the simple case of a particle described by a Gaussian wave function at rest in flat space-time, for which we explicitly obtain the probability that the particle is a black hole, we recover the GUP and a minimum measurable length, and estimate the decay rate of a black hole with mass around the Planck scale; in Sect. 4, we comment on our findings and outline future applications
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