A clock containing a massive object in a superposition of states; what makes Penrosian wavefunction collapse tick?

TL;DR

This paper explores how superpositions of massive objects in a gravitational field can lead to wavefunction collapse, using a thought experiment involving a hypothetical clock to analyze the role of time dilation and phase differences.

Contribution

It introduces a conceptual framework linking time dilation effects in superposed massive objects to wavefunction collapse, providing a derivation of Penrose's collapse time estimate.

Findings

Time ambiguity affects the evolution of superposed states.

Collapse occurs when phase difference reaches order unity.

Penrose's estimate is recovered through energy considerations.

Abstract

Penrose has been advocating the view that the collapse of the wave function is rooted in the incompatibility between general relativity and quantum mechanics. On the basis of conceptual analysis, he arrived at an estimate for the collapse time. To better understand his estimate, in this paper we present a thought experiment, which singles out the role of time-dilations in massive superpositions. First we investigate the behavior of a hypothetical clock containing a component which can be in a superposition of states. The clock contains a massive object, whose only purpose is to introduce a curvature of space time into the problem. We find that a state of this massive object with a smaller radius, but with the same mass, experiences a larger time dilation. Considering a coherent superposition of the large and small object, introduces an ambiguity in the definition of a common time for both states. We assert that this time ambiguity can be thought to affect the time evolution of a state in different ways and that the relative phase difference between these different interpretations can be calculated. We postulate that the wave function collapse will occur when this phase difference becomes of order unity. An absolute energy scale enters this equation and we recover Penrose's estimate for the collapse time by equating the absolute energy scale to the rest mass of the object.