The Collapse of Supertasks

Then it is again clear that nothing will remain, but it will be all gone...

Simplicius.

Abstract

A supertask consists in the performance of an infinite number of actions in a finite time. I show that any attempt to carry out a supertask will produce a divergence of the curvature of spacetime, resulting in the formation of a black hole. I maintain that supertaks, contrarily to a popular view among philosophers, are physically impossible. Supertasks, literally, collapse under their own weight.

Appendix: Quantum Indetermination Theorems

The relation (4) given above is different from the usual inequality presented in many textbooks:

$$\begin{aligned} \Delta E \Delta t \ge \frac{\hbar }{2}. \end{aligned}$$

(10)

This inequality cannot be derived from the framework of quantum mechanics, since, as noted by Bunge (1967) and Perez Bergliaffa et al. (1993) among others, there is no time operator in the theory. The inequalities between properties represented by non-commuting operators are derived using the Schwarz inequality (Weyl 1928), i.e. if $\hat{A}\hbox { and }\hat{B}$ are two operators representing the quantum properties $A\hbox { and }B$ such that

$$\begin{aligned}{}[\hat{A}, \; \hat{B}]=i\hbar \end{aligned}$$

(11)

then, through $(\Delta A)^{2}(\Delta B)^{2}\ge \frac{1}{2}\left| \hat{A}\hat{B}\right|$, we get

$$\begin{aligned} \Delta A \Delta B \ge \frac{1}{2} \hbar , \end{aligned}$$

(12)

where $\Delta$ indicates the root–mean–square deviation of the corresponding operators. In particular, since the position and momentum operators satisfy $[\hat{q_{i}}, \; \hat{p_{j}}]=i \hbar \delta _{ij}$, the usual relation $\Delta q_{r} \Delta p_{r}\ge \hbar /2$ obtains.

This procedure fails when applied to the Hamiltonian that represents the energy of the system and time, since time is a parameter. In 1945, Mandelshtam and Tamm obtained the correct form for an energy-time relation, where the time involved was the timescale of an evolving operator. This relation is given by inequality (4).