On the Possibility of Supertasks in General Relativity

Abstract

Malament-Hogarth spacetimes are the sort of models within general relativity that seem to allow for the possibility of supertasks. There are various ways in which these spacetimes might be considered physically problematic. Here, we examine these criticisms and investigate the prospect of escaping them.

Using Isabelle/HOL to Verify First-Order Relativity Theory

“…In his seminal analysis of computation, Turing [6] discussed the nature of human computation, and showed that certain tasksmost famously, the Halting Problem (HP) -are not decidable by computational means. Subsequent theoretical investigation by various researchers suggests, however, that physical systems may exist which can in fact decide HP by exploiting cosmological phenomena [7][8][9][10][11][12]. We focus here on one particular scheme for cosmological hypercomputation [9,13], and consider the extent to which it rests on secure logical foundations.…”

Relativistic Computation

“…Another worry in the literature concerns communication between the computer and the programmer. The new-relative to [12, p.118], [26] or to [17]-feature we propose in section 4.1 is that we send the message not via light-signals but via a postman (not inertial and slower than the speed of light). It is a future task to check whether in the Kerr spacetime there is a sequence of worldlines for postmen starting from a cofinal set of events in the worldline of the computer such that there is a common bound for the acceleration in these worldlines.…”

“…In anti-de-Sitter spacetime each event is an MH event. The spacetime of an electrically charged black hole (called a Reissner-Nordström spacetime) is MH, spacetimes with closed timelike curves are MH, and there are many other examples [12, sec.4], [22,26]. Figure 4: Conformal (or Penrose) diagram of a slowly rotating black hole.…”

“…We say that we have a bounded blue-shift for an MH event together with the infinite worldline of C if there is an arrangement for P's worldline together with a series of possible light-signals from a cofinal set of events of C to P such that the rate of frequency of a light-signal in this arrangement as observed by C and P, respectively, is bounded. Otherwise, we say that we have a divergent blue-shift [12,26].…”

“…Earman [12, Lemma 4.2 on p.111 and its converse in sec. 4.5] proves that, among the stably causal spacetimes, the MH ones are exactly those with a divergent blue-shift, if we assume the following hypothesis: The family of light-signals that C can send to P forms a 2-dimensional integral submanifold in which the order of omission from C matches the order of reception at P. However, Manchak [26] constructs a stably causal MH spacetime with bounded blue-shift, and his construction exhibits many other desirable properties for a relativistic computer. He thus shows that in order to avoid the blue-shift problem we do not have to give up even the existence of a "good" global time-function, we only have to give up the above-cited hypothesis about a strong concert between the light-signals.…”