Malament-Hogarth spacetime
Encyclopedia
A Malament–Hogarth (M-H) spacetime, named after David B. Malament
and Mark Hogarth, is a relativistic
spacetime
that possesses the following property: there exists a worldline
and an event 
such that all events along 
are a finite interval in the past of 
, but the proper time
along
is infinite. The event 
is known as an M-H event. The significance of M-H spacetimes is that they allow for the implementation of certain non-Turing computable tasks (hypercomputation
). The idea is for an observer at some event in
's past to set a computer (Turing machine) to work on some task and then have the Turing machine
travel on
, computing for all eternity. Since 
lies in 
's past, the Turing machine can signal (a solution) to 
at any stage of this never-ending task. Meanwhile, the observer takes a quick trip (finite proper time) through spacetime to 
, to pick up the solution. The set-up can be used to decide the halting problem
, which is known to be undecidable by an ordinary Turing machine. All the observer needs to do is to prime the Turing machine to signal to
if and only if the Turing machine halts.
The Kerr metric
, which describes empty spacetime around a rotating black hole
, possesses these features: a computer can orbit the black hole indefinitely, while an observer falling into the black hole experiences an M-H event as they cross the inner event horizon
. (This, however, neglects the effects of Black Hole Evaporation.)
David B. Malament
David B. Malament is an American philosopher of science.He attended Stuyvesant High School and received a B.A. in mathematics 1968 at Columbia College and Ph.D. in philosophy 1975 at Rockefeller University...
and Mark Hogarth, is a relativistic
Theory of relativity
The theory of relativity, or simply relativity, encompasses two theories of Albert Einstein: special relativity and general relativity. However, the word relativity is sometimes used in reference to Galilean invariance....
spacetime
Spacetime
In physics, spacetime is any mathematical model that combines space and time into a single continuum. Spacetime is usually interpreted with space as being three-dimensional and time playing the role of a fourth dimension that is of a different sort from the spatial dimensions...
that possesses the following property: there exists a worldline
and an event such that all events along are a finite interval in the past of , but the proper timeProper time
In relativity, proper time is the elapsed time between two events as measured by a clock that passes through both events. The proper time depends not only on the events but also on the motion of the clock between the events. An accelerated clock will measure a smaller elapsed time between two...
along
is infinite. The event is known as an M-H event. The significance of M-H spacetimes is that they allow for the implementation of certain non-Turing computable tasks (hypercomputationHypercomputation
Hypercomputation or super-Turing computation refers to models of computation that are more powerful than, or are incomparable with, Turing computability. This includes various hypothetical methods for the computation of non-Turing-computable functions, following super-recursive algorithms...
). The idea is for an observer at some event in
's past to set a computer (Turing machine) to work on some task and then have the Turing machineTuring machine
A Turing machine is a theoretical device that manipulates symbols on a strip of tape according to a table of rules. Despite its simplicity, a Turing machine can be adapted to simulate the logic of any computer algorithm, and is particularly useful in explaining the functions of a CPU inside a...
travel on
, computing for all eternity. Since lies in 's past, the Turing machine can signal (a solution) to at any stage of this never-ending task. Meanwhile, the observer takes a quick trip (finite proper time) through spacetime to , to pick up the solution. The set-up can be used to decide the halting problemHalting problem
In computability theory, the halting problem can be stated as follows: Given a description of a computer program, decide whether the program finishes running or continues to run forever...
, which is known to be undecidable by an ordinary Turing machine. All the observer needs to do is to prime the Turing machine to signal to
if and only if the Turing machine halts.The Kerr metric
Kerr metric
The Kerr metric describes the geometry of empty spacetime around an uncharged axially-symmetric black-hole with an event horizon which is topologically a sphere. The Kerr metric is an exact solution of the Einstein field equations of general relativity; these equations are highly non-linear, which...
, which describes empty spacetime around a rotating black hole
Black hole
A black hole is a region of spacetime from which nothing, not even light, can escape. The theory of general relativity predicts that a sufficiently compact mass will deform spacetime to form a black hole. Around a black hole there is a mathematically defined surface called an event horizon that...
, possesses these features: a computer can orbit the black hole indefinitely, while an observer falling into the black hole experiences an M-H event as they cross the inner event horizon
Event horizon
In general relativity, an event horizon is a boundary in spacetime beyond which events cannot affect an outside observer. In layman's terms it is defined as "the point of no return" i.e. the point at which the gravitational pull becomes so great as to make escape impossible. The most common case...
. (This, however, neglects the effects of Black Hole Evaporation.)