General Relativity + Computation
Malament-Hogarth Spacetimes
Some solutions to Einstein's equations would allow computers to run for infinite time while you wait only moments for the result - solving the unsolvable.
The Core Idea
In certain curved spacetimes, one observer can experience infinite proper time while remaining in the causal past of another observer who experiences only finite time. This would allow "supertask" computations that solve problems no ordinary computer ever could.
In 1992, mathematician Mark Hogarth and philosopher David Malament identified a remarkable class of spacetimes permitted by general relativity. These "Malament-Hogarth spacetimes" have a peculiar property: they contain pairs of worldlines where one has infinite proper time but stays within the causal past of the other.
The implications for computation are extraordinary. A computer following the infinite-time worldline could run forever, checking all possible cases of any mathematical conjecture. Meanwhile, the waiting observer receives the result after only a finite wait.
“With an M-H spacetime, you could solve the halting problem by actually running programs forever and waiting for the result.”- Mark Hogarth
This would violate the Church-Turing thesis - the foundation of computer science stating that anything "computable" can be computed by a Turing machine. Problems proven mathematically impossible to solve would become trivially solvable.
Visualizing the Worldlines
A spacetime diagram shows how objects move through space and time. In an M-H spacetime, two observers can have dramatically different experiences of time while remaining causally connected.
Observer A waits at a safe distance. Observer B falls into a region of extreme spacetime curvature, experiencing vastly more proper time before sending a signal back to A.
Higher values increase the proper time difference between worldlines
Observer A (Finite Time)
10.0 units
Waits finite time for result
Observer B (Infinite Time)
20.0 units
Experiences infinite computation
The key insight: Observer B's worldline has infinite proper time, yet stays within Observer A's causal past. This means B can compute forever and send the result to A in finite time.
The key requirement: Observer B's entire worldline (including the "infinite" part) must lie within Observer A's causal past. This means signals from any point on B's worldline can reach A. The computation result can be transmitted.
The Supertask Computer
A "supertask" is the completion of an infinite number of operations in finite time. In an M-H spacetime, this becomes physically possible: the computer experiences infinite time to complete the operations, while you wait only moments.
Try sending different computations to the supertask computer and see how infinite iterations compress into seconds of your time.
Calculate n! by multiplying all integers from 1 to n
Iterations Sent
—
Your Elapsed Time
0.0s
Computer Time
—
Solving the Halting Problem
In 1936, Alan Turing proved that no algorithm can determine, for all possible programs and inputs, whether the program will eventually halt or run forever. This halting problem is undecidable - fundamentally unsolvable by any Turing machine.
But an M-H computer can solve it trivially: just run the program! If it halts, send a signal. If after infinite time no signal was sent, it loops forever. The waiting observer knows the answer in finite time either way.
Turing proved this is impossible for any ordinary computer. But in an M-H spacetime, we can solve it by actually running the program forever.
for i in range(100): print(i)
How it works: The M-H computer runs the program. If it halts, it sends "HALTS" immediately. If it never halts, it never sends a message. After infinite time, if no "HALTS" signal arrived, we know it loops. All of this happens in finite time for the waiting observer.
What Would It Take?
Creating an M-H spacetime is not easy. The extreme curvature required demands either exploiting existing cosmic objects (like rotating black holes) or creating artificial spacetime geometry with exotic matter.
Calculate the requirements for different computation lengths and see why physical realizability remains highly questionable.
Rotating black hole - paths through inner horizon may have infinite proper time
Requires crossing Cauchy horizon
Inner horizon unstable
Likely destroyed by perturbations
Time Ratio
100:1
Required Curvature
20.0 Rs
Energy Density
-1.0e+16
J/m^3 (NEGATIVE)
Black Hole Mass
4.0 Msun
Key problem: Creating an M-H spacetime requires negative energy density - exotic matter that violates the weak energy condition. No known matter has this property, though quantum effects like the Casimir effect produce tiny amounts of negative energy.
Kerr Black Holes: Nature's M-H Candidates
Rotating black holes (described by the Kerr metric) have been proposed as potential natural M-H spacetimes. Unlike non-rotating black holes, they have a ring singularity and an inner "Cauchy" horizon that might allow worldlines with infinite proper time.
Explore the structure of a Kerr black hole and see where an M-H path might exist.
Outer Horizon
1.436M
Cauchy Horizon
0.564M
Time Dilation
1.54x
Ring Radius
0.900M
M-H potential: A worldline that crosses the Cauchy horizon and orbits near the ring singularity could have infinite proper time while remaining in the causal past of an external observer. However, the inner horizon is believed to be unstable - perturbations create an infinite blueshift that destroys anything crossing it.
Implications for Computability
If M-H spacetimes were physically realizable, the entire landscape of theoretical computer science would change. Problems proven mathematically impossible to solve would become tractable.
M-H spacetimes would collapse the computability hierarchy. Problems that are fundamentally unsolvable by Turing machines become trivially solvable:
Halting Problem
Determine if a Turing machine halts on a given input
Classical Computer
Undecidable
M-H Computer
Decidable
How M-H solves it:
Run the machine for infinite time; if it halts, send signal
The Church-Turing thesis would be violated. The class of physically computable functions would be strictly larger than Turing-computable functions. This has profound implications for the philosophy of mathematics and computation.
Why This Probably Does Not Work
While M-H spacetimes are mathematically valid solutions to Einstein's field equations, most physicists believe they cannot be physically realized. The obstacles are formidable.
While M-H spacetimes are valid solutions to Einstein's equations, their physical realizability faces serious challenges:
The Takeaway
Malament-Hogarth spacetimes reveal a profound connection between the structure of spacetime and the limits of computation:
Mathematically Valid
Solutions to Einstein's equations
Probably Not Physical
Multiple realizability obstacles
The question remains: Does physics fundamentally respect the Church-Turing thesis, or are there loopholes in the laws of nature?
Explore More Physics Explainers
Malament-Hogarth spacetimes connect to other deep questions about time, computation, and the nature of physical law. Explore our other interactive explainers.
References: Hogarth (1992), Earman & Norton (1993), Etesi & Nemeti (2002)