Malament–Hogarth Supertask Computer Calculator

Introduction

This calculator explores one of the strangest thought experiments in relativity and theoretical computer science. A Malament–Hogarth spacetime is a setting in which one observer can, at least in principle, receive the outcome of an unbounded computation while experiencing only a finite amount of personal time. The page below does not claim that a real machine can literally solve infinity-sized tasks. Instead, it gives a concrete numerical model for the central tradeoff: as an external computer sits closer to a black hole horizon, gravitational time dilation lets ever more external time and ever more computational steps fit into the interval available to an infalling observer.

The Malament–Hogarth spacetime concept reveals that general relativity admits bizarre scenarios in which an observer can witness the outcome of infinitely many computations executed by another agent in finite proper time. In such a spacetime, a computer stationed outside a black hole performs an unbounded sequence of steps while an infalling observer experiences only a finite duration before crossing the event horizon. By tuning the orbital radius of the external computer arbitrarily close to the horizon, the redshift factor becomes so extreme that the computer's run time, as measured by a distant clock, diverges toward infinity even though its own proper time remains bounded.

The philosophical appeal of this setup is obvious: it hints at a route to hypercomputation, meaning forms of computation that go beyond an ordinary Turing machine. The physical obstacles are equally obvious, which is why it remains a thought experiment rather than an engineering proposal. Even so, it is a valuable teaching example because the ingredients are familiar enough to calculate: black hole mass, orbital radius, time dilation, and a machine clock rate measured in operations per second.

How to Use the Calculator

The form asks for three inputs. First, enter the black hole mass in solar masses. Larger black holes have larger Schwarzschild radii, and in this simplified model they also give the infalling observer more proper time between horizon crossing and the singularity. Second, enter the fractional horizon offset $\epsilon$, which says how close the computer orbits to the horizon. Smaller values place the computer closer to the horizon, producing larger time dilation. Third, enter the computer speed in operations per second. You can use scientific notation such as 1e-6 or 1e12 in the numeric fields.

The calculator's interface requests three inputs. First is the black hole mass in solar masses. Larger black holes yield a longer proper time before the infaller is destroyed, because the radius and hence interior crossing time scale linearly with mass. Second, the user chooses the fractional offset $\epsilon$ describing how close the external computer orbits to the horizon. Finally, the computer speed sets the number of operations performed per second of its proper time. The output includes the Schwarzschild radius, the proper time to the singularity, the dilated external time, and the total operations executed.

After you press Compute Supertask, the results appear as four values. The Schwarzschild radius tells you the event horizon size of the chosen black hole. The proper time to singularity is the approximate interval available to the infaller once they have crossed the horizon. The dilated external time converts that interval into the much larger coordinate time available to the near-horizon computer. The final line multiplies that dilated time by the computer speed to estimate how many operations could be completed before communication becomes impossible in this idealized picture.

Logging Your Scenario

After running a supertask estimate, click the copy button to record horizon offsets, time dilations, and operation counts for further study. Keeping a log of different mass and epsilon combinations helps researchers compare how exotic geometries influence computational possibilities. If you try several values in sequence, a useful pattern quickly emerges: changing the mass changes the scale linearly, while pushing $\epsilon$ smaller can change the answer by many orders of magnitude.

Formula and Model

The calculator simulates this situation using a simplified Schwarzschild geometry. The event horizon of a non-rotating black hole of mass M occurs at the Schwarzschild radius $r_s=\frac{2GM}{c^2}$. We imagine a computer orbiting at radius $r=r_s(1+\epsilon )$, where $\epsilon$ is a tiny positive parameter supplied by the user. The closer $\epsilon$ is to zero, the closer the orbit is to the horizon and the greater the gravitational time dilation.

Gravitational redshift modifies the relationship between the proper time $d\tau$ experienced by the computer and the coordinate time $dt$ measured by a distant observer. In Schwarzschild coordinates, the factor is $\frac{dt}{d\tau }=\frac{1}{\sqrt{1-\frac{r_s}{r}}}$. Substituting $r=r_s(1+\epsilon )$ yields $\frac{dt}{d\tau }=\frac{1}{\sqrt{\frac{\epsilon }{1+\epsilon }}}$. This expression demonstrates how the dilation can be made arbitrarily large by choosing an extremely small $\epsilon$.

An infalling observer who dives through the horizon after waiting for a while outside will experience a finite proper time before reaching the central singularity. For simplicity we approximate this proper time as $\tau \approx \frac{\pi r_s}{2c}$, which gives about $5\times {10}^{-6}s$ per solar mass. The calculator multiplies this estimate by the mass entered by the user to obtain the duration available to the infaller from horizon entry to the singularity. During this interval, the external computer continues to run, executing operations according to its own clock rate but producing a vastly longer coordinate time because of redshift.

The number of operations accessible before communication becomes impossible is therefore $N=ft$, where $f$ is the computer's speed in operations per second, and $t$ is the dilated external time $t=\frac{\tau }{\sqrt{\frac{\epsilon }{1+\epsilon }}}$. As $\epsilon$ approaches zero, $t$ diverges and $N$ grows without bound. The computer can, in principle, complete an infinite loop and send a signal to the infaller indicating the result before they cross the horizon.

In practical terms, the calculator uses the black hole mass to compute the horizon size, uses the horizon offset to compute the redshift factor, converts the infaller's proper time into a far longer external time, and then multiplies by the supplied clock speed. That chain is simple enough to evaluate numerically, but it still captures the feature that makes the thought experiment famous: the closer the machine sits to the horizon, the faster the accessible operation count grows.

The following table summarizes the key constants used in the calculations:

By manipulating the inputs, you can probe how these constants interact to produce staggering temporal expansions. Setting the offset to ${10}^{-12}$ yields external times exceeding the age of the universe even for modest black hole masses, an illustration of the theoretical possibility of completing supertasks—procedures involving infinitely many steps—in finite personal time.

Worked Example

To illustrate, consider a ten solar mass black hole with a computer operating at ${10}^{12}$ operations per second and an offset $\epsilon ={10}^{-6}$. The proper time to reach the singularity is roughly $0.0005s$. The redshift factor at this radius turns that into an external time of approximately $500s$. The computer thus executes about $5\times {10}^{14}$ operations before the infaller reaches the singularity. By choosing a smaller $\epsilon$, the external time and operation count can be made arbitrarily large.

This worked example is useful because it shows the scale of the result in ordinary language. A fraction of a millisecond of proper time for the infaller becomes several minutes of external time for the computer, even before pushing the orbit especially close to the horizon. If you reduce the offset further, the operation count rises explosively. That does not prove that infinity is physically achievable, but it does show why relativistic time dilation is the key ingredient in the Malament–Hogarth argument.

Limitations and Assumptions

Although these numbers are mind-boggling, there are significant caveats. Quantum gravity might forbid the precise Schwarzschild geometry required. Tidal forces could shred the computer and the infaller. Moreover, the act of performing unbounded computation might consume enough energy to appreciably alter the spacetime, violating the assumptions of the model. Nevertheless, the Malament–Hogarth scenario remains a fruitful playground for exploring the limits of computation in relativistic settings.

Yet performing such a feat raises thorny issues. Communication from the external computer to the infaller must arrive before the horizon crossing, but signals emitted from arbitrarily close to the horizon suffer infinite redshift, diminishing their energy to zero by the time they escape. Engineers of this fantastical computer must therefore balance the orbital offset to permit both enormous computation and successful communication. The setup also presumes perfect isolation from perturbations that could knock the computer into the black hole prematurely.

In the philosophy of computation, such setups challenge the Church–Turing thesis, which posits that any physically realizable computation can be performed by a Turing machine. Malament–Hogarth spacetimes hint that if general relativity allows these extreme geometries, an observer could decide the halting problem for ordinary Turing machines by delegating the computation to a near-horizon computer. However, whether such spacetimes can exist in a realistic universe remains controversial.

Malament–Hogarth spacetimes have inspired logicians to classify tasks according to their computational strength. Some problems undecidable by Turing machines become decidable by supertasks, but others remain out of reach. The hierarchy of hypercomputation includes ω-machines, accelerating Turing machines, and other exotic constructs. While these models broaden our theoretical understanding, whether any correspond to physical reality remains an open question.

Finally, note that the calculator does not literally compute an infinite number of steps; it merely extrapolates the finite quantities required for the Malament–Hogarth argument. In a real experiment, the falling observer would inevitably encounter quantum gravitational effects before infinite computation could be realized. Nonetheless, the ability to stretch finite proper times into unbounded external durations underscores the profound interplay between relativity and computation.