Introduction to the Malament–Hogarth black-hole supertask model
This Malament–Hogarth supertask calculator turns an abstract idea from relativity and theoretical computer science into a concrete numerical estimate. The thought experiment asks whether spacetime geometry could let one observer receive the result of an unbounded computation while experiencing only a finite amount of proper time. Rather than claiming that real engineers can build an infinite computer beside a black hole, this page models the central mechanism behind the idea: a machine placed extremely close to a Schwarzschild event horizon can accumulate a huge amount of externally measured time while an infalling observer still has only a short, finite interval left on their own clock.
That distinction between proper time and external time is the heart of the calculation. Proper time is what a local clock experiences along its own path. External time, in this simplified presentation, is the stretched coordinate time associated with a near-horizon location when compared with a far-away frame. If the computer hovers or orbits at radius just above the Schwarzschild radius, its clock relation to that external frame picks up an enormous redshift factor. The infaller still reaches the singularity in finite proper time, yet the model says more and more outside time fits into that window as the computer's radius approaches the horizon.
The philosophical interest is obvious. If a computer can keep checking, step after step, whether an ordinary program ever halts, then in a genuine Malament–Hogarth spacetime the infalling observer might receive a signal announcing the answer before their journey ends. In practice, however, this remains a teaching model rather than a laboratory proposal. The calculator is valuable because it isolates a few quantities that people can actually reason about: black hole mass, a tiny offset from the horizon, a clock rate measured in operations per second, and the redshift factor linking one clock to another.
Used that way, the page helps you see what really drives the headline result. The mass sets the scale of the event horizon and the approximate proper time available to the infaller after crossing it. The fractional offset from the horizon determines how extreme the time dilation becomes. The computer speed translates that time budget into an operation count. When you vary those three inputs, you can watch the model move from modest relativistic stretching to mind-bendingly large values that illustrate why Malament–Hogarth spacetimes occupy such an unusual place in discussions of hypercomputation.
How to Use the Calculator for a near-horizon supertask estimate
This Malament–Hogarth input form asks for exactly the three parameters needed by the simplified Schwarzschild model. First, enter the black hole mass in solar masses. A larger mass means a larger Schwarzschild radius and, in this approximation, a longer proper time between horizon crossing and the singularity. Second, enter the fractional horizon offset . This value tells the calculator how close the computer sits to the horizon. Smaller positive values place the machine closer to the horizon and therefore produce larger time-dilation factors. Third, enter the computer speed in operations per second. Scientific notation such as 1e-6 or 1e12 is supported by the form and by the calculation script.
For the offset, it helps to think in relative rather than absolute distance. The model places the computer at radius , so is not measured in meters. It is the fractional amount by which the orbital radius exceeds the event-horizon radius. Choosing = 10-6 means the computer is one part in a million above the Schwarzschild radius. Choosing 10-12 pushes the model dramatically closer, which is why the result can jump by many orders of magnitude even when the mass and computer speed stay fixed.
After you press Compute Supertask, the result box reports four quantities. The Schwarzschild radius gives the event-horizon size of the chosen black hole in meters. The proper time to singularity is the approximate time available to the infaller after crossing the horizon. The dilated external time applies the redshift factor to that proper time to estimate how much external time the near-horizon computation can exploit in this model. Finally, the calculator multiplies that external time by the computer speed to estimate a total number of operations.
When you interpret the output, look for scale rather than false precision. These numbers are not predictions for a real mission plan; they are a way to compare scenarios. If doubling the mass roughly doubles both the horizon size and the infaller's proper time, the page is reflecting the linear mass dependence built into the approximation. If shrinking makes the accessible external time explode, the page is showing the near-horizon divergence that motivates the Malament–Hogarth discussion in the first place.
Logging your Malament–Hogarth scenario
This Malament–Hogarth calculator includes a copy button so you can record the exact combination of mass, horizon offset, and computer speed that produced a result. That is helpful when you want to compare several runs side by side, especially because the model has two different kinds of scaling. Changing mass mostly rescales the horizon and proper-time window in a straightforward way, while pushing smaller can transform the redshift factor far more dramatically. A saved log of several scenarios makes those contrasting trends easy to see.
Formula and Model for the Schwarzschild supertask estimate
This Malament–Hogarth formula section spells out the exact simplifications used by the calculator so you can see where each output comes from. The page assumes a non-rotating Schwarzschild black hole, not a rotating Kerr black hole and not a full causal proof of a true Malament–Hogarth spacetime. Within that restricted setup, the event horizon for mass M lies at the Schwarzschild radius . We then imagine a computer orbiting at radius , where is a tiny positive parameter supplied by the user.
Time dilation enters through the Schwarzschild redshift factor. Gravitational redshift modifies the relationship between the proper time experienced by the computer and the coordinate time measured by a distant observer. In Schwarzschild coordinates, the factor is . Substituting yields . This is the mathematical reason the output grows so quickly when becomes tiny.
The infalling observer still has only finite time left, and the calculator approximates that remaining proper time with . In the useful rule of thumb quoted on the page, that is about per solar mass. So a stellar-mass black hole gives milliseconds or less, while a supermassive black hole gives much longer proper times in direct proportion to mass. The model uses the user-entered mass to compute , then computes , and finally applies the redshift factor.
The operation count follows from simple multiplication. The number of operations accessible before communication becomes impossible is therefore , where is the computer's speed in operations per second, and is the dilated external time . As approaches zero, diverges and grows without bound in the idealized model. That does not prove a machine can really complete an infinite task, but it does show exactly where the temptation toward hypercomputation arises.
In practical reading, the chain of reasoning is straightforward. Mass determines the basic horizon scale. The proper-time approximation turns that scale into a finite personal timeline for the infaller. The near-horizon offset converts that proper time into a much larger external duration through the redshift factor. The supplied clock rate then converts time into total operations. Because each stage corresponds to a visible input or output on the page, you can trace how the calculator arrives at its result rather than treating the final numbers as a black box.
The following table summarizes the key constants used in the calculations:
| Constant | Symbol | Value |
|---|
| Gravitational constant | | 6.674×10-11 m3kg-1s-2 |
| Speed of light | | 2.998×108 m/s |
| Solar mass | | 1.989×1030 kg |
Those constants are ordinary, but the combination is extraordinary. Even a modest black hole mass paired with a minute horizon offset can yield external times that dwarf everyday intuition. Setting the offset to can push the model to enormous durations, which is why the script also checks for overflow and warns you when the numeric range becomes too extreme for a clean floating-point result.
Interpreting a Malament–Hogarth result
This Malament–Hogarth result should be read as an idealized scaling estimate, not as evidence that infinity has been domesticated. If the page reports a huge number of operations, the important lesson is that the near-horizon geometry is amplifying the external time available to the computation. If the number changes by many orders of magnitude after only a small change in , that is exactly what the Schwarzschild redshift formula predicts. In other words, the calculator is best used to understand sensitivity, divergence, and tradeoffs rather than to make claims about a physically realizable supercomputer.
Worked Example: a 10-solar-mass black hole with ε = 10-6
This Malament–Hogarth worked example uses the calculator's default values so you can check what the page is doing with realistic-looking scientific notation. Start with a ten solar mass black hole, a computer operating at operations per second, and an offset . The proper time to reach the singularity is roughly . The redshift factor at this radius turns that into an external time of approximately . The computer thus executes about operations before the infaller reaches the singularity.
Read that result in everyday language and the contrast becomes easier to grasp. From the infaller's perspective, the remaining trip is almost over immediately; the proper-time budget is only a fraction of a millisecond. Yet the near-horizon computer, viewed through the redshift model, effectively gains several minutes of externally measured time in which to run. That is already enough for a trillion-operations-per-second machine to execute hundreds of trillions of steps. If you reduce the offset further, the model makes the accessible computation grow explosively.
The example also highlights why the inputs play different roles. Holding the mass fixed while shrinking attacks the denominator of the redshift term and therefore increases the external time nonlinearly. Holding fixed while increasing the mass mostly stretches the horizon size and infaller proper time linearly. Finally, increasing the computer speed does not change the geometry at all; it simply scales the final operation count. This separation of effects makes the calculator a useful classroom example because you can change one variable at a time and immediately see what part of the story it controls.
Limitations and Assumptions of the Malament–Hogarth horizon model
This Malament–Hogarth limitations section matters just as much as the formulas because the calculator is intentionally idealized. The page assumes a simple Schwarzschild geometry, ignores the engineering problem of keeping a computer in a near-horizon orbit, and treats the redshift factor as the main quantity of interest. A real black hole environment would introduce tidal forces, radiation hazards, finite hardware tolerance, orbital instability, and potentially severe back-reaction from any machine powerful enough to perform vast computation. The script therefore illustrates the scaling argument, not a buildable device.
Communication is another major caveat. In the thought experiment, it is not enough for the computer to perform an enormous number of steps; it must also send a usable signal to the infalling observer at the right time. Signals emitted from extremely close to the horizon suffer severe redshift, and any realistic setup would have to worry about whether the message still carried enough energy and arrived on the correct causal path. The optional mini-game on this page visualizes that tension by rewarding close approaches to the horizon while making successful relays harder to maintain.
The calculator also uses a very compressed proper-time estimate for the infaller. That approximation is convenient for a fast browser tool, but a full general-relativistic treatment would require specifying worldlines, initial conditions, and causal structure more carefully. Many discussions of true Malament–Hogarth spacetimes focus on rotating or otherwise more exotic geometries rather than the stripped-down Schwarzschild picture used here. So although the output captures the famous divergence of near-horizon time dilation, it should not be mistaken for a rigorous proof that the exact scenario on the page is a genuine physically realized Malament–Hogarth spacetime.
There are also foundational limits outside classical relativity. Quantum gravity may alter the interior picture before the singularity is reached. Hawking radiation, quantum instability, or cosmic censorship considerations may change the long-term story. If the machine stores memory, dissipates heat, or consumes energy in any realistic way, those resource requirements could become dominant well before the model's operation count becomes spectacular. In that sense, the calculator is closer to a conceptual map than a forecast.
Even with those warnings, the model remains useful. It shows why debates about hypercomputation keep returning to relativity: the mathematics of time dilation can make finite personal time coexist with arbitrarily large external time in idealized settings. That coexistence is enough to sharpen philosophical questions about the Church–Turing thesis, the meaning of physically possible computation, and the difference between mathematical consistency and engineering feasibility. The numbers on this page are best understood as a quantitative guide to that debate rather than as evidence that an infinite loop can be turned into a practical algorithm.