Primordial Black Hole Evaporation Calculator

What this calculator shows

This calculator estimates three linked properties of a non-rotating, uncharged black hole from a single input mass: Hawking temperature, evaporation lifetime, and Schwarzschild radius. It is aimed especially at primordial black holes, which are hypothetical black holes that may have formed in the early universe rather than from dying stars. Primordial black holes are interesting because, unlike stellar-mass black holes, they could in principle exist at much smaller masses. In that low-mass regime, Hawking radiation is no longer just a tiny correction. It can dominate the long-term evolution of the object and eventually cause the black hole to evaporate away.

The central idea is simple even though the physics behind it is deep. In classical general relativity, a black hole is defined by an event horizon from which nothing escapes. In semiclassical gravity, however, quantum fields near that horizon lead distant observers to detect thermal radiation. That radiation is called Hawking radiation. Once a black hole emits energy, it loses mass. As the mass drops, the temperature rises. As the temperature rises, the emission becomes stronger. The result is a feedback loop in which smaller black holes evaporate faster and faster.

That is why mass is the only input needed here. For a Schwarzschild black hole, mass determines the horizon size, the surface gravity, the Hawking temperature, and the approximate lifetime. The calculator uses standard SI constants and the textbook formulas most often quoted for introductory and intermediate discussions of black hole evaporation. The outputs are therefore best read as clean baseline estimates. They are not a full astrophysical simulation, but they are very useful for intuition, teaching, and quick order-of-magnitude checks.

If you are exploring cosmology, dark matter ideas, or the observational consequences of Hawking radiation, these estimates help answer practical questions. Is a black hole of a given mass still around today, or would it already have evaporated? Is its temperature tiny, moderate, or extremely high? Is its event horizon merely small, or astonishingly microscopic? Putting all three answers together gives a compact picture of the physical regime you are studying.

How to use the calculator

Enter a black hole mass in kilograms and press the compute button. The calculator then returns the Hawking temperature in kelvin, the evaporation lifetime in years, and the Schwarzschild radius in meters. It also compares the lifetime with the approximate age of the universe and labels the object as either already evaporated or still evaporating. The result area updates immediately below the form.

Scientific notation is often the most practical way to enter values. For example, you can type 1e11 for 10¹¹ kilograms, 5e11 for 5 × 10¹¹ kilograms, or 1e15 for 10¹⁵ kilograms. These are convenient test cases because they span very different physical regimes. A mass near 5 × 10¹¹ kg is often used as a rough benchmark for a lifetime comparable to the age of the universe. Much larger masses live vastly longer, while much smaller masses evaporate much more quickly.

When you read the output, remember the scaling. Temperature falls as mass increases. Lifetime rises as the cube of the mass. Radius rises linearly with mass. Those three trends explain most of the behavior you will see. If you increase the mass by a factor of ten, the radius becomes ten times larger, the temperature becomes ten times lower, and the lifetime becomes one thousand times longer. That dramatic contrast is one reason primordial black holes are so useful in thought experiments and classroom examples.

It is often helpful to try several masses separated by powers of ten. Doing that makes the scaling laws obvious. You can start with 10⁵ kg, then 10¹¹ kg, then 10¹⁵ kg, and compare how strongly the lifetime changes relative to the radius. The calculator is especially good for this kind of quick exploration because it keeps the input simple while still reporting physically meaningful outputs.

Core formulas preserved in MathML

The Hawking temperature is commonly written as $T_H$. For a Schwarzschild black hole, the calculator uses the standard relation

In this expression, $\hslash$ is the reduced Planck constant, $c$ is the speed of light, $G$ is Newton's gravitational constant, $k_B$ is Boltzmann's constant, and $M$ is the black hole mass. The most important feature is the inverse dependence on mass. Double the mass and the temperature is cut in half. Reduce the mass and the temperature rises accordingly.

The approximate evaporation lifetime is given by

This formula shows why lifetime changes so dramatically across different masses. The dependence on $M^3$ means that a tenfold increase in mass produces a thousandfold increase in lifetime. That cubic scaling is the main reason large black holes survive for inconceivably long times while sufficiently small primordial black holes could evaporate within the current age of the universe.

The Schwarzschild radius is

This is the radius of the event horizon for a non-rotating, uncharged black hole. It is a classical quantity rather than a quantum one, but it provides a useful sense of scale. Even when the mass is enormous by everyday standards, the radius can still be microscopic. That contrast between huge mass and tiny size is one of the most striking features of black hole physics.

The calculator also relies on a simple energy interpretation of temperature through $E=k_{B}T$. This relation helps explain why hotter black holes emit more energetic quanta. As the black hole shrinks and the temperature rises, the characteristic energy of emitted particles rises as well. Near the end of evaporation, the process becomes extremely energetic in the semiclassical picture.

For readers who want the scaling summarized compactly, the formulas imply $T_H\propto M^{-1}$, $\tau \propto M^3$, and $r_s\propto M$. Those proportionalities are often the fastest way to build intuition before looking at exact numerical values.

Another useful comparison is between the lifetime and the age of the universe, which this page treats as approximately $4.35\times {10}^{17}\text{s}$. If $\tau$ is smaller than that benchmark, the black hole would have evaporated by now in this simplified model. If it is larger, the object could still exist today.

Because the formulas are evaluated in SI units, the calculator keeps the interpretation straightforward. Mass is entered in kilograms, temperature is returned in kelvin, lifetime is converted to years for readability, and radius is reported in meters. That consistency makes it easier to compare outputs across a wide range of masses without doing extra unit conversions by hand.

Worked example

Suppose you enter a mass of 5 × 10¹¹ kg. This is a classic example because it lies near the rough threshold where the evaporation lifetime is on the order of the present cosmic age. The calculator will return a tiny Schwarzschild radius, a relatively high Hawking temperature compared with ordinary astrophysical black holes, and a lifetime that is close enough to the age of the universe to make the cosmological interpretation interesting. In plain language, this is the kind of mass scale where primordial black holes become potentially relevant to present-day searches for evaporation signatures.

Now compare that with 10¹⁵ kg. The radius becomes larger in direct proportion to the mass, but the temperature drops sharply because temperature is inversely proportional to mass. The lifetime becomes enormous because of the cubic dependence. A black hole in this range would survive far longer than the current age of the universe. Its Hawking radiation would be weak, and the object would behave as a very long-lived relic on any human or even ordinary cosmological timescale.

At the opposite extreme, try 10⁵ kg. The calculator will show an extremely high temperature and a very short lifetime. This illustrates the runaway nature of evaporation. Once the mass is small enough, the black hole becomes so hot that it radiates intensely and disappears almost immediately in the semiclassical approximation. The radius is also extraordinarily tiny, reinforcing how compact such an object would be.

These examples show how to read the result block. The temperature tells you how energetic the radiation is. The lifetime tells you whether the black hole could plausibly survive to the present day. The Schwarzschild radius gives a geometric sense of size. The status line then summarizes the cosmological interpretation by comparing the lifetime with the age of the universe used in the script.

How to interpret the results

The evaporation lifetime is often the first output people care about. If the computed lifetime is shorter than the age of the universe, then a primordial black hole with that initial mass would not still be present today in this simplified isolated model. If the lifetime is longer, then the black hole could still exist. This does not prove that such an object exists in nature, but it does tell you whether survival to the present is even possible under the standard Hawking evaporation estimate.

The Hawking temperature is equally important because it indicates the characteristic energy scale of the emitted radiation. Large black holes are cold. Very small black holes are hot. For stellar-mass black holes and larger, the temperature is so low that Hawking radiation is negligible compared with the surrounding cosmic microwave background and other environmental effects. For sufficiently small primordial black holes, however, the temperature can become high enough that the emitted particles are energetic and potentially relevant to observational searches.

The Schwarzschild radius often surprises people most. Even a black hole with a mass far beyond any ordinary object can have an event horizon much smaller than an atom. That is one reason black holes feel so counterintuitive: enormous mass can be compressed into an incredibly small region. Seeing the radius next to the temperature and lifetime helps connect the geometry of the horizon with the thermodynamics of evaporation.

It is also worth noticing that the three outputs tell a single story rather than three unrelated facts. A larger mass means a larger horizon. A larger horizon corresponds to lower surface gravity. Lower surface gravity means lower Hawking temperature. Lower temperature means weaker emission, which in turn means a much longer lifetime. The calculator is useful because it makes that chain of reasoning visible with one input and one result panel.

Reference values and physical context

The following reference values are not a substitute for the live calculator, but they help anchor the scale of the problem. They show how quickly the physics changes as the mass moves across a few representative orders of magnitude.

The smallest example in the table is fantastically hot and short-lived. The middle example sits in the broad neighborhood where the lifetime becomes cosmologically interesting. The largest example is still tiny by astronomical standards, yet it already survives for a period vastly longer than the current age of the universe. That contrast is exactly what makes primordial black holes such a rich topic in cosmology and high-energy astrophysics.

For stellar-mass black holes, the Hawking temperature is far below the temperature of the cosmic microwave background. In practice, such black holes absorb more energy from their environment than they emit through Hawking radiation. Primordial black holes are different because they may occupy mass ranges where evaporation is not merely a theoretical curiosity but a process with possible observational consequences. Depending on the mass, they have been discussed as dark matter candidates, probes of early-universe density fluctuations, and possible sources of high-energy particles or transient signals.

Assumptions and limitations

This calculator uses the standard semiclassical formulas for a Schwarzschild black hole. That means it assumes the black hole is non-rotating and electrically neutral. Real black holes can carry angular momentum, and rotation changes the horizon structure and emission details. Charge is usually expected to be negligible in astrophysical settings, but it is still excluded here. So the outputs should be interpreted as clean Schwarzschild estimates, not exact predictions for every possible black hole.

The lifetime formula is also an approximation. It captures the dominant mass scaling very well, but it does not explicitly model the full set of gray-body factors or the changing number of particle species that can be emitted as the temperature rises. In more detailed treatments, the evaporation rate depends on which particles are available at a given temperature and on how efficiently they escape the gravitational potential. Those refinements matter for precision work, especially near the final stages.

The calculator also assumes the black hole evaporates in isolation. In reality, a black hole can absorb radiation and matter from its surroundings. For large black holes, environmental absorption can easily dominate over Hawking emission. For small primordial black holes in sparse environments, the isolated approximation is often more reasonable, but it remains an assumption. The result should therefore be read as an idealized estimate rather than a full environmental evolution model.

Finally, the late end of evaporation is uncertain because the semiclassical description is expected to break down near the Planck scale. Some theories suggest a final explosive phase, while others allow for remnants or more subtle quantum-gravity effects. None of those possibilities are included here. The calculator is best used as an educational and exploratory tool for the standard regime where the familiar Hawking formulas are most informative.

Even with those caveats, the page remains useful because it captures the main physical relationships in a transparent way. If you want a quick answer to whether a chosen mass corresponds to a cold, long-lived object or a hot, rapidly evaporating one, this calculator gives that answer immediately. It is a practical starting point for students, teachers, researchers, and curious readers who want to connect black hole thermodynamics with concrete numerical scales.