Planck Star Bounce Time Calculator
Quantum-Gravitational Rebounds Inside Black Holes
In classical general relativity, a black hole that collapses under its own gravity is predicted to form a singularity: a point of effectively infinite density where the known laws of physics break down. Many physicists suspect that a complete theory of quantum gravity will avoid such singularities by modifying spacetime at extremely small (Planck) scales.
One proposal from loop quantum gravity is the Planck star. Instead of ending in a true singularity, the collapsing core of a black hole reaches a finite but extremely small radius, where quantum geometry effects become strongly repulsive. The core then undergoes a bounce, eventually re-expanding. Because of extreme gravitational time dilation, this internal bounce can unfold over a very long time as seen by a distant observer.
This calculator illustrates how two simple scaling relations, often used in qualitative discussions of Planck stars, translate into orders of magnitude for:
- External bounce time: how long a distant observer would wait between collapse and re-emergence.
- Minimum bounce radius: the smallest radius that the core reaches before rebounding.
The results are not predictions for real astrophysical observations. They are back-of-the-envelope estimates based on highly idealized, speculative models. Their main purpose is to show how extreme the relevant time and length scales are if such Planck star scenarios were realized in nature.
Planck Star Bounce Time Scaling Relation
In simplified loop quantum gravity-inspired treatments, the characteristic external bounce time for a Planck star is taken to scale quadratically with mass. A common heuristic relation is
External bounce time
where:
- is the bounce time measured by a distant observer.
- is a dimensionless bounce coefficient that captures model uncertainties, usually assumed to be of order 0.1–1.
- is the Planck time, approximately 5.39 × 10−44 s.
- is the black hole mass.
- is the Planck mass, about 2.18 × 10−8 kg.
The calculator asks for the mass in solar masses (M☉). Internally, this mass is converted to kilograms and then to Planck units before applying the relation above. Because of the quadratic dependence on , even modest increases in mass lead to an enormous growth in the external bounce time.
To provide more intuitive output, the tool typically reports bounce times in human-scale units (such as years) after converting from seconds.
Minimum Bounce Radius Scaling Relation
A separate scaling relation is often used for the minimum radius reached at the bounce. A simple choice consistent with some loop quantum gravity arguments is
Minimum Planck star radius
with:
- the minimum bounce radius.
- a dimensionless radius coefficient, typically assumed to be of order 1.
- the Planck length, about 1.62 × 10−35 m.
- and as defined above.
The dependence means that the minimum radius grows only slowly with mass. For astrophysical black holes the resulting values are still many orders of magnitude smaller than atomic scales, emphasizing how deeply quantum-gravitational the bounce region would be.
How to Use This Planck Star Bounce Time Calculator
The form above requires three inputs:
- Black Hole Mass (solar masses) – The mass expressed in units of the Sun's mass. Typical astrophysical black holes span roughly 3–100 M☉, while supermassive black holes range from 106 to 109 M☉ or more.
- Bounce Coefficient α – A dimensionless parameter that scales the external bounce time. Values between 0.1 and 1 are often quoted in the literature as order-of-magnitude examples, but there is no consensus value.
- Radius Coefficient β – A dimensionless factor that scales the minimum radius. Values around 1 are commonly assumed for illustrative calculations.
After entering your chosen values and clicking Compute Bounce, the calculator applies the scaling relations to estimate:
- External bounce time in seconds and in more intuitive units (such as years).
- Minimum bounce radius in meters.
Because of the enormous ratios between astrophysical masses and the Planck mass, you should expect numbers that vastly exceed everyday or even cosmological scales.
Interpreting the Results
The outputs of this tool are designed to convey the sheer magnitude of the scales involved in quantum gravity thought experiments. When interpreting the results, keep in mind:
- Even for masses well below one solar mass, the predicted bounce times can already exceed the current age of the Universe (~13.8 billion years).
- The minimum radii are so small that they lie far below the size of atomic nuclei, indicating that any bounce would occur in an ultra-compact region where our usual geometric intuition breaks down.
- For stellar-mass and supermassive black holes, the timescales may be so long that any bounce would be effectively unobservable on cosmic timescales.
Viewed this way, the calculator is best used as a way to develop intuition about how Planck units and black hole masses relate to each other, rather than as a tool for direct comparison with astronomical data.
Worked Example: 1 Solar-Mass Black Hole
Consider a hypothetical black hole of 1 M☉, with coefficients set to α = 0.1 and β = 1.
- Convert the mass to Planck units. One solar mass is about 1.99 × 1030 kg, while the Planck mass is about 2.18 × 10−8 kg. The ratio is therefore on the order of 1038.
- Estimate the bounce time. Squaring this ratio yields a factor of ~1076. Multiplying by the Planck time (~5.39 × 10−44 s) and α = 0.1 gives an external bounce time on the order of 1032 seconds, or about 1024 years.
- Estimate the minimum radius. Taking the cube root of the mass ratio gives a factor of roughly 1012. Multiplying this by the Planck length (~1.62 × 10−35 m) yields a minimum radius around 10−23 m, still vastly smaller than a proton (~10−15 m).
These approximate numbers are in line with the sort of outputs you should expect from the calculator for a 1 M☉ object with α ≈ 0.1 and β ≈ 1. The exact values will depend on the numerical constants and conversions implemented, but the key lesson is the same: the predicted bounce is unimaginably slow and extremely compact.
Sample Bounce Times and Radii
The table below sketches example scales for different masses using α = 0.1 and β = 1. Values are rounded and intended only as order-of-magnitude illustrations.
| Mass (M☉) | Bounce Time (years) | Minimum Radius (m) |
|---|---|---|
| 0.01 | ~1020 | ~10−24 |
| 1 | ~1024 | ~10−23 |
| 10 | ~1026 | ~5 × 10−23 |
| 106 | ~1036 | ~10−21 |
Moving down the table, you can see how quickly the bounce time inflates with mass (because of the M² scaling), while the minimum radius grows much more slowly (like M1/3). This contrast reflects the different physical roles of time dilation and spatial compression in these simplified models.
Assumptions and Limitations
This calculator is based on highly idealized, theoretical ideas rather than established, experimentally verified physics. When using or citing its outputs, it is important to keep the following points in mind:
- Speculative framework – Planck stars arise from certain approaches to loop quantum gravity. Neither the existence of Planck stars nor the detailed dynamics of black hole bounces have been observationally confirmed.
- Simplified scaling laws – The formulas implemented here are scaling relations meant for order-of-magnitude reasoning. They ignore many complications (spin, charge, realistic matter content, dynamic spacetime geometry, etc.).
- Model-dependent coefficients – The parameters α and β bundle together substantial theoretical uncertainty. Different authors may choose different values, and some models may imply a different mass dependence altogether.
- No predictive power for observations – The outputs are not intended to match any specific astrophysical signal (such as fast radio bursts or gamma-ray transients). They should not be interpreted as observational predictions or constraints.
- Extreme extrapolations – The calculations extrapolate well beyond experimentally tested energy scales. The validity of such extrapolations is uncertain.
- Educational purpose – The primary goal is to illustrate how Planck units, black hole masses, and speculative quantum gravity corrections might interplay, and to provide a sense of the extreme time and length scales involved.
Because of these limitations, this tool should be used only for conceptual exploration, pedagogy, or qualitative discussion—not for quantitative research conclusions.
Further Context and Comparison
To put the calculated numbers in perspective, it is useful to compare them to more familiar physical scales. The table below gives rough benchmarks.
| Quantity | Typical Scale | How Results Compare |
|---|---|---|
| Age of the Universe | ~1.4 × 1010 years | Bounce times for ~1 M☉ and above can exceed this by many orders of magnitude. |
| Human lifetime | ~102 years | Even the smallest masses considered here produce bounce times utterly beyond human timescales. |
| Proton radius | ~10−15 m | Minimum Planck star radii are often many orders of magnitude smaller, around 10−23 m or below. |
| Planck length | ~1.6 × 10−35 m | The bounce radius in this model remains larger than the Planck length but still extremely close to that fundamental scale. |
These comparisons underscore the main message of the calculator: if Planck star bounce scenarios are realized in nature with these simple scalings, they would involve time delays and length scales that are so extreme that direct verification would be extraordinarily challenging.