Quantum-Gravitational Rebounds inside Black Holes
The concept of a Planck star arises from loop quantum gravity, a theoretical approach that attempts to reconcile general relativity with quantum mechanics. In classical physics, a black hole collapsing under its own gravity is expected to reach a singularity, a point of infinite density where known laws break down. Loop quantum gravity suggests a different outcome: as the density approaches the Planck scale, quantum geometry exerts a repulsive force, halting collapse and causing a bounce. The interior undergoes a dramatic reversal, expanding outward, while to an external observer the process is obscured by gravitational time dilation. The collapsed object behaves like a compressed core that stores information and eventually releases it as the bounce propagates to the exterior. This speculative mechanism has been invoked to address the black hole information paradox and to conjecture astrophysical signatures such as fast radio bursts.
Although Planck stars remain hypothetical, their proposed dynamics can be summarized by simple scaling relations. The external bounce time—the time measured by a distant observer between collapse and re-emergence—is on the order of , where is the Planck time and is the Planck mass. The coefficient α encapsulates model-dependent uncertainties and is commonly assumed to be around 0.1–1. The bounce is unimaginably slow compared to human timescales; even for microscopic masses the predicted delay can exceed cosmic epochs.
The minimum radius at which the bounce occurs is also expected to scale with a simple power law, , where ℓP is the Planck length and β is a dimensionless factor of order unity. If loop quantum gravity's discretized geometry is correct, the spacetime volume at these scales is quantized, preventing the formation of singularities. Instead, the core reaches a finite size and rebounds.
This calculator implements these relations. By entering a black hole mass in solar masses and choosing coefficients α and β, you can explore how long a Planck star would remain hidden and how tiny its bounce radius would be. The goal is not to predict real astrophysical events but to familiarize oneself with the scales involved in quantum gravity scenarios. The numbers produced are extreme: bounce times dwarf the age of the universe for stellar-mass objects, while radii shrink to subatomic lengths. Such results emphasize the speculative nature of the idea and highlight the challenge of testing it observationally.
Sample Bounce Times
The table provides example outputs assuming α = 0.1 and β = 1. Times are given in years, and radii in meters. Even tiny masses lead to astonishingly long delays.
| Mass (M☉) | Bounce Time (years) | Radius (m) |
|---|---|---|
| 10-6 | 3.7×1011 | 2.0×10-24 |
| 1 | 3.7×1065 | 2.0×10-14 |
| 103 | 3.7×1071 | 9.3×10-13 |
How to Use the Calculator
Provide the mass of the black hole in solar masses and optionally adjust the coefficients α and β. Press “Compute Bounce” to obtain the bounce time and radius. The output expresses time in both seconds and years for convenience. Because the numbers quickly exceed standard notation, results are presented in scientific form.
Derivation of the Scaling Relations
Loop quantum gravity posits that areas and volumes are quantized in units of the Planck scale. The Planck time is defined as and the Planck length as . The Planck mass is . In models of Planck stars, the exterior time dilation relates internal dynamics to the observer frame such that collapse and bounce take roughly . The radius scaling arises from requiring that the interior reaches Planckian density before rebounding, leading to .
While these equations capture the essence of the proposal, they overlook numerous complexities. The collapse is highly dynamical; quantum pressure might depend on more than just density; and the translation between interior proper time and exterior coordinate time involves the black hole's spacetime geometry. Nevertheless, the simple power-law forms convey the dramatic separation between microscopic and macroscopic scales inherent in quantum gravity models.
Implications and Speculative Applications
If Planck stars exist, they could resolve the black hole information paradox by releasing information during the bounce, perhaps in the form of a brief burst of high-energy radiation. Some researchers have suggested that observed fast radio bursts might hint at such events, though this interpretation remains highly contentious. Another implication is the possibility of primordial black holes formed in the early universe: if they were small enough, some might be bouncing today, potentially contributing to cosmic backgrounds or strange transient signals.
From a theoretical standpoint, the existence of Planck stars would imply that spacetime has a granular structure at the Planck scale. This granularity could manifest in other phenomena, such as deviations from Lorentz invariance or modified dispersion relations for high-energy particles. Experimental confirmation of any such effect would revolutionize physics, but current observations have yet to yield conclusive evidence.
The calculator underscores the difficulty of observing Planck stars directly. Even a black hole with a mass comparable to a mountain would have a bounce time far exceeding the age of the universe. Consequently, if Planck stars do occur, they are effectively stable on observable timescales. Only extremely tiny primordial black holes could have bounced within cosmic history, and even then their signals might be buried in astrophysical noise.
Limitations
The formulas implemented here arise from heuristic arguments rather than complete solutions of quantum gravity. Parameters α and β are placeholders for ignorance; they may vary with mass, spin, or other properties. The scaling laws assume spherical symmetry and neglect Hawking radiation, accretion, and environmental interactions. Additionally, some researchers question whether a bounce would occur at all, proposing instead that black holes evaporate via Hawking radiation without ever reversing collapse. The calculator therefore serves as an educational toy rather than a predictive tool.
Using the Results
Despite their speculative nature, Planck stars offer fertile ground for thought experiments. By plugging in different masses, you can gauge how quantum gravity might intervene in gravitational collapse. If you set the mass to 10-5 solar masses and keep α = 0.1, the predicted bounce time is already greater than 1051 years, rendering the event practically eternal. Such mind-bending figures invite reflection on the scale of the universe and the frontiers of theoretical physics.
All calculations are performed locally in your browser. The constants used are the solar mass M☉ = 1.98847×1030 kg, Planck time tP = 5.391×10-44 s, Planck length ℓP = 1.616×10-35 m, and Planck mass mP = 2.176×10-8 kg. The equations apply those constants directly without approximation. By experimenting with the inputs, you can see how sensitive the predictions are to even tiny variations in mass or coefficients, highlighting the challenge of probing quantum gravity experimentally.