### Nature has its own book of records

Every quantity in nature is limited. This page presents the most famous of these limits. These extreme values cannot be exceeded: they are nature's record values.

In fact, almost every physical quantity has *two* limits: a corrected
Planck limit and a cosmological limit.

In the domain of quantum gravity, when the limits include all three
constants c, ℏ and G, the Planck limits *cannot be achieved or even
approached*. In those cases, there also exist real experimental
records.

This is a further page in the collection of true statements about nature and
physics that merit to be more widely known. `True' means checked by
experiments. References to such experiments can be found via Google Scholar.
A preprint collecting many of the recent references is given at the end.

### Corrected Planck limits

The limits are called *corrected* because they contain 4G instead of
G.

Physical quantity |
Limit and value | Record holders and consequences |

Speed of matter, energy and signals | Upper limit c ≈ 3 · 10^{8} m/s |
Achieved by light or gravitational waves. Implies special relativity, including the relativity of time and length. Implies the twin paradox. Measured regularly. |

Action, or change | Lower limit ℏ ≈ 10^{-34} Js |
Achieved by any elementary process, such as a photon absoption or a spin flip. Implies quantum theory, including indeterminacy relation or uncertainty relations, and the existence of atoms. Implies lasers, electronics and life. |

Entropy | Lower limit k ln2 ≈
10^{-23} J/K |
Achieved by quantum systems. Implies statistical thermodynamics. Implies thermodynamic indeterminacy (uncertainty) relations. |

Mass per length | Upper limit c^{2}/4G ≈
3 · 10^{24} kg/m |
Achieved by black hole horizons. Implies curvature and general relativity, as shown here. Regularly observed in our galaxy and in numerous other galaxies. |

Mass rate | Upper limit c^{3}/4G ≈ 10^{35} kg/s |
Achieved by black hole horizons. Implies curvature and general relativity, as shown here. |

Force | Upper limit c^{4}/4G ≈ 3 · 10^{43} N |
Achieved by black hole horizons. Implies curvature and general relativity, as shown here. Implies maximum gravitational fields and maximum electromagnetic fields. |

Power and Luminosity | Upper limit c^{5}/4G ≈ 9 ·
10^{51} W |
Achieved by black hole horizons. Implies curvature and general relativity, as shown here. Implies maximum luminosity. The most powerful observed source, a black hole merger, was dimmer by more than a factor 100. |

Length | Lower limit (4 G ℏ / c^{3})^{1/2} ≈
3 · 10^{-35} m |
The essence of quantum gravity. Space is continuous and
has a smallest measurable distance at the same time. Follows from the
previous limits. Not achieved anywhere. Implies the lack of singularities and
of point particles. Implies a smallest area and smallest volume.
Measurements of the electron dipole moment are
about a factor 10^{3} away. |

Time | Lower limit (4 G ℏ / c^{5})^{1/2} ≈
10^{-43} s |
Another way to put the essence of quantum gravity. Time
is continuous despite having a smallest measurable time interval. Follows from the
previous limits. Not achieved anywhere. Implies the lack of sudden jumps.
Also implies a maximum frequency and a maximum acceleration. Measurements
of time intervals are about a factor 10^{20} away. |

Probability density | Upper limit (4 G ℏ/c^{3})^{-3/2} ≈
3 · 10^{103} m^{-3} |
Follows from smallest volume. Limits wave functions in
quantum gravity. Consistent with other quantum gravity limits. Not
achieved anywhere. Measurements
of probability densities are more than a factor 10^{50} away. |

Elementary particle energy | Upper limit (ℏ c^{5}/4 G )^{1/2} ≈
1 GJ |
Follows from general relativity and quantum theory. Not achieved by any
elementary particle, anywhere. Implies a temperature limit. Measurements
of particle energies are more than a factor 10^{7} away. Measured
temperatures are more than a factor 10^{12} away. |

Elementary particle momentum | Upper limit (ℏ c^{3}/4 G )^{1/2} ≈
3.8 kg m/s |
Follows from general relativity and quantum theory. Not achieved by any
elementary particle, anywhere. Implies a pressure limit. Measured
temperatures are more than a factor 10^{12} away. |

Elementary particle mass | Upper limit (ℏ c/4 G )^{1/2} ≈
11 μg |
Follows from general relativity and quantum theory. Not achieved by any
elementary particle, anywhere. The most massive elementary particle
is more than a factor 10^{15} away from the limit. Implies,
with the smallest volume, a limit on mass density. |

The above limits are absolute. They are extremes, valid for every physical
system (or for single elementary particles, as mentioned).
These extreme bounds cannot be exceeded.

The above extremes are valid across the universe, at all times, across nature, across all sciences.

Note that the above table of correcetd Planck limits is not complete. There is a corrected Planck limit for almost every variable.

Electric charge is not in the list. Charge is quantized, but not limited. Still, charge quantization leads to limits for electromagnetic fields.

The limits restrict what can happen, what can be, and what can be achieved.

The limits are invariant: they are the same for every observer.

The limits *define* modern physics, as shown on
the page on Bronshtein's limit cube - when they are added to the principle of least action.
In that case, the limits describe everything that can happen.

When the limits are used to define physics, only a few are needed, as shown on the page summarizing physics in 9 lines.

The limits of special relativity, of general relativity and of quantum theory can be realized by physical systems. The limits of quantum gravity cannot.

No known physical system approaches the limits of quantum gravity – i.e., the limits containing ℏ, c and 4G – by several orders of magnitude. It is expected that this will never be possible.

Find an unknown limit - and publish it.

Find any exception to any corrected Planck limit - and become famous.

### Cosmological limits

Every quantity in nature has a second, opposite limit.

Quantity |
Limit |
Details |

Distance |
Upper limit: the diameter of the universe | Makes sky dark at night. Implies cosmology. |

Time |
Upper limit: the age of the universe | Makes sky dark at night. Implies cosmology. |

Force |
`Practical' lower limit: gravity between two neutrinos separated by the universe's diameter | Maybe implies dark energy. |

Mass |
Upper limit: sum of all matter mass | Determines the fate of the universe. |

Luminosity |
Upper limit c^{5}/4G ≈ 9 ·
10^{51} W |
Limits the brightness of the night sky. |

These cosmological limits are valid for the present universe.

Most cosmological limits change over time.

Have fun completing the list. Some missing limits are still unpublished.

### Summary

Everything in nature is limited. Nothing in nature is infinitely large
or infinitely small. One of the two limits is always a corrected Planck
limit.

### Reference

More detailed arguments and references are found in C. Schiller, From maximum force to physics in 9 lines and
towards relativistic quantum gravity,

published in Zeitschrift für Naturforschung A (2022).

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