Extreme physics: records of nature

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, whenever a limit includes 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 page in the collection of correct statements about nature and physics that merit to be more widely known. `Correct' 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 · 10⁸ m/s	Achieved by light or gravitational waves. Implies special relativity, including the relativity of time and length, as shown here.. Implies the twin paradox. Measured regularly.
Action, or change	Lower limit ħ ≈ 10^-34 Js	Achieved by any elementary process, such as a photon absorption or a spin flip. Implies quantum theory, including the indeterminacy relation or uncertainty relation, and the existence of atoms, as shown here.. Implies lasers, electronics and life.
Entropy	Lower limit k ln2 ≈ 10^-23 J/K	Achieved by quantum systems. Implies statistical thermodynamics, as shown here.. Implies thermodynamic indeterminacy (uncertainty) relations.
Mass per length	Upper limit c²/4G ≈ 3 · 10²⁴ 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 c³/4G ≈ 10³⁵ kg/s	Achieved by black hole horizons. Implies curvature and general relativity, as shown here.
Force	Upper limit c⁴/4G ≈ 3 · 10⁴³ 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 c⁵/4G ≈ 9 · 10⁵¹ W	Achieved by same black hole horizons. Limits the brightness of the night sky. 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.
Angular momentum per mass squared	Upper limit G/c ≈ 2.2 · 10^-19 m²/skg	Valid for large matter objects kept together by gravity. Achieved by extremal black holes. Limits the angular momentum of objects. Observed values are very close to G/c.
Angular momentum per mass squared	Lower limit G/c ≈ 2.2 · 10^-19 m²/skg	Valid for elementary particles.
Length	Lower limit (4 G ħ / c³)^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 10³ away.
Time	Lower limit (4 G ħ / c⁵)^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 10²⁰ away.
Probability density	Upper limit (4 G ħ/c³)^-3/2 ≈ 3 · 10¹⁰³ 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 10⁵⁰ away.
Elementary particle energy	Upper limit (ħ c⁵/4 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 10⁷ away. Measured temperatures are more than a factor 10¹² away.
Elementary particle momentum	Upper limit (ħ c³/4 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 10¹² 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 10¹⁵ away from the limit. Implies, with the smallest volume, a limit on mass density.
Black hole mass	Lower limit (ħ c/4 G )^1/2 ≈ 11 μg	Follows from general relativity and quantum theory. Not achieved by any black hole, anywhere.

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 all three constant 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.
Mass	Upper limit: sum of all matter masses	Determines the fate of the universe.
Force	`Practical' lower limit: gravity between two neutrinos separated by the universe's diameter	May be related to dark energy.

These cosmological limits are valid for the present universe.

The cosmological limits change over time.

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

Curious limit

No human has jumped so much that it stayed more than about 1s in the air.

Summary

Every observable 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).