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
|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. 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 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 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 / 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.|
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.
Every quantity in nature has a second, opposite limit.
||Upper limit: the diameter of the universe||Makes sky dark at night. Implies cosmology.|
||Upper limit: the age of the universe||Makes sky dark at night. Implies cosmology.|
||`Practical' lower limit: gravity between two neutrinos separated by the universe's diameter||Maybe implies dark energy.|
||Upper limit: sum of all matter mass||Determines the fate of the universe.|
||Upper limit c5/4G ≈ 9 · 1051 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.
Everything in nature is limited. Nothing in nature is infinitely large
or infinitely small. One of the two limits is always a corrected Planck
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).