● Consequences for general relativity
● Different views
● Consequences for unification
● Bibliography on maximum force
● Personal publication pdfs on maximum force
● Personal preprints on maximum force
50 years of maximum force
There is no way, in nature, to measure a force value at a point that is larger than c4/4G or a power value larger than c5/4G. This is neither possible with black holes, nor by combining force or power values from different sources. For an overview of the topic that includes the most recent research, see my publication that appeared as Physical Review D 104 (2021) 124079.
The first published statement on the maximum force c4/G was by Elizabeth Rauscher in 1973; her result dates from 1968, in an internal report of the Lawrence Livermore National Laboratory. Unfortunately, the report is lost. Also in 1973, Dennis Sciama wrote about maximum power c5/G. For stars, maximum power implies a maximum luminosity.
About a dozen people independently rediscovered maximum force after Elizabeth Rauscher, including De Sabata and Sivaram. The first statement on maximum force in a research paper that included the correct factor 1/4 in c4/4G was by Gary Gibbons in 2002. Published statements on maximum force c4/4G also appeared around 2000 in my Motion Mountain Physics Textbook, and in 2003, in the related preprint arxiv.org/abs/physics/0309118. That preprint includes the principle of maximum force, i.e., the statement that general relativity follows from maximum force. In the past twenty years, maximum force, maximum power and maximum mass flow rate c3/4G became a topic of research around the world.
Astonishingly, Einstein, Wheeler, Hawking, Dyson and many others overlooked the force and power/luminosity limits. In 2011, I exchanged emails with Freeman Dyson, explained the limits, and asked whether he had explored them as well. In his answer he wrote: "It is not true that I proposed the formula c5/G as a luminosity limit for anything. I make no such claim." On the other hand, he did not contradict the limit either. In 2021, when I discovered that Elizabeth Rauscher was the first to have written about maximum force, she had already passed away.
There are at least two ways to show that maximum force implies general relativity (as explained in detail in the mentioned PRD). The first: Maximum force implies the first law of black hole mechanics, and vice versa. The first law in turn implies the field equations, and vice versa.
The second, developed with C. Sivaram and A. Kenath: Maximum force also describes the elasticity of space. Thus, maximum force determines the maximum shear strength of space. In other terms, maximum force determines the elastic limit of space. Space elasticity in turn implies the field equations, and vice versa.
Consequences of maximum force for general relativity
Maximum force is a simple statement. The simplicity strongly suggests that general relativity, which follows form maximum force, is correct – without any modification. Indeed, the principle of maximum force predicts – at least since 2002 – the lack of any deviation from general relativity for strong gravitational fields. (I also propose a bet on the topic.)
Two recent experimental papers, "Strong-field Gravity Tests with the Double Pulsar" https://arxiv.org/abs/2112.06795 (with 30 authors) and "Tests of General Relativity with GWTC-3" https://arxiv.org/abs/2112.06861 (with over 1600 authors), confirmed the lack of modifications of general relativity.
Maximum force allows an extremely simple derivation of the inverse square law of gravity from general relativity. This is told in the mentioned PRD and in a preprint below.
Maximum force implies, as told in a preprint below, the hoop conjecture. Also in this case it appears that the reverse is correct. Here are the steps.
1. The field equations imply a maximum force value c⁴/4G. This idea is, in its origin, about 50 years old.
2. Maximum force is only achieved at horizons. Force values are always smaller than the maximum at locations away from horizons. This is analogous to maximum speed: it only arises for massless particles; it does not arise in other cases.
3. Any attempt to produce a force larger than the maximum value c⁴/4G produces a horizon that prevents a larger value to appear. This is analogous to maximum speed: if you try to exceed it, things get difficult.
4. If no horizon would or could appear, a force value higher than c⁴/4G could be achieved. This would be in contrast with the field equations. This is because the field equations can also be deduced from maximum force. This result is 20 years old.
5. When a mass is concentrated as tightly as possible, in particular, smaller than its own Schwarzschild radius, this is an attempt to exceed the maximum force.
6. A horizon always arises in such a case; a lack of horizon would contradict maximum force and thus would contradict the field equations. This yields (and proves) the hoop conjecture.
For the same reasons, the equivalence of general relativity and maximum force also proves the weak cosmic censorship conjecture.
There are several physicists who disagree completely with the idea of maximum force. One counter-argument arises by adding force or power values at different points or coordinates. This allows to exceed the force limit; however, also the speed limit c can be exceeded this way. The force and power limits make no statement at all for sums of forces at different points.
Some people dislike the use of force in general relativity. However, despite hugely successful internet videos stating the opposite, gravity remains a force in relativity. (Since Newton, force is defined as momentum change with time.)
So far, most of those who are against maximum force are starting from the idea that G is not a fundamental constant, but that it is an effective quantity due to a different, deeper, underlying interaction or mechanism.
In that case, G would depend on energy, and gravity would differ at different length or energy scales. At present, there is no hint that G depends on energy: neither experimental nor theoretical. Several theoretical arguments even speak against the option. They are discussed and cited in the PRD. To make varying G into a sufficient counter-argument, experimental evidence against a constant G is needed. So far, there is none.
On the other hand, arguing that general relativity is simple and beautiful because it follows from maximum force, is not sufficient to prove general relativity right either. Experimental evidence is needed. So far, all experiments confirm maximum force.
Also arguing that general relativity and maximum force imply black hole entropy is not an argument. Experimental evidence is needed. So far, there is none about black hole entropy. (There is supporting evidence about sonic analogies.)
One can say that the real issue is the following: is general relativity an approximation or not? Only experimental evidence can decide. So far, all experiments confirm general relativity.
If experiments ever find that G varies with energy or distance, then general relativity and maximum force are falsified. If experiments ever find a deviation from general relativity, then maximum force and constant G are falsified. If experiments ever find a deviation from maximum force, then general relativity and constant G are falsified. In any of these cases, I would loose several bets.
Consequences for unification
The principle of maximum force completes the proof that physics can be summarized in 9 lines, as shown in the dedicated page. The 9 lines describe all of nature.
The principle of maxmimum force transforms the Bronshtein cube into a Bronshtein limit cube.
Maximum force might well be the "last" law of physics to be discovered. The argument is strengthened by the past decades of precision experiments. If there is really nothing beyond the standard model and general relativity, then maximum force completes the laws of physics. If, instead, any one effect beyond general relativity or beyond the standard model is found, then, of course, maximum force is not the last law of nature.
Maximum force implies, together with maximum speed and the quantum of action, that there is no trans-Planckian physics.
Maximum force is predicted, like maximum speed and the quantum of action, to hold also in any theory of quantum gravity.
Maximum force is predicted, like maximum speed and the quantum of action, to hold also in any unified theory.
Maximum force was the trigger that led to the description of fundamental motion with the strand conjecture. The strand conjecture predicts that there is no physics beyond general relativity and the standard model, but that the fundamental constants of nature can be calculated.
Maximum force is a simple statement. It agrees with all observations.
Maximum force is expected and predicted to remain valid, like maximum speed and the quantum of action, in quantum gravity and in the final theory.
No statement in physics is ever definitively correct. In the case that you have an argument against maximum force or against maximum power, publish it and get in touch.
Bibliography on maximum force
E.A. Rauscher, Einstein’s Field Equations and the Quantal Force, Lawrence Livermore National Laboratory, UCRL-71435, October 1968. (This internal report seems lost, even at LLNL. Please help find a copy!)
E. A. Rauscher, The Minkowski metric for a multidimensional geometry, Lett. Nuovo Cim. 7S2, 361-367 (1973). doi.org/10.1007/BF02735134. In it, she wrote: "F can be considered an upper bound on force".
R.H. Castellano, A Modified Theory of Gravitation, Transactions of the Nebraska Academy of Sciences and Affiliated Societies (1976) 398.
H.-J. Treder, The planckions as largest elementary particles and as smallest test bodies, Foundations of Physics 15 (1985) 161-166.
R. J. Heaston, Identification of a superforce in Einstein field equations, Journal of the Washington Academy of Sciences, 80 (1990) 25-36.
V. de Sabbata & C. Sivaram, On limiting field strengths in gravitation, Found. Phys. Lett. 6, 561-570 (1993). doi.org/10.1007/BF00662806.
L. Kostro & B. Lange, Is c4/G the greatest possible force in nature?, Phys. Essays 12, 182-189 (1999).
G.W. Gibbons, The maximum tension principle in general relativity, Found. Phys. 32, 1891-1901 (2002). doi.org/10.1023/A:1022370717626.
C. Schiller, Maximum force and minimum distance: physics in limit statements, arXiv:physics/0309118.
C. Schiller, General relativity and cosmology derived from principle of maximum power or force, Int. J. Theor. Phys. 44, 1629-1647 (2005). doi.org/10.1007/s10773-005-4835-2.
M.P. Dabrowski & H. Gohar, Abolishing the maximum tension principle, Phys. Lett. B 748, 428-431 (2015). doi.org/10.1016/j.physletb.2015.07.047. The paper shows that in other models of gravity, the maximum force does not hold.
V. Cardoso, T. Ikeda, C.J. Moore and C.-M. Yoo, Remarks on the maximum luminosity, Physical Review D 97 (2018) 084013.
Numerous additional papers have treated the topic of maximum force, mostly confirming it. The latest attempts to construct counter-examples were published by V. Faraoni, Phys. Rev. D 103, 124010 (2021) and by A. Jowsey & M. Visser, arXiv:2102.01831. However, because these publications added forces acting at different locations in space, the counter-examples turn out to be only apparent. A detailed exploration shows that, in the claimed configurations of those papers, local maximum force is never exceeded, as explained in C. Schiller, Comment on "Maximum force and cosmic censorship", Physical Review D 104 (2021) 068501.
An general update on the topic, with more references, is C. Schiller, Tests for maximum force and maximum power, Physical Review D 104 (2021) 124079, 10.1103/PhysRevD.104.124079. Also at arxiv.org/abs/2112.15418.
The first paper treating maximum mass flow rate c3/4G is Li-Ming Cao, Long-Yue Li and Liang-Bi Wu, A Bound on the Rate of Bondi Mass Loss, Physical Review D 104 (2021) 124017. Also at arxiv.org/abs/2109.05973.
See also the different view by G. E. Volovik, Negative Newton constant may destroy some conjectures, Modern Physics Letters A 37, 2250034 (2022). https://doi.org/10.1142/S0217732322500341.
The papers by Naresh Dadhich explore the topic in other theories of gravity. See Naresh Dadhich, Maximum Force for Black Holes and Buchdahl Stars, arxiv.org/abs/2201.10381.
Another interesting addition is: S. Di Gennaro, M.R.R. Good and Y.C. Ong, Black hole Hookean law and thermodynamic fragmentation: Insights from the maximum force conjecture and Ruppeiner geometry, Physical Review Research 4 (2022) 023031.
The 2021 hoop formulation by G. Liu and Y. Peng, A conjectured universal relation for black holes and horizonless compact stars, Nuclear Physics B (https://www.sciencedirect.com/science/article/pii/S0550321321001826) can be seen as a confirmation that in nature, maximum force limits energy per diameter.
See also R.G. Torromé, Maximal acceleration and black hole evaporation, arxiv.org/abs/2203.09483.
Personal publication pdfs on maximum force
C. Schiller, From maximum force via the hoop conjecture
to inverse square gravity, (2021), accepted for publication.
Download the preprint here (3 pages).
The equivalence of maximum force c⁴/4G and the field equations of general relativity provides a simple derivation of inverse square gravity. The derivation confirms the hoop conjecture and suggests a lack of gravitational physics beyond general relativity. Possible loopholes are pointed out.
Two ways to deduce the equivalence of the field equations of general relativity and the principle of maximum force c⁴/4G – or the equivalent maximum power c⁵/4G – are presented. A simple deduction of inverse square gravity directly from maximum force arises. Recent apparent counter-arguments are refuted. New tests of the principle in astronomy, cosmology, electrodynamics, numerical gravitation and quantum gravity are proposed.
Despite suggestions to the contrary, no counterargument to the principle of maximum force or to the equivalent principle of maximum power has yet been provided.
C. Schiller, Simple
derivation of minimum length, minimum dipole moment and lack of space–time
continuity, International Journal of Theoretical Physics 45
Download it at doi.org/10.1007/s10773-005-9018-7.
Read the published paper online for free at rdcu.be/cdG3E.
Download the preprint here.
C. Schiller, General relativity and
cosmology derived from principle of maximum power or force,
International Journal of Theoretical Physics 44 (2005)
1629–1647. Download it at
the published paper online for free at rdcu.be/cdG3C.
Download the preprint here.
Personal preprints on maximum force
The submission to the Gravity Research Foundation 2022 Awards for Essays on Gravitation
was selected for honorable mention: A. Kenath, C. Schiller and C. Sivaram, From
maximum force to the field equations of general relativity - and implications, preprint (2022)
Download the preprint here (12 pages).
A paper with C. Sivaram and A. Kenath is in work. An early preprint is found here https://www.preprints.org/manuscript/202109.0318/v1.
C. Schiller, Present physics in 9 lines - and implications for its future, preprint (2022)
Download the preprint here (9 pages).
C. Schiller, The quadruple gravitational constant, the Bronshtein cube of limits, and the future of
fundamental physics, preprint (2022)
Download the preprint here (5 pages)
The preprint explained – probably for the first time – how the force limit F≤c⁴/4G implies general relativity. As a result, it became possible to state that the speed limit v≤c yields special relativity, the action limit W≥ℏ yields quantum theory, and the force limit F≤c⁴/4G yields general relativity.