### A maximum force c4/4G and a maximum power c5/4G hold in nature – in addition to maximum speed c.

Maximum speed c ≈ 3⋅108 m/s defines and implies special relativity. In the same way, maximum force c4/4G ≈ 3⋅1043 N – or, equivalently, maximum power c5/4G ≈ 9⋅1051 W or maximum mass flow rate c3/4G ≈ 1035 kg/s – together with maximum speed c, define and imply general relativity. This page introduces the arguments that led to the result, presents the consequences and discusses various tests.

### 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.

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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.

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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.

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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.

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For the same reasons, the equivalence of general relativity and maximum force also proves the weak cosmic censorship conjecture.

### Different views

There are several physicists who disagree completely with the idea of maximum force. Some think that the maximum does not exist; others disagree on the value of the maximum.

A statement that no maximum value exists is always at the border of science, as it is not testable. Nevertheless, it is often heard. So far, nobody ever published a statement that 100 or 1000 times c^4/4G (or c^5/5G) is found in nature. No system that achieves such a large value has ever been proposed. Various papers, including my own, state that this is impossible; all use the argument that gravitational horizons prevent this. Nevertheless, many scholars disagree.

One counter-example against a maximum 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, power and speed limits make no statement at all for sums of values at different points in space.

Regularly, people suggest that oppositely charged black holes attract each other with a larger force than the limit value. A common trap! No, the force never exceeds c^4/4G. The force limit even applies to rotating black holes.

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.)

Many scholars who are against maximum force start 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 my 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. And 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 three cases, I would loose several bets.

In short, nobody has ever provided evidence that a physical system can exceed c^4/4G or c^5/4G.

A further argument in favour of these limits is the following. If c^4/4G or c^5/4G could be exceeded, this would imply that c^2/4G could so as well. This would imply, that black holes are not, for a given mass, the most compact objects. But this latter statement does not have any experimental or theoretical support.

### 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.

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The principle of maxmimum force transforms the Bronshtein cube into a Bronshtein limit cube.

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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. In fact, in that case, it is not a law of nature at all.

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Maximum force implies, together with maximum speed and the quantum of action, that there is no trans-Planckian physics. For example, in nature, action W, length l and force F are related by W/l = Fl/c. Inserting the force limit F ≤ c^4/4G and the action limit W ≥ ℏ implies l ≥ (4Gℏ/c^3)^(1/2). In other words, twice the Planck length is the smallest length in nature.

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Maximum force is predicted, like maximum speed and the quantum of action, to hold also in any theory of quantum gravity.

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Maximum force is predicted, like maximum speed and the quantum of action, to hold also in any unified theory.

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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.

### Hilbert's sixth problem: physics as a whole has no axioms

1. Maximum force implies the existence of a smallest length in nature. The smallest length follows from maximum force (general relativity) when it is combined with maximum speed (special relativity) and the quantum of action (quantum theory). The smallest length is twice the Planck length, about 3 · 10-35 m. This tiny value is both the smallest possible length measurement result and the smallest possible length measurement error.

2. In nature, there is not always a point between two other points. The smallest length implies the lack of points in nature as we usually think about them. We have to throw overboard the idea that points exist in nature. We do use points to describe nature, but they are approximations. Points and instants of time have no basis in nature.

3. The same limitation holds for any other observable quantity: measurement results without errors are impossible.

4. Measurement results are not real numbers. Space is not made of points. Time is not made of instants. Coordinates are approximations.

5. Also point particles do not exist in nature. Singularities do not exist in nature. Both concepts are incompatible with the smallest length. Locality is an approximation. (Local operators are approximations.)

6. The lack of points implies the lack of (mathematical) continuity. Continuous space and time are approximations. (Continuity, Euclidean space, space-time, Riemannian space, vector spaces and Hilbert spaces are all mathematical concepts based on points. Given that points do not describe nature, all these concepts are approximations that do not apply to nature.)

7. The lack of exact mesurement results of points implies the lack of sharp boundaries. There is no way to separate space into separate regions. There are no exact boundaries in nature. ("Nature is fuzzy.") There is no way to draw exact distinctions. There is no way to distinguish (exactly).

8. The lack of precise measurements and of sharp boundaries implies the lack of sets. It is impossible to distinguish with precision what is inside and outside a set. Sets cannot be defined in nature.

9. The impossibility to measure more precisely than the minimum length implies the impossibility to distinguish elements from each other. This implies the impossibility to define elements of sets in nature.

10. Because all axioms are based on distinctions, on elements and on sets, the minimum length implies the lack of axioms.

11. In conclusion, physics as a whole cannot be based on axioms. Hilbert's sixth problem, where he asked for an axiomatic system for all of physics, is unsolvable, when gravity and quantum theory are combined.

12. However, parts of physics can be based on axioms - just not physics as a whole. Axioms are possible in parts of physics because in parts of physics (such as quantum theory without gravity, or gravity without quantum theory), the minimum length does not arise. In parts of physics, measurements can be infinitely precise. Only the combination of quantum theory with gravity eliminates the possibility of infinite precision and thus eliminates the possibility to define axioms.

13. Even though space and time are approximations, they must be used to talk about nature. There is no way to do physics without them. One proof: all the fundamental constants, c, hbar, G and k, and thus all measured quantities, have metre and second in their units. Another proof: We must use space and time to distinguish and to think; there is no way to do otherwise. Or again: we must use space and time to talk and to communicate; there is no way to do otherwise.

14. Despite the lack of axioms in physics, a complete description of physics and of nature remains possible. For a recent status report in simple language, see C. Schiller, From maximum force to physics in 9 lines - and implications for quantum gravity, arxiv.org/abs/2208.01038. A future, complete description of nature will be logically circular, at least in parts, but it will be possible.

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P.S. This chain of arguments is neither original nor new, but is obvious and old. For example, it was obvious already in 2003, as shown in arXiv:physics/0309118.

### Conclusion

Maximum force c4/4G is a simple statement. It agrees with all observations.

Maximum force defines general relativity, in the same way that maximum speed defines special relativity.

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.

S.G. Low, State space relativity: an analysis of relativity from the Hamiltonian point of view (1982).

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 (2003).

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.

Yu L. Bolotin et al., An ideal quantum clock and Principle of maximum force, arXiv:1604.01945 (2016).

Yu L. Bolotin, V. A. Cherkaskiy and V. V. Yanovsky, Limit values as an universal method of description of physical reality, Odessa Astronomical Publications 30 (2017) 6-12.

Yu L. Bolotin and V. V. Yanovsky, Modified Planck units, arXiv:1701.01022 (2016).

V. Cardoso, T. Ikeda, C.J. Moore and C.-M. Yoo, Remarks on the maximum luminosity, Physical Review D 97 (2018) 084013.

B. Mirza, Z. Mirzaiyan, and H. Nadi, Maximum rate of entropy emission. Annals of Physics 415 (2020) 168117.

Yu.L. Bolotin, and V.V. Yanovsky, How the Limit Values Work, East European Journal of Physics 1 (2021) 5-12.

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.

See also A. Loeb, Four Novel Observational Tests of General Relativity, arxiv.org/abs/2205.02746.

### Personal publications on maximum force

There are at least two ways to deduce Einstein’s field equations from the principle of maximum force c⁴/4G or from the equivalent principle of maximum power c⁵/4G. Tests in gravitational wave astronomy, cosmology, and numerical gravitation confirm the two principles. Apparent paradoxes about the limits can all be resolved. Several related bounds arise. The limits illuminate the beauty, consistency and simplicity of general relativity from an unusual perspective.

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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.

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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.

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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.

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### Personal preprints on maximum force

C. Schiller, From maximum force to physics in 9 lines and towards relativistic quantum gravity, arxiv.org/abs/2208.01038. Download the updated preprint here (12 pages).

The preprint shows what can be learned from maximum force about all of physics and about the future theory of relativistic quantum gravity.

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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 explores a new aspect of Bronshtein's physics cube and explores its consequences.

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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≥ yields quantum theory, and the force limit F≤c⁴/4G yields general relativity.

In 2003, the preprint did not pass peer review. Two decades later, the same idea received a honorable mention, and dozens of people in various research groups across the world are using it in their work.

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