### A maximum force c^{4}/4G and a maximum power
c^{5}/4G hold in nature – at black hole
horizons – in addition to maximum speed c.

Ute Kraus computed this spectacular image of a black hole:

The maximum speed c ≈ 3⋅10^{8} m/s defines and implies
*special* relativity. In the same way, *general* relativity is
defined and implied by the maximum force c^{4}/4G ≈ 3⋅10^{43} N
– or, equivalently, the maximum power c^{5}/4G ≈
9⋅10^{51} W, the maximum mass flow rate c^{3}/4G ≈
10^{35} kg/s, or the maximum per length ratio c^{2}/4G.
These limits are all realized by black holes. This page introduces the arguments
that led to the result, presents the consequences and discusses various
tests.

● Consequences for general
relativity

● Different views

● Consequences for
unification

● Hilbert's sixth
problem: physics has no axioms

● Conclusion

● Bibliography on maximum force

● Personal publications 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
c^{4}/4G or a power value larger than c^{5}/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 c^{4}/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 c^{5}/G.
For stars and lamps, 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 c^{4}/4G was by Gary Gibbons in 2002.
Published statements on maximum
force c^{4}/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 c^{3}/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 c^{5}/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 a simple derivation of the *inverse
square law of gravity*. This is told in the
mentioned PRD, for example.

The energy E contained in a sphere of area A is limited by E/A = Fmax/Cmin, where Fmax is the maximum force and Cmin is the minimum circumference.

Now substitute E=mc^{2}, A=4πr^{2},
Fmax=c^{4}/4G and, from special relativity,
the relation
Cmin= π^{2}c/a. The result is

a = MG/r^{2}.

Gravity's inverse square law thus arises when the maximum force is added to special relativity. In other words, by letting a stone fall and measuring the speed of light, you have proven the existence of a maximum force.

*

Maximum force also implies, as told in a paper below,
the *hoop conjecture*. The reverse is also correct.
Here are the steps.

1. 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 is only achieved by massless particles; it is not achieved in other cases.

2. Any attempt to produce a force larger than the maximum value c⁴/4G produces a horizon that prevents a larger value from appearing. This is analogous to maximum speed: if you try to exceed it, things get difficult.

3. 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.

4. 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.

5. A horizon always arises in such an attempt; 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*.

### Different views

Nature limits force to c^{4}/4G, a value of about 3⋅10^{43} N.
Several physicists disagree completely with the idea of maximum force.
Some think that the maximum does not exist; others disagree on the value of
the maximum. However, their arguments are mistaken.

The planet Jupiter contains more than 10^{53} hydrogen
atoms. Inside each atom, the force between the proton and the electron is
about 10^{-7} N.
The total force is thus *higher* than the maximum force. Besides,
there are many more hydrogen atoms in the universe...

Maximum force can also be exceeded without quantum systems. Just take the force on a small mass near a black hole horizon. Then take a sufficient number of copies of this small mass, and place them around a horizon. (Typically just a handful are enough.) The total force will be larger than the maximum force.

Speed can help to understand the fallacy.
Nature limits speed to c, a value of about 3⋅10^{8} m/s.
But simply adding all speeds of the atoms in the air of the room you are sitting in.
The total value is *higher* than c.
Besides, there are many more atoms in the universe...

Like maximum speed c, also the maximum force c^{4}/4G is the maximum
value *at a single point*.
Like maximum speed, also the maximum force applies to
*a sufficiently small region of space.*

Maximum force c^{4}/4G is *equivalent* to a maximum mass to length
ratio c^{2}/4G (the "hoop conjecture"):
maximum force simply states that there is no way to compress matter or energy to
less than a black hole.

Exceeding maximum force implies acting across a horizon. Exceeding maximum force implies avoiding the creation of a black hole when compressing matter to less than the Schwarzschild radius. Exceeding maximum force contradicts Einstein's field equations. All this contradicts observations.

It is simple to construct systems with speed sums larger than the maximum speed.
But these systems do not contradict the *local* maximum speed value.
It is simple to construct systems with force sums larger than the maximum force.
But these systems do not contradict the *local* maximum force value.
Local maximum speed agrees with all observations.
Local maximum force agrees with all observations.

A statement that *no* maximum value exists at all is always at the border of
science, as it is not testable and contradicts every observation.
Nevertheless, it is sometimes heard.

So far, nobody ever published a statement that 100 or 1000 times
c^{4}/4G (or c^{5}/4G)
is observed at some point 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, several 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 statements 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, and to charged 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 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 (like the fine structure constant), 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 paper. To make varying G into a sufficient counter-argument, experimental evidence against a constant G is needed, as always in physics. 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 indeed, 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 state 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 an exception to maximum force c^{4}/4G,
then general relativity and constant G are falsified.
In any of these three cases, I would loose several bets.

If the existence of singularities is ever shown, then general relativity and maximum force are falsified. It is often stated that the mass of a black hole is concentrated in a singularity at its centre. This is wrong. If that were true, a black hole would have no moment of inertia. (And the horizon would not be a horizon.) The mass of a black hole is distributed over its horizon.

If the mass flow rate limit c^{3}/4G is ever exceeded,
then general relativity and maximum force are falsified.

In short, nobody has ever provided evidence that a physical system
does or can exceed c^{4}/4G or c^{5}/4G at a single
point.

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 be exceeded as well. This would imply, that black holes are not, for a
given mass, the most compact objects.
This would imply that singularities exist.
But none of these consequences has any experimental support.
Unlimited local force is a personal wish, not a reality.

### 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 maximum 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 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 that case, maximum force is not a law of nature at all.

*

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.

*

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

*

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 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 measurement 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.

*

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

### Conclusion

Maximum force c^{4}/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 c^{4}/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 c^{3}/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

C. Schiller, Testing a microscopic model for black
holes deduced from maximum force, book chapter in "A Guide to Black Holes"
(Nova Science) January 2023.

Download the preprint here.

The chapter shows how maximum force leads to the strand model of black holes and of general relativity.

*

C. Schiller, From maximum force to physics in 9 lines and towards relativistic quantum gravity, published in Zeitschrift für Naturforschung A (2022) https://doi.org/10.1515/zna-2022-0243.

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

*

C. Schiller, From the Bronshtein cube of limits to the degrees of
freedom of relativistic quantum gravity, preprint (2023) arxiv:2302.12485

Download
the preprint here.

The preprint explores the limit aspect of Bronshtein's physics cube and deduces the consequences.

*

A. Kenath, C. Schiller and C. Sivaram, From
maximum force to the field equations of general relativity - and
implications,
*International Journal of Modern Physics D* **31**
(2022) 2242019, 10.1142/S0218271822420196.
Won an honourable mention by the Gravity Research
Foundation 2022 Awards for Essays on Gravitation.
Download
the pdf here.

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.

*

C. Schiller, From
maximum force via the hoop conjecture to inverse square gravity,
*Gravitation and Cosmology* 28 (2022) 305–307,
10.1134/S0202289322030082.

Download the pdf
here.

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.

*

C. Schiller, Tests for maximum force and maximum power,
*Physical Review D* 104 (2021) 124079,
10.1103/PhysRevD.104.124079.

Download
the pdf here.

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.

*

C.
Schiller, Comment on "Maximum force and cosmic censorship", *Physical Review
D* 104 (2021) 068501,
10.1103/PhysRevD.104.068501.

Download
the pdf here.

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
(2006) 213–227.
Download it at doi.org/10.1007/s10773-005-9018-7.
Read the published paper online for free at rdcu.be/cdG3E.

Download
the pdf 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
doi.org/10.1007/s10773-005-4835-2. Read
the published paper online for free at rdcu.be/cdG3C.

Download
the pdf here.

*

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.

Since 2003, the ideas bore a lot of fruit. A paper received a honorable mention, and dozens of people in various research groups across the world are using it in their work.

* *