Content uploaded by Christoph Schiller

Author content

All content in this area was uploaded by Christoph Schiller on Dec 28, 2021

Content may be subject to copyright.

Tests For Maximum Force and Maximum Power

Christoph Schiller ID ∗

Motion Mountain Research, 81827 Munich, Germany

(Dated: 16 December 2021)

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, nu-

merical gravitation and quantum gravity are proposed.

I. INTRODUCTION

Special relativity is based on an invariant maximum

speed cvalid for all physical systems. It is less known

that general relativity can be based on a maximum in-

variant force valid for all physical systems, given by

Fmax =c4

4G≈3.0·1043 N.(1)

In the following, two arguments show that the ﬁeld equa-

tions of general relativity follow from maximum force,

and vice versa, that the maximum force value follows

from the ﬁeld equations. Maximum force helps in get-

ting an overview of the features and eﬀects of gravity, in-

cluding the inverse square law, curvature, horizons, black

holes and gravitational waves. Recent criticisms of max-

imum force and maximum power are addressed. Above

all, several possible tests in experimental and theoreti-

cal research ﬁelds are presented. Finally, the limits are

placed in a wider context that spans all of fundamental

physics.

II. HISTORY AND EXPERIMENTS

The ﬁrst person to mention maximum force in writing

was Rauscher, in 1973 [1]. She was followed by Treder [2],

Heaston [3], de Sabbata and Sivaram [4] and others [5,6].

When the topic was explored in more detail, the factor

1/4, which is the force limit in natural units, was deduced

by Gibbons [7] and others [8], and studied further [9–16].

Maximum force is a consequence of the deﬁnition F=

ma. In relativity, the acceleration of (the front of) a body

of length lis known to be limited by a≤c2/l [17]. As

a result, the force on a body of mass mand length lis

limited by F≤c2(m/l). The largest ratio m/l arises for

a black hole, with a value c2/4G. This yields a maximum

force value Fmax =c4/4G, independently of the mass and

the length of the body.

Force is also energy per length: a force acting along

a path deposes an energy along its length. The highest

energy per length ratio is achieved when a Schwarzschild

∗fb@motionmountain.net

black hole of energy Mc2is deposed over a length given

by its diameter 4GM/c2. This again yields a maximum

force of c4/4G.

Another derivation of the limit arises when considering

the force produced by a Schwarzschild black hole on a

test mass. When a mass mis lowered, using a string,

towards the horizon of a Schwarzschild black hole, the

force of gravity Fat a radial distance r– for a vanishing

cosmological constant – is known to be given [18,19], to

ﬁrst order, by

F=GMm

r2q1−2GM

rc2

.(2)

At ﬁrst sight, the expression diverges when the test mass

approaches the horizon, and thus seems to contradict

maximum force. However, every test mass mis extended

in space. To generate a measurable force, the whole test

mass needs to be located outside of the horizon. The

test mass itself has a minimum size given by its own

Schwarzschild radius 2Gm/c2. Neglecting spacetime ef-

fects due to the test mass by assuming mM, the

minimum size yields a smallest possible value for the dis-

tance between the centers of both masses. This minimum

distance is given by r= 2G(m+M)/c2. Inserting this

distance – which is slightly larger than the black hole ra-

dius – the force of gravitation on the test mass mobeys

F=c4

4G

M√m

(M+m)3/26c4

4G.(3)

In other terms, the force of gravity felt by a test mass

never exceeds the maximum force. This upper limit re-

mains valid if force is calculated to second order using

the results of LaHaye and Poisson [19].

Physically, a maximum force c4/4Gis equivalent to a

maximum power, or a maximum luminosity, given by

Pmax =c Fmax =c5

4G≈9.1·1051 W,(4)

corresponding to about 50 700 solar masses per second.

For comparison, the most massive known star, R136a1,

has about 315 solar masses, whereas black holes, such as

the ones in TON68 or in Holm15A, can be as massive as

4 to 6 ·1010 solar masses.

The ﬁrst to investigate maximum power seems to have

been Sciama, also in 1973 [20,21]. Others followed

2

[15,22–27]. The factor 1/4 – again specifying maximum

power in natural units – arose together with maximum

force.

The maximum force and maximum power values are

not well-known. First of all, both values are so large that

they do not arise in everyday life, nor under the most

extreme experimental situations. In fact, both limit val-

ues are only relevant in strong gravitational regimes near

black hole event horizons and thus hard to reach. Sec-

ondly, maximum force was only deduced several decades

after the development of general relativity, so that it

is not found in textbooks. Thirdly, many people are

hesitant to use ‘force’ in general relativity. However,

force, with its usual deﬁnition as change of momentum,

F=dp/dt, can be freely used also in general relativ-

ity. Finally, maximum force leads to several apparent

counter-arguments. They are discussed below.

Experimentally, no force value close to the maximum

force has ever been measured. The literature is silent

on this topic, including the canonical overview of general

relativity tests by Will [28]. However, in the last few

years, checks for the maximum power value are in sight.

The most powerful known energy sources in the uni-

verse are black hole mergers. So far, the most powerful

events detected by the LIGO and Virgo facilities have

reached an instantaneous power of 0.46 ±0.16 % of the

maximum value, namely 230±80 solar masses per second

[29]. The well-known 2019 black hole merger radiated up

to 207 ±50 solar masses per second [30]. Thus, observa-

tions with gravitational waves (and simulations) are just

2 orders of magnitude away from potential experimental

falsiﬁcation. Future space-based detectors will do better.

Also the luminosity of the full universe did not and

does not exceed the value c5/4G. This can be tested in

more detail in the future, as shown below.

III. A SHORT DERIVATION OF THE FIELD

EQUATIONS

Observations during solar eclipses, the constancy of the

speed of light, and also the force increase given by expres-

sion (2) imply that space is curved around a mass. For

example, only taking curvature into account can expres-

sion (2) be deduced with the dust ball method of Baez

and Bunn [31]. In short, maximum force implies that

vacuum bends and is elastic.

The elasticity of a material can be described with the

shear modulus. The shear modulus also determines the

shear strength, i.e., the maximum shear that a material

can support (before breaking). The two quantities are

related by a factor of order O(1). Likewise, the elastic

constant of the vacuum, c4/8πG, determines, within a

factor O(1), the maximum force c4/4Gthat the vacuum

can support.

Vacuum elasticity suggests a simple heuristic way to

reach the ﬁeld equations of general relativity starting

from maximum force [32]. The energy density εin vac-

uum is a force per area. A maximum force c4/4Gthat

also describes the elasticity of vacuum implies

c4/4G

A=ε . (5)

This is the maximum energy density for a spherical sur-

face. For a spherical surface of radius rand curvature

R= 1/r2, the area is related to curvature by A= 4π/R.

The relation between curvature Rand energy density ε

then becomes

R=16πG

c4ε . (6)

This is the maximum possible curvature for a sphere. For

ageneral observer, the curvature R/2 is replaced by the

Einstein tensor Gµν =Rµν −gµν R/2, and the energy

density εis replaced by the energy–momentum tensor

Tµν . This yields

Gµν =8πG

c4Tµν .(7)

This form of the ﬁeld equations does not yet incorporate

the cosmological constant; but it can be extended to do

so [32]. In short, using a line of reasoning inspired by

vacuum elasticity, the ﬁeld equations can be intuitively

deduced from maximum force.

IV. A LONGER DERIVATION OF THE

COMPLETE FIELD EQUATIONS

Maximum force arises at event horizons. Among other

properties, all event horizons show energy ﬂow. Now,

maximum force limits the energy ﬂow through an event

horizon. This limit allows deriving the ﬁeld equations.

The simplest ﬁnite event horizon is a sphere, character-

ized by its radius ror, equivalently, by its surface gravity

a=c2/2r. Event horizons arise from matter or energy

in permanent free fall. Any falling system at a horizon

is characterized by its energy Eand its proper length L.

When the fall is perpendicular through the horizon, the

momentum change or force measured by an observer at

the horizon is given by dp/dt =F=E/L. For a spherical

event horizon, the maximum force value and the horizon

area 4πr2imply

E/L

A=c4/4G

4πr2.(8)

Horizons being extreme conﬁgurations, the left hand side

limits the amount of energy Eof a system with length

Lﬂowing through an event horizon of surface A. Now,

when a system falls into a horizon, it is accelerated. The

geometry of the black hole limits the length Lto a max-

imum value given by the radius

L≤r=c2

2a.(9)

3

Combining the last two expressions yields the fundamen-

tal relation for every horizon:

E=c2

8πG a A . (10)

This horizon equation relates (and limits) the energy ﬂow

Ethrough an area Aof a horizon with surface gravity

a. The horizon equation thus follows from and is equiv-

alent to the observation that event horizons are surfaces

showing maximum force at every point.

One notes that the horizon equation also arises if one

starts with maximum power instead of maximum force.

One further notes that the horizon equation is based on

test bodies whose speed, acceleration and length are lim-

ited by special relativity.

The next step is to generalize the horizon equation

from the static and spherical case to the general case. For

a horizon whose curvature varies over space and time, the

horizon equation (10) becomes

δE =c2

8πG a δA . (11)

This diﬀerential horizon equation is called the ﬁrst law

of black hole mechanics [33,34]. Equating the surface

gravity awith temperature and the area Awith entropy

is a common procedure. In this case, the equation is

called the ﬁrst law of black hole thermodynamics.

The ﬁrst law (11) describes how a changing horizon

area δA induces a changing horizon energy δE for a given

surface gravity a. In other words, the ﬁrst law describes

the dynamics of every horizon. In particular, the ﬁrst law

shows that the dynamics of every horizon is determined

by the maximum force. The situation is analogous to

special relativity, where the dynamics for light x=ct is

determined by maximum speed.

The ﬁrst law (11) is known to be equivalent to general

relativity at least since 1995, when this equivalence was

shown by Jacobson [35]. The equivalence was conﬁrmed

by Padmanabhan [36,37], by Ashtekar et al. [38], by

Hayward [39], and by Oh, Park and Sin [40]. The general

argument is the following: using a suitable coordinate

transformation, or frame of reference, it is possible to

position a horizon at any desired location in space-time.

This possibility implies that the dynamics of horizons

contains and is equivalent to the dynamics of space-time.

In other words, the ﬁrst law contains the ﬁeld equations.

To see in detail how the dynamics of horizons imply the

dynamics of space-time, the ﬁrst law needs to be formu-

lated for arbitrary observers and coordinate systems. To

achieve this formulation, one introduces the general sur-

face element dΣ and the local boost Killing vector ﬁeld k

that generates the horizon (with a suitable norm). These

two quantities allow rewriting the left hand side of the

ﬁrst law (11) as

δE =ZTab kadΣb,(12)

where Tab is the energy-momentum tensor. This relation

describes horizon energy for an arbitrary coordinates.

The right hand side of the ﬁrst law (11) can be written

a δA =c2ZRab kadΣb,(13)

where Rab is the Ricci tensor describing space-time curva-

ture. This relation describes how the area change of the

horizon, given the local acceleration, depends on the local

curvature. The rewriting [35–37] makes use of the Ray-

chaudhuri equation, which is a purely geometric equa-

tion for curved manifolds. (The Raychaudhuri equation

is comparable to the expression that links the curvature

radius of a curve to its second and ﬁrst derivative. In

particular, the Raychaudhuri equation does not contain

any physics of space-time or of gravitation.)

Combining the generalizations of both sides of the ﬁrst

law (11) yields the equation

ZTabkadΣb=c4

8πG ZRabkadΣb.(14)

This equation is thus the ﬁrst law for general coordinate

systems and describes the horizon dynamics in the gen-

eral case. Making use of local conservation of energy (i.e.,

of the vanishing divergence of the energy-momentum ten-

sor), one ﬁnds that this equation is only satisﬁed if

8πG

c4Tab =Rab −R

2+ Λgab .(15)

Here, R=Rc

cis the Ricci scalar. The cosmological con-

stant Λ arises as an unspeciﬁed constant of integration.

These are Einstein’s ﬁeld equations of general relativity.

In short, maximum force or maximum power, together

with the maximum speed, imply the ﬁrst law of horizon

mechanics. The ﬁrst law in turn implies the ﬁeld equa-

tions. One notes that the derivation only requires the

existence of a Riemannian space-time with 3+1 dimen-

sions, and no further conditions.

V. THE PRINCIPLE OF MAXIMUM FORCE

Each step in the previous derivation of the ﬁeld equations

can be reversed: one can return from the ﬁeld equations

(15) to the ﬁrst law (11) and then, using maximum speed,

to the maximum force used in equation (8).

Also the short derivation of the ﬁeld equations given

above using equations (5) to (7) can be reversed. Again,

maximum force arises from the ﬁeld equations, when

maximum speed is taken into account.

In short, the ﬁeld equations and maximum force or

power are equivalent. It is therefore acceptable to speak

of the principle of maximum force or power in general

relativity. This is akin to speak of the principle of max-

imum speed in special relativity and its equivalence to

the Lorentz transformations.

4

The equivalence of general relativity and of maximum

force implies that every test of general relativity near a

horizon is, at the same time, a test of maximum force.

Deviations from general relativity near horizons can be

searched for in double pulsars, in black hole mergers, in

collisions between neutron stars and black holes, and pos-

sibly in other systems [28]. So far, no deviations arose.

VI. DERIVATION OF UNIVERSAL GRAVITY

In the absence of a horizon, equation (8) still holds. It

limits the energy inside a general surface A. However,

instead of equation (9), special relativity now implies L≤

2r=c2/a. Equation (10) then becomes E=aA c2/4πG.

Inserting E=Mc2and A= 4πr2results in a=M G/r2.

Inverse square gravity thus follows from maximum force.

An even simpler deduction starts with the energy limit

per enclosed area

E

A=Fmax

Cmin

.(16)

Then one inserts the area A= 4πr2, the maximum

force Fmax =c4/4Gand, from special relativity, energy

E=Mc2and minimum circumference Cmin =πLmin =

πc2/a. Together, this yields a=MG/r2, as a direct

consequence of maximum force in ﬂat space.

This derivation of the inverse square law does not seem

to have been published before. The lack of the constant

cin the inverse square law is thus as natural consequence

of the maximum force c4/4G.

VII. COUNTER-ARGUMENTS

The statement of a maximum force has led to many at-

tempts to exceed the limit. First of all, it has to be

checked whether Lorentz boosts allow one to exceed the

maximum force. Since a long time, textbooks show that

this is not possible, because both the acceleration and

the force values in the proper frame of reference are not

exceeded in any other frame [41–43]. (For the simple

one-dimensional case, the boosted acceleration value is

the proper acceleration value divided by γ3, while the

boosted force value is the same as the proper force value.)

As a consequence, maximum force is observer-invariant.

What happens if one adds two forces whose sum is

larger than the maximum? If the forces act at diﬀer-

ent points, their sum is not limited by the principle of

maximum force. Any force is a momentum ﬂow; the

principle does not limit the sum of ﬂows at diﬀerent lo-

cations. If, instead, the forces in question all act at a

single point, the principle states that their sum cannot

exceed the maximum value. In the same way that adding

speeds at diﬀerent points in space can give results that

exceed the speed of light, also adding forces at diﬀer-

ent points in space can give values exceeding the limit.

The speed and force limits are local. (An incorrect state-

ment on locality is also found in reference [8].) Recent

proposals for exceeding maximum force by Jowsey and

Visser [44] explicitly disregarded locality. Nevertheless,

they were taken up [45]. A refutation was ﬁrst given in

reference [46] and lead to reference [47]. Whenever one

tries to exceed maximum force at a speciﬁc location, a

horizon appears that prevents doing so.

How can gravitation be the weakest interaction and yet

determine the maximum force value? Because gravity

has only charges of one sign, it is easiest to experience

in everyday life. However, gravity’s “weakness” is due to

the smallness of typical elementary particle masses, and

not to an intrinsic eﬀect [48]. In fact, all interactions

lead to space-time curvature. The maximum force value

relates curvature to energy density, independently of the

type of interaction.

Another potential counter-argument arises from the

topic of renormalization of Gin quantum ﬁeld theory.

The study goes back to the work of Sakharov [49]. Var-

ious approaches to this issue suggest that Gchanges

with increasing energy, and in particular that Gincreases

when approaching Planck energy. This is argued in the

papers by Frolov, Fursaev and Zelnikov [50], Visser [51],

Volovik and Zelnikov [52], and Hamber and Williams [53].

In contrast, reasons for a fundamental impossibility that

Gis renormalized were given by Anber and Donoghue

[54,55]. So far, no hint for a change of Gwith energy

has been found. If, however, future experiments do ﬁnd

such such a change, maximum force would be falsiﬁed.

A further potential counter-example is still subject of

research. Exact calculations on the force between two

black holes on the line connecting their centers yield an

expression that diverges when horizons touch, thus al-

lowing larger force values at ﬁrst sight [56]. However,

it appears that those expressions disregard the overall

shape changes of the horizons [57]; these shape changes

make the horizons touch on a circle around the straight

connecting line before they touch on the line. Whether

this eﬀect prevents exceeding the force limit is still open.

At least four papers have claimed that the factor in

maximum force or power is 1/2 instead of 1/4, namely

references [23], [15], [26] and [27]. In those papers, the

missing factor 1/2 shows up either when distinguishing

radius and diameter, or when the factor 2 in the expres-

sion E= 2T S, valid for black hole thermodynamics, is

taken into account.

Maximum power has its own paradoxes. At ﬁrst sight,

it seems that the maximum power can be exceeded by

combining two (or more) separate power sources that add

up to a higher power value. However, at small distance

from the sources, their power values cannot be added.

And at large distance, the power limit cannot be ex-

ceeded, because the sources will partially absorb each

other’s emission.

A recent theoretical attempt, again by Jowsey and

Visser [58], to invalidate the power limit in explosions

makes use of an expansion front speed larger than c.

5

However, the front speed is a signal speed and an energy

speed; such speeds are never larger than c. Equation (4)

and maximum power remain valid.

In short, no conﬁrmed counter-example to maximum

force or maximum power has yet been found.

VIII. FURTHER GRAVITATIONAL LIMITS

The limits c5/4Gand c4/4Gare not the only ones in

general relativity. An equivalent bound limits mass ﬂow

rate by dm/dt =c3/4G≈1.0·1035 kg/s: nature does

not allow transporting more mass per time. Again, this

is a local limit, valid at each point in space-time. And

again, the limit is realized only by horizons. For example,

the maximum mass ﬂow rate value limits the speed of a

Schwarzschild black hole to the speed c. Again, boosts

do not allow exceeding the limit.

The maximum mass rate limit c3/4Gsuggests the pos-

sibility of future tests, both during the merger of black

holes and in numerical simulations. However, no dedi-

cated studies seem to have been published yet.

Maximum force also limits mass to length ratios by

c2/4G≈3.4·1026 kg/m. Again, this limit is realized by

horizons of Schwarzschild black holes. The limit states

that for a given mass, nothing is denser than a black hole.

Also this limit cannot be exceeded by a boost: spheri-

cal objects, including Schwarzschild black holes, do not

Lorentz-contract. The maximum force thus appears to

include the hoop conjecture. Again, any counter-example

would invalidate maximum force.

In cosmology, more limits arise. Maximum power im-

plies a maximum energy density for the universe. Inte-

grating the maximum power c5/4Gover the age t0of

the universe and dividing by half the Hubble volume

(2π/3)(ct0)3yields an upper mass density limit of

%max =3

8πG (t0)2.(17)

This is the usual critical density. In cosmology, the crit-

ical density can thus be seen as due to the maximum

power c5/4G. Indeed, the value is not exceeded in the

ΛCDM cosmological model, nor in measurements.

In cosmology, expression (17) for the critical density

has further consequences. Within a factor O(1), the

quantity c/4G≈1.1·1018 kg s/m2appears to limit the

product % RHTHof matter density, Hubble radius and

Hubble time [59]. Similarly, within a factor O(1), the

quantity 1/4G≈3.7·109kg s2/m3appears to limit

the product % T 2

Hof matter density and (Hubble) time

squared. Precision tests are under way.

In short, all limits cn/4Gwith 0 ≤n≤5 hold. They

can be tested further with measurements and with sim-

ulations.

Any one of the six gravitational limits cn/4Gcan be

seen as fundamental. This also applies to their inverse

values. All these limits are equivalent. As a result, also

4Gis a limit, even though it is not usually seen as one.

Despite this equivalence, speaking of the smallest pos-

sible value for the inverse of mass density times time

squared – usually called 4G– is somewhat less incisive

than speaking of the maximum force c4/4Gor of the

maximum power c5/4G.

IX. ALTERNATIVE THEORIES OF GRAVITY

Does the maximum force hold in alternative theories

of gravity? Because general relativity is equivalent to

maximum force, the question leads to additional tests.

Dabrowski and Gohar [26] have shown that maximum

force does not apply in theories with varying constants

Gand c. However, even the most recent experiments

[60–62] show no such eﬀect. Dabrowski and Gohar also

argue that, similarly, a running of Gwith energy would

invalidate maximum force. Furthermore they show, as

did Atazadeh [63], that any volume term in black hole

entropy invalidates maximum force. Atazadeh also ex-

plains that quintessence is likely to invalidate the maxi-

mum force limit, and so is Gauss-Bonnet gravity. Also,

maximum force might not be valid in higher spatial di-

mensions or in conformal gravity.

It is unclear whether maximum force is invalidated by

modiﬁed Newtonian dynamics [64]. It is seems that not,

but the issue is still a topic of research.

In short, maximum force seems to be closely tied to

general relativity – at least near horizons. If an alterna-

tive theory of gravity is found to describe systems with

high curvature, maximum force will be falsiﬁed.

X. ELECTROMAGNETIC LIMITS

Electric charge is quantized in multiples of the down

quark charge −e/3. Electric ﬁeld is deﬁned as force per

charge. As a result, a maximum force and a minimum

charge imply maximum values for electric and magnetic

ﬁelds given by Emax = 3c4/4Ge = 5.7·1062 V/m and

Bmax = 3c3/4Ge = 1.9·1054 T.

Unfortunately, the electromagnetic ﬁeld limits cannot

be tested experimentally: in practice, observed ﬁeld val-

ues are limited by the Schwinger ﬁeld limit, at which pair

production arises. The Schwinger ﬁeld is many orders of

magnitude lower than the Planck-scale limit. For this

reason, maximum power is not in reach of electromag-

netic sources [65]. Only sources of gravitational waves

can achieve values near the power limit.

Could the force between two charged black holes be

larger than the maximum force? No; the charge reduces

horizon radius, but the force limit for test particles re-

mains valid even if the test particle is charged. Explicit

calculations of this conﬁguration have been performed

in reference [19], and more tests will be possible in the

future.

Maximum force also implies a limit on the ratio be-

tween the magnetic moment and the angular momentum,

6

as deduced by Barrow and Gibbons [66]. They showed

that the ratio is limited by O(1)√G/c, a purely relativis-

tic limit that does not contain ~. So far, this and all other

electromagnetic limits thus allow only theoretical tests.

XI. CONSEQUENCES FOR QUANTUM

GRAVITY

Maximum force and power hold independently of quan-

tum theory. Therefore, the limits can be combined with

quantum theory to produce additional insights. For ex-

ample, general relativity alone does not limit curvature,

energy density, or acceleration. However, limits for these

quantities do arise if quantum theory is included.

Combining the limits on speed v, force Fand action

Wusing the general relation F vt =W/t leads to a limit

on time measurements given by

t≥r4G~

c5≈1.1·10−43 s,(18)

i.e., twice the Planck time. Shorter times cannot be mea-

sured or observed. Similarly, for acceleration, the rela-

tion W a =F v 3/a leads to the limit a≤pc7/4G~≈

2.8·1051 m/s2, or half the Planck acceleration. Higher

accelerations do not arise in nature.

Using the mixing of space and time yields a limit for

length given by l≥p4G~/c3≈3.2·10−35 m, twice the

Planck length. (It thus seems that the existence of actual

points in space, which contradicts a smallest measurable

length, should at least be put into question.) The mini-

mum length in turn leads to limits on area, volume and

curvature. Similar algebra also allows deducing a limit on

mass density given by ρ≤c5/(16G2~)≈3.3·1095 kg/m3,

and a corresponding limit on energy density.

The quantum gravity limits just deduced are direct

consequences of the three basic limits on speed, force and

action. Because the limits prevent the existence of inﬁ-

nite density, inﬁnite curvature and negligible size, they

suggest that singularities are not possible, at least for the

case of 3 spatial dimensions discussed here. This con-

clusion rises for time-like, space-like, naked and conical

singularities. (In more dimensions, the situation might

diﬀer [13].) For example, the brightest black holes are

those with highest density and thus with smallest possi-

ble mass: their mass is half the Planck mass. But again,

during their evaporation, no power larger than c5/4Gis

ever emitted.

Another direct consequence of the three fundamental

limits arises from the relation F l =W/t, namely the

limit t l ≥4G~/c4. This yields an uncertainty relation

relating clock precision and clock size [67] given by

∆t∆l≥~

c4/4G.(19)

Various analogous uncertainty relations in quantum grav-

ity can be deduced.

A particle is elementary – thus not composed – if it

is smaller than its own reduced Compton length λ=

~/mc. Combining this condition with the limits on force,

speed and action yields limits on mass, momentum and

energy that are valid only for elementary particles: E≤

p~c5/4Gor half the Planck energy, p≤p~c3/4Gor

half the Planck momentum, and m≤p~c/4Gor half the

Planck mass (thus the opposite limit of that for black hole

mass). These well-known limits for elementary particles

thus also arise from the limits on speed, force and action.

And indeed, no higher values have ever been observed –

in cosmic rays or anywhere else.

Combining the limits of this section with the limit on

electric charge leads to limits for charge density and for

all other electric quantities. For example, the limits for

acceleration and jerk also apply to charged particles. The

jerk limit therefore limits the Abraham-Lorentz-Dirac

force [68]. Indeed, the force limit is smaller than the

maximum force by a factor given by the ﬁne structure

constant and a number of order O(1).

Also the emission of radiation by an accelerated mirror

can be investigated [69–71]. Inserting the limit on accel-

eration derived above into the expression for the emitted

power P=~a2/6πc2yields a value that never larger than

the maximum power divided by 6π.

Maximum force, together with the quantum of action

~, also implies a limit on jerk j, given by

j=a/t ≤c6/(4G~)≈2.6·1094 m/s3.(20)

It seems that a jerk limit has not been discussed in the

literature yet. It is known that in the dynamical Casimir

eﬀect, the jerk limit implies a power limit. Using the

usual expression [72], the power limit for the dynamical

Casimir eﬀect turns out to be given by c5/4G, as ex-

pected. This shows again that Planck-scale limits form

a consistent set, independent of the speciﬁc physical ef-

fect under investigation. In particular, the limits appear

independently of whether the physical eﬀect explicitly

incorporates gravitation or not.

In short, maximum force allows deducing the limits

and uncertainty relations usually explored in quantum

gravity, including uncommon ones. No contradictions

with experiments or with expectations arise.

XII. THERMODYNAMIC LIMITS

This rapid overview of quantum gravity did not cover

thermodynamic limits that arise by including the Boltz-

mann constant k. In 1929, Szilard [73] argued that there

is a smallest observable entropy of the order of kin na-

ture. (With its invariance and limit property, the small-

est observable entropy kresembles the smallest observ-

able action ~.) Including the Boltzmann constant allows

deducing an upper temperature limit p~c5/(4Gk2)≈

7.1·1031 K given by half the Planck temperature.

Black hole entropy, being a horizon entropy, is the up-

per limit for the entropy of a physical system with surface

7

A, where the surface is a multiple of the smallest surface

Amin = 4G~/c3. The factor 4 in the minimum surface is

the same factor 4 appearing in the maximum force oc-

curring at horizons. In turn, the factor 4 in the smallest

surface appears in black hole entropy, which also occurs

at horizons. In short, the factor 1/4 in black hole entropy

is related to the factor 1/4 in maximum force.

The Fulling-Davies-Unruh eﬀect and Hawking radia-

tion can also be deduced and allow additional tests. For

example, even an evaporating black hole in its ﬁnal mo-

ments is never hotter than the temperature limit.

XIII. EXCEEDING AND APPROACHING THE

LIMITS

What would happen if maximum force or maximum

power would be exceeded? Exceeding the force limit

would mean the ability to aﬀect systems behind a hori-

zon. The issue is akin to the ability to circumvent causal-

ity by exceeding the speed of light. Both are impossible.

Given that maximum force describes the elastic prop-

erties of the vacuum, what happens if one gets close to

the limit? Just before a material loses its elastic proper-

ties, defects arise. Similarly, just before the vacuum loses

its elastic properties, defects arise; and vacuum defects

are particles. Indeed, whenever one approaches maxi-

mum force by approaching a horizon, particles arise, e.g.,

in the form of Hawking or thermal radiation. Exploring

the microscopic aspects of maximum force and gravita-

tion is subject of ongoing research in quantum gravity.

XIV. SEARCHING FOR A NEW EFFECT

Given that maximum force or power are equivalent to

general relativity, one does not expect an eﬀect that is

speciﬁc to maximum force and that is still unknown.

Nevertheless, one candidate might exist.

Maximum speed cimplies a (purely classical) uncer-

tainty relation between frequency and wavelength in wave

phenomena given by ∆f∆λ&c. Minimum action ~im-

plies an uncertainty relation between position and mo-

mentum in quantum phenomena given by ∆x∆p&~.

This suggests that an uncertainty relation might exist

between observables related by maximum force or power.

An example is

∆E

∆l.c4

4G.(21)

All known systems, such as a typical rock or the Sun, ap-

pear to fulﬁl the inequality. The gravitational uncertainty

relation (21) – if valid generally – implies that length un-

certainties cannot be zero, but are limited from below

by energy uncertainties. As a consequence, a quantum

vacuum, with its energy ﬂuctuations, cannot be perfectly

smooth and ﬂat. For a similar reason, due to quantum

eﬀects, black hole geometry cannot be perfectly smooth

and classical. Vacuum and horizons must be cloudy. All

this is as expected.

More such gravitational uncertainty relations can be

derived. They allow further tests of maximum force and

power.

XV. WHAT IF MAXIMUM FORCE OR POWER

WOULD NOT EXIST?

The question about non-existence of maximum force can

be compared to that about the non-existence of maxi-

mum speed c. In the latter case, special relativity would

not be valid, light would not be the fastest moving sys-

tem, and, without a natural invariant standard, speeds

could not be measured. Similarly, if force or power would

not be bounded, the ﬁeld equations would not be valid:

curvature and energy-momentum tensors would not be

connected. Also, there would be no way to measure

force, power, luminosity, mass rate, or mass to length

ratio because no natural, invariant standards for them

would exist.

XVI. THE FUNDAMENTAL STATUS OF

MAXIMUM FORCE

One way to state the above results is the following: gen-

eral relativity results from maximum force – in the same

way that special relativity results from maximum speed.

At ﬁrst sight, this can seem surprising, because physi-

cists are used to think that Gand care fundamental,

but not c4/4G. However, as argued in Section VIII,

there are various possible choices for the fundamental

constant of gravity. In particular, it is also possible to

take the constants cand Fmax as fundamental and think

of c4/4Fmax =Gas a derived constant that appears in

inverse square gravity. In fact, if desired, one can even

take 1/4Gas a fundamental maximum value of a suit-

ably deﬁned observable, namely mass density times time

squared. These – and other – choices are all equally fun-

damental.

Many arguments about maximum force c4/4G(or any

other of its equivalent limits) and maximum speed ccan

be extended to the elementary quantum of action ~. In

all three cases, the limit is invariant, cannot be overcome

experimentally, leads to apparent paradoxes (as explored

for ~in the debate between Bohr and Einstein), and yield

a speciﬁc description of natural phenomena.

The three limits can be used to express the Bronshtein

cube of physical theories – introduced in the 1930s [74]

– even more incisively, by using a limit at every cor-

ner of the cube. The three upper limits 1/4G,cand

1/~respectively deﬁne non-relativistic gravity, special

relativity and quantum theory. Upper limits from com-

binations, such as c4/4G,c/~and 1/4G~, respectively

deﬁne general relativity, quantum ﬁeld theory and non-

relativistic quantum gravity. Finally, fully combined up-

8

per limits such as c/4G~deﬁne relativistic quantum grav-

ity. In short, one gets a Bronshtein limit cube of theo-

ries. If desired, the inverse Boltzmann constant 1/k can

be added, thus yielding a limit hypercube of physical the-

ories [74].

The three (or four) fundamental limits also have con-

ceptual consequences. Special relativity predicts the lack

of physical systems exceeding the speed limit c. Likewise,

general relativity predicts the lack of physical systems ex-

ceeding the force limit c4/4G(or any other limit equiva-

lent to it). For example, there are no objects denser than

black holes. Finally, quantum theory predicts the lack

of physical systems below the action limit ~. Because

maximum force allows deﬁning limits in every domain

of nature, it predicts the lack of any trans-Planckian ef-

fect. Numerous consistency tests, in addition to the ones

above, are possible. So far, are all positive.

As a ﬁnal consequence, all invariant limits – including

c,c4/4G,~, 1/k, etc. – are predicted to hold also in a

future uniﬁed theory. This prediction will be testable in

the future.

XVII. CONCLUSION

In summary, the principle of maximum force and the

principle of maximum power allow deducing general rel-

ativity and inverse square gravity. The limits are con-

sistent across physics and are useful for teaching and

research. Searching for counter-examples leads to new

experimental tests in black hole mergers and cosmology,

and to new theoretical tests in numerical relativity, elec-

trodynamics, quantum gravity and uniﬁcation. So far,

no test failed.

ACKNOWLEDGMENTS

The author thanks Michael Good for an intense and pro-

ductive exchange and Ofek Birnholtz, Barak Kol, Shahar

Hadar, Pavel Krtouˇs, Andrei Zelnikov, Grigory Volovik,

Eric Poisson, Gary Gibbons, Chandra Sivaram, Arun Ke-

nath, Saverio Pascazio, Britta Bernhard, Isabella Bor-

gogelli Avveduti, Steven Carlip and an anonymous ref-

eree for fruitful discussions.

[1] E. A. Rauscher, Lett. Nuovo Cim. 7S2, 361 (1973).

[2] H. J. Treder, Found. Phys. 15, 161 (1985).

[3] R. J. Heaston, Journal of the Washington Academy of

Sciences 80, 25 (1990).

[4] V. de Sabbata and C. Sivaram, Found. Phys. Lett. 6, 561

(1993).

[5] C. Massa, Astrophys. Space Sci. 232, 143 (1995).

[6] L. Kostro and B. Lange, Phys. Essays 12, 182 (1999).

[7] G. W. Gibbons, Found. Phys. 32, 1891 (2002),arXiv:hep-

th/0210109.

[8] C. Schiller, Int. J. Theor. Phys. 44, 1629 (2005),

arXiv:physics/0607090.

[9] J. Barrow and G. Gibbons, Mon. Not. Roy. Astron. Soc.

446, 3874 (2014).

[10] M. R. R. Good and Y. C. Ong, Phys. Rev. D 91, 044031

(2015),arXiv:1412.5432 [gr-qc].

[11] S. Di Gennaro, M. R. R. Good, and Y. C. Ong, (2021),

arXiv:2108.13435 [gr-qc].

[12] J. D. Barrow, Class. Quant. Grav. 37, 125007 (2020),

arXiv:2002.10155 [gr-qc].

[13] J. D. Barrow, Int. J. Mod. Phys. D 29, 2043008 (2020),

arXiv:2005.06809 [gr-qc].

[14] C. Schiller, Int. J. Theor. Phys. 45, 221 (2006).

[15] Y. C. Ong, Phys. Lett. B 785, 217 (2018),

arXiv:1809.00442 [gr-qc].

[16] J. D. Barrow and N. Dadhich, Phys. Rev. D 102, 064018

(2020),arXiv:2006.07338 [gr-qc].

[17] E. F. Taylor and A. P. French, American Journal of

Physics 51, 889 (1983).

[18] H. C. Ohanian and R. Ruﬃni, Gravitation and Space-

time, 3rd ed. (Cambridge University Press, 2013).

[19] M. LaHaye and E. Poisson, Phys. Rev. D 101, 104047

(2020),arXiv:2001.00430 [gr-qc].

[20] D. W. Sciama, Phys. Bull. 24, 657 (1973).

[21] D. W. Sciama, J. Phys. Colloques 34, C7 (1973).

[22] J. W. C.W. Misner, K.S. Thorne, Gravitation (W. H.

Freeman, San Francisco, 1973).

[23] C. J. Hogan, NATO Sci. Ser. C 565, 283 (2001),

arXiv:astro-ph/9912110.

[24] L. Kostro, Phys. Essays 13, 143 (2000).

[25] V. Cardoso, T. Ikeda, C. J. Moore, and C.-M. Yoo, Phys.

Rev. D 97, 084013 (2018),arXiv:1803.03271 [gr-qc].

[26] M. P. Dabrowski and H. Gohar, Phys. Lett. B 748, 428

(2015),arXiv:1504.01547 [gr-qc].

[27] V. G. Gurzadyan and A. Stepanian, (2021),

arXiv:2108.07407 [astro-ph.CO].

[28] C. M. Will, Living Rev. Rel. 17, 4 (2014),

arXiv:1403.7377 [gr-qc].

[29] B. P. Abbott et al. (LIGO Scientiﬁc, Virgo), Phys. Rev.

X9, 031040 (2019),arXiv:1811.12907 [astro-ph.HE].

[30] R. Abbott et al. (LIGO Scientiﬁc, Virgo), Astrophys. J.

Lett. 900, L13 (2020),arXiv:2009.01190 [astro-ph.HE].

[31] J. C. Baez and E. F. Bunn, Am. J. Phys. 73, 644 (2005),

arXiv:gr-qc/0103044.

[32] C. Sivaram, A. Kenath, and C. Schiller, preprint (2021).

[33] J. M. Bardeen, B. Carter, and S. W. Hawking, Commun.

Math. Phys. 31, 161 (1973).

[34] R. M. Wald, in Directions in General Relativity: An

International Symposium in Honor of the 60th Birth-

days of Dieter Brill and Charles Misner (1993) arXiv:gr-

qc/9305022.

[35] T. Jacobson, Phys. Rev. Lett. 75, 1260 (1995),arXiv:gr-

qc/9504004.

[36] T. Padmanabhan, Gen. Rel. Grav. 40, 2031 (2008).

[37] T. Padmanabhan, Rept. Prog. Phys. 73, 046901 (2010),

arXiv:0911.5004 [gr-qc].

[38] A. Ashtekar, S. Fairhurst, and B. Krishnan, Phys. Rev.

D62, 104025 (2000),arXiv:gr-qc/0005083.

[39] S. A. Hayward, R. Di Criscienzo, M. Nadalini, L. Vanzo,

and S. Zerbini, (2009), arXiv:0911.4417 [gr-qc].

9

[40] E. Oh, I. Y. Park, and S.-J. Sin, Phys. Rev. D 98, 026020

(2018),arXiv:1709.05752 [hep-th].

[41] W. Pauli, Relativit¨atstheorie (Vieweg+Teubner Verlag,

1921).

[42] C. Møller, The Theory of Relativity (Clarendon Press,

1952).

[43] E. Rebhan, Theoretische Physik, Bd. 1: Mechanik

(Springer Verlag, 1999).

[44] A. Jowsey and M. Visser, Universe 7(2021),

arXiv:2102.01831 [gr-qc].

[45] V. Faraoni, Phys. Rev. D 103, 124010 (2021),

arXiv:2105.07929 [gr-qc].

[46] C. Schiller, Phys. Rev. D 104, 068501 (2021),

arXiv:2109.07700 [gr-qc].

[47] V. Faraoni, Phys. Rev. D 104, 068502 (2021).

[48] F. Wilczek, Phys. Today 54N6, 12 (2001).

[49] A. D. Sakharov, Dokl. Akad. Nauk Ser. Fiz. 177, 70

(1967).

[50] V. P. Frolov, D. V. Fursaev, and A. I. Zelnikov, Nucl.

Phys. B 486, 339 (1997),arXiv:hep-th/9607104.

[51] M. Visser, Mod. Phys. Lett. A 17, 977 (2002),arXiv:gr-

qc/0204062.

[52] G. E. Volovik and A. I. Zelnikov, JETP Lett. 78, 751

(2003),arXiv:gr-qc/0309066.

[53] H. W. Hamber and R. M. Williams, Phys. Rev. D 75,

084014 (2007),arXiv:hep-th/0607228.

[54] M. M. Anber and J. F. Donoghue, Phys. Rev. D 85,

104016 (2012),arXiv:1111.2875 [hep-th].

[55] J. F. Donoghue, Front. in Phys. 8, 56 (2020),

arXiv:1911.02967 [hep-th].

[56] P. Krtouˇs and A. Zelnikov, Phys. Rev. D 102, 024065

(2020),arXiv:1910.04290 [gr-qc].

[57] M. S. Costa and M. J. Perry, Nucl. Phys. B 591, 469

(2000),arXiv:hep-th/0008106.

[58] A. Jowsey and M. Visser, (2021), arXiv:2105.06650 [gr-

qc].

[59] L. Kostro, AIP Conference Proceedings 1316, 165

(2010).

[60] K. Wang and L. Chen, Eur. Phys. J. C 80, 570 (2020),

arXiv:2004.13976 [astro-ph.CO].

[61] A. Vijaykumar, S. J. Kapadia, and P. Ajith, Phys. Rev.

Lett. 126, 141104 (2021),arXiv:2003.12832 [gr-qc].

[62] T. D. Le, Gen. Rel. Grav. 53, 37 (2021).

[63] K. Atazadeh, Phys. Lett. B 820, 136590 (2021).

[64] M. Milgrom, Mon. Not. Roy. Astron. Soc. 437, 2531

(2014),arXiv:1212.2568 [astro-ph.CO].

[65] W. Lu and P. Kumar, Mon. Not. Roy. Astron. Soc. 483,

L93 (2019),arXiv:1810.11501 [astro-ph.HE].

[66] J. D. Barrow and G. W. Gibbons, Phys. Rev. D 95,

064040 (2017),arXiv:1701.06343 [gr-qc].

[67] Y. L. Bolotin, V. A. Cherkaskiy, A. V. Tur, and V. V.

Yanovsky, (2016), arXiv:1604.01945 [gr-qc].

[68] O. Birnholtz, S. Hadar, and B. Kol, Phys. Rev. D 88,

104037 (2013),arXiv:1305.6930 [hep-th].

[69] A. Myrzakul, C. Xiong, and M. R. R. Good, (2021),

arXiv:2101.08139 [gr-qc].

[70] M. R. Good, Y. C. Ong, A. Myrzakul, and

K. Yelshibekov, Gen. Rel. Grav. 51, 92 (2019),

arXiv:1801.08020 [gr-qc].

[71] A. Zhakenuly, M. Temirkhan, M. R. R. Good, and

P. Chen, Symmetry 13, 653 (2021),arXiv:2101.02511 [gr-

qc].

[72] M. R. R. Good and E. V. Linder, Phys. Rev. D 96, 125010

(2017),arXiv:1707.03670 [gr-qc].

[73] L. Szilard, Zeitschrift f¨ur Physik 53, 840 (1929).

[74] L. B. Okun, in Multiple Facets Of Quantization And

Supersymmetry: Michael Marinov Memorial Volume,

edited by M. Olshanetsky and A. Vainshtein (World Sci-

entiﬁc, 2002) pp. 670–675, arXiv:hep-ph/0112339.