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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.
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