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From maximum force to physics in 9 lines and towards relativistic quantum gravity
Christoph Schiller ID ∗
Motion Mountain Research, 81827 Munich, Germany
(Dated: 7 November 2022)
A compact summary of present fundamental physics is given and evaluated. Its 9 lines describe
all observations exactly and contain both general relativity and the standard model of particle
physics. Their precise agreement with experiments, in combination with their extreme simplicity
and their internal consistency, suggest that there are no experimental effects beyond the two theories.
The 9 lines imply a smallest length in nature and make concrete suggestions for the microscopic
constituents in a complete theory of relativistic quantum gravity. It is shown that the microscopic
constituents cannot be described by a Lagrangian or by an equation of motion. Finally, the 9 lines
specify the only decisive tests that allow checking any specific proposal for such a theory.
Keywords: Gravitation; general relativity; maximum force; quantum gravity
In fundamental physics, a world-wide search for the the-
ory of relativistic quantum gravity is under way. Despite
intense attempts in experiment and theory, the search is
still ongoing. So far, all experiments ever performed and
all observations ever made can be described with general
relativity and with the standard model of particle physics
(with massive Dirac neutrinos, as always implied in this
article [1]). The world-wide search for observations be-
yond general relativity was unsuccessful [2–4], and so was
the world-wide search for observations beyond the stan-
dard model [5].
The present article first summarizes general relativity
and the standard model in a way that is as simple and as
compact as possible, while keeping the precision that the
two theories provide. The summary consists of 9 short
lines, each of them decades old: five general principles
and four lines of specific choices. Evaluating the 9-line
summary shows that it contains all equations of funda-
mental physics. The 9 lines also highlight the open issues
in the foundations of physics. The simplicity of the sum-
mary yields explicit experimental predictions. The 9 lines
also define the requirements that any theory of relativis-
tic quantum gravity must fulfil, in particular about the
type and behaviour of the microscopic constituents of na-
ture. In particular, the 9 lines lead to a limited number
of decisive tests. The deduced requirements and tests ex-
plain why relativistic quantum gravity has not yet been
achieved, and provide guidance for future searches.
I. LEAST ACTION
In nature, all motion can be described by the principle
of least action: motion minimizes action. More precisely,
this applies to microscopic motion; on large scales, action
can also be stationary.
In everyday life, action is the time integral of the La-
grangian, i.e., of the difference between kinetic and po-
tential energy [6]. In the general case, the action is de-
fined as the integral of a Lagrangian density based on and
∗cs@motionmountain.net
built with observable fields. The equations of motion fol-
low from the requirement that action is minimized – or
stationary.
The history of the principle of least action is compli-
Table 1. Nine lines describe all observations about nature.
Nr. Line Details
(1) dW= 0 Action W=RLdtis minimized in local
motion. The lines below fix the two fun-
damental Lagrangians L.
(2) v≤cEnergy speed vis limited by the speed
of light c. This invariant implies special
relativity and restricts the possible La-
grangians.
(3) F≤c4/4GForce Fis limited by cand by the gravita-
tional constant G. This invariant implies
general relativity and, together with lines
1 and 2, fixes its Hilbert Lagrangian.
(4) W≥ℏAction Wis never smaller than the quan-
tum of action ℏ. This invariant implies
quantum theory and restricts the possi-
ble Lagrangians.
(5) S≥kln2 Entropy Sis never smaller than ln 2 times
the Boltzmann constant k. This invariant
implies thermodynamics.
(6) U(1) is the gauge group of the electromagnetic
interaction. It yields its Lagrangian when
combined with lines 1, 2 and 4.
(7) SU(3) and
broken SU(2)
are the gauge groups of the two nuclear
interactions, yielding their Lagrangians
when combined with lines 1, 2 and 4.
(8) 18 particles – gauge bosons, the Higgs boson, quarks,
leptons, and the undetected graviton –
with all their quantum numbers [5], make
up everything and, with the interactions,
fix the standard model Lagrangian.
(9) Finally, 27
numbers
– dimensions, cosmological constant, cou-
pling constants, particle mass ratios, mix-
ing matrices [5] – complete the two fun-
damental Lagrangians. They determine
all observations, including all colours.
2
cated and long. After the first precise description of mo-
tion by Galileo, researchers took about 150 years to com-
plete the definition of ‘action’. In physics, action Wis a
scalar quantity measuring the change occurring in a sys-
tem. Measurement of action is based on the ability to
measure length and time intervals.
Experimental validation of the principle of least (or
stationary) action occurs every day – in classical physics,
in quantum theory and in general relativity. Action min-
imization describes every type of motion. Action mini-
mization is valid for the motion of machines, molecules,
animals, electricity and light, for the motion of planets
and stars, for the motion of particles and fields, and for
the change of curvature of empty space. (Also the path
integral formulation of quantum field theory can be taken
as following from a Lagrangian.) Falsification of least (or
stationary) action requires finding an exception in an ob-
servation. In principle, this is possible, but the probabil-
ity is low. In fact, no non-equivalent alternative to the
principle of least (or stationary) action appears to have
ever been proposed.
It is important to note that the principle of least (or
stationary) action also implies and contains the principle
of observer invariance, because the action for an evolving
system is defined to have the same value for all observers.
Depending on the set of observers being studied, the in-
variance and the underlying symmetry differs, as shown
in the remaining lines.
In short, on a small scale, all motion follows the prin-
ciple of least (or stationary) action
dW= 0 .(1)
Microscopic motion minimizes action W=RLdt, i.e.,
minimizes the integral of the Lagrangian L. The two fun-
damental Lagrangians of nature, the Hilbert Lagrangian
of general relativity and the Lagrangian of the standard
model of particle physics, are defined in the following.
II. MAXIMUM SPEED
Special relativity is based on the principle of an invariant
maximum speed with a value c≈3.0·108m/s. In nature,
energy cannot move faster than c. The maximum speed
itself is only achieved by massless radiation, such as elec-
tromagnetic or gravitational waves. Maximum speed is
the origin of the Lorentz transformations, the mixing of
space and time, the equivalence of energy and mass, the
relativity of time, the relativity of length, and the speed
addition formula. The invariant limit property of cthus
goes beyond a conversion factor between length and time.
Maximum speed was discovered in the years from 1860
to 1890. In 1905, Einstein deduced the Lorentz transfor-
mations from maximum speed [7]. In particular, max-
imum speed cdetermines the form of any Lagrangian
that complies with special relativity. In particular, be-
cause action is observer-invariant, it must be a Lorentz
scalar.
In a vacuum, light from a moving lamp has the same
speed as light from a lamp at rest. Experimentally, this
holds in all directions [8]. Furthermore, even the compar-
atively light electrons cannot be accelerated faster than
light, even using the largest amounts of energy. This
speed limit is found to apply also to protons, neutri-
nos, rockets, radio waves, X-rays and gravitational waves.
The speed limit is so fundamental that it is used to define
the meter as the path of light during a given interval of
time. No type of matter and no type of radiation moves
faster than c. The speed limit is a local limit: it is valid
for energy speeds at a single point. Sums of speeds at
different locations can exceed the limit. This aspect is of
importance in the next section.
Experimental validation of maximum speed is fre-
quent. Every electric motor confirms the existence of a
maximum speed. No known example of motion of energy
contradicts maximum speed. Maximum speed is valid in
classical physics, in quantum theory and in general rela-
tivity. Falsification means finding a system in which en-
ergy moves faster than c. Such an observation is possible
in principle, but the probability is low. Despite high po-
tential rewards, nobody has found a way to move energy
faster than light in vacuum. Likewise, attempts to find
a description of nature without maximum energy speed
have not been successful.
Again, it is important to note that the principle of
maximum speed, together with the principle of least (or
stationary) action, implies and contains Lorentz invari-
ance.
In short, special relativity can be deduced from the
principle of least (or stationary action) together with the
principle of maximum speed
v≤c . (2)
There is an energy speed limit in nature. Among others,
the principle requires that Lagrangians must be Lorentz-
scalars.
III. MAXIMUM FORCE
In 1973, Elizabeth Rauscher discovered that general rel-
ativity implies a limit to force: she assumed that is was
given by the force F=c4/G [9]. She was followed by
many other researchers [10–41]. In 2002, Gary Gibbons
and, independently, Schiller deduced the factor 1/4 and
showed that force at a point is never larger that the max-
imum value c4/4G≈3.0·1043 N [15,16]. The maximum
value is realized on black hole horizons. At that time, it
also became clear that the field equations of general rel-
ativity and the Hilbert action can be deduced from the
invariant maximum force c4/4G[16,17,33,34,40].
The maximum force value c4/4Gis due to the maxi-
mum energy per distance ratio appearing in general rel-
ativity. Indeed, for a Schwarzschild black hole, the ratio
between its energy Mc2and its diameter D= 4GM/c2is
given by the maximum force value, independently of the
3
size and mass of the black hole. Also the force on a test
mass that is lowered with a rope towards a gravitational
horizon – whether charged, rotating or both – never ex-
ceeds the force limit, but only when the minimum size of
the test mass is taken into account. All apparent counter-
examples to maximum force disappear when explored in
detail [29–33,42,43].
A maximum force implies that space is curved. The
maximum force value is realized at horizons. In fact,
maximum force c4/4Gimplies Einstein’s field equations
of general relativity. There are at least two ways to de-
duce the field equations from maximum force [16,17,33,
34,40]. Maximum force also implies the cosmological
constant term, but does not fix its value. As a con-
sequence, the maximum force limit can be seen as the
defining principle of general relativity. The situation re-
sembles special relativity, of which the maximum speed
limit can be seen as the defining principle. The invariant
limit property of c4/4Gthus goes beyond a conversion
factor between curvature and energy density.
Because maximum force implies general relativity with
the cosmological constant, also the usual big-bang cos-
mology follows from maximum force. Maximum force
implies all observed aspects of gravitation.
The maximum force principle for general relativity is
not the only possible principle. Other maximum quanti-
ties combining cand G, such as maximum power c5/4G
[13,20,23,24,33,37,38,44–46] or maximum mass flow
rate c3/4G[33,35], can also be taken as principles of
relativistic gravity. Also the length to mass limit c2/4G,
realized by black holes, can be taken as defining general
relativity. Each of these equivalent limits can be taken as
starting principle of general relativity. Maximum force is
chosen here only because it is the most striking of these
limits.
Attempts to find counterexamples to maximum force
(or the other equivalent limits) are not successful. In
flat space and at low speeds, the maximum force value
implies inverse square gravity [47], which is well estab-
lished experimentally. Because the force limit is local,
an observer cannot add forces acting at different location
and claim that their sum exceeds the local limit c4/4G.
(Such examples are easily found.) The value c4/4Gis also
the largest possible gravitational force between two black
holes. Maximum force also implies the hoop conjecture
[47–50]. Furthermore, maximum force eliminates most,
but not all, alternative theories of gravity [33]. However,
it is unclear whether modified Newtonian dynamics re-
mains possible or is eliminated.
No counterexample to the maximum luminosity and
power value c5/4G≈9·1051 W has been found. Even
the most recent observations of black hole mergers fail
to exceed the luminosity limit; the highest instantaneous
luminosity observed so far is about 0.5% of the maximum
value. Also in cosmology, no power value exceeding the
limit is observed [33,41].
Falsification of the limits is possible. It is sufficient to
observe or to point out a value for local force, power or
luminosity that exceeds the respective limit. The prob-
ability is low. Every day, maximum force and general
relativity are confirmed by the position determination
performed by mobile phones with satellites.
Again, it is important to note that the principle of
maximum force, together with the principle of maximum
speed and the principle of least (or stationary) action,
implies and contains diffeomorphism invariance.
In short, general relativity can be deduced from the
principle of least (or stationary action), the principle of
maximum speed, and the principle of maximum force:
F≤c4/4G . (3)
There is a force limit in nature. More precisely, the
Hilbert action, Einstein’s field equations of general rel-
ativity, and diffeomorphism invariance can be deduced
from the principle of maximum force combined with the
principle of maximum speed and the principle of least ac-
tion. The principle of maximum force was the last build-
ing block that allowed summarizing physics in 9 simple
lines.
IV. THE QUANTUM OF ACTION
Quantum theory is based on the invariant smallest action
ℏ≈1.1·10−34 Js. It is not possible to measure action
values – i.e., changes – smaller than ℏ, a constant of na-
ture that is called the elementary quantum of action. (In
fact, the smallest change is h= 2πℏ, but often the two
quantities are used interchangeably.) The quantum of
action is the origin of the indeterminacy relation. Above
all, the quantum of action explains photons and atoms.
Planck discovered the quantum of action ℏin the
1890s, when studying light. The term ‘quantum’ was in-
troduced by Galileo, who explained that matter is made
of ‘piccolissimi quanti’, tiny quanta, that are not divis-
ible. In 1906, following Einstein, Planck took over the
term [51].
In nature, action is quantized. An action value, or
change, smaller than ℏis never measured [52–55]. In ad-
dition, every action value – every measured change – is
a multiple of ℏ. This property also implies the quanti-
zation of angular momentum. The quantum of action is
so fundamental that it is used to define the kilogram in
the international system of units. The limit ℏrequires
to introduce wave functions, Hilbert spaces and opera-
tors. This leads to the Schr¨odinger equation, the Dirac
equation and all of quantum theory, including probabil-
ities and entanglement [52]. The quantum of action ℏ
modifies the principle of least action in the microscopic
domain: it determines the mathematical structure of La-
grangians using operators and quantum states that cor-
rectly describe the probabilistic outcomes of experiments
[52–54].
A straightforward attempt to falsify smallest action is
to measure a system’s or a particle’s energy Etwice, once
at the start and once at the end of an interval δt. Even
4
though in the classical approximation action is given by
the product W=E δt and can get as small as desired,
in nature – and in quantum theory – the action value W
remains finite when δt gets small: the measured energy
(difference) increases when δt decreases. The reason is
the uncertainty relation: it prevents that the measured
action value approaches zero when δt does so.
Other attempts at finding a counter-example to the
quantum of action use spin. Because action is quantized
in multiples of ℏ, there is no spin smaller than 1/2: de-
tecting a spin 1/2 flip requires an action ℏ. There is no
way to detect a spin flip with a smaller amount of action.
A further attempt is the detection of light. But even
detecting even the dimmest light requires an action ℏ.
Light consists of photons. In nature, there is no way to
detect one half or one hundredth of a photon. Photons
are elementary quanta: they cannot be split. If ℏwere
not the smallest action value, photons would not exist.
Also atoms would not exist without the lower limit set
by ℏ. The invariant limit property of ℏthus goes beyond
a conversion factor between angular velocity and energy,
or between wave number and momentum.
Action quantization is confirmed by all experiments
ever performed. The discovery of ℏled to the devel-
opment of electronics, lasers, computers and the inter-
net. Indeed, no (non-equivalent) alternative description
of quantum physics has ever been proposed. Neverthe-
less, falsification remains possible, by measuring a smaller
action value than the quantum of action ℏ. It is unlikely
that this will happen.
In short, combining the principle of least action with
the quantum of action
W≥ℏ(4)
implies quantum theory. In line with the above state-
ments one can state: quantum theory can be deduced
from the principle of quantized action. The quantum of
action implies the Lagrangian of quantum theory. In par-
ticular, when the speed limit cis included into quantum
theory, antiparticles, the Dirac equation and quantum
field theory arise.
V. THE BOLTZMANN CONSTANT
Whether thermodynamics is part of fundamental physics
or not has been a subject of debate. Cohen-Tannoudji,
Okun, and Oriti are among those in favour [56–58].
Therefore, it is included here.
Classical thermodynamics can be seen, to a large ex-
tent, as a consequence of the principle of least action.
Similarly, statistical physics can be seen as following from
quantum theory. Indeed, there are uncertainty relations
for thermodynamic properties. As an example, tempera-
ture Tand energy Uobey ∆(1/T ) ∆U≥k/2. This rela-
tion was first given by Bohr; it was discussed by Heisen-
berg and other scholars [59–61]. It suggests that entropy
is similar to action, with the Boltzmann constant ktimes
O(1) taking the role of ℏ.
Planck introduced and named the Boltzmann constant
k≈1.4·10−23 J/K together with ℏ. Is ka just unit con-
version factor between energy and temperature or is it re-
lated to a fundamental limit? In 1929, Szilard suggested
[62] that there is a smallest entropy in nature. Since then,
the concept of a ‘quantum of entropy’ has been explored
by many authors [56,63–88]. Entropy is observed to be
quantized in various systems: in electromagnetic radia-
tion [74,75], in the entropy of two-dimensional electron
gases [83] and in low temperature thermal conductance
[84–88]. These investigations conclude that there is a
smallest entropy value, which is given by a multiple of k.
Often, but not always, the smallest entropy is given as
kln 2, as done by Szilard. In modern terms, this numer-
ical factor expresses that the smallest possible entropy is
related to a single bit.
The concept of a smallest entropy was explored in de-
tail by Zimmermann [67–71] and by Lavenda [89]. They
deduced statistical mechanics from the existence of such
a smallest entropy value in nature. The invariant limit
property of the smallest entropy thus goes beyond a con-
version factor between temperature and energy. (As a
note, combining statistical mechanics with quantum the-
ory yields and explains decoherence.)
Entropy quantization is confirmed by all experiments
ever performed. Every time a thermometer is read out
and every time hot air rises, the relevance of the Boltz-
mann constant is confirmed. Nevertheless, falsification is
possible, by measuring a smaller value than the quantum
of entropy. Also in this case, it is unlikely that this will
happen.
It has to be stressed that the quantum of entropy does
not imply a smallest value for the entropy per particle,
but a smallest entropy value for a physical system. For
interacting systems of particles, entropy values per par-
ticle can be much lower than the limit. In Bose-Einstein
condensates, measured values for the entropy per particle
can be as low as 0.001k[90].
In short, there is a smallest entropy value in nature.
Continuing the above collection of limits, one can state:
statistical thermodynamics can be deduced from
S≥kln 2 .(5)
This is the principle of smallest entropy.
VI. ELECTROMAGNETISM
The theory of quantum electrodynamics is based on
the U(1) gauge symmetry (or U(1) gauge invariance) of
electromagnetism. The gauge symmetry determines the
(minimal) coupling of the Dirac equation to the electro-
magnetic field. The vector potential in the Dirac equa-
tion has a local phase freedom that is called gauge free-
dom [91]. The U(1) gauge group explains the vanishing
mass of the photon, Coulomb’s law, magnetism and light.
5
When particle properties (of line 8) are included, U(1)
implies charge conservation, Maxwell’s equations [92,93],
stimulated emission, Feynman diagrams, and perturba-
tive quantum electrodynamics. This in turn yields the
change or ‘running’ of the fine structure constant and of
the electron mass, as well as all other observations in the
domain, without any exception.
The description provided by quantum electrodynamics
and the corresponding experiments match to high preci-
sion. Deviations between calculation and experiments
are possible, but have not been found yet. Clever mea-
surement set-ups for the well-known g-factor of the elec-
tron yield results with 13 to 14 significant digits that all
agree with calculations [94]. Even in the case of the muon
g-factor, there is still no confirmed deviation between ex-
periment and calculation [95,96]. In everyday life, every
laser confirms quantum electrodynamics.
In short, combining least action, the quantum of ac-
tion, maximum speed and the
U(1) gauge group (6)
with the particle properties and the fine structure con-
stant of line 8 and 9 below, fully specifies and describes
electromagnetism, both in the quantum and the macro-
scopic domain. The Dirac equation for charged particles
and the Lagrangian of QED arise in this way. For exam-
ple, the QED Lagrangian explains all observed material
properties.
VII. THE NUCLEAR INTERACTIONS
The strong and the weak nuclear interactions are based
on an SU(3) and a broken SU(2) gauge symmetry – or on
the corresponding gauge invariances (broken in the case
of the weak interaction). They define strong charge and
weak charge, as well as all their properties and effects.
For example, the gauge groups explain the burning of the
Sun, radioactivity, and the history of the atomic nuclei
found on Earth.
The verification of the two non-Abelian gauge theo-
ries – with all their detailed particle properties, particle
reactions, and consequences for nuclear physics – took
many decades [5]. The verification was completed when
accelerator experiments confirmed the existence of the
Higgs boson in 2012. Both gauge groups also imply the
running of the fundamental constants with energy. At-
tempts at falsification or even just at extension of the
gauge description – such as the search for a fifth force,
grand unification, more gauge bosons, etc. – were not
successful, despite intense research all over the world [5].
Also the recent W boson mass measurement is not a con-
firmed deviation [97].
In short, the combination of least action, the quantum
of action, the speed limit and the gauge groups
SU(3) and broken SU(2) (7)
fully specifies and describes the nuclear interactions, in-
cluding the Lagrangians of QCD and of the weak interac-
tion, provided the particle spectrum and the fundamental
constants given in the following are included.
VIII. THE PARTICLE SPECTRUM
The world around us is made of elementary fermions and
bosons. All matter consists of fermions: six types of
quarks and six types of leptons. All radiation is made of
gauge bosons – the photon, the W, the Z and gluons –
and of the predicted graviton. The Higgs boson, giving
mass to all particles, completes the list. The Higgs boson
also explains the breaking of SU(2) gauge symmetry.
Each elementary particle is described by mass, spin,
electric charge, weak charge, colour charge, parities,
baryon number, lepton number and the flavour quantum
numbers. No other particle property has been detected.
All the known particle properties and their conservation
laws have been explored in great detail. Every two years,
the Particle Data Group documents the status and ex-
perimental progress across the world [5].
In short, everything observed is made of
18 elementary particles. (8)
Nature specifies these particles and their properties. One
can also speak of 18 fundamental fields. The particle
number 18 arises if all gluons are counted as one particle,
and if the coloured quarks and all the antiparticles are
not counted separately. The essence of the statement is
that the 18 fermions and bosons just mentioned suffice to
build everything observed in nature, and that they fix the
full mathematical expression for the Lagrangian of the
standard model – together with the last line. Therefore,
these elementary particles and their properties need to
appear in Table 1.
IX. THE FUNDAMENTAL CONSTANTS
The standard model is specified with 25 characterizing
numbers. They include 15 elementary particle masses
(or more precisely, the ratios to the Planck mass), 3 cou-
pling constants, as well as 6 mixing angles and 2 CP
phases both in the CKM (Cabibbo-Kobayashi-Maskawa)
and in the PMNS (Pontecorvo-Maki-Nakagawa-Sakata)
mixing matrices for quarks and neutrinos [5]. One pa-
rameter is redundant. Because the constants run with
energy, the precise statement is that the standard model
is described by 25 fundamental constants at some defined
energy value. Two further characterizing numbers, the
cosmological constant and the number of spatial dimen-
sions, determine the expansion of space-time. In accor-
dance with all present experiments, nature is thus de-
scribed by 27 fundamental constants. Together, these 27
specific numerical values determine the remaining details
6
of the Hilbert Lagrangian and of the standard model La-
grangian.
The last fundamental constants in the standard model
Lagrangian have been introduced in the 1970s. All the
values are being measured with a precision that usually
increases when new experiments are performed [5]. At
present, the fundamental properties of the neutrinos are
the least precisely known. The cosmological constant in
the Lagrangian of general relativity has been introduced
more than a century ago. After a complicated history,
its value was first measured in the 1990s.
Neither general relativity nor the standard model ex-
plain the values of the fundamental constants. Explain-
ing these values – which include the mass of the electron
and the fine structure constant 1/137.036(1) – remains
an open issue. These two particular constants almost
completely determine the colours in nature. As long as
the numbers are unexplained, colours are not fully un-
derstood.
Various attempts to reduce the number of fundamen-
tal constants have been proposed. Most attempts predict
new effects that have not been observed. Other propos-
als, such as certain kinds of supersymmetry, require ad-
ditional fundamental constants. However, no additional
fundamental constant has yet been discovered [5].
In short, nature somehow chooses
27 fundamental constants (9)
that, together with the previous lines, completely de-
termine the Hilbert Lagrangian of general relativity and
the Lagrangian of the standard model of particle physics
(with massive neutrinos).
X. THE SUMMARY OF PRESENT PHYSICS
Lines 1, 2, 3 and 9 fully determine the Hilbert La-
grangian, including the cosmological constant. The
derivation is found in references [33] and [40]. Line 5 de-
termines thermodynamics, as shown in references [67–71]
and [89]. All lines except 3 and 5 fully determine the La-
grangian of the standard model of particle physics: they
determine the elementary particle spectrum, the parti-
cle mixing matrices, the particle masses, their couplings,
the interaction terms, the kinetic terms, and, as a result,
the full Lagrangian. The complete expression of the La-
grangian of the standard model is derived in references
[98] (for vanishing neutrino mass) and [5] (for massive
neutrinos). The corresponding lines in Table 1 have ex-
actly the same physical and mathematical content, while
avoiding writing down the algebraic details that are im-
plied by them.
While the number of lines in Table 1 is subjective, the
content is not. The number could easily be expanded or
reduced by one or two lines, while keeping the same con-
tent. Whatever form is chosen, the content of the lines in
Table 1 agrees with all experiments. Only standard text-
book physics is included. No part of standard textbook
physics is missing.
Table 1 resulted from the work of many thousands of
scientists and engineers during over 400 years. Galileo
started around the year 1600, with the first-ever mea-
surements of the dynamics of moving bodies. Line 1,
the principle of least action, was fully formulated around
1750. Line 5, on thermodynamics, arose from 1824 to
1929, and line 6, on electrodynamics, arose around 1860.
Line 2, on maximum speed came around 1890, and line
4, about the quantum of action, around 1900. Line 3, on
maximum force, was implicitly given in the year 1915,
and formulated in 2002. As a result, the Hilbert La-
grangian of general relativity agrees with experiments
since more than 100 years. The remaining lines 7 to 9,
on the standard model, arose in the years from 1936 to
1973. The standard model Lagrangian of particle physics
thus agrees with experiments since about 50 years.
Given that no observation contradicts the two La-
grangians and thermodynamics, one can say that the
9 lines contain all present knowledge about nature, in-
cluding all textbook physics and all observations ever
made. The 9 lines also contain chemistry, material sci-
ence, biology, medicine, geology, astronomy and engi-
neering. This is the conclusion of a world-wide and
decade-long effort to evaluate the 9 lines. The simplicity
of the 9 lines and their vast domain of validity form an
intriguing contrast.
The 9 lines contain five general principles and four
lines of specific choices taken from an infinity of possi-
bilities. The five principles define the framework of mod-
ern physics. The four lines of choices specify the everyday
world and, at the same time, contain what is unexplained
about modern physics.
In short, the 9 lines of Table 1, five principles and
four sets of choices, contain the evolution equations of
the standard model, of general relativity and of thermo-
dynamics. The 9 lines describe all of nature, without
any deviation between theory and experiment. This con-
troversial summary leads to several questions, challenges
and predictions.
XI. WHAT ADVANTAGE DO THE 9 LINES
PROVIDE?
The formulation of physics given in Table 1 is compact,
but provides no new content. Is it still useful? It turns
out that the 9 lines do imply a shift in perception about
several important issues in relativistic quantum gravity.
It is regularly suggested that general relativity and
quantum theory are incompatible. Often, the incompat-
ibility is even called a contradiction. Table 1 suggests
that this is not the case, and that, instead, the princi-
ples complement each other. The five principles appear
to address different aspects of nature, and all of them
in a similar way. Indeed, experiments have never found
7
a contradiction between general relativity and the stan-
dard model. In fact, such contradictions only arise when
extrapolations into inaccessible domains are made. This
conclusion can be stated with precision.
Combining the experimental limits on speed v, force
Fand action Wusing the general relation for energy
E=F vt =W/t leads to a limit on measurements of
time tgiven by
t≥p4Gℏ/c5≈1.1·10−43 s.(10)
The five principles thus eliminate instants of time and
introduce a minimum time interval, given by twice the
Planck time. In the same way, the principles also elim-
inate points in space and introduce a minimum length
[99] given by
l≥p4Gℏ/c3≈3.2·10−35 m,(11)
twice the Planck length. As a consequence, continuity of
space and time is not intrinsic to nature, but due to an
averaging process.
Conversely, the assumption of fully continuous space
implies a vanishing Planck length. This implies a vanish-
ing quantum of action or a vanishing gravitational con-
stant, or both. Full continuity thus implies the lack of
most physical measurement units. In other words, if con-
tinuity is taken as an exact property of nature, it contra-
dicts modern physics.
All issues disappear if space and time are seen as (ef-
fectively) continuous only for all intervals larger than the
Planck limits. The existence of a smallest length and of
a smallest time interval – and of all Planck limits in gen-
eral – ensures that quantum theory and general relativity
never actually contradict each other. No accessible do-
main of nature yields a contradiction between the two
theories. This conclusion is confirmed by studies from
particle physics, such as reference [100].
A smallest observable length value also implies that
there is no way to ever detect or measure additional spa-
tial dimensions – be they microscopic or macroscopic. In
other words, higher spatial dimensions cannot arise in
nature. Indeed, there are no experimental hints for such
additional dimensions.
The minimum length also implies that space has no
scale symmetry, no conformal symmetry, and no twistor
structure. The length measurement limit also prevents
the observation and the existence of lower dimensions. A
smallest observable length further implies, together with
the observed isotropy and boost invariance, that space is
neither discrete nor a lattice. The length measurement
limit prevents the observation and the existence of any
additional spatial structure at the Planck scale, including
non-commutativity or additional symmetries.
In short, the 9 lines imply that continuity is approx-
imate, that space has three curved dimensions without
additional structure, that the minimum length eliminates
most past unification attempts, and that general relativ-
ity and the standard model are not contradictory or mu-
tually exclusive, but that they complement each other.
XII. ARE THERE MORE THAN 9 LINES?
Candidates for disagreement between the 9-line summary
and experiment arise regularly. Examples are W mass
measurements, muon g−2 measurements, dark energy
differing from the cosmological constant, dark matter,
the rotation curves of galaxies, or table-top quantum
gravity. It could be that a future experiment will re-
quire changes in the 9 lines. (It has to be stressed that
elementary dark matter particles have not been detected
yet.) Therefore, these and other candidates for disagree-
ment are being explored around the world in great detail.
Even though no confirmed observation is unexplained by
the 9 lines, the experimental quest for such an effect will
never be over.
In particular, all the experiments that confirmed lines
6 to 9 make a further statement. The specific choices
contained in these four lines imply that additional in-
teractions, additional particles, or additional constants
would greatly increase the complexity of the table – in
contrast to observations.
Are unexplained observations possible at all? In other
terms, are additional lines necessary to describe nature?
The simplicity and consistency of the 9 lines suggest a
negative answer. So far, proposals for physics beyond
the standard model either require more lines, like super-
symmetry, or, if they don’t, like grand unified theories,
they disagree with experiment [101]. Some proposals,
like additional elementary dark matter particles, require
more lines and disagree with experiment. Nevertheless,
a future unexplained observation cannot be excluded.
In short, the 9 lines suggest the lack of new physics.
Any observation or experiment unexplained by the 9 lines
will create a sensation and would falsify most of the con-
clusions presented in this article.
XIII. WHAT EXPERIMENTAL PREDICTIONS
FOLLOW?
The 9 lines summarizing physics can and should be tested
in as many future experiments as possible.
Prediction 1. Lines 2 to 5 imply that no trans-
Planckian quantities or effects arise in nature. This is
valid for length, time and for every other physical ob-
servable. The precise limits are given by the (corrected)
Planck limits, such as the minimum length p4Gℏ/c3, the
minimum time p4Gℏ/c5or the maximum force c4/4G.
Here, the factor 4 from maximum force corrects the com-
monly used Planck units. (For some cases, such as Planck
energy or Planck momentum, the derivation of the limit
is only valid for a single elementary particle.) For ex-
ample, a limit on length measurements also implies that
no experiment will observe singularities, discrete space-
time, additional space-time structures, or additional di-
mensions. These predictions are in agreement with all
experiments. They also confirm the prediction of the
8
lack of infinitely large and of infinitely small observables
made by Hilbert in 1935 [102].
Prediction 2. A continuous space-time in spite of
the existence of a minimum length implies that locality,
continuity, causality and three-dimensionality are valid
at all observable scales, i.e., at all scales larger than the
Planck scale. The lack of trans-Planckian effects prevents
the observability, the influence and the existence of other
dimensions or other structures in space-time – both mi-
croscopic and macroscopic. This prediction agrees with
all data so far.
Preditction 3. The 9 lines predict that there is
no physics beyond special relativity, beyond general rel-
ativity, beyond thermodynamics, beyond quantum theory
and beyond the standard model. The nine lines predict
the lack of any additional symmetries, structures or ef-
fects whatsoever, at any length or energy scale: the lines
predict the so-called high-energy desert. In the past
centuries, mistaken predictions about the lack of new
physics have been made several times. At present how-
ever, there is a difference: the prediction agrees with all
high-precision observations since over five decades.
In short, the nine lines predict a lack of new physics.
Interestingly, this prediction does not completely exclude
the observation of new relativistic quantum gravity ef-
fects. (The lines 6 to 9 can be seen as known relativistic
quantum gravity effects.) Also the observation of non-
relativistic quantum gravity effects remain possible [103].
However, if any such new effects are detected, they will
not contradict Table 1.
XIV. ARE THERE FEWER THAN 9 LINES?
Each of the 9 lines in Table 1 generates a question about
its origin. In particular, one can ask for the origin of
the five principles listed in the lines 1 to 5. So far, no
accepted explanation for the origin of the principle of
least action nor for the other limit principles has been
proposed. It is unclear how nature enforces its five prin-
ciples. In modern terms, it is unclear how the principles
emerge from an underlying description.
The lack of explanation is especially evident in lines
6 to 9. These four lines contain all the specific choices
that fix the details of the standard model and of general
relativity. At present, the origins of the force and particle
spectra are unknown, as is the origin of each fundamental
constant. One can say that so far, these four lines are
the only known observations beyond the standard model
and beyond general relativity. However, despite multiple
and intense efforts, no explanation for the four lines of
specific choices has been successful and accepted. The
mechanism of their emergence is unclear.
The lack of explanations despite the successful descrip-
tion of nature with Table 1 leads to a related question:
can the 9 lines be deduced from a smaller set? All these
queries are part of the other, theoretical quest being pur-
sued in fundamental physics.
The five principles of lines 1 to 5 are not good candi-
dates to shorten Table 1, because they are independent
of each other. It appears impossible to reduce the num-
ber of principles in lines 1 to 5 in a simple way. Only
a radical explanation based on emergence might have a
chance to reduce their number.
In contrast, reducing the number of specific choices
given in lines 6 to 9 should be possible. The specific
choices out of an apparent infinity of options are so par-
ticular that they cannot be fundamental. The four lines
must be due to a deeper, emergent explanation. In the
past five decades, various proposals to reduce the num-
ber of lines 6 to 9 – or simply their details – have been
made. However, no proposal agrees with observations to
full precision.
In short, a description of nature with less than 9 lines
must exist. Such a description must be emergent. So
far, it has not been found. Nevertheless, Table 1 yields
several suggestions.
XV. IMPLICATIONS FOR THE MICROSCOPIC
CONSTITUENTS OF NATURE
The 9 lines provide clear hints for any description of na-
ture based on emergence. The 9 lines describe the be-
haviour of curvature, radiation and matters, thus of space
and particles.
In any complete description of nature, space and par-
ticles must emerge from some common description. The
horizon of a black hole makes this especially clear: it can
bee seen as due to collapsing matter and can be seen as
curved space. The common description of matter and
space must be based on common microscopic degrees of
freedom, or microscopic constituents. These microscopic
constituents describe extended space and curvature, lo-
calized particles, probabilistic quantum motion, and the
minimum length.
To realize the minimum length, the constituents must
be of minimum size, i.e., of Planck size, in at least one di-
mension. To realize the macroscopic extension of space,
the constituents must have at least one macroscopic di-
mension. To reproduce probabilities, particles must be
made of fluctuating constituents. To allow both fluctua-
tions, particle localization and particle motion, the con-
stituents must be of codimension two: they must have
at most one macroscopic dimension. As a result, the
fluctuating constituents must resemble thin fluctuating
strands of macroscopic length and Planck-size radius. In
other terms, the constituents of space and particles must
be filiform, with Planck radius.
More precisely, the microscopic constituents of space
and particles differ from points in two ways: they are
filiform and they are discrete, i.e., countable. Their dis-
creteness implies and confirms the finiteness of black
hole entropy, the Bekenstein entropy bound [104] and the
maximum entropy emission rate [105]. In fact, their dis-
9
creteness implies all Planck limits. Their filiform struc-
ture explains, as shown by Dirac’s trick [106], the spin
1/2 behaviour of fermions, i.e., their different behaviour
under rotations by 2πand by 4π. The filiform struc-
ture also explains the surface-dependence of black hole
entropy [107].
In short, the 9 lines of Table 1 imply that the descrip-
tion of space and of the known particles must emerge
from fluctuating constituents that are filiform and have
Planck radius. This conclusion agrees with decades-old
arguments and allows a number of predictions on how to
achieve an even shorter summary of physics.
XVI. PREDICTIONS ABOUT THE THEORY OF
RELATIVISTIC QUANTUM GRAVITY
The 9 lines and the discrete filiform constituents of space
and matter imply a number of theoretical predictions.
Prediction 4. It was shown above that the limits
c,c4/4G,ℏand kln 2 define special relativity, general
relativity, quantum theory and thermodynamics. In the
same way, the limits arising when combining those theo-
ries – i.e., when combining the four limits, such a mini-
mum length or minimum time – define the complete the-
ory of relativistic quantum gravity. More precisely, the
9 lines imply that relativistic quantum gravity is already
known in all its experimental and theoretical effects: rel-
ativistic quantum gravity implies the five principles of
line 1 to 5 and it fixes the choices of lines 6 to 9.
Prediction 5. Continuity emerges from averaging
fluctuating filiform constituents. This must apply to
space, to fields, and to wave functions. Because continu-
ity arises in all settings that allow measurements – and
despite the existence of a smallest length and time inter-
vals – space and time can be used to describe nature. In
fact, because all the limits c,c4/4G,ℏ, and kln 2 contain
meter and second in their units, space and time must be
used to describe nature. There is no chance to find a
back-ground-free description of motion, nor to describe
nature without time. The limits imply that relativis-
tic quantum gravity must continue to use – as is done
by general relativity and by the standard model – lo-
cally one-dimensional time and locally three-dimensional
space. The use of space and time remains necessary de-
spite the impossibility to define distances and time inter-
vals below the Planck limits.
Prediction 6. In contrast to the situation in special
relativity, in general relativity and in quantum theory,
no simple physical system realizes any limit of relativis-
tic quantum gravity – i.e., any limit containing c,Gand
ℏbut not the Boltzmann constant k. Light moves with c,
electron spin flips allow to measure ℏ, black hole horizons
realize c4/4G. In contrast, the microscopic constituents
of nature are out of experimental reach: no simple physi-
cal system realizes the Planck length, the Planck energy,
the Planck time or any other limit of relativistic quan-
tum gravity. As a result, no evolution equation for the
microscopic constituents of relativistic quantum gravity
can be defined. Such an equation would not be testable.
Equations in relativistic quantum gravity can only be
found for large numbers of microscopic constituents, such
as in black holes, where the smallest length arises in the
expressions for the entropy or for the temperature, and
the Boltzmann constant kis therefore used. In relativis-
tic quantum gravity, only statistical effects can be ex-
pected to be calculated or to be observed. Apart from
black hole entropy, the Bekenstein bound and the Un-
ruh effect are examples of calculations. Curvature, wave
functions and particle masses are examples of observa-
tions that are due to large numbers of microscopic con-
stituents. In contrast, single microscopic constituents
cannot be described nor observed.
In other terms, there will never be an evolution equa-
tion or a Lagrangian for relativistic quantum gravity.
This result requires a change in thinking habits. Evolu-
tion equations and Lagrangians have been seen as essen-
tial part of physics for over four centuries. In stark con-
trast, the complete theory of relativistic quantum gravity
is defined completely by the limits c,c4/4G,ℏ,kln 2 and
by the statistics of the microscopic constituents – without
additional or new equations of motion. The conclusion
confirms the statement that no Lagrangian can explain
the appearance of Lagrangians – in the same way that
turtles cannot support turtles all the way down.
In short, Table 1 implies that in the complete the-
ory of relativistic quantum gravity, all nine lines follow
from large numbers of filiform constituents of Planck ra-
dius that fluctuate in 3+1 dimensions. No equations
of motion can describe the constituents. Only a statisti-
cal description of their behaviour is possible. This result
confirms the expectations for any emerging and complete
description of nature. The lack of equations of motion
also explains the limited use of mathematical formulae
in this article.
XVII. OTHER APPROACHES
The theoretical predictions about the complete theory
of relativistic quantum gravity that were just given are
unusual. Many approaches differ.
A number of past approaches use concepts that con-
tradict the smallest length. Examples are approaches
that use conformal symmetry, scaling symmetry, non-
commutative space, higher or lower spatial dimensions,
space-time foam, vortices, twistors, tetrahedra, or addi-
tional group structures.
A large part of past approaches contradict the small-
est time value: this applies to all approaches based on
Lagrangians. A further group of approaches attempts to
avoid space-time completely and explores background-
independent descriptions.
Other past approaches use different microscopic con-
stituents, such as strings, membranes, loops, spins, sets,
lines, triangles, or bands.
10
Various past approaches to relativistic quantum grav-
ity predict new observations in particle physics, such
as new energy scales, new particles, new symmetries,
new interactions and reactions, or new fundamental con-
stants. Other approaches predict deviations from general
relativity [108].
However, none of these approaches fully explains the
four lines of choices in Table 1, despite intense efforts.
This also holds for proposals that extend the content of
Table 1. At the same time, none of these approaches
is based on fluctuating filiform constituents with Planck
radius.
In short, the approaches to a complete theory of rela-
tivistic quantum gravity that were explored in the past
appear less promising than the direction suggested by the
9 lines summarizing physics.
XVIII. TOWARDS A THEORY OF
RELATIVISTIC QUANTUM GRAVITY
The specific choices in lines 6 to 9 must emerge from the
theory of relativistic quantum gravity. This is a demand-
ing requirement.
So far, the requirement is not fully realized by any pro-
posed kind of microscopic constituents. In other terms,
the statistical behaviour of fluctuating microscopic con-
stituents will and must explain the gauge groups, the
spectrum of elementary particles, their mass values, their
mixing angles and CP phases, the values of the coupling
constants, the cosmological constant and the number of
spatial dimensions.
As long as Table 1 remains valid, checks of lines 6 to 9
are also the only tests possible at present for the correct-
ness of any proposed model of microscopic constituents.
In fact, counting the lines of Table 1 that are explained
and tested successfully by a given research approach is a
practical way to quantify its achievements. Among the
required tests, explaining the last line of Table 1, line 9,
is the decisive one.
Line 9 specifies the masses of elementary particles (or,
equivalently, the mass ratios to the Planck mass), the
coupling constants and the mixing matrices. These pure
numbers are fundamental constants that describe the
world around us. These fundamental constants are not
yet explained by any research approach that also agrees
with the first 5 lines. A way to calculate the fundamen-
tal constants needs to be found. (References [109–114]
explore a possible starting point.) In addition, line 9
specifies the number of dimensions and the cosmologi-
cal constant. Also these two numbers describe the world
around us and must be explained.
In short, in the search for relativistic quantum grav-
ity, the most productive way forward appears to be the
following. First, propose a specific microscopic model of
space and matter that is based on filiform constituents of
Planck radius and that realizes the five principles. After-
wards, check its consequences for the four choices of lines
6 to 9. In the decisive test, the fundamental constants
need to be calculated and compared with experiment.
XIX. CONCLUSION AND OUTLOOK
Present physics – experiment and theory – can be con-
densed in 9 lines that describe all observations to full pre-
cision and determine both the Lagrangian of general rel-
ativity and the Lagrangian of the standard model. The 9
lines consist of the five principles of least action, of max-
imum speed, of maximum force, of action quantization
and of smallest entropy, plus four lines of specific choices
for the gauge interactions, the elementary particles, and
the fundamental constants.
The main experimental prediction of the 9 lines is the
lack of any effect beyond general relativity and beyond
the standard model of particle physics, with massive neu-
trinos. The main theoretical prediction is that some dis-
crete, filiform and fluctuating constituents of Planck ra-
dius in 3 + 1 dimensions will explain all 9 lines, not with
an equation of motion or a Lagrangian, but instead only
with statistical arguments.
The complete theory of relativistic quantum gravity –
the “theory of everything” or “final theory” – is predicted
to have only two experimental effects: general relativity
and the standard model of elementary particle physics,
with massive neutrinos. Therefore, the decisive test of
any proposed complete theory is the calculation of the
values of the elementary particle masses, of the mixing
matrices and of the coupling constants.
ACKNOWLEDGMENTS AND DECLARATIONS
The author thanks Chandra Sivaram, Arun Kenath, Erik
Baigar, Lucas Burns, Thomas Racey, Michael Good,
Peter Woit, Louis Kauffman, Michel Talagrand, Luca
Bombelli, Isabella Borgogelli Avveduti, Peter Schiller,
Steven Carlip and an anonymous referee for discussions.
Part of this work was supported by a grant of the Klaus
Tschira Foundation. The author declares that he has no
conflict of interest and no competing interests. There is
no additional data available for this manuscript (if all
experiments ever made are put aside).
[1] J. D. Wells, The once and present standard model of ele-
mentary particle physics, in Discovery Beyond the Stan-
dard Model of Elementary Particle Physics (Springer,
2020) pp. 51–69.
11
[2] C. M. Will, The Confrontation between General Rela-
tivity and Experiment, Living Rev. Rel. 17, 4 (2014),
arXiv:1403.7377 [gr-qc].
[3] R. Abbott et al. (LIGO Scientific, VIRGO, KAGRA),
Tests of General Relativity with GWTC-3, preprint
(2021), arXiv:2112.06861 [gr-qc].
[4] M. Kramer et al., Strong-Field Gravity Tests with
the Double Pulsar, Phys. Rev. X 11, 041050 (2021),
arXiv:2112.06795 [astro-ph.HE].
[5] R. Workman et al. (Particle Data Group), Review of
Particle Physics, Prog. Theor. Exp. Phys. 2022, 083C01
(2022).
[6] L. D. Landau and E. M. Lifshitz, Mechanics: Volume
1, Course of Theoretical Physics, Vol. 1 (Butterworth-
Heinemann, Oxford, 1976).
[7] A. Einstein, Zur Elektrodynamik bewegter K¨orper, An-
nalen der Physik 322, 891 (1905).
[8] P. Antonini, M. Okhapkin, E. Goklu, and S. Schiller,
Test of constancy of speed of light with rotating cryo-
genic optical resonators, Phys. Rev. A 71, 050101(R)
(2005),arXiv:gr-qc/0504109.
[9] E. A. Rauscher, The Minkowski metric for a multidi-
mensional geometry, Lett. Nuovo Cim. 7S2, 361 (1973).
[10] H. J. Treder, The planckions as largest elementary par-
ticles and as smallest test bodies, Found. Phys. 15, 161
(1985).
[11] R. J. Heaston, Identification of a Superforce in the
Einstein Field Equations, Journal of the Washington
Academy of Sciences 80, 25 (1990).
[12] V. de Sabbata and C. Sivaram, On limiting field
strengths in gravitation, Found. Phys. Lett. 6, 561
(1993).
[13] C. Massa, Does the gravitational constant increase?, As-
trophys. Space Sci. 232, 143 (1995).
[14] L. Kostro and B. Lange, Is c**4/G the greatest possible
force in nature?, Phys. Essays 12, 182 (1999).
[15] G. W. Gibbons, The Maximum Tension Principle in
General Relativity, Found. Phys. 32, 1891 (2002),
arXiv:hep-th/0210109.
[16] C. Schiller, Maximum force and minimum dis-
tance: physics in limit statements, preprint (2003),
arXiv:physics/0309118 [physics.gen-ph].
[17] C. Schiller, General Relativity and Cosmology Derived
From Principle of Maximum Power or Force, Int. J.
Theor. Phys. 44, 1629 (2005),arXiv:physics/0607090.
[18] C. Schiller, Simple Derivation of Minimum Length, Min-
imum Dipole Moment and Lack of Space-time Continu-
ity, Int. J. Theor. Phys. 45, 221 (2006).
[19] J. Barrow and G. Gibbons, Maximum tension: with
and without a cosmological constant, Mon. Not. Roy.
Astron. Soc. 446, 3874 (2014).
[20] M. P. Dabrowski and H. Gohar, Abolishing the maxi-
mum tension principle, Phys. Lett. B 748, 428 (2015),
arXiv:1504.01547 [gr-qc].
[21] M. R. R. Good and Y. C. Ong, Are black holes spring-
like?, Phys. Rev. D 91, 044031 (2015),arXiv:1412.5432
[gr-qc].
[22] Y. L. Bolotin, V. A. Cherkaskiy, A. V. Tur, and V. V.
Yanovsky, An ideal quantum clock and Principle of max-
imum force, preprint (2016), arXiv:1604.01945 [gr-qc].
[23] V. Cardoso, T. Ikeda, C. J. Moore, and C.-M. Yoo, Re-
marks on the maximum luminosity, Phys. Rev. D 97,
084013 (2018),arXiv:1803.03271 [gr-qc].
[24] Y. C. Ong, GUP-corrected black hole thermodynamics
and the maximum force conjecture, Phys. Lett. B 785,
217 (2018),arXiv:1809.00442 [gr-qc].
[25] J. D. Barrow, Non-Euclidean Newtonian cos-
mology, Class. Quant. Grav. 37, 125007 (2020),
arXiv:2002.10155 [gr-qc].
[26] J. D. Barrow, Maximum force and naked singularities
in higher dimensions, Int. J. Mod. Phys. D 29, 2043008
(2020),arXiv:2005.06809 [gr-qc].
[27] J. D. Barrow and N. Dadhich, Maximum force in modi-
fied gravity theories, Phys. Rev. D 102, 064018 (2020),
arXiv:2006.07338 [gr-qc].
[28] K. Atazadeh, Maximum force conjecture in Kiselev, 4D-
EGB and Barrow corrected-entropy black holes, Phys.
Lett. B 820, 136590 (2021).
[29] A. Jowsey and M. Visser, Counterexamples to
the maximum force conjecture, Universe 7(2021),
arXiv:2102.01831 [gr-qc].
[30] V. Faraoni, Maximum force and cosmic censorship,
Phys. Rev. D 103, 124010 (2021),arXiv:2105.07929 [gr-
qc].
[31] C. Schiller, Comment on “Maximum force and cos-
mic censorship”, Phys. Rev. D 104, 068501 (2021),
arXiv:2109.07700 [gr-qc].
[32] V. Faraoni, Reply to “Comment on ‘Maximum force and
cosmic censorship”’, Phys. Rev. D 104, 068502 (2021).
[33] C. Schiller, Tests for maximum force and maxi-
mum power, Phys. Rev. D 104, 124079 (2021),
arXiv:2112.15418 [gr-qc].
[34] C. Sivaram, A. Kenath, and C. Schiller, From Maxi-
mal Force to the Field Equations of General Relativity,
preprint (2021).
[35] L.-M. Cao, L.-Y. Li, and L.-B. Wu, Bound on the rate
of Bondi mass loss, Phys. Rev. D 104, 124017 (2021),
arXiv:2109.05973 [gr-qc].
[36] N. Dadhich, Maximum force for black holes and
Buchdahl stars, Phys. Rev. D 105, 064044 (2022),
arXiv:2201.10381 [gr-qc].
[37] C. J. Hogan, Energy Flow in the Universe, NATO Sci.
Ser. C 565, 283 (2001),arXiv:astro-ph/9912110.
[38] V. G. Gurzadyan and A. Stepanian, Hubble tension and
absolute constraints on the local Hubble parameter, As-
tron. Astrophys. 653, A145 (2021),arXiv:2108.07407
[astro-ph.CO].
[39] 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 Rup-
peiner geometry, Phys. Rev. Res. 4, 023031 (2022),
arXiv:2108.13435 [gr-qc].
[40] A. Kenath, C. Schiller, and C. Sivaram, From maximum
force to the field equations of general relativity - and
implications, to appear in Int. J. Mod. Phys. D (2022),
arXiv:2205.06302 [gr-qc].
[41] A. Loeb, Six novel observational tests of general relativ-
ity, preprint (2022), arXiv:2205.02746.
[42] A. Jowsey and M. Visser, Reconsidering maximum lu-
minosity, Int. J. Mod. Phys. D 30, 2142026 (2021),
arXiv:2105.06650 [gr-qc].
[43] G. E. Volovik, Negative Newton constant may destroy
some conjectures, Mod. Phys. Lett. A 37, 2250034
(2022),arXiv:2202.12743 [gr-qc].
[44] D. W. Sciama, Gravitational radiation and general rel-
ativity, Phys. Bull. 24, 657 (1973).
12
[45] C. Sivaram, A general upper limit on the mass and en-
tropy production of a cluster of supermassive objects,
Astrophysics and Space Science 86, 501 (1982).
[46] L. Kostro, The quantity c**5/G interpreted as the
greatest possible power in nature, Phys. Essays 13, 143
(2000).
[47] C. Schiller, From maximum force via the hoop conjec-
ture to inverse square gravity, Gravitation and Cosmol-
ogy 28, 305 (2022).
[48] K. S. Thorne, Nonspherical gravitational collapse–a
short review, in Magic Without Magic: John Archibald
Wheeler, edited by J. Klauder (Freeman, 1972) pp. 231–
258.
[49] S. Hod, Introducing the inverse hoop conjecture
for black holes, Eur. Phys. J. C 80, 1148 (2020),
arXiv:2101.05290 [gr-qc].
[50] G. Liu and Y. Peng, A conjectured universal relation for
black holes and horizonless compact stars, Nucl. Phys.
B970, 115485 (2021).
[51] M. Planck, Vorlesungen ¨uber die Theorie der
W¨armestrahlung (J. A. Barth, 1906).
[52] C. Cohen-Tannoudji, B. Diu, and F. Lalo¨e, Quantum
Mechanics, Volume 1: Basic Concepts, Tools, and Ap-
plications (Wiley, 2019).
[53] M. Bartelmann, B. Feuerbacher, T. Kr¨uger, D. L¨ust,
A. Rebhan, and A. Wipf, Theoretische Physik 3 —
Quantenmechanik (Springer Berlin Heidelberg, 2018).
[54] A. Zagoskin, Quantum Theory: A Complete Introduc-
tion (Mobius, 2015).
[55] J. L´evy-Leblond and F. Balibar, Quantique: rudiments ,
Enseignement de la physique (Masson, 1998).
[56] G. Cohen-Tannoudji, Les constantes universelles, Ques-
tions de sciences (Hachette, Paris, 1991).
[57] L. B. Okun, Cube or hypercube of natural units, in
Multiple Facets Of Quantization And Supersymmetry:
Michael Marinov Memorial Volume, edited by M. Ol-
shanetsky and A. Vainshtein (World Scientific, 2002)
pp. 670–675, arXiv:hep-ph/0112339.
[58] D. Oriti, The Bronstein hypercube of quantum gravity,
in Beyond Spacetime, edited by N. Huggett, K. Mat-
subara, and C. W¨uthrich (Cambridge University Press,
2020) pp. 25–52, arXiv:1803.02577 [physics.hist-ph].
[59] J. Uffink and J. van Lith, Thermodynamic uncertainty
relations, Foundations of Physics 29, 655 (1999).
[60] A. E. Shalyt-Margolin and A. Y. Tregubovich, Gener-
alized uncertainty relation in thermodynamics, preprint
(2003), arXiv:gr-qc/0307018.
[61] Y. Hasegawa, Thermodynamic bounds via bulk-
boundary correspondence: speed limit, thermodynamic
uncertainty relation, and Heisenberg principle, preprint
(2022), arXiv:2203.12421 [cond-mat.stat-mech].
[62] L. Szilard, ¨
Uber die Entropieverminderung in einem
thermodynamischen System bei Eingriffen intelligenter
Wesen, Zeitschrift f¨ur Physik 53, 840 (1929).
[63] L. Brillouin, Science and Information Theory, Dover
Books on Physics (Dover Publications, 2013).
[64] H. W. Zimmermann, Die Entropie von Teilchen und
ihre Quantisierung, Zeitschrift f¨ur Physikalische Chemie
195, 1 (1996).
[65] H. W. Zimmermann, Plancks Strahlungsgesetz und die
Quantisierung der Entropie, Berichte der Bunsenge-
sellschaft f¨ur physikalische Chemie 91, 1033 (1987).
[66] H. W. Zimmermann, ¨
Uber die Quantisierung der En-
tropie und die Verteilungsfunktionen von Boltzmann,
Bose-Einstein und Fermi-Dirac, Berichte der Bunsenge-
sellschaft f¨ur physikalische Chemie 92, 81 (1988).
[67] H. W. Zimmermann, Particle Entropies and Entropy
Quanta. I. The Ideal Gas, Zeitschrift f¨ur Physikalische
Chemie 214, 187 (2000).
[68] H. W. Zimmermann, Particle Entropies and Entropy
Quanta II. The Photon Gas, Zeitschrift f¨ur Physikalis-
che Chemie 214, 347 (2000).
[69] H. W. Zimmermann, Particle Entropies and Entropy
Quanta. III. The van der Waals Gas, Zeitschrift f¨ur
Physikalische Chemie 216, 615 (2002).
[70] H. W. Zimmermann, Particle Entropies and Entropy
Quanta: IV. The Ideal Gas, the Second Law of Thermo-
dynamics, and the P–t Uncertainty Relation, Zeitschrift
f¨ur Physikalische Chemie 217, 55 (2003).
[71] H. W. Zimmermann, Particle Entropies and Entropy
Quanta V. The P–t Uncertainty Relation, Zeitschrift
f¨ur Physikalische Chemie 217, 1097 (2003).
[72] J. Garcia-Bellido, Quantum black holes, preprint
(1993), arXiv:hep-th/9302127.
[73] L. Liao and P. Shou-Yong, Sommerfeld's quantum
condition of action and the spectra of quantum
schwarzschild black hole, Chinese Physics Letters 21,
1887 (2004).
[74] A. Kirwan Jr, Intrinsic photon entropy? The darkside of
light, International Journal of Engineering Science 42,
725 (2004).
[75] M. Meschke, W. Guichard, and J. P. Pekola, Single-
mode heat conduction by photons, Nature 444, 187
(2006).
[76] D. Kothawala, T. Padmanabhan, and S. Sarkar, Is grav-
itational entropy quantized?, Phys. Rev. D 78, 104018
(2008),arXiv:0807.1481 [gr-qc].
[77] Y.-X. Liu, S.-W. Wei, R. Li, and J.-R. Ren, Quanti-
zation of black hole entropy from quasinormal modes,
Journal of High Energy Physics 2009, 076 (2009).
[78] J.-R. Ren, L.-Y. Jia, and P.-J. Mao, Entropy quantiza-
tion of d-dimensional Gauss–Bonnet black holes, Mod-
ern Physics Letters A 25, 2599 (2010).
[79] Y. V. Dydyshka and A. E. Shalyt-Margolin, Black Hole
Quantum Entropy and Its Minimal Value, Nonlin. Phe-
nom. Complex Syst. 16, 255 (2013), arXiv:1303.1444
[physics.gen-ph].
[80] A. Bakshi, B. R. Majhi, and S. Samanta, Gravitational
surface hamiltonian and entropy quantization, Physics
Letters B 765, 334 (2017).
[81] L. Yu and D.-J. Qi, Spectroscopy of the Rotating
Kaluza-Klein Spacetime via Revisited Adiabatic In-
variant Quantity, International Journal of Theoretical
Physics 56, 2151 (2017).
[82] Q.-Q. Jiang, Revisit emission spectrum and entropy
quantum of the Reissner–Nordstr¨om black hole, The
European Physical Journal C 72, 2086 (2012).
[83] A. A. Varlamov, A. V. Kavokin, and Y. M. Galperin,
Quantization of entropy in a quasi-two-dimensional elec-
tron gas, Phys. Rev. B 93, 155404 (2016).
[84] K. Schwab, E. A. Henriksen, J. M. Worlock, and M. L.
Roukes, Measurement of the quantum of thermal con-
ductance, Nature 404, 974 (2000).
[85] K. Schwab, J. Arlett, J. Worlock, and M. Roukes,
Thermal conductance through discrete quantum chan-
nels, Physica E: Low-dimensional Systems and Nanos-
13
tructures 9, 60 (2001), proceedings of the Eleventh
International Winterschool on New Developments in
Solid State Physics, ”New Frontiers in Low-Dimensional
Physics”.
[86] K. Schwab, Information on heat, Nature 444, 161
(2006).
[87] M. Partanen, K. Y. Tan, J. Govenius, R. E. Lake,
M. K. M¨akel¨a, T. Tanttu, and M. M¨ott¨onen, Quantum-
limited heat conduction over macroscopic distances, Na-
ture Physics 12, 460 (2016).
[88] F. M´arkus and K. Gamb´ar, Minimum entropy pro-
duction effect on a quantum scale, Entropy 23,
10.3390/e23101350 (2021).
[89] B. Lavenda, Statistical Physics: A Probabilistic Ap-
proach, Dover Books on Physics (Dover Publications,
2016).
[90] R. Olf, F. Fang, G. E. Marti, A. MacRae, and D. M.
Stamper-Kurn, Thermometry and cooling of a Bose gas
to 0.02 times the condensation temperature, Nature
Physics 11, 720 (2015).
[91] W. Pauli, Relativistic Field Theories of Elementary Par-
ticles, Rev. Mod. Phys. 13, 203 (1941).
[92] J. A. Heras, Can Maxwell’s equations be obtained from
the continuity equation?, American Journal of Physics
75, 652 (2007).
[93] L. Burns, Maxwell’s equations are universal for locally
conserved quantities, Advances in Applied Clifford Al-
gebras 29, 1 (2019).
[94] G. Gabrielse, S. E. Fayer, T. G. Myers, and
X. Fan, Towards an Improved Test of the Standard
Model’s Most Precise Prediction, Atoms 7, 45 (2019),
arXiv:1904.06174 [quant-ph].
[95] S. Borsanyi, Z. Fodor, J. N. Guenther, C. Hoelbling,
S. D. Katz, L. Lellouch, T. Lippert, K. Miura, L. Parato,
K. K. Szabo, F. Stokes, B. C. Toth, C. Torok, and
L. Varnhorst, Leading hadronic contribution to the
muon magnetic moment from lattice qcd, Nature 593,
51 (2021).
[96] B. Abi et al. (Muon g-2), Measurement of the Positive
Muon Anomalous Magnetic Moment to 0.46 ppm, Phys.
Rev. Lett. 126, 141801 (2021),arXiv:2104.03281 [hep-
ex].
[97] T. Aaltonen et al. (CDF), High-precision measurement
of the W boson mass with the CDF II detector, Science
376, 170 (2022).
[98] M. Veltman, Diagrammatica: the path to Feynman dia-
grams (Cambridge University Press, 1994) pp. 249–272.
[99] C. A. Mead, Possible connection between gravitation
and fundamental length, Physical Review 135, B849
(1964).
[100] J. Ellis, J. Espinosa, G. Giudice, A. Hoecker, and A. Ri-
otto, The probable fate of the standard model, Physics
Letters B 679, 369 (2009).
[101] M. Shifman, Musings on the current status of hep, arXiv
preprint arXiv:2001.00101 (2019).
[102] D. Hilbert, Naturerkennen und Logik, in Dritter Band:
Analysis ·Grundlagen der Mathematik ·Physik ·Ver-
schiedenes (Springer, 1935) pp. 378–387.
[103] S. Hossenfelder, C. Marletto, and V. Vedral, Quantum
effects in the gravitational field, Nature 549, 31 (2017).
[104] J. D. Bekenstein, Universal upper bound on the entropy-
to-energy ratio for bounded systems, Phys. Rev. D 23,
287 (1981).
[105] B. Mirza, Z. Mirzaiyan, and H. Nadi, Maximum rate of
entropy emission, Annals of Physics 415, 168117 (2020).
[106] M. Gardner, Riddles of the Sphinx and Other Math-
ematical Puzzle Tales (Mathematical Association of
America, 1987) p. 47.
[107] G. Weber, Thermodynamics at boundaries, Nature 365,
792 (1993).
[108] A. Addazi, J. Alvarez-Muniz, R. A. Batista,
G. Amelino-Camelia, V. Antonelli, M. Arzano,
M. Asorey, J.-L. Atteia, S. Bahamonde, F. Bajardi,
et al., Quantum gravity phenomenology at the dawn of
the multi-messenger era—a review, Progress in Particle
and Nuclear Physics 125, 103948 (2022).
[109] C. Schiller, A conjecture on deducing general relativity
and the standard model with its fundamental constants
from rational tangles of strands, Phys. Part. Nucl. 50,
259 (2019).
[110] C. Schiller, Testing a conjecture on the origin of the
standard model, Eur. Phys. J. Plus 136, 79 (2021).
[111] C. Schiller, Testing a conjecture on the origin of space,
gravity and mass, Indian Journal of Physics 96, 3047
(2022).
[112] C. Schiller, Testing a conjecture on quantum electrody-
namics, J. Geom. Phys. 178, 104551 (2022).
[113] M. Botta Cantcheff, Spacetime geometry as statistic en-
semble of strings, arXiv:1105.3658 (2011).
[114] S. Carlip, Dimension and dimensional reduction in
quantum gravity, Classical and Quantum Gravity 34,
193001 (2017).
