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 ). 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 .
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 . In the general case, the action is de-
ﬁned as the integral of a Lagrangian density based on and
built with observable ﬁelds. The equations of motion fol-
low from the requirement that action is minimized – or
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-
(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-
(5) S≥kln2 Entropy Sis never smaller than ln 2 times
the Boltzmann constant k. This invariant
(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
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 , make
up everything and, with the interactions,
ﬁx the standard model Lagrangian.
(9) Finally, 27
– dimensions, cosmological constant, cou-
pling constants, particle mass ratios, mix-
ing matrices  – complete the two fun-
damental Lagrangians. They determine
all observations, including all colours.
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 . 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
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 . 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-
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-
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 . 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
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
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
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 , 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 . 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
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
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 . 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
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
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
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
 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  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 . 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
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.
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.
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 . The U(1) gauge group explains the vanishing
mass of the photon, Coulomb’s law, magnetism and light.
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 . 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
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 . 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 .
Also the recent W boson mass measurement is not a con-
ﬁrmed deviation .
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 .
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 . 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
of the Hilbert Lagrangian and of the standard model La-
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 . 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-
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 .
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  and . Line 5 de-
termines thermodynamics, as shown in references [67–71]
and . 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
 (for vanishing neutrino mass) and  (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
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
XI. WHAT ADVANTAGE DO THE 9 LINES
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
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
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
 given by
twice the Planck length. As a consequence, continuity of
space and time is not intrinsic to nature, but due to an
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 .
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
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 . 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
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
lack of inﬁnitely large and of inﬁnitely small observables
made by Hilbert in 1935 .
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 .
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
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
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
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
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  and the
maximum entropy emission rate . In fact, their dis-
creteness implies all Planck limits. Their ﬁliform struc-
ture explains, as shown by Dirac’s trick , 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
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-
Other past approaches use diﬀerent microscopic con-
stituents, such as strings, membranes, loops, spins, sets,
lines, triangles, or bands.
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
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
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-
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
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
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