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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

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