Content uploaded by Christoph Schiller
Author content
All content in this area was uploaded by Christoph Schiller on Oct 31, 2021
Content may be subject to copyright.
Testing a conjecture on the origin of space,
gravity and mass
Christoph Schiller *
July 23, 2021
Abstract
A Planck-scale model for the microscopic degrees of freedom of space and gravity,
based on a fundamental principle that involves fluctuating one-dimensional strands,
is tested. Classical and quantum properties of space and gravitation, from the field
equations of general relativity to gravitons, are deduced. Predictions include the lack
of any change to general relativity at all sub-galactic scales, the validity of black
hole thermodynamics, the lack of singularities and the lack of unknown observable
quantum gravity effects. So far, all predictions agree with observations, including the
validity of the maximum luminosity or power value c5/4Gfor all processes in nature,
from microscopic to astronomical. Finally, it is shown that the strand conjecture
implies a model for elementary particles that allows deducing ab-initio upper and
lower limits for their mass values.
Keywords: general relativity; quantum gravity; particle mass; strand conjecture.
PACS numbers: 04.20.-q (classical general relativity), 04.60.-m (quantum gravity).
*Motion Mountain Research, 81827 Munich, Germany, cs@motionmountain.net
ORCID 0000-0002-8188-6282.
1
1 The quest to uncover the microscopic aspects of space and gravity
What are the microscopic degrees of freedom of black holes, the microscopic nature of the vac-
uum, and the microscopic details of curvature? Many possibilities have been proposed and ex-
plored [1, 2, 3, 4, 5, 6, 7, 8, 9, 10]. In order to show that any proposed microscopic degrees of
freedom are candidates for a description of nature, it is necessary to show that they reproduce
space, curvature, mass and gravitation in all their macroscopic and microscopic aspects. It is
equally necessary to show that strands provide additional results about gravitation that go beyond
the usual description of space as a continuous manifold made of points.
The so-called strand conjecture proposes a microscopic model for black holes, particles, space
and gravity that is based on one-dimensional fluctuating constituents that are called strands.The
model is based on a single fundamental principle that describes nature at the Planck scale. It will
appear that the strand conjecture agrees with all observations about general relativity and quantum
gravity at all sub-galactic scales. Strands also provide a model for elementary particles and their
gauge interactions, and suggest a way to estimate their mass values.
2 The origin of the strand conjecture
When Max Planck discovered the quantum of action in 1899, he found the underlying quantity
that explains the observation of all quantum effects in nature [11]. Bohr described quantum theory
as consequence of the minimum observable action value [12]. Heisenberg introduced the canon-
ical commutation relation and Schrödinger introduced the wave function. Pauli included spin and
Dirac the maximum energy speed c. From around 1929 onwards, Dirac regularly made use of the
so-called string trick or belt trick in his lectures. The trick, illustrated below in Figure 9, assumes
that fermions are connected to spatial infinity by tethers that are unobservable, but whose cross-
ings are observable. With help of the trick, Dirac used to describe spin 1/2behaviour as result of
tethered rotation. Nevertheless, he never published anything about this connection. Answering a
letter from Gardner, Dirac wrote that the trick demonstrates that angular momenta below /2are
2
not possible in nature [13]. A smallest angular momentum /2still implies a smallest observable
action value .
Historically, tethers were the first hint that nature might be built from unobservable extended
constituents. It took several decades to understand that also the complete Dirac equation could
be deduced from unobservable tethers. This was first achieved by Battey-Pratt and Racey in 1980
[14]. Independently, in 1987, Kauffman conjectured a direct relation between the canonical com-
mutation relation – and thus Planck’s constant – and a crossing switch of tethers [15]. Again,
without stating so explicitly, the assumption was that tethers are unobservable, whereas their cross-
ings are. In the early twenty-first century, independently of the work by Battey-Pratt and Racey
and of that by Kauffman, Dirac’s trick again led to the discovery of the relation between crossing
switches of unobservable tethers, , wave functions, and the Dirac equation [16]. In short, Dirac’s
trick of invisible tethers implies Dirac’s equation. It thus appeared that every quantum effect can
be thought as being due to unobservable extended constituents – called strands in the following –
whose crossings are observable.
A question arises naturally: can unobservable strands also explain gravity? The finite value
of black hole entropy and its surface dependence provided first hints [17, 18]. Indeed, it turns
out that both the properties of black holes and Einstein’s field equations can be deduced from
crossing switches of unobservable strands. This deduction is repeated below. It thus appeared
that every gravitational effect can be thought as being due to unobservable extended constituents
whose crossings are observable.
The strand conjecture for fundamental physics appears promising also from another perspec-
tive. The central parts of quantum field theory can be summarized by the statements that all
observable action values obey W≥and that all observable energy speeds obey v≤c. General
relativity can be summarized by the statement that all observable power values obey P≤c5/4G,
as discussed in various publications [19, 20, 21, 22]. This suggests that nature is fundamentally
simple and that a complete and unified description of motion could be based on inequalities.
3
Strand conjecture:
The fundamental Planck-scale principle of the strand conjecture
Observation:
A fundamental
event in space
tt+Δ
W = ℏ
Δl ≥ √4Gℏ/c³
Δt ≥ √4Gℏ/c⁵
S = k ln 2
Figure 1: The fundamental principle of the strand conjecture specifies the simplest ob-
servation – a simplified version of Dirac’s trick, shown in Figure 9, taking place at the
Planck scale – that is possible in nature: the almost point-like fundamental event results
from a skew strand switch,orcrossing switch, at a position in three-dimensional space.
The strands themselves are not observable. They are impenetrable and are best imag-
ined as having Planck size radius. The observable switch defines the action unit .The
double Planck length limit and the double Planck time limit arise, respectively, from
the smallest and from the fastest crossing switch possible. The paper plane represents
background space, i.e., the local tangent Euclidean space defined by the observer.
3 The fundamental principle and the claims of the strand conjecture
The strand conjecture states: all physical systems found in nature – matter, radiation, space and
horizons – are made of strands that fluctuate at the Planck scale but remain unobservable.
Astrand is defined as smooth curved line – a one-dimensional, open, continuous,
everywhere infinitely differentiable subset of R3or of a curved 3-dimensional Rie-
mannian space, with trivial topology and without endpoints – that is surrounded by
a perpendicular disk of Planck radius G/c3at each point of the line, and whose
shape is randomly fluctuating over time.
The strand conjecture is then formulated in the following way:
Strands are unobservable. However, crossing switches of skew strands – exchanges
of over- and underpasses – are observable. Crossing switches determine the Planck
units G,cand ; this fundamental principle is illustrated in Figure 1.
4
The defining Figure 1 thus combines the essence of Dirac’s trick with the Planck limits.
The strand conjecture claims: the fundamental principle of Figure 1 contains all observations,
all equations of motions and all Lagrangians. In particular, the fundamental principle implies:
Physical space is a (three-dimensional) network of fluctuating strands – i.e., of strands
that are neither woven nor tangled nor knotted, as illustrated in Figure 2).
Horizons are (two-dimensional) weaves of fluctuating strands – i.e., similar to a fabric
made of woven threads, and illustrated in Figure 3.
Particles are (localized) rational tangles of fluctuating strands – using the term from
topological knot theory, defined and illustrated in Figure 8.
Physical motion minimizes the number of observable crossing switches of fluctuating
unobservable strands.
Using Figure 1, the strand conjecture appears to imply general relativity, fermions, bosons and
the gauge interactions, with all their observed properties. The figure also illustrates the most
fundamental event and the most fundamental process in nature, from which all other processes
are built, including all motion in nature. The following sections check these claims in detail for
gravitation at sub-galactic scales. After all the checks are passed successfully, a number of new
results are presented.
The implications of the strand conjecture for particle physics, gauge interactions, and the stan-
dard model have been explored elsewhere [16, 23]. Tangles of strands imply the particle spectrum,
Reidemeister moves imply the three gauge groups and couplings, and tether exchanges imply par-
ticle mixings. The complete Lagrangian of the modern standard model follows from strands. In
the following, however, only gravitation is explored.
The fundamental principle of the strand conjecture of Figure 1 states that action, length, time
and entropy are limited from below:
W≥,Δl≥4G/c3,Δt≥4G/c5,S≥kln 2 .(1)
5
Strands realize and visualize these inequalities. Together with Figure 1, they contain everything
that is needed to deduce the rest of the present work. The number ln 2 in the minimum entropy is
due to the 2 strand configurations. One notes that the minimum length and time are given by twice
the Planck values. In order to visualize the minimum length, it is easiest to visualize strands as
having Planck-size radius.
Apart from their crossings in space – a strand segment passing over another – strands have no
observable properties. Strands have no colour, no tension, no mass, and no energy. Due to the
impossibility of observing strands, strands have no meaningful equation of motion. Indeed, all
results in the following are independent of the detailed fluctuating motion one might imagine for
strands – as long as crossing switches reproduce observations. This aspect eliminates any apparent
arbitrariness of the description of space, horizons and matter with fluctuating strands.
Strands cannot be cut; they are not made of parts. Strands cannot interpenetrate; they never
form an actual crossing. When the term ‘crossing’ is used in the present context, only the two-
dimensional projection shows a crossing. In three dimensions, strands are always at a distance.
Like in Dirac’s trick, a crossing switch – the change from an overpass to an underpass – cannot
arise through strand interpenetration, but only via strand deformation.
In the strand conjecture, all physical observables – action, momentum, energy, mass, velocity,
length, surface, volume, tension, entropy, field intensities, quantum numbers, etc. – arise from
combinations of crossing switches. No physical observable is a property of strands themselves;
all physical observables arise from shape configurations of several strands. In short: all physical
observables emerge from strand crossings.
4 Deducing physical space from strands
Because strands are unobservable, it is not possible to describe them with differential equations,
and it is not possible to speak about their motion or their dynamics. The only observable aspect of
strands – as in Dirac’s trick – are their crossing switches, and thus, for example, the distribution of
crossing switches. To relate strands to physics, it is important to deduce the behaviour of crossing
6
Figure 2: A simplified and idealized illustration of the strand conjecture for a flat vac-
uum, i.e., for flat physical space. The space of the picture is background space. Physical
space is generated by strand crossings. Strands fluctuate in all directions. (Typical strand
distances are many orders of magnitude larger than their diameters.) For sufficiently long
time scales, the lack of crossing switches leads to a vanishing energy density; for short
time scales, particle–antiparticle pairs, i.e., rational tangle–antitangle pairs, arise in the
vacuum due to the shape fluctuations of the strands, as illustrated in Figure 12 below. The
difference between background space and physical space is discussed in Appendix A.
switches from the fundamental principle. This is done now, starting with physical space.
In the strand conjecture, a network of fluctuating strands is conjectured to yield physical flat
empty space. A strand network is illustrated in Figure 2. The picture uses background space to
define physical space. Background space is what is needed to talk about nature. Physical space is
everything that can be measured about space: curvature, vacuum energy, entropy, temperature etc.
The circularity issues that arise are discussed in Appendix A.
Anetworkofuntangled,unwoven and unknotted strands models empty and flat physical
space. The time-average of the fluctuations, on a scale of a few Planck times or more, yields
three-dimensional physical flat space, including its continuity, homogeneity, isotropy and Lorentz-
invariance. On sufficiently long time scales, there are (on average) no crossing switches, and thus
neither matter nor energy – just empty space. Strands thus imply that no deviation from the con-
tinuity, homogeneity, isotropy, dimensionality and Lorentz-invariance of (physical) flat space can
be observed – at any energy – despite the existence of a smallest length lmin =4G/c3.
7
smallest
area
neighbouring
strand
with
additional
crossing
2 lPl
Black hole horizon in the strand conjecture, side view top view
2 lPl
Observed
horizon:
a thin spherical cloud first ring (black) first ring (black)
n=1
additional
crossing
Figure 3: The strand conjecture illustrated for a Schwarzschild black hole, as seen by a
distant observer at rest: the horizon is a cloudy or fuzzy surface produced by the crossing
switches of the strands woven tightly into it. Due to the additional crossings above the
horizon, the number of microstates per smallest area is larger than 2, and given by the
base eof the natural logarithms (see text). This yields the entropy of black holes.
5 Deducing horizons and black holes from strands
In the strand conjecture, woven fluctuating strands define horizons. More precisely, the strand
conjecture implies that
Horizons are one-sided, tight weaves.
In this statement, one-sided means that all strands leave the horizon on one side, the side of the
observer. One-sidedness means that there is ‘nothing’, not even an unobservable strand, on the
other side of the horizon. A schematic illustration of a Schwarzschild black hole, shown both as a
cross section and as a top view for a distant observer at rest, is given in Figure 3. For a black hole,
and for any other horizon, all strands come in from far away, are woven into the horizon, and leave
again to far away. If strands are imagined as having Planck radius, the weave of strands forming a
horizon is as tight as possible: seen from above, there is one crossing for each smallest area.
At a larger scale, a weave becomes a two-dimensional surface. For a distant observer at rest, a
one-sided weave also implies that no space and no events are observable behind it. The weave thus
8
actsasalimit to observation. For a falling observer, the strands do not form a weave, but continue
on the other side and form a (distorted) network, i.e., curved vacuum. Such an observer does not
notice anything special when approaching the horizon, as seen by an observer at spatial infinity,
or when crossing it, in its own reference frame. A one-sided weave thus shows the qualitative
properties that characterize a horizon.
The strand conjecture for horizons allows determining the energy and thus the mass of a spher-
ical, non-rotating horizon. Energy Ehas the dimension action per time. Because every crossing
switch is associated with an action , the horizon energy is found by determining the number Ncs
of crossing switches, multiplied by , that occur per unit time. This number will depend on the
surface area of the horizon. In a horizon, crossing switches propagate from one crossing to the
next, over the surface of the whole (tight) weave. Since the horizon weave is tight, the propaga-
tion speed is one smallest crossing per shortest switch time: switch propagation thus occurs at the
speed of light c. Since the horizon weave is tight, each crossings has the size of the minimum
length squared, given by AcPl =4G/c3. In the time Tneeded to circumnavigate a spherical,
non-rotating horizon of area A=4πR2at the speed of light, all crossings of the horizon switch.
This yields:
E=Ncs
T=A/(4G/c3)
2πR/c =c4
2GR. (2)
The woven strand model of a horizon thus reproduces the relation between the energy – or mass –
and the radius of a Schwarzschild black hole.
Strands also determine the number of microstates per horizon area. Figure 3 shows that for a
smallest area on the horizon, i.e., for an area that contains just one strand crossing, the effective
number Nof microstates above that smallest area is larger than 2. The number would be two if
each smallest area would contain just one crossing, with its 2 possible signs. However, a number
larger than 2 occurs because also fluctuating neighbouring strands sometimes cross above that
smallest area.
The probability for a neighbouring strand to cross above a given (central) smallest area will
depend on the distance at which the neighbouring strand leaves the lowest woven layer of the
9
horizon. To calculate the probability, one imagines the central crossing surrounded by an infinite
series of rings, each with a smallest area value AcPl =4G/c3. As illustrated in Figure 3, the
rings are numbered with a number n. The central crossing corresponds to n=0. Ring number n
therefore encloses ntimes the smallest area AcPl. The probability that a strand from ring 1 reaches
the centre, forming an additional crossing above it, is
p1=1
2=1
2! .(3)
The probability that a strand from ring nreaches the centre and forms an additional crossing is
pn=1
n+1pn−1=1
(n+1)! ,(4)
because the strand has to continue in the correct direction above every ring on its way to the
centre. This expression is a result of the extension of strands; it would not arise if the fundamental
constituents of the horizon would not be extended – in short, if they would not be strands. The
expression yields an effective number Nof microstates above the central crossing given by
N=2 + 1
2! +1
3! +1
4! +... +1
n!+... =e=2.718281... (5)
In this expression, the term 2is due to the two options at the central point; the term 1/2! arises
from the first ring around it, as shown in Figure 3; the following terms are due to the subsequent
rings. Expression (5) implies that the average number Nof strand microstates for each smallest
area, i.e., for each corrected Planck area AcPl =4G/c3on the black hole horizon, is given by
N=e. In the strand conjecture, every corrected Planck area therefore contains more than 1 bit of
information (which would correspond to N=2).
The calculation of the entropy of the complete black hole horizon starts with the usual defini-
tion
S=kln Ntotal ,(6)
10
where kis the Boltzmann constant and Ntotal is total number of microstates of the complete
horizon. Because the full horizon area Acan be seen as composed of many corrected Planck
areas, the total number of microstates is given by the product of the number of states for every
corrected Planck area:
Ntotal =NA/(4 G/c3).(7)
So far, only standard thermodynamics was used. The next step is to insert the result (5) due to
strands. This yields
Ntotal =e
A/(4 G/c3).(8)
This total number of horizon microstates can be inserted into expression (6) for the entropy. The
horizon entropy Sof a black hole with surface Ais
S
k=A
4G/c3.(9)
This is the expression discovered by Bekenstein [17]. In the strand conjecture, the finiteness of the
entropy is thus due to the discreteness of the microscopic degrees of freedom. Strands also imply
that both the surface dependence of the entropy and the factor 1/4– including the lack of factors
like ln 2 or of a Barbero-Immirzi parameter – are due to the extension of the microscopic degrees
of freedom.
In short, strands imply the energy Eand the entropy Sof Schwarzschild black holes. As usual,
the ratio E/2Sdetermines the temperature of such black holes [24]:
TBH =c
4πk
1
R=
2πkc a. (10)
In the last equality, the surface gravitational acceleration a=GM/R2=c2/2Rwas introduced,
using expression (2). In short, black holes are warm.
The finite temperature value implies that black holes radiate. As a consequence, strands re-
produce black hole evaporation. Radiation and evaporation are due to strands detaching from the
11
horizon. If a single strand detaches, a photon is emitted. If a tangle of two or three strands de-
taches, a graviton or a fermion is emitted. When all strands have detached, the complete black
hole has evaporated.
The expressions (1) and the fundamental principle of Figure 1 contain a further result of inter-
est. The gravitational acceleration on the surface of a black hole is a=GM/R2=c2/2R;thisis
the maximum value possible. The value of black hole energy (2) implies that the black hole mass
is given by M=Rc2/2G. Taken together, this yields a limit on force F=Ma given by
F≤c4
4G=3.0·1043 N.(11)
This is the maximum force that can be observed at a single point. The existence of a maximum
force is inextricably tied and equivalent to the minimum size of masses in nature. All deriva-
tions of its value make use of this connection; for example, c4/4Gis also the maximum possible
gravitational force between two black holes [19, 20, 21, 22].
6 Deducing general relativity from thermodynamics
In 1995, in a path-breaking paper, Jacobson showed that the thermodynamic properties of the
microscopic degrees of freedom of space and of black holes imply Einstein’s field equations of
general relativity [25]. He started with three thermodynamic properties:
the entropy–area relation S=Akc
3/4G,
the temperature–acceleration relation T=a/2πkc,
the relation between heat and entropy δQ =TδS.
Using these three properties, the basic thermodynamic relation
δE =δQ , (12)
12
which is valid only in case of a horizon, yields the first principle of horizon mechanics
δE =c2
8πGaδA . (13)
This expression can be rewritten, using the energy–momentum tensor Tab ,as
Tab kadΣb=c2
8πGaδA , (14)
where dΣbis the general surface element and kis the Killing vector that generates the horizon.
The Raychaudhuri equation [26] – a purely geometric relation – allows rewriting the right-hand
side as
Tab kadΣb=c4
8πG Rab kadΣb,(15)
where Rab is the Ricci tensor that describes space-time curvature. This equality between integrals
implies that the integrands obey
Tab =c4
8πG Rab −(R
2+Λ)gab,(16)
where Ris the Ricci scalar and Λis an undetermined constant of integration. These are Ein-
stein’s field equations of general relativity. The value of the cosmological constant Λis thus not
determined by the thermodynamic properties of horizons.
As Jacobson explained, the field equations are valid everywhere and for all times, because a
suitable coordinate transformation can position a horizon at any point in space and at any instant
of time. Achieving this just requires a change to a suitable accelerating frame of reference.
Given that horizons and black holes are thermodynamic systems, so is curved space. In other
words, the field equations result from thermodynamics of space. Jacobson’s argument thus shows
that space is made of microscopic degrees of freedom, and that gravity is due to the same micro-
scopic degrees of freedom.
13
7 Deducing general relativity from strands
As explained in Section 5 above, strands imply the existence of horizons and of black holes. Above
all, strands imply their thermodynamic properties: strands reproduce the entropy relation (6) of
black holes, the temperature (10) of black holes, and their heat–entropy relation from (2). These
are the three conditions for using Jacobson’s argument to derive general relativity. Strands thus
fully reproduce the argument. Therefore,
Fluctuating strands lead to general relativity.
However, the result must be taken with caution. Jacobson’s deduction of the field equations is
independent of the details of the fluctuations and independent of the microscopic model of space,
as long as the three thermodynamic properties given at the start are valid. After Jacobson’s result,
various kinds of microscopic degrees of freedom for space have been conjectured, including those
found in references [1, 2, 3, 4, 5, 6, 8, 9, 10]. These explorations have shown that finding the
correct microscopic degrees of freedom of physical space among all the proposals in the literature
is not possible using arguments from quantum gravity alone.
Any promising candidate for the microscopic degrees of freedom of space and gravitation
must also reproduce the standard model of particle physics and, above all, explain particle masses
and the other fundamental constants. This seems the only way to differentiate between the various
microscopic models of gravitation. Given that strands appear to reproduce the Lagrangian of the
standard model – as argued in references [16] and [23] – it is worth exploring them also in the
domain of gravitation. Any experiment finding a deviation from the standard model would falsify
the strand conjecture.
In summary, in the strand conjecture, the field equations – and thus the Hilbert action – ap-
pear as consequences of fluctuations of impenetrable, featureless, unobservable strands. The first
prediction of strands in the domain of gravitation is:
Pr. 1 No deviations between general relativity and the strand conjecture arise.
14
Figure 4: An illustration of the strand conjecture for a curved vacuum. The strand and
crossing configuration is not homogeneous and is midway between that of a flat vacuum
and that of a horizon. Strands in black differ in their configuration from those in a flat
vacuum. The value of the curvature is inversely proportional to the distance d.
As a result, all processes described by general relativity are reproduced by strands. This includes
1/r2gravity, as illustrated in Figure 5. Therefore, the smallest deviation between general relativity
and observations would falsify the strand conjecture. Below, in Section 11, the lack of deviations
is made more precise: the prediction is limited to sub-galactic distances. The validity for galactic
and cosmological distances will be explored in a separate paper.
8 Deducing curvature from strands
Strands not only visualize flat space; strands also visualize curvature. The fundamental principle
of the strand conjecture implies:
Flat space is a homogeneous network of fluctuating strands.
Curvature is an inhomogeneous crossing (switch) density in the vacuum network.
An illustration of spatial curvature is given in Figure 4. The strand configuration differs from
that of flat space: certain strands break the isotropy and homogeneity. The main curvature value
depends on the configuration of the strands leading to the inhomogeneity. The curvature can
15
evolve over time. This strand model for curved space implies that curved space-time is, locally, a
Minkowski space. Thus, strands lead to a pseudo-Riemannian space-time.
In short, strands visualize space, black holes, gravity and curvature. It is now time to test the
strand conjecture in detail.
9 Strand predictions about physical space
Pr. 2 Because tangling of strands is not possible in other dimensions, strands predict that flat
physical space is three-dimensional,unique and well-behaved at all scales. Flat physical
space is a three-dimensional continuum that is homogenous and isotropic, without observ-
able deviations.
So far, these predictions about physical space agree with expectations [27] and with the
most recent observations [28, 29]. Any evidence for other dimensions, other topologies,
quantum foam, different vacuum states, different vacuum states, cosmic strings, domain
walls, regions of negative energy, ‘space-time noise’, ‘particle diffusion’, ‘space viscos-
ity’ or crystal behaviour of space, or any other deviation from a well-behaved pseudo-
Riemannian space-time manifold would directly falsify the strand conjecture.
Pr. 3 As a consequence of the fundamental principle in Figure 1 and of the expressions (1),
the maximum local energy speed in nature is c. This applies at all energy scales, in all
directions, at all times, at all positions, for every physical observer. In short, the strand
conjecture predicts no observable violation of Lorentz-invariance, for all energies and all
physical systems. It predicts no variable speed of light, no time-dependent speed of light,
no time-dependent energy of light, i.e., no ‘tired’ light, no energy-dependent speed of light,
no helicity-dependent speed of light, no ‘double’ and no ‘deformed special relativity’.
Strands predict the lack of dispersion, birefringence and opacity of the vacuum. So far, all
this agrees with observations [30].
Pr. 4 The strand conjecture for the vacuum illustrated in Figure 2 predicts the lack of trans-
16
gravitational
interaction
~ 1 / r 2
first mass
distance
~ r
~ r
~ r
r
second mass
The strand conjecture for universal 1/r² gravity
virtual
gravitons
Figure 5: Gravitational attraction results from strands. More precisely, everyday gravi-
tation is due to tether pair twists and their influence on tether fluctuations. When speeds
are low and spatial curvature is negligible, as illustrated here, twisted tether pairs – i.e.,
virtual gravitons – from any mass lead to a 1/r2attraction of other masses. The average
length of twisted pairs of tethers scales with r. As a consequence, the curvature around
such a mass scales as 1/r3. These results are valid for infinite, approximately flat space.
Planckian effects. For example, the existence of a minimal measurable length given by the
corrected Planck length
lmin =4G/c3(17)
is predicted. If any effect due to space intervals smaller than the minimal length can be
observed – for example in electric dipole moments [31], in higher order effects in quantum
field theory, or in the discreteness of space – the strand conjecture is falsified. The same
holds for time intervals shorter than the corrected Planck time. So far, no observation
exceeds the corrected Planck limits.
Pr. 5 Strand crossings resemble fermionic or anti-commuting coordinates as used in supergrav-
ity, resemble non-commutative space [32, 33], resemble Clifford algebras, and even re-
semble the internal spaces of the aikyon approach based on octonions [34]. A crossing,
17
as illustrated in Figure 10, can also be seen as a four-dimensional subspace, spanned by
the four angles describing the crossing, specific to a point in background space, and thus
resembles twistor space [35]. Though strands resemble these internal spaces, they do so
only at certain points in space and at certain instants in time, because strands fluctuate. In
short, Figure 2 implies that strands do not produce fixed internal spaces. This result agrees
with data so far, but could be falsified in the future.
Pr. 6 An untangled strand network generates flat physical space. In contrast, a weakly tangled
network, as illustrated in Figure 4, generates curved physical space. As a result, also
curved space has three dimensions, at all measurable distance scales, in all directions, at
all times, at all positions, for every physical observer. So far, this is observed.
Pr. 7 The value of spatial curvature κaround a mass is due to the tether crossing switch density
induced by the mass. In the strand conjecture, the crossing switch density decreases with
distance rfrom the mass. So does the strand inhomogeneity. As illustrated in Figure 5,
this yields the proportionality
κ∼1
r3.(18)
This relation agrees with expectations. A factor 1/r2is due to Gauss’ law, and a factor
1/r is due to the average size of twisted pairs of tethers – the virtual gravitons due to
Dirac’s trick – around the mass. The third power in the decrease of the curvature around a
mass is thus due to the three dimensions of space and to the extension of strands. Without
extended constituents, an explanation of the 1/r3dependence does not seem possible.
Pr. 8 Strands and expressions (1) imply a limit to curvature κ. The limit is given by the inverse
of the smallest length:
κ≤1
lmin
=c3
4G.(19)
This limit implies the lack of singularities in nature, of any kind. So far, this prediction is
not in contrast with observations.
18
10 Strand predictions about black holes
The fundamental principle of the strand conjecture – Figure 1 and expressions (1) – allows drawing
numerous testable conclusions about black holes. If any of the following predictions is wrong, the
strand conjecture is falsified.
Pr. 9 The fundamental principle of the strand conjecture – in particular the expressions (1) –
implies that in all processes, near or far from horizons, the power or luminosity limit
P≤c5/4G=9.0709(3) ·1051 W(20)
and the force or momentum flow limit
F≤c4/4G=3.0257(2) ·1043 N(21)
are always valid, at every point in space. These limits – one quarter Planck mass per Planck
time, or 50 756(12) solar masses per second, times cand times c2– are predicted to hold
for every local process in nature [19, 20, 21, 22]. (A solar mass of 1.9885(5) ·1030 kg is
used.)
No Earth-bound process approaches the force and power limit, by far. Astrophysical ob-
servations are necessary to check the limits. Galaxies, quasars, galaxy clusters, and blazar
jets all emit below 10−5solar masses per second. In supernovae and hypernovae, both
accretion and emission are below 10−2solar masses per second. Gamma ray bursts emit
at most 1 solar mass per second. The fastest observed and simulated accretion processes
achieve 10 solar masses per second. The highest observed luminosities so far are those
observed in black hole mergers by LIGO and VIRGO [36]. At present, the highest peak
powers were observed for the events GW170729 and GW190521. They showed values
of 4.2(1.5) ·1049 Wor 230 ±80 solar masses per second [37] and of 3.7(9) ·1049 Wor
207 ±50 solar masses per second [38]. All these values are well below the (corrected)
19
Planck limit of 50 756(12) solar masses per second.
Present data therefore does not yet allow distinguishing between the corrected Planck lu-
minosity limit P≤c5/4Gand the conventional Planck limit P≤c5/G that is four
times larger. Future discoveries might allow a direct test of this aspect common to general
relativity and the strand conjecture.
Similar limits arise for mass flow dm/dt≤c3
4Gand for energy density E/V ≤c7
16 G2,
again preventing singularities.
Pr. 10 The strand conjecture implies that a rotating black hole realizes a belt trick that involves
ahuge number of tethers. Animations illustrating such a process were available on the
internet [39] before the strand conjecture formulated this equivalence, programmed by
Jason Hise. Figure 6 shows such a configuration during rotation. In this description, the
ergosphere is the region in which the crossing switches take place during the belt trick. In
contrast to the figure, the horizon of a rotating black hole is flattened at the poles, and so
is its ergosphere.
11 Strand predictions about quantum gravity and gravitons
In the derivation of general relativity in Sections 6 and 7, the cosmological horizon was not taken
into account. Strands thus imply that for sub-galactic distances, when the horizon has no influence,
general relativity holds exactly. This is the precise version of the first prediction given above in
Section 7. But strands also make predictions on quantum gravity.
Pr. 11 Gravity is due to the exchange of virtual gravitons. The tangle model of the graviton,
a twisted pair of strands, is illustrated in Figure 7. Indeed, gravitons surround masses:
twisted strand pairs arise in Dirac’s trick. The model also predicts that gravitons have spin
2, because gravitons return to their original state after a rotation of the tangle core by π.
Gravitons are predicted to be massless, because their core is not localized. Gravitons are
predicted to be bosons, because cores can swap positions along the strands. As a result,
20
Figure 6: The strand conjecture for a rotating black hole rotating about the vertical axis
(© Jason Hise). The flattening of the horizon at the poles is not shown. For an animation,
see the online video at reference [39].
The strand conjecture for the graviton
wavelength
Figure 7: The strand conjecture for the graviton: a twisted pair of strands has spin 2,
boson behaviour and vanishing mass. A gravitational wave is a coherent superposition
of a large number of gravitons.
coherent gravitons are predicted to yield gravitational waves with spin 2 and velocity c.,
as observed.
Pr. 12 In the strand conjecture, single gravitons cannot be detected, for two reasons. First, strands
imply the indistinguishability between graviton observation from any other quantum fluc-
tuation of or at the detector. Equivalently, in the strand conjecture, graviton absorption
21
does not lead to particle emission. Secondly, even if gravitons were detectable, in the
strand conjecture, they have an extremely small cross section, of the order of the square
of the Planck length. The low cross section is due to the topology of the graviton tangle.
This implies a low detection probability, as expected [40, 41].
Pr. 13 Strands and expressions (1) imply that the gravitational constant Gdoes not run when
energy is increased from everyday values to higher values. In the language of perturbative
quantum field theory, Gis not renormalized. This prediction agrees with expectations and
with data, though the available data is sparse.
Pr. 14 Strands imply that no quantum superposition effects for gravity are observable at experi-
mentally accessible scales, because graviton exchange destroys entanglement. This agrees
with expectations [42].
Pr. 15 The strand conjecture implies that in a double-slit experiment with electrons, the electrons
pass both slits at the same time, because the core splits in two pieces during passage –
though in different fractions at every passage. Therefore strands predict that the grav-
itational field of an electron arises on both slits, for every passage, though in different
fractions at every passage. Such an experiment might be possible one day.
Pr. 16 The impossibility to detect single gravitons and single strands implies that there are no
unknown, observable quantum corrections to general relativity. This prediction is in con-
trast with many expectations, and may well be the most contentious prediction in this
list. Equivalently, strands predict the lack of observable quantum effects in semiclassical
gravity.
So far, these predictions are not contradicted by any experiment. The future discovery of any new
deviation from general relativity at sub-galactic scale, including any non-trivial quantum gravity
effect, would falsify the strand conjecture.
22
Figure 8: In the strand conjecture, elementary particles are modelled as rational tan-
gles of strands. Strand segments, including tethers, are not observable; only crossing
switches are observable. Together with their fluctuating shape, tangles lead to the ob-
servation that particles are localized in the region of the tangle core. Tangles are called
rational when they are formed by switching tethers. Only rational tangles model the ob-
served behaviour of elementary particles. Fermion tangles, such as the one in the figure,
automatically have spin 1/2.
12 Strand predictions about elementary particle masses
So far, the strand conjecture has not predicted anything new. However, new predictions are possi-
ble, and in particular, predictions about elementary particle masses.
Black holes are made of large numbers of woven strands. It is natural to assume that elemen-
tary particles are made of afewwoven strands. Indeed, in the strand conjecture, all elementary
particles are rational tangles – i.e., woven, unknotted tangles – made of one, two or three strands
[16, 23]. Tangles made of four or more strands are composed, not elementary. An example of an
elementary rational tangle is shown in Figure 8.
Among tangles made of a few strands, those made of one strand are bosons; more precisely,
23
they are photons. Massive elementary particles tangles are made of two or three strands.
Every fermion tangle, being a tethered structure that is tangled in the region of its tangle core,
has non-vanishing mass. Every fermion tangle reproduces spin 1/2 behaviour under rotations –
using Dirac’s belt trick – and fermion behaviour under the exchange of positions of tangle cores.
All tangles reproduce the gauge groups U(1), SU(2) and SU(3) as the result of Reidemeister moves
on their tangled cores.
Only rational tangles – i.e., tangles that arise through the motion or braiding of their tethers
– allow reproducing the transformation of particles observed in experiments. And only rational
tangles allow a classification into a finite number of families that correspond to the observed
elementary particles. These arguments are summarized in Appendix B and are worked out in
detail in references [16] and [23].
Mass is the property of tangles that creates virtual gravitons around them. This implies:
The particle mass (in corrected Planck units) is the probability of strand crossing
switches occurring, per Planck time, in spontaneous belt tricks of the particle tangle.
Rational tangles directly allow deducing a number of predictions about mass values of elementary
particles.
Pr. 17 Strands promise, through the analogy between thermodynamic effects and gravitational
attraction, to allow calculating the gravitational mass of quantum particles. The value of
gravitational mass is predicted to depend on the tangle shape of the particle – and on
nothing else.
Research has shown that the average shape of a fluctuating tangle is the same as the shape
of a tight tangle [43, 44]. Therefore, the mass of an elementary particle is determined by
its tight tangle shape.
Since particle mass is due to their (tight) tangle shape, the mass values of all elementary
particles are predicted to be zero or positive, equal to that of their antiparticles, fixed,
24
unique, calculable and constant in time and space. This agrees with data. If particle masses
would be found to vary over space or time, the strand conjecture would be falsified.
Pr. 18 In the strand conjecture, all fermions are localizable (i.e., not trivial) tangles. Thus,
fermions have positive mass. The model of the graviton implies that gravitational charge,
or mass, of a fermion is defined by the (tight) shape of the fermion tangle core. In the
strand conjecture, mass values are automatically discrete, but are not integer multiples of a
smallest value. In the strand conjecture, mass thus automatically differs from the charges
of the gauge interactions, which are integer multiples of a smallest value. This agrees with
observation.
In the strand conjecture, as shown in references [16, 23], only particles with positive mass
can have electric and weak charge. In addition, it was shown that all mass values are
due to Yukwawa coupling to the Higgs. Only those particles that couple to the Higgs are
observed to be massive. All this agrees with observation.
Pr. 19 The tangle model of elementary particles implies that both the gravitational and the inertial
mass of elementary particles are due to tether fluctuations. Gravitational mass describes
the virtual gravitons around a mass: they arise in the tethers due to the belt trick. Inertial
mass describes how a rotating mass advances through the vacuum with the belt trick, as
described in reference [16]. In the strand conjecture, these two processes are exactly the
same: both involve tether fluctuations around the core, and in particular, both involve the
belt trick. Therefore, inertial and gravitational mass are equal – for infinite, flat space.
Strands thus imply that the equivalence principle holds, in its weak and strong forms –
at least for sub-galactic scales, when there is no effect of the cosmological horizon. This
agrees with observations [45].
Pr. 20 Strands imply that elementary particle mass values run with four-momentum. The reason
is that the tangles completely reproduce quantum field theory, as summarized in Appendix
B: elementary particles are surrounded by virtual particle pairs; thus their mass values run
25
with four-momentum. This agrees with observations – e.g., [46] – and expectations.
Pr. 21 Because spontaneous tangle fluctuations leading to the belt trick are rare, the gravitational
mass mof elementary particles is predicted to be positive and to be much smaller than the
corrected Planck mass:
0<mc/4G. (22)
This inequality agrees with observations and agrees with old arguments [47]. Strands thus
provide a general answer to the mass hierarchy problem.
Pr. 22 Strands imply that falling particles are fluctuating and diffusing tangles. This implies
that more complex particle tangles have higher gravitational mass (for equal number of
tethers). The same connection has already been deduced for inertial mass in a different
way [16]. The connection yields the correct mass sequences for all hadrons and predicts
normal mass ordering for neutrinos. If neutrino masses would not obey normal ordering,
the strand conjecture would be falsified.
The tangle model also explains that neutrinos mix and that their mass values are stable
under renormalization, as shown in references [16, 23]. Strands thus allow non-vanishing
neutrino mass in the standard model of particle physics.
Pr. 23 Strands allow deducing approximations for the mass values of elementary particles. As
mentioned, the mass is given by the number of crossing switches per time that occur around
the particle. For a fermion, the crossing switches are generated by the tethered rotation of
the particle, illustrated in Figure 9. The figure yields
m≈p·f·n, (23)
where pis the probability for the initial double rotation of the core, fis the probability or
frequency of the subsequent belt trick, and nis the number of crossing switches arising.
The factor pdescribes the process from the first to the second configuration in Figure 9.
26
move
all
tethers
particle,
i.e.,
tangle core
(plays role
of belt
buckle)
move
lower
tethers
move
upper
tethers
move
tethers
sideways
like
start
The belt trick or string trick: double tethered rotation is no rotation
Observation: probability density and phase for unobservable tethers with observable crossings
rotate
particle
twice in
any
direction
phase
Figure 9: The belt trick or string trick, as popularized by Dirac, shows that a rotation
by 4πof a tethered particle, such as a belt buckle or a tangle core, is equivalent to
no rotation – when the tethers are allowed to fluctuate and untangle as shown. This
equivalence, illustrated here in six configurations, allows the tethered particle to rotate
forever. Untangling is impossible after a rotation by 2πonly. The trick illustrates spin
1/2– if one assumes that tethers are not observable, but crossing switches are. The
belt trick works for any number of tethers or belts. In contrast to this illustration, in the
strand conjecture, leptons have six tethers, nor four; the tangle core topology determines
the particle type. The belt trick also allows estimating particle mass, if the probabilities
for the six configurations are explored (see text).
For a symmetric core, the rotation probability, whatever the orientation of the axis, is ex-
pected to be equal in clockwise and anticlockwise direction. In other terms, pvanishes
for symmetric tangle cores. For slightly non-symmetric tangle cores, as is the case for
tangles, the factor pis thus expected to be quite small. Its value will depend on the (aver-
aged, three-dimensional, geometric) asymmetry of the tangle core. The asymmetry is the
27
quantity that couples to the Higgs braid. A non-zero asymmetry leads to a non-zero mass.
The belt trick frequency ffor the process that changes the second configuration in Figure 9
into the sixth configuration will also be small, as it competes with the inverse rotation
of the tangle core. Interestingly, this small frequency is expected to be roughly scale
independent: the size of the tangle core does not play an important role.
Finally, the average number nof crossing switches per belt trick plays a role. The number
ncounts the crossing switches among tethers and also those between the tangle core and
the tethers. This number will depend on the size of the tangle core.
The strand explanation (23) for particle mass mcan be checked before any calculation or
estimate is performed. As mentioned above, the resulting particle mass value is equal for
particle and antiparticles, constant over space and time, and not quantized in multiples of
some basic number. Gravitational and inertial mass are equal. Mass values run with en-
ergy, i.e. with the looseness of the tangle core. Mass values, via p, depend on the Yukawa
coupling to the Higgs boson. Particle mass values, due to the factor f, are much smaller
that the Planck mass. Above all, as expected, particle mass values, due to the factors pand
n, increase for more complex tangles, as large tangles are also more asymmetric.
At present, a direct calculation of m, even an approximate one, is still elusive. However,
some numerical estimates can be deduced.
Pr. 24 Expression (23) allows estimating the mass ratio between the most massive and the least
massive leptons.
Leptons tangles are made of three strands and thus have 6 tethers [16, 23]. For a neutrino,
the belt trick frequency ffor the subsequent configuration change in Figure 9 results from
the probability that the belt trick arises instead of the backwards rotation of the core. To
occur, the tether configuration has to form six circles around the tangle core, all with the
same orientation. The size of the six circles is not important. For each tail, the probability
is roughly given by the probability to form a circle divided by the number of possible
28
rotation axes. Thus one gets the rough estimate
f≈e−2π
6·4·26
≈3·10−27 .(24)
where the exponent is due to the six tethers of the leptons. An error of a few orders of
magnitude is expected. In the mass expression (23), the number nfor neutrinos is surely
larger than 24, which just counts the crossing switches in the tethers.
For the most massive lepton, the estimate n=24for the neutrinos will be approximately
doubled. The estimate for fwill increase for tangle cores that are elongated; the value (6 ·
4·2)−6will be replaced by a number of the order O(1/10). The asymmetry pfor massive
leptons is surely larger than that for neutrinos. As a result, the mass ratio rbetween the
most massive and the least massive leptons will be
r≈2O(1/10) (6 ·4·2)6pmm
pν
.(25)
The lepton mass ratio ris thus surely larger than 2·109. The observed value for the lepton
mass hierarchy ris above 109. Better estimates for rrequire to determine pmm /pν.This
is not yet possible. In short, only a precise determination of the neutrino mass – both from
experiments and from strand calculations – will allow a definite test of this prediction.
Pr. 25 Strands will allow deducing a lower limit for the (bare) mass values of leptons.
The rotation probability pfor a neutrino results from the averaged asymmetry of its tangle
core. For an electron neutrino, the asymmetry that results from the geometric chirality of
the tangle is negligibly small. It is expected that the asymmetry arises only through the
mixing with the other two neutrinos, or through the Yukawa term, or both. The averaged
asymmetry is hard to estimate. It is expected to be larger than one part per million, thus
p≈10−6.(26)
29
A systematic error of a few orders of magnitude is expected. This estimate yields a lower
mass bound mll for leptons of
mll
c/4G=p·f·n≈10−31 ,(27)
which is of the order of meV/c2, though with a large error margin. So far, this lower limit
is not in contrast with the present experimental limit on neutrino mass, which is below
0.9 eV [48]. Again, the difficulties of deriving a reliable lower mass bound for leptons are
evident.
Pr. 26 Strands will also allow deducing an upper limit for the mass value of leptons. For heavy
leptons, the probability pdue to the asymmetry of the tangle core is expected to be of order
O(1). As a result, the upper mass limit for leptons will be
mul ≈1017 ·mll ,(28)
again with a large error margin. The most massive lepton, the tau lepton, has an observed
mass of 1.7 GeV/c2. It indeed is several orders of magnitude below the upper bound.
More precise strand estimates of particle masses will require the development of better approxi-
mations and of suitable computer simulation programs. The failure to reproduce the correct mass
value of a single particle, at any single energy value, would falsify the strand conjecture.
13 Results and discussion
In summary, strands reproduce general relativity and quantum gravity at sub-galactic scales. Strands
also predict the lack of observable deviations.
On the one hand, it is not easy to think about nature as made of strands. It is also unusual
to describe physical processes as made of fundamental events. On the other hand, the conjecture
has the charm of deriving all observations about general relativity (at sub-galactic scales) directly
30
from the Planck scale. Also, the complete standard model of particle physics, with its Lagrangian,
arises from the Planck scale, as argued elsewhere [16, 23]. So far, no deviations from these two
descriptions have been observed in any experiment. The discovery of any new deviation from
general relativity at sub-galactic scales would invalidate the strand conjecture.
The possibility that additional quantum gravity effects are unobservable has already been ex-
plored in the past [49, 50, 51]. Strands confirm the result. They can make such a prediction
because they incorporate general relativity, quantum physics and the standard model exactly.
As a new result in the domain of quantum gravity, strands propose a solution to the mass hi-
erarchy problem and promise that the gravitational mass of elementary particles can be calculated
ab initio from their tangle details. Such calculations will allow the most stringent tests of the con-
jecture. As long as the strand predictions about space, gravity, particles and mass are not falsified,
strands remain a candidate for a complete description of nature.
Acknowledgments and declarations
The author thanks Jason Hise for his animations and Thomas Racey, Gary Gibbons, John Barrow,
Eric Rawdon, Yuan Ha, Lou Kauffman, Clifford Will, Steven Carlip, Isabella Borgogelli Avve-
duti, and the anonymous referee for discussions and suggestions. The present work was partly
supported by a grant of the Klaus Tschira Foundation.
Appendix A On the circularity of the fundamental principle
On the one hand, the crossing switch of Figure 1 is assumed to take place in space. On the other
hand, space, distances and physical observables are assumed to arise from strands. The apparent
circularity can be avoided – to a large degree, but not completely – by increasing the precision of
the formulation.
Crossing switches take place in background space. In the strand conjecture, background space
is defined by the observer. In contrast, physical space, physical distances and physical observables
arise from strands and their crossing switches. When space is flat, background space and physical
31
space coincide. Otherwise, they do not; in that case, background space is (usually) the local
tangent space of physical space. A similar situation arises for the concept of time.
In nature, any observation of a change implies the use of (background) time; any observation
of difference between objects or systems implies the use of separation in (background) space. In-
deed, a local background space – observer-defined and usually observer-dependent – is required
to describe any observation, or simply, to talk about nature. In the strand conjecture, it is equally
impossible to define crossing switches or any Planck unit without a background. The strand con-
jecture asserts that a description of nature without a background space and time is impossible.
Every use of the term ‘observation’ or ‘observable’ or ‘physical’ implies and requires the use
of a background space and time. All the illustrations of the present work are drawn in background
space. In contrast, physical space – an observable in general relativity, dynamical and pseudo-
Riemannian – arises through crossing switches of strands. The local background space agrees
with physical space only locally, where the crossing switches being explored are taking place. In
fact, the need for a background space to describe nature is rooted in a deeper issue.
Background space is what is needed to talk about nature. Physical space is everything that can
be measured about space: curvature, vacuum energy, entropy, temperature etc.
There is a fundamental contrast between nature and its precise description. The properties of
nature itself and the properties of a precise description differ and contradict each other. A precise
description of nature requires axioms, sets, elements, functions, and in particular continuous back-
ground space, continuous background time, and points in background space and background time.
In contrast, due to the uncertainty relations, at the Planck scale, nature itself does not provide the
possibility to define points in physical space or time; physical space and time are not continuous
at smallest scale, and in fact, physical)space and time are emergent. In short, observer space, or
background space, differs in its properties from physical space.
Any precise description of nature thus requires a limited degree of circularity in its definition
of physical time and space with the help of background time and background space. Therefore,
an axiomatic description of all of nature is impossible. An axiomatic description is only possible
32
for those parts of nature that avoid the fundamental circularity, such as quantum theory, or special
relativity, or quantum field theory, or electromagnetism, or general relativity. Even though Hilbert
asked for an axiomatic description of physics in his famous sixth problem, no claim for an ax-
iomatic description of all of nature (all of physics) has ever appeared in the literature. Any unified
description of nature must be circular. The strand conjecture, like or any other unified model, can
be tested by asking whether it is a consistent,complete and correct description of nature. So far,
this appears to be the case for strands.
An important example for the difference between an axiomatic description and a consistent,
complete, correct – but somewhat circular – description is the dimensionality of space. The num-
ber of dimensions of (background and physical) space is not a consequence of the fundamental
principle or of some axiom; the number of dimensions is assumed in the fundamental principle
right from the start. Tangles only exist in three dimensions. Only three dimensions allow a de-
scription of nature that is consistent, complete, and correct: only three dimensions allow crossing
switches, particle tangles, spin 1/2, Dirac’s equation and Einstein’s field equations.
Appendix B From strands to quantum theory and the standard model Lagrangian
This appendix provides an extremely short summary of references [16] and [23], which explain
how quantum theory, quantum field theory and the full Lagrangian of the standard model arise
from strands. The tangle model for massive quantum particles is illustrated in Figure 10 and
Figure 11. The figures visualize that crossings have properties similar to those of wave functions,
and that time-averaged crossing switches have the same properties as probability densities.
Starting from the fundamental principle and Dirac’s belt trick, tangles of fluctuating strands in
flat (physical) space indeed describe matter particles and wave functions: the wave function of a
particle is the strand crossing density of its fluctuating tangle. In other words, wave functions arise
as local time averages of strand crossings. More specifically, to get the value of the wave function
at a certain position in space, the local time average of the strand crossings at that position is taken,
averaging over a time scale of (at least) a few Planck times. In this way, a density and a phase can
33
shortest
distance
with
midpoint
orientation
phase
Strand crossings have the same properties as wave functions
Figure 10: A configuration of two skew strands, called a strand crossing in the present
context, allows defining density, orientation, position, and a phase. These are the same
properties that characterize a wave function. The freedom in the definition of phase is at
the origin of the choice of gauge. For a complete tangle, the density, the phase, and the
two (spin) orientation angles define, after spatial averaging, the two components of the
Dirac wave function Ψof the particle and, for the mirror tangle, the two components of
the antiparticle.
The tangle model for a fermion
tether
tangle
core
phase
time average
of crossing
switches
probability
density
Observation:
crossing phase
spin spin
tether
Figure 11: In the strand conjecture, the wave function and the probability density are
due, respectively, to crossings and to crossing switches at the Planck scale. The wave
function arises as time average of crossings in fluctuating tangled strands. The proba-
bility density arises as time average of the crossing switches in a tangle. The tethers –
strand segments that continue up to large spatial distances – generate spin 1/2behaviour
under rotation and fermion behaviour under particle exchange. The tangle model also
ensures that fermions are massive and move slower than light (see text).
34
be defined, for each ‘position’ in space. As usual for quantum theory, also in the strand conjecture
physical space and time have to be defined before defining the concept of wave function. The
probability density for a particle is the local time average of the crossing switch density of its
fluctuating tangle. A detailed exploration [16, 23] shows that strands produce a Hilbert space,
the quantum phase, interference, contextuality and freedom in the definition of the absolute phase
value.
Moving particles are advancing rotating tangles. Antiparticles are mirror tangles rotating in
the opposite direction. Fluctuating rational tangles made of two or more strands imply spin 1/2
behaviour under rotation and, above all, Dirac’s equation [14]. For systems of several particles,
tangles reproduce fermion behaviour and entanglement. Tangles of strands are fully equivalent to
textbook quantum theory and predict the lack of any extension or deviation, up to Planck energy.
For example, the principle of least action is the principle of fewest crossing switches.Inthisway,
strands also explain the origin of the principle of least action [23].
No new physics arises in the domain of quantum theory. Strands only visualize quantum
theory; they do not modify it. Every quantum effect is due to crossing switches – and vice versa.
The visualization of quantum effects with strands requires that strands remain unobservable in
principle, whereas their crossing switches are observable.
Tangles also allow deducing quantum field theory. Exploring all possible tangles, it appears
that rational, i.e., unknotted tangles reproduce the known spectrum of elementary particles and
their properties [16, 23]. Every massive elementary particle is represented by an infinite family
of rational tangles made of either two or three strands. Quarks are made of two strands; all other
massive elementary particles are made of three strands. Three generations for quarks and for
leptons arise. The Higgs itself is represented by a braid. The family members for each elementary
particle differ among them only by the number of attached braids. The structure of each elementary
particle tangle explains the spin value, parity, charge and all other quantum numbers.
Models for the massless bosons also arise. In particular, a photon is a single, twisted strand.
Photons are emitted or absorbed by topologically chiral tangles, i.e., by fermion tangles that are
35
time average
of crossing
switches
photon
charged
fermion
time
Observed
Feynman diagram:
Fermion-antifermion annihilation
t1
charged
antifermion
photon
photonphoton
charged
fermion
charged
antifermion
time average
of crossing
switches
Observed
Feynman diagram:
V
irtual fermion-antifermion pair creation
t3
time
t1
photon
t2
photon
vacuum
vacuum
vacuum
charged
fermion
charged
anti-
fermion
charged
fermion
charged
antifermion
t2
vacuum
vacuum
Figure 12: An illustration of two Feynman diagrams of quantum electrodynamics in the
tangle model.
36
electrically charged. Figure 12 illustrates the strand conjecture for quantum electrodynamics.
Only three kinds of massless bosons arise, each kind due to one Reidemeister move. The boson
generator algebras turn out to be the well-known U(1), broken SU(2) and SU(3) of the three gauge
interactions [16, 23]. The violation of parity in the weak interaction and the way that the massless
bosons of SU(2) acquire mass are also explained.
A detailed investigation shows that tangles reproduces every propagator and every Feynman
vertex observed in nature – and no other ones. Particle mixing appears naturally. The correct
couplings between fermions and bosons also arise. As a result, the full Lagrangian of the modern
standard model arises, term by term, including PMNS mixing of Dirac neutrinos, without any
addition or modification [16, 23].
In short, strands predict the lack of any physics beyond the standard model. Discovering
such an effect or any new influence between quantum field theory and gravitation at sub-galactic
scales – apart from the cosmological constant, the particle masses and the other constants of the
standard model, including their running with energy – would falsify the strand conjecture. This
terse summary of the implications of strands for quantum field theory allows proceeding with the
exploration of space and gravity.
References
[1] G ’t Hooft Int. J. Mod. Phys. A 24 (2009) 3243.
[2] L Susskind arxiv.org/abs/1311.3335.
[3] C Rovelli arxiv.org/abs/1102.3660.
[4] T Padmanabhan AIP Conf. Proc. 861 (2006) 179.
[5] T Padmanabhan Rep. Prog. Phys. 73 (2010) 046901.
[6] T Padmanabhan J. Phys.: Conf. Series 701 (2016) 012018.
[7] H Nicolai arxiv.org/abs/1301.5481.
[8] G Amelino-Camelia Living Rev. Relat. 16 (2013) 5.
37
[9] S Carlip Class. Quant. Gravity 34 (2017) 193001.
[10] M Botta Cantcheff arxiv.org/abs/1105.3658.
[11] M Planck Sitzungsber. kön. preuß. Akad. Wiss. Berlin (1899) 440.
[12] N Bohr Atomtheorie und Naturbeschreibung, Springer (1931).
[13] M Gardner Riddles of the sphinx and other mathematical puzzle tales, Math. Assoc. America
(1987) page 47.
[14] E P Battey-Pratt and T Racey Int. J. Theor. Phys. 19 (1980) 437.
[15] L H Kauffman On Knots, Princeton University Press (1987), beginning of chapter 6.
[16] C Schiller Phys. Part. Nucl. 50 (2019) 259.
[17] J D Bekenstein Phys. Rev. D 7(1973) 2333.
[18] S W Hawking Comm. Math. Phys. 43 (1975) 199.
[19] V de Sabbata and C Sivaram Found. Phys. Lett. 6(1993) 561.
[20] G W Gibbons Found. Phys. 32 (2002) 1891.
[21] C Schiller Int. J. Theor. Phys. 44 (2005) 1629.
[22] J D Barrow and G W Gibbons Monthly Notices Roy. Astr. Soc. 446 (2014) 3874.
[23] C Schiller Eur. Phys. J. Plus 136 (2021) 79.
[24] L Smarr Phys. Rev. Lett. 30 (1973) 71; erratum: Phys. Rev. Lett. 30 (1973) 521.
[25] T Jacobson Phys. Rev. Lett. 75 (1995) 1260.
[26] A K Raychaudhuri Phys. Rev. 98 (1955) 1123.
[27] D Christodoulou and S Klainerman The global nonlinear stability of the Minkowski space,
Princeton University Press (1993).
[28] J W Richardson et al. arxiv.org/abs/2012.06939.
[29] R Cooke et al. Monthly Notices Roy. Astr. Soc. 494 (2020) 4884.
[30] P Laurent, et al. Phys. Rev. D 83 (2011) 121301.
[31] C Schiller Int. J. Theor. Phys. 45 (2006) 213.
38
[32] A H Chamseddine arxiv.org/abs/1805.08582.
[33] A H Chamseddine, A Connes and M Marcolli Adv. Theor. Math. Phys. 11 (2007) 991.
[34] T P Singh Zeitschr. f. Naturf. A73 (2020) 733.
[35] R Penrose Rep. Math. Phys. 12 (1977) 65.
[36] V Cardoso, T Ikeda, C J Moore and C-M Yoo Phys. Rev. D 97 (2018) 084013.
[37] LIGO Scientific Collab. and Virgo Collab. Phys. Rev. X 9(2019) 031040.
[38] LIGO Scientific Collab. and Virgo Collab. Astrophys. J. Lett. 900 (2020) L13.
[39] J Hise youtube.com/watch?v=LLw3BaliDUQ, youtube.com/watch?v=eR9ZCwYPhhU.
[40] T Rothman and S Boughn Found. Phys. 36 (2006) 1801.
[41] F Dyson Int. J. Mod. Phys. A 28 (2013) 1330041.
[42] S Rijavec et al. arxiv.org/abs/2012.06230.
[43] V Katritch et al. Nature 384 (1996) 142.
[44] E J Janse van Rensburg et al. J. Knot Theory Ramif. 6(1997) 31.
[45] C M Will Living Rev. Relat. 17 (2017) 4.
[46] CMS collaboration Phys. Lett. B 803 (2020) 135263.
[47] V A Berezin Phys. Part. Nucl. 29 (1998) 274.
[48] M Aker et al., arxiv.org/abs/2105.08533.
[49] L Rosenfeld Nucl. Phys 40 (1963) 353.
[50] S Carlip Class. Quant. Gravity 25 (2008) 154010.
[51] S Boughn Found. Phys 39 (2009) 331.
39
