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The Finite Universe

Jean Louis Van Belle, Drs, MAEc, BAEc, BPhil

30 March 2021

Abstract

This paper offers a few reflections on the Universe and cosmological theories, including a possible

explanation of dark matter (which we think of antimatter), plus an explanation of SRT/GRT based on an

analysis of the argument of the quantum-mechanical wavefunction. Last but not least, we also present

some fun thoughts on space travel (see Annex III).

Contents

Introduction: forces and scales ..................................................................................................................... 2

The geometric approach to gravity ............................................................................................................... 4

The closed Universe ...................................................................................................................................... 5

The mass of the Universe .............................................................................................................................. 6

Dark matter is antimatter ............................................................................................................................. 6

Beginnings ..................................................................................................................................................... 8

An oscillating Universe? ................................................................................................................................ 9

Life – on Earth and elsewhere… .................................................................................................................. 10

The cosmological constant .......................................................................................................................... 13

Annex I: The wavefunction and special relativity ....................................................................................... 14

Annex II: The wavefunction and general relativity ..................................................................................... 20

Annex III: Thoughts on space travel ............................................................................................................ 22

1

Post- or pre-scriptum (added on 29 April 2023):

In this paper, we felt the mystery of dark matter should and can be explained by the chirality of the

electromagnetic force: the electromagnetic force is right-handed. The force is one, but when we break it

down into a magnetic and electric field (both depend on your frame of reference), then we know the

magnetic field will always lag the electric field (with a phase shift of 90 degrees). We think an antiforce

can be imagined, which would differ from the force we know because, in a decomposition of electric

and magnetic field, this antiforce would be left-handed: the magnetic field would precede the electric

field by 90 degrees.

We have not worked out the math yet, but we think this amounts to an opposite spacetime signature

when using four-vector algebra to represent Maxwell’s equations and/or a description of

electromagnetics in terms of the scalar and vector potential. We talk about that in this YouTube video,

and the idea is also described in our last paper, which focuses on the question of dark matter. It is this:

we now think dark matter is not antimatter. A positron is the antimatter counterpart of an electron in

our world, and a dark electron would have an antimatter counterpart as well, but both the dark electron

and its antimatter counterpart would be held together by this lefthanded antiforce. That makes them

very different from our electrons and positrons. The same reasoning applies to protons and to whatever

other particle would exist in the dark world out there.

To make the argument somewhat more real, let us think of an electron and a dark positron: would they

annihilate each other in a big flash, just like electrons and positrons do? We are not sure, but we think it

would be logical to assume that⎯just like dark photons and dark neutrinos (i.e., antiforce photons and

neutrinos) do not interact with our measurement instruments on Earth and in space

1

⎯ dark matter

would also not interact with matter here⎯be it ordinary matter or antimatter. Hence, we would have

the following fourfold classification:

Maer

Anmaer

Ordinary maer

Electrons, protons, etcetera

Positrons, anprotons, etcetera

Dark maer

Dark electrons, dark protons, etc.

Dark positrons, dark anprotons, etc.

We did not want to produce another version of this paper, so we ask the reader kindly to read the

section on dark matter as anti-matter in this paper (see page 6) taking into account that we have a

different view on this now.

1

To be sure, I am talking about all of the huge ground stations and all of our space telescopes here. If dark matter

and dark energy exists in the quantity that scientists presume, then we should also have discovered some dark

radiation coming out of those corners of space. We did not. Hence, the only logical conclusion is that such

radiation may exist but that we cannot detect it because of its ‘lefthanded’ signature.

2

Introduction: forces and scales

All of the energy of the Universe adds up to a constant. Singularities – points of infinite potential – are

mathematical abstractions: they are not real. Particles are finite quanta: their energy/mass is finite, and

they pack a finite amount of physical action. Stable particles pack one or multiple units of ħ (angular

momentum): E = ħ = hf = h/T. For unstable particles, the Planck-Einstein relation is not valid. The

wavefunction of unstable particles involves an additional decay factor :

The sign of the coefficient A models matter/antimatter, while the sign of the complex exponent

(iEt/ħ) captures the spin direction of matter/antimatter particles.

2

Light-particles differ from matter-

particles because they carry no charge. Their oscillation (if photons are electromagnetic oscillations,

then neutrinos must be nuclear oscillations) is, therefore, not local: they effectively travel at the speed

of light.

The relevant distance scale for nuclear interactions (e.g., the motion of charges within the deuteron

nucleus) is about 1 to 100 fm (10−15 m). The size of atomic electron orbitals – whose energy is

electromagnetic – is of the order of the Bohr radius, whose order of magnitude (10−11 m) differs with a

factor of 10,000 from the picometer scale. We may, therefore, say that, roughly speaking, the fine-

structure constant () separates nuclear from electromagnetic and atomic distance scales.

The fine-structure constant also separates the scales of the classical electron radius (re = rC = 2rB), its

Compton radius (rC = ħ/mec = rB), and the electron’s atomic orbital radius (rn = n2rC/),with n the

principal quantum number):

Mass is, quite simply, inertia to a change in the state of motion. The equations below model mass-

without-mass particles: all mass is electromagnetic (mC) or nuclear mass (mN).

The order of magnitude of the electromagnetic and nuclear force may be compared using the standard

electromagnetic and nuclear parameters in the orbital energy equations above.

3

However, to compare

forces, we must use the same numerical values for mC = mN. Evaluating the functions above at r = a, we

get:

2

See Annex IV, V and VI of our paper on ontology and physics.

3

The plus or minus sign is there because the nuclear charge keeps like charges together, while the Coulomb force

repels them. The sign also depends on the reference point for the potential energy: the U = 0 point may be chosen

at r = or at r = 0. We often make abstraction of the sign when comparing the magnitudes of forces, energy, or

distance scales.

3

This should not surprise us: we define the range parameter here as the distance r = a for which the

magnitude of the two forces (whose direction is opposite) is the same. The form factor and, hence, the

nature of the two forces, is very different, though.

Figure 1 shows the electromagnetic and nuclear potential functions: the nuclear potential yields an

inverse-cube law for the nuclear force, which explains why it is apparent at very short range only: for

larger r, the combined nuclear and electromagnetic potential function approaches the electromagnetic

potential.

Figure 1: The nuclear potential well arising from 1/r and a/r2 functions

4

The geometric approach to gravity

The energy equation for gravitational orbitals follow from Kepler’s laws for the motion of the planets

4

:

The kinetic and potential energy (per unit mass) add up to zero instead of c2 (nuclear and

electromagnetic orbitals), which is why a geometric approach to gravity makes eminently sense: massive

objects simply follow a geodesic in space, and there is no gravitational force in such geometric approach.

We may also compare the standard parameters by equating m to mC (in practice, this means using the

mass and charge of the electron in the equation below) and r to a

5

:

Hence, the force of gravity – if considered a force – is about 1042 weaker than the two forces we know

(electromagnetic and nuclear).

6

Gravity acts on matter-particles (matter-particles carry charge) as well

as light-particles (photons

7

, neutrinos) – no matter their energy – which is why we think efforts to model

gravity as a residual force must fail: a force acts on a charge⎯but photons/neutrinos carry no charge!

General relativity establishes an equivalence between the description of gravity as a force and its

description as curved space. A (large) mass results in a curvature of space, but the (relativistic) mass of

photons must curve space as well and we must, therefore, rely on the superposition principle to arrive at

a complete description. The description in terms of curved space and geodesics might feel more natural

but is difficult in terms of the math: it is easier to just superpose forces to arrive at the equations of

motion.

8

4

See, for example, the MIT OCW reference course on orbital motion. We believe the law is relativistically correct

because the velocity is an orbital/tangential velocity (the concept of escape velocity is often used as a synonym

but, for elliptical orbitals, acquires a slightly different meaning) and because – in accordance with relativity theory

– we use the relativistic mass concept m = m0.

5

The physical dimension of the gravitational constant is usually written as m3kg−1s−2. The dimensional equation

below shows this is equivalent to the Nm2kg−2 dimension:

6

Feynman (Lectures, I-2, Table 2-3) gives a factor of the order 1040. Perhaps he uses photon masses and energies:

the fine-structure constant effectively also appears as a photon-electron coupling constant (see our paper on the

meaning of the fine-structure constant, section V).

7

See the Eddington observation (29 May 2019), which unequivocally proves light is bent by large masses.

8

The analysis becomes complicated when considered not two but three bodies in motion (cf. the three-body

problem).

5

The closed Universe

The Universe is finite and all matter/energy in the Universe revolves around a (mathematical) center-of-

mass. Its (global) curvature must, therefore, be positive. Photons and neutrinos traveling outwards

must, therefore, also follow elliptical orbitals: what goes around, must come around.

These far-out light-particles may or may not have been created because of large-scale matter-antimatter

annihilation events when the Universe came into existence. We think there is a lot of antimatter in the

Universe because we think of dark matter/energy as antimatter. We are not very used to thinking of

antimatter. However, in our matter-world, matter-antimatter pairs are created routinely

9

: it is only

because of the predominance of matter in our world, that antimatter is not very present.

The closed Universe expands and may, at some point in time, collapse back into itself. Its expansion –

and its possible collapse – means the finite energy in the Universe gets distributed over an ever-

increasing volume. Hence, the Universe’s energy density E = E/V (measured in J/m3 or N/m2) goes down.

The relation between the concepts of energy, temperature and entropy are complicated and we will not

dig into them here. As for now, the decreasing energy density of the Universe may be equated with a

decreasing density of matter (including antimatter

10

), and temperature may be defined as the (average)

kinetic energy of the (stable) particles inside of a volume: two volumes with different mass densities –

think of a different number of particles here – are at the same temperature if the (average) kinetic

energy of the particles is the same. The current average temperature in the Universe is measured from

cosmic microwave background radiation and is about 2.73 K (−270.42 °C).

The age of the Universe is currently estimated to be about 13.8 billion (109) years. The Universe

expands, and the bits and pieces that sent us radiation from the limits of the observable Universe moved

further out in the meanwhile, and at pretty impressive speed. Astronomers think the end-to-end

distance across the Universe is of the order of 46 billion lightyears. A diameter of 46 billion (lightyears) is

about 1.666… times the diameter of the observable Universe (27.6 lightyears), and the estimated speed

of expansion (the radial velocity vr) of the Universe must, therefore, be equal to:

Two thirds of the speed of light? How can we know such things? Relativistic redshift: photons emitted by

stars who are moving away from us still travel at the speed of light, but their wavelength shifts to a

lower-energy spectrum.

If we think of radial velocity, we should think of a tangential velocity too: the Universe might have

angular momentum too, right? Yes, but there is no agreement on the question of whether or not the

Universe has an actual center. More generally speaking, measurements depend on the choice of the

9

Carl Anderson (1932) discovered the positron while analyzing the ion trail of cosmic rays, using a cloud chamber

and ordinary photographs.

10

Antimatter is not to be equated with negative matter. Negative matter does not exist (negative energy does not

exist either: negative potential energy is only measured as negative because of the choice of the U = 0 reference

point at infinity).

6

reference frame: if we choose the reference frame to be that of the Universe itself, all of the angular

momentum of the bits and pieces inside of it should add up to zero.

The mass of the Universe

The Wikipedia article on the observable Universe states that the total mass of ordinary matter in the

universe can be calculated using the critical density and the diameter of the observable universe (whose

radius should correspond to the above-mentioned 13.8 billion lightyears) to be about 1.5 × 1053 kg.

The article distinguishes ordinary matter (about 74% of which (in mass) is hydrogen) from dark matter

and dark energy. In fact, it states that ordinary matter – which is understood to be matter-particles –

might constitute only 4.9% of all matter, with the rest being either dark matter (26.8%) or dark energy

(68.3%).

11

We, therefore, need to clarify what dark matter/energy might actually be.

Dark matter and dark energy is matter/energy which is assumed to be out there, but which cannot be

detected. Black holes? No! Even black holes emit radiation as they evaporate (Hawking radiation).

12

We

also have all of the light-particles at the edge of our Universe orbiting out and which – who knows

when? – might come back in again.

13

We think dark energy/dark matter is antimatter, so let us explain that.

Dark matter is antimatter

[See our pre-scriptum (20 April 2023): we would rewrite this section now. Dark matter cannot be simply

reduced to, or be viewed as, anti-matter. Text below should, therefore, be revised if ever we would write

a new version of this paper.]

The electromagnetic force has an asymmetry: the magnetic field lags the electric field. The phase shift is

90 degrees. We can use complex notation to write the E and B vectors as functions of each other.

Indeed, the Lorentz force on a charge is equal to: F = qE + q(v×B). Hence, if we know the (electric field) E,

then we know the (magnetic field) B: B is perpendicular to E, and its magnitude is 1/c times the

magnitude of E. We may, therefore, write:

B = –iE/c

The minus sign in the B = –iE/c expression is there because we need to combine several conventions

here. Of course, there is the classical (physical) right-hand rule for E and B, but we also need to combine

the right-hand rule for the coordinate system with the convention that multiplication with the imaginary

unit amounts to a counterclockwise rotation by 90 degrees. Hence, the minus sign is necessary for the

11

The Wikipedia article on dark matter distinguishes between dark matter and dark energy based on the

matter/energy density in space.

12

Of course, black holes also swallow matter/energy and, if there is enough of such matter around, they keep

growing.

13

As a child, I thought a lot of the radiation that comes from far away galaxies and star systems might be absorbed

by galaxies and star systems that are closer to us, and that this might explain why we cannot see (detect) dark

matter. That idea is not valid, however, because energy in must, over the long run, equal energy out. Another

objection is that, while the idea that galaxies and star systems may hide or eclipse (other) far-away galaxies and

star systems from our line of sight is valid, such eclipses should, of course, be temporary only (different rotational

velocities).

7

consistency of the description. It ensures that we can associate the aeiEt/ħ and ae–iEt/ħ functions with left

and right-handed spin (angular momentum), respectively.

Now, we can easily imagine an antiforce: an electromagnetic antiforce would have a magnetic field

which precedes the electric field by 90 degrees, and we can do the same for the nuclear force (EM and

nuclear oscillations are 2D and 3D oscillations respectively). It is just an application of Occam’s Razor

principle: the mathematical possibilities in the description (notations and equations) must correspond to

physical realities, and vice versa (one-on-one). Hence, to describe antimatter, all we have to do is to put

a minus sign in front of the wavefunction. [Of course, we should also take the opposite of the charge(s)

of its antimatter counterpart, and please note we have a possible plural here (charges) because we think

of neutral particles (e.g., neutrons, or neutral mesons) as consisting of opposite charges.] This is just the

principle which we already applied when working out the equation for the neutral antikaon

14

:

The point is this: matter and antimatter are each other opposite, literally: the wavefunctions aeiEt/ħ and –

aeiEt/ħ add up to zero, and they correspond to opposite forces too! Of course, we also have lightparticles,

so we have antiphotons and antineutrinos too.

We think this explains the rather enormous amount of so-called dark matter and dark energy in the

Universe. Dark matter is called dark because it does not appear to interact with the electromagnetic

field: it does not seem to absorb, reflect, or emit electromagnetic radiation, and is, therefore, difficult to

detect. That should not be a surprise: antiphotons would not be absorbed or emitted by ordinary

matter. Only anti-atoms (i.e., think of an antihydrogen atom as an antiproton and a positron here)

would do so.

15

14

See Annex IV, V and VI of our paper on ontology and physics.

15

The opposite spacetime signature of antimatter is, obviously, equivalent to a swap of the real and imaginary

axes. This begs the question: can we, perhaps, dispense with the concept of charge altogether? Is geometry enough

to understand everything? We are not quite sure how to answer this question, but we do not think so: a positron is

a positron, and an electron is an electron⎯the sign of the charge (positive and negative, respectively) is what

distinguishes them! We also think charge is conserved, at the level of the charges themselves (see our paper

on matter/antimatter pair production and annihilation). We, therefore, think of charge as the essence of the

Universe. But, yes, everything else is sheer geometry.

8

Beginnings

It is tempting to think all began with matter/antimatter creation but, as far as we know, protons and

electrons are the only stable elementary particles. We also have neutrons, but they are unstable outside

of the nucleus. We also think our model of the neutron as a composite proton-electron equilibrium state

inside of a nucleus makes a lot of sense.

16

In short, we are left with an enormous amount of protons and electrons – an overall electrically neutral

ensemble – which combined to form hydrogen in one of its two stable isotopes (1H and 2H

17

). Protons

are nuclear oscillations of the positive (pointlike) Zitterbewegung (zbw) charge, while electrons are

electromagnetic oscillations of the negative zbw charge. Two forces, two particles. We see no need to

complicate history by invoking asymmetries or other ad hoc hypotheses.

We may, therefore, think of the conditions that were prevalent at the time of the Big Bang as being the

same as those that are prevalent in a star: thermonuclear fusion reactions generated heat and

lightparticles (photons as well as neutrinos

18

). The thermonuclear reactions first burn hydrogen and,

because of gravity, the more massive nuclei are drawn to the center, where nuclear fusion produces

even more massive elements but with the balance going from producing to consuming heat, as a result

of which the core eventually collapses. The implosion is then followed by a huge explosion, with non-

homogeneous fragments going everywhere.

Does this answer the question of where protons and electrons – or the positive and negative charges at

their core – came from? No. Should we be interested in the original distribution of protons and

neutrons? Probably, but as Feynman (III-2-6) notes:

“The tiniest irregularities are magnified […], so that we get complete randomness. […] Given an

arbitrary accuracy, no matter how precise, one can find a time long enough that we cannot

make predictions valid for that long a time. […] This length of time is not very large. It is not that

the time is millions of years if the accuracy is one part in a billion. The time goes, in fact, only

logarithmically with the error, and it turns out that in only a very, very tiny time we lose all our

information.”

One may also think of Einstein’s statistical analysis of the random walk.

19

Let us quickly illustrate this.

16

See our paper on the mass-without-mass model of electrons, protons, and neutrons. Also see our paper on

(matter/antimatter) pair production and annihilation as a nuclear process, in which we show charge, energy/mass,

and (linear and angular) momentum are being preserved, always.

17

The deuterium nucleus consists of a neutron and a proton and, therefore (n = p + e), of two protons and two

electrons.

18

We think of neutrinos as the equivalent of photons for the nuclear force: photons carry electromagnetic energy,

while neutrinos carry nuclear energy. Neither carry charge: only matter-particles do.

19

Einstein’s analysis of the random walk is part of one of his 1905 (Annus Mirabilis) papers. It has a rather

impossible title: Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden

Flüssigkeiten suspendierten Teilchen (On the Motion of Small Particles Suspended in a Stationary Liquid, as

Required by the Molecular Kinetic Theory of Heat).

9

Figure 2: The random walk (illustration taken from Feynman I-6-3)

The idea here is that we start at point x = 0 and that we move by either take a step forward (i.e. away

from our starting point x = 0) or backward (toward x = 0) but that the direction of each step is random.

20

The resulting motion is described by (i) the horizontal coordinate N, which is the total number of steps

taken and (ii) the vertical coordinate DN, which is the net distance moved from the starting position. The

illustration (Figure 2) shows three possible outcomes after 30 steps, two of which involve the same final

position but different paths. Now, it is impossible to reconstruct what happened – which path was

actually taken – from analyzing the final position only because the information about the individual

steps got lost.

It is fairly to show that, after N steps, we will have covered an expected distance from the x = 0 point

that is equal to

21

:

As you can see, we might expect to be at x = 2 after four steps, but progress quickly slows: after 100

steps, the expected distance of travel is only x = 10, and for N = 1000, it is equal to 1000 31.6 only.

The good news here is that, if ever you get lost, you are usually not too far from the point where you got

lost.

An oscillating Universe?

Was there one Big Bang, or were there several at different distances in space and in time? We do not

know. It might help to try to think about a contracting Universe: hydrogen – but also more massive

elements now – contracting into local clusters of mass and energy⎯forming stars, that first expand and

then collapse/explode again.

We think the coming-into-being of the Universe was probably not a linear process and, therefore, we

should not exclude there were several Big Bangs at different places and times. The ultimate re-collapse

20

We should not be killing Schrödinger’s cats to determine where we go, but the logic and the math are very

similar.

21

These are notations the reader should familiarize him- or herself with. We also recommend the reader reviews

first-order approximations of functions, including complex functions. We refer to one of our blog posts on Euler’s

formula here. Einstein’s approach to the random walk is a good way to familiarize oneself not only with the basic

notions of statistics but also to get a more intuitive feeling of physical scales and ratios.

10

of the Universe will not be a linear process either! But both the Universe’s expansion as well as its

collapse are governed by the same laws and should, therefore, not be too different.

Life – on Earth and elsewhere…

It is quite remarkable how life evolved: atoms formed more stable – or less energetic – combinations by

sharing or exchanging valence electrons, and molecules then arranged into organisms: vulnerable to

high-energy radiation (but protected from it by atmospheric layers on planets⎯cool matter) but

adaptable to changing circumstances. How crawling insects and conscious Homo Sapiens came into

being is weird but explainable: insects are more robust to changing circumstances, while man adapted

its environment – or searched other environments – to ensure better chances of survival and successful

reproduction.

We too obey the laws of the Universe: we burn a lot of energy to get our cars rolling, or produce mobile

phones. Data and information that is not immediately useful is stored for future purposes, or discarded

as useless.

Is there life on Venus, or on Mars? Or in other planetary systems? Maybe. Probably. Is it useful for us?

Most probably not⎯on the contrary: other life forms are probably likely to harm us We should take

Stephen Hawking’s warnings in that regard very seriously, I think!

22

We should also add that the COVID

crisis showed we are not very good at dealing with new viruses, even if – amazingly – pharmaceutical

companies were able to develop a vaccine against it in less than a year.

23

Of course, we should add that communication could be difficult. Indeed, the nearest star system to that

of our Sun is the Alpha Centauri system, and it is about 4.3 lightyears away. Hence, messages from here

to there – or vice versa – would take 4.3 lightyears to get there. Perhaps they are trying to reach us, but

the intensity of the signals they send might be too weak to detect. However, after reading the Wikipedia

article on communication with extraterrestrial intelligence, my sense is that, if there would be intelligent

life out there – within reasonable communication distances (say, 5 to 50 lightyears from here), we

should have conclusively detected signals already.

Just to add to your sense of wonder, we may add that NASA’s Voyager 2 is currently only 18.7 billion km

away, so that is only about 0.002 lightyears away.

24

:-/

22

“If aliens visit us, the outcome would be much as when Columbus landed in America, which didn't turn out well

for the Native Americans.” (Hawking, 2010, quoted in the Wikipedia article on extraterrestrial life, which also

mentions other thinkers such (such Jared Diamond) expressing similar concerns)

23

We are writing mid-March 2021 in Belgium here, almost one year after the government-imposed lock-down

measures.

24

NASA launched the Voyager 2 space probe was back in 1977 (almost 44 years ago!), and it has left the Solar

system now. It yielded valuable data on the outer planets Jupiter, Saturn, Uranus, and Neptune when it passed

these planets in 1979, 1981, 1986 and 1989, respectively. Another spacecraft designed to leave the Solar system –

NASA’s New Horizons probe – was launched in 2006, and explored the Kuiper belt, including the dwarf planets

Pluto and Charon (one of Pluto’s ‘moons’, but its diameter is about half that of Pluto). It will take a while before

any of these interstellar probes will reach a nearby star: Voyager 1 is scheduled to enter the Sirius system 40,000

years from now, while Voyager 2 is headed for star AC +79 3888. We will probably have lost contact with these

probes by then.

11

12

So, what is left? Wonder! In one of the introductory chapters of his Lectures on Physics (I-3-7), Feynman

writes this:

“A poet once said, “The whole universe is in a glass of wine.” We will probably never know in

what sense he meant that, for poets do not write to be understood. But it is true that if we look

at a glass of wine closely enough we see the entire universe. There are the things of physics: the

twisting liquid which evaporates depending on the wind and weather, the reflections in the

glass, and our imagination adds the atoms. The glass is a distillation of the earth’s rocks, and in

its composition we see the secrets of the universe’s age, and the evolution of stars. What

strange array of chemicals are in the wine? How did they come to be? There are the ferments,

the enzymes, the substrates, and the products. There in wine is found the great generalization:

all life is fermentation. Nobody can discover the chemistry of wine without discovering, as did

Louis Pasteur, the cause of much disease. How vivid is the claret, pressing its existence into the

consciousness that watches it! If our small minds, for some convenience, divide this glass of

wine, this universe, into parts—physics, biology, geology, astronomy, psychology, and so on—

remember that nature does not know it! So let us put it all back together, not forgetting

ultimately what it is for. Let it give us one more final pleasure: drink it and forget it all!”

He wrote that in the early 1960s. In the meanwhile, we have the worldwide web enhancing

intelligence

25

and, effectively, we are currently witnessing the introduction of artificial intelligence itself.

We have also entered the areas of nanotechnology and biotechnology. I summarized these advances as

follows in one of my blog posts

26

:

“I just wrapped up my writings on physics (quantum physics) with a few annexes on the

(complex) math of it. And then a friend of mine sent me this image of the insides of a cell. There

is more of it on where it came from. Just admit it: it is truly amazing, isn't? I suddenly felt a huge

sense of wonder - probably because of the gap between the simple logic of quantum physics

and this incredible complex molecular machinery.

25

I would not have been able to develop all this knowledge without it!

26

See: Reading Feynman, All of Physics, 23 February 2021.

13

I quote: "Seen are Golgi apparatus, mitochondria, endoplasmic reticulum, cell wall, and

hundreds of protein structures and membrane-bound organelles. The cell structure is of a

Eukaryote cell i.e., a multicellular organism which means it can correspond to the cell structure

of humans, dogs, or even fungi and plants." These images were apparently put together from

"X-ray, nuclear magnetic resonance (NMR) and cryoelectronic microscopy datasets."

I think it is one of those moments where it feels great to be human.”

We have nothing to add to this, except for two annexes on special and general relativity, respectively.

The cosmological constant

All of the above was written before I checked the Wikipedia article on the cosmological constant,

according to which the expansion of the universe is actually accelerating! I quote:

“In 1998 two teams of astrophysicists, one led by Saul Perlmutter, the other led by Brian

Schmidt and Adam Riess, carried out measurements on distant supernovae and show that the

speed of galaxies recession in relation to the Milky Way increases over time. The universe is in

accelerated expansion. […] The universe would contain a mysterious dark energy producing a

repulsive force that counterbalances the gravitational braking produced by the matter

contained in the universe (see standard cosmological model). For this work, Perlmutter

(American), Schmidt (American-Australian), and Riess (American) jointly received the Nobel

Prize in physics in 2011.”

I am sure there must be a rational explanation for this. Perhaps the Universe is just a blob in a larger

cluster of Universes, most of which would also consist of antimatter (our explanation of dark matter)?

Brussels, 30 March 2021

14

Annex I: The wavefunction and special relativity

Particles are finite quanta: their energy/mass is finite, and they pack a finite amount of physical action.

Stable particles pack one or multiple units of ħ (angular momentum): E0 = ħ = hf = h/T. For unstable

particles, the Planck-Einstein relation is not valid. The wavefunction of unstable particles involves an

additional decay factor :

The sign of the coefficient A captures the difference between matter and antimatter, while the sign

of the complex exponent (iEt/ħ) captures the direction of spin (angular momentum).

27

Light-particles

differ from matter-particles because they carry no charge. Their oscillation (if photons are

electromagnetic oscillations, then neutrinos must be nuclear oscillations) is, therefore, not local: they

effectively travel at the speed of light.

The energy in the wavefunction is the rest energy of the particle, which we think of as a wavicle: its

essence is an oscillating pointlike charge. We, therefore, think of the elementary wavefunction to

represents the motion of the pointlike charge by interpreting r = A·eiθ = A·ei·(E·t − k·x)/ħ as its position

vector. The coefficient A is then, equally obviously, nothing but the Compton radius A = rC = ħ/mc. The r

= A·eiθ = A·ei·(E·t − k·x)/ħ expression shows how classical motion adds a linear component to the argument of

the wavefunction (see Figure 3).

Figure 3: The Compton radius must decrease with increasing velocity

28

The relativistic invariance of the argument of the wavefunction is then easily demonstrated by noting

that the position of the pointlike particle in its own reference frame will be equal to x’(t’) = 0 for all t’.

We can then relate the position and time variables in the reference frame of the particle and in our

frame of reference by using Lorentz’s equations

29

:

27

See Annex IV, V and VI of our paper on ontology and physics.

28

We borrow this illustrations from G. Vassallo and A. Di Tommaso (2019).

29

We can use these simplified Lorentz equations if we choose our reference frame such that the (classical) linear

motion of the electron corresponds to our x-axis. See Feynman’s Lectures, I-15-2.

15

When denoting the energy and the momentum of the electron in our reference frame as Ev and p =

m0v, the argument of the (elementary) wavefunction a·ei can be re-written as follows

30

:

Besides proving that the argument of the wavefunction is relativistically invariant, this calculation also

demonstrates the relativistic invariance of the Planck-Einstein relation when modelling elementary

particles.

31

Needless to say, the plane of the local oscillation is not necessarily perpendicular to the direction of

(linear) motion, nor must we assume the local oscillation is necessarily planar. For a proton, one must

apply an extra factor (4) to calculate its Compton radius:

The 4 factor is the 4 factor which distinguishes the formula for the surface area of a sphere (A = 4πr2)

from the surface area of a circle (A = πr2).

32

We effectively think of an oscillation in three rather than just

two dimensions only here: the oscillation is, therefore, driven by two (perpendicular) forces rather than

just one, and the frequency of each of the two oscillations is equal to = E/2ħ = mc2/2ħ: each of the two

perpendicular oscillations would, therefore, pack one half-unit of ħ only

33

, and – applying the

equipartition theorem – each of the two oscillations packs half of the total energy of the proton. This

spherical view of neutrons (and protons) – as opposed to the planar picture of an electron – fits nicely

with packing models for nucleons.

34

Let us analyze the argument of the wavefunction more in detail. We wrote it as:

30

We use the relativistically correct p = mv equation, and substitute m for m = E/c2.

31

The relativistic invariance of the Planck-Einstein relation emerges from other problems, of course. However, we

see the added value of the model here in providing a geometric interpretation: the Planck-Einstein relation

effectively models the integrity of a particle here.

32

Cf. the 4π factor in the electric constant, which incorporates Gauss’ Law (expressed in integral versus differential

form).

33

This explanation is similar to our explanation of one-photon Mach-Zehnder interference, in which we assume a

photon is the superposition of two orthogonal linearly polarized oscillations (see p. 32 of our paper on basic

quantum physics, which summarizes an earlier paper on the same topic).

34

We think a neutron consists of a positive and a negative charge, and combines an electromagnetic as well as a

nuclear oscillation. See the above-mentioned paper on ontology and physics.

16

The momentum of a photon (and, we must assume, a neutrino

35

) is equal to p = mc = mc/c2 = E/c, with

E = Ev = Ec. The equation above is, then, equal to:

We can, therefore, see that the argument of the wavefunction for a particle traveling at the speed of

light vanishes! This is not easy to interpret. It is not like time has no meaning anymore but relativistic

time dilation becomes absolute: in our frame of reference, we think of the clock as the photon as

standing still. To put it differently, all of its energy is in its motion, and it derives all of its energy from its

momentum.

For particles that are not traveling at the speed of light, we still have the two terms:

The dimensional analysis of the Ev/ħ and the p/ ħ is rather instructive and shows the argument (of

phase) of the wavefunction has no physical dimension:

This makes sense because the phase of the wavefunction is measured in radians which can be used both

as distance as well as time units. One can appreciate this idea when re-writing the phase as:

The p = mv = Ev/c2 relation allows us to rewrite the argument of the wavefunction also as:

This relation, too, can be easily verified

36

:

The point is this: an elementary particle packs one unit of physical action (ħ) – per oscillation cycle, that

is – and, when in motion, we think of this as expressing itself as a combination of (i) angular momentum

(and, therefore, rotational energy) and (ii) linear momentum.

Now, the functional behavior of the t’ = (t − vx/c2) function may not be immediately obvious: goes

from 1 to infinity () as v goes from 0 to c, and time dilation may, therefore, not be immediately

understood. Hence, a graph may be useful. To produce one, we write x as a function of t: x(t) = vt. The t’

function can, therefore, be rewritten as:

35

We think of the neutrino as the light-particle of the nuclear force: just like a photon, it does not carry charge, but

it carries nuclear energy.

36

We use the

equation here.

17

The −1 factor is the inverse Lorentz factor, and its function (for positive v) is the arc of the first quadrant

of the unit circle, as illustrated below. It is, therefore, easy to see that, for any velocity v (0 < v < c), t’ will

be smaller than t, which illustrates the point.

Figure 4: The inverse Lorentz factor (−1) as a function of

Likewise, the behavior of the = (Evt − px)/ħ function may also not be immediately obvious, but

rewriting it as = (E0t)/ħ – and taking what we wrote about the t’ = −1t function – shows that the

phase of the wavefunction shows the same time dilation.

Note: The reader should not think we established a non-heuristic logical proof of special relativity based

on the reality of the wavefunction. If anything, we only showed that quantum mechanics is fully

consistent with special relativity (and, as we will show in the following annex, with general relativity).

We do think, however, that we did show what the relativistic invariance of the argument of the

wavefunction actually means, and that quantum mechanics and relativity theory mutually confirm each

other.

That does not amount to an intuitive understanding of special relativity, of course. Understanding

(special) relativity theory intuitively may not be possible, but the following considerations may or may

not help the reader to play some more with it.

When observing an object which is moving sideways with velocity v, we may think of its velocity v as a

tangential velocity.

Figure 5: Tangential velocity

Of course, you will say that most objects are not moving sideways only, but also towards or away from

us. However, such motion along the line of sight (which we will refer to as the radial velocity) can be

determined from the red- or blueshift of the light we use to determine the position of the object (in

order for us to able to track the position of an object – in what we refer to as the inertial reference

18

frame – it has to emit or reflect light). Hence, if we can determine both the tangential as well as the

radial velocity, we can add the two velocity components to get the combined velocity vector.

It is good to specify what is relative and what is not here: the distance between us, the observer, and the

object is not relative: there is no length contraction along the line of sight. Also, in the reference frame

of the object (which we will refer to as the moving reference frame), the (tangential) velocity of our

reference frame will be measured just the same: v. Finally, the speed of light does not depend on the

reference frame, either. Clock speeds, however, will depend on the reference frame, which gives rise to

the distinction between t and t’.

Because there is no length contraction along the line of sight, its length will be measured the same in

the inertial and moving reference frame. Lightspeed is used as the yardstick in both reference frames,

and we must, therefore, conclude this distance must be measured using non-moving clocks. In other

words, we must assume the same clock is used here.

37

In contrast, the relative velocity of the reference

frames is measured using moving clocks:

When combining this with the t’ = −1t relation (which establishes time dilation

38

), we get the relativistic

length contraction equation:

We get the same graph (Figure 4): for any velocity v (0 < v < c), ds’ will be smaller than ds, and s’ will,

therefore, be smaller than s

39

, which illustrates the point.

There is little to add, except for a few remarks on geometry perhaps:

1. If the distance between the origin of the inertial reference frame and the s = s’ = 0 point is equal to a

(the same in both reference frames, remember!), then we may measure that distance in equivalent time

units by dividing it by the speed of light. This amounts to measuring the distance a as a time distance. Of

course, we can always go back to measuring a as a distance by multiplying the time distance by c again:

we then get the distance expressed in light-seconds, i.e., as a fraction or multiple of 299792458 m.

37

This is not a matter of synchronization: we must assume the clock that is used to measure the distance from A to

B does not move relative to the clock that is used to measure the distance from B to A. It is one of these logical

facts which makes it difficult to understand relativity theory intuitively: clocks that are moving relative to each

other cannot be made to tick the same. An observer in the inertial reference frame can only agree to a t = t’ = 0

point (or, as we are talking time, a t = t’ = 0 instant, we should say). From an ontological perspective, this entails

both observers can agree on the notion of an infinitesimally small point in space and an infinitesimally small instant

of time. Indeed, both observers also have to agree on the s’ = s = 0 point!

38

We get the time dilation equation from writing s as a function of t: s(t) = vt and substituting in the Lorentz

transformation:

39

See footnote 37: observers need to agree both on the t = t’ = 0 as well as on the s = s’ = 0 point!

19

In fact, we think a good understanding of the absolute nature of the speed of light, and a deeper

understanding of the equivalence of using time and spatial distances may be all what can be provided in

terms of a more intuitive understanding of relativity theory. Indeed, when everything is said and done,

we are always measuring things in one specific reference frame: swapping back and forth between

reference frames is a rather academic exercise which does not clarify all that much: the laws of physics

(mass-energy equivalence, Planck-Einstein relation, force law, etcetera) are the same in every reference

frame and, hence, students should probably consistently focus on understanding these rather than

relativity, as relativity is just a logical consequence of these laws!

In any case, let us agree on writing a – which is, of course, the length of the base of the triangle in Figure

5 – as a spatial distance but assume all spatial distances are measured in light-seconds. This also implies

that we can write the velocities v, vt, and vr as relative velocities , t, and r, respectively.

Let us, indeed, introduce the radial velocity again now. We can then write the velocity vector as = t +

r, with t = ds/dt = ds’/dt’. The length of the hypotenuse will, therefore, be equal to a + rt.

Pythagoras’s Theorem then gives us the following equation:

(a + rt)2 = a2 + (tt)2

a2 + r2t2 + 2art = a2 + t2t2

(t2 − r2)t = 2ar

Multiplying both sides with c2, yields an equation in terms of the usual velocities measured in m/s:

(vt2 − vr2)t = 2acvr

It is a nice equation, but there is probably not all that much we can do with it.

40

2. Figure 5 introduces the concept of the phase (), which we measure in radians, and the angular

frequency , whose dimension is s−1. The two are related through the = t equation and, also using

the v = a equation, it will be easy for the reader to verify the following relation:

We leave it to the reader to establish the relations for the variables in the moving reference frame.

40

The reader will probably know Pythagoras’s Theorem does not apply to curved spacetime, but here we are

talking about special relativity only. Note that the ac factor gives us a radial distance expressed in meter again (not

in light-seconds). We are a little bit puzzled to what this expression might mean geometrically, so any suggestion

and/or correction of our readers is most welcome!

20

Annex II: The wavefunction and general relativity

We know a clock goes slower when placed in a gravitational field. To be precise, the closer the clock is to

the source of gravitation, the slower time passes. This effect is known as gravitational time dilation.

41

This cannot be explained by writing the argument of the wavefunction as a function of its energy Ev and

its momentum p. We will, therefore, distinguish (i) the rest energy of the particle outside of the

(gravitational) field (E0) and (ii) the potential energy it acquires in the field (Eg). The total energy as

measured in the equivalent of the inertial frame of reference (which is the reference frame without

gravitational field, i.e., empty space), and the argument of the wavefunction, can therefore be written

as:

E = E0 + Eg E0 = E − Eg

This effectively shows the frequency of the oscillation is lower in a gravitational field. At first, the

analysis looks somewhat counterintuitive because the convention is to measure potential energy (PE) as

negative (the reference point for PE = 0 is usually taken at infinity, i.e., outside of the gravitational field).

However, when noting extra energy must be positive (i.e., when taking the reference point for PE = 0 at

the center of the gravitational field, or as close to the source as possible

42

), all makes sense.

We hope this provides a more intuitive understanding of gravitational time dilation based on the

elementary wavefunction.

The reader should note this analysis is also valid for an electromagnetic or nuclear potential, or for any

potential (which may combine two or all three of the forces

43

). We may refer the reader here to

Feynman’s rather excellent analysis of potential energy in the context of quantum physics in his

Lectures, in which he also explains the nature of quantum tunneling.

44

However, we think Feynman’s

analysis suffers from a static view of the potentials involved.

We think one should have a dynamic view of the fields surrounding charged particles. Potential barriers

– or their corollary: potential wells – should, therefore, not be thought of as static fields: they vary over

time. They result from two or more charges moving around and creating some joint or superposed field

which varies in time. Hence, we think a particle breaking through a ‘potential wall’ or coming out of a

potential ‘well’ is just using a temporary opening corresponding to a very classical trajectory in space

and in time. We, therefore, think there is no need to invoke an Uncertainty Principle.

41

See, for example, the Wikipedia article on gravitational time dilation.

42

A gravitational field comes with a massive object which is usually taken to have a (finite) radius.

43

We are not aware of any successful attempt proving the gravitational force may be analyzed as some residual

force resulting from asymmetries or other characteristics of the two forces which we consider to be fundamental

(electromagnetic and nuclear). The jury is, therefore, still out on the question of whether or not we should think of

the gravitational force as a pseudoforce. We, therefore, still think of Einstein’s geometric approach to gravity

(curved spacetime) as an equivalent analysis. The question may be entirely philosophical: it should be possible to

also come up with a geometric interpretation of the electromagnetic and nuclear forces but, because of their

multidimensional character (2D/3D, respectively), this may not be easy.

44

See: Feynman’s Lectures, Potential energy and energy conservation (III-7-3).

21

22

Annex III: Thoughts on space travel

Note: this annex was added on 5 March 2023. It offers no new thinking, but we think of it as some easy reflections on a rather

remote future.

Those of you who read my papers, know that I think we know it all and that the science of physics is

complete now. There are no mysteries left to be uncovered: we now know what matter-particles and

light-particles actually are, and the explanation does not need fluffy concepts such as gluons, quarks, or

other virtual particles. All can be explained in terms of charge oscillations. Reality is charge in motion.

We also wrote about dark matter as anti-matter: we do not think of dark matter as some question to

which there is no answer. No. All that is left is engineering, and we are doing that. Let us, therefore,

think about the future of those spaceships that we may or may not build to escape Earth when nuclear

holocaust or some other catastrophe unfolds. So, what planets could we possibly reach beyond Mars?

You know that the university is pretty large. Hence, we must build ships that can travel fast. Fast means

the order of magnitude of lightspeed. Is that possible? Maybe. Maybe not. The most obvious problem is

propulsion. We can, theoretically, keep accelerating our ship to ever higher speeds by letting some force

act on it. The problem, as you know, is that, as our ship reaches higher velocities, its mass will increase.

There is nothing mysterious about that. The young Louis de Broglie offered what probably still stands as

the best interpretation or explanation of Einstein’s mass-energy equivalence relation when he wrote this

back in 1924:

“We may, thus, conceive that, because of some grand law of Nature, a periodic phenomenon of

frequency ν0 would be associated with each energy packet with rest mass m0, such that hν0 =

m0c2. The frequency ν0 is, of course, to be measured, in the rest frame of the energy packet. This

hypothesis is the basis of our theory: it is, just like all hypotheses, worth only as much as the

consequences that can be deduced from it.”

45

The assumption that mass is nothing but the inertia of energy – and that energy is, ultimately, charge in

motion – has proved to be worth a lot. It is worth everything as far as I am concerned. Hence, I value his

hypothesis as much as the consequences that can be deduced from it. Of course, I should now,

perhaps, explain what this frequency is all about, and how some linear velocity fits into this picture of an

elementary particle (think of an electron or a proton here) as a charged oscillation, but I did that

elsewhere rather convincingly (my opinion, of course), and so I will not be repeating myself here.

46

45

See the de Broglie’s Recherches sur la Théorie des Quanta (Ann. de Phys., 10e série, t. III (Janvier-Février 1925).

We translated this, somewhat freely, from the original: « On peut donc concevoir que par suite d’une grande loi de

la Nature, à chaque morceau d’énergie de masse propre m0, soit lié un phénomène périodique de fréquence ν0

telle que l’on ait : hν0 = m0c2, ν0 étant mesurée, bien entendu, dans le système lié au morceau d’énergie. Cette

hypothèse est la base de notre système : elle vaut, comme toutes les hypothèses, ce que valent les conséquences

qu’on en peut déduire. »

46

See my paper on de Broglie’s matter-wave. It interprets the de Broglie frequency – when an (elementary)

particle is at rest – as an orbital rather than a linear frequency: think of a pointlike charge going around and around

or Schrödinger’s Zitterbewegung interpretation of an electron here. It is only when the (elementary) particle

acquires some linear momentum that we have to think of some linear oscillation as well. These two oscillations

then combine. In the same paper, you will find a rather nice geometrical interpretation of this combined oscillation

and, therefore, a geometrical interpretation of the de Broglie’s frequency under all circumstances (i.e., not only in

the rest frame but in moving reference frames as well).

23

Some may still find it counterintuitive that kinetic energy – the energy that our spaceship acquires as we

keep accelerating it to go travel ever faster

47

- has an equivalent mass as well, but they should rather

think of the absurdities related to the opposite assumption: i.e., the assumption infinite velocities might

be possible. Something traveling at infinite speed is everywhere and nowhere along its trajectory: now

that is not a paradox (a seemingly contradictory statement) but a truly absurd proposition.

48

Again, if you keep in mind that the concept of mass is nothing but a measure of inertia to a change in

the state of motion of an object but that, for all other practical matters, energy is the only concept that

really matters (because all objects are just a ‘morceau d’énergie’ (a piece of energy), as de Broglie

famously wrote), then things all fall into place.

Now, the Lorentz transformation formulas result in the Lorentz formula for relativistic mass increase,

and that formula tells us only lightlike particles (these are photons, in practice, and – possibly but not

likely – neutrinos

49

, and their anti-matter counterparts, of course) can – and must

50

– actually travel at

the speed of light. All other objects can only achieve velocities that may be close to c but not quite equal

to c, unless we could marshal infinite forces, which we cannot. Even pushing our spaceship to, say, c/2

or c/3 will require engines we do not have right now. Just to give you an idea of proportions, current

rockets must (and do) reach escape velocities of about 30,000 km/h to bring something into orbit, and

those new hypersonic rockets you might have heard about reach velocities of the same order of

magnitude. Now, the velocity of light is 1,079,252,848.8 km/h⎯not approximately but exactly.

51

The

difference between both numbers is a factor of about 36,000. Hence, our current rocket engines will

need a lot of improvement.

However, let us assume we can, perhaps, use gravitational fields or the enormous energy that is being

released when matter and anti-matter annihilate each other or, somewhat less fictional, perhaps, in

nuclear fusion or fission reactions?

52

In short, let us assume that, a few hundred years from now,

47

Do not worry, I will not write about the twin paradox or weird time travel stories here. The Wikipedia article on

that is quite clear on that: the twin paradox (a seemingly absurd or contradictory statement or proposition) is

easily explained using special relativity theory only.

48

We once wrote a rather intuitive introduction to the Lorentz formula in a manuscript of ours: see the

introductory chapter and, more in particular, the section on Newton’s force law and (special) relativity (see pp. 24

ff. of the document).

49

Neutrinos probably have a very tiny rest mass. The argument is related to the rather special structure of the

‘nuclear’ force. Indeed, we think of a proton as a rather special oscillation in three rather than two dimension only,

and the oscillation in the dimension of the direction of motion may give the neutrino some apparent rest mass. We

quote from one of our papers on this: “Because neutrinos must result from transitions (of energy states) between

the 3D nuclear oscillations that generate them, they must be three-dimensional oscillations as well, and the

component of the oscillation in the direction of motion must have some energy and, therefore, must have an

equivalent mass, which must appear as some kind of rest mass of the lightlike neutrino.” See point 9 of our paper

on nuclear oscillations and neutrinos for more details.

50

Newton’s force formula – even in its relativistic form – implies the slightest force on a massless particle must

give it an infinite acceleration and, therefore, push it to lightspeed.

51

You may or may not know that lightspeed (c) is defined as a constant in the SI system of units: our time and

distance units depend on it. If you did not know this, you should look up the history of the SI system of units, and

look into the latest revision of it (2019).

52

This is true science fiction, of course⎯with the emphasis on fiction. Fission or fusion energy may be a more likely

energy candidate, but the problem is the same: how do you get directional energy out of nuclear or matter-

antimatter explosions?

24

perhaps, we can tap into new energy sources to fuel our spaceships and we can effectively start thinking

beyond colonizing the Moon and Mars. Where would or could we go next?

[…]

As we wrote elsewhere, astronomers broadly agree the end-to-end distance across the Universe is of

the order of 46 billion lightyears. This diameter of 46 billion (lightyears) is different from the diameter of

the observable Universe, which is only 27.6 lightyears because of the Universe’s continuous expansion,

and the speed of that expansion is a rather formidable fraction of c. Indeed, based on the two numbers

provided here, we can calculate the estimated speed of expansion (i.e., the radial velocity vr) of the

Universe as:

Two thirds of the speed of light? How can we know such things? Relativistic redshift: photons emitted by

stars who are moving away from us still travel at the speed of light, but their wavelength shifts to a

lower-energy spectrum. We must assume all this is hard science so we will not question it and just

accept that we can never hope to reach the outskirts of our Universe: that would require 23 billion years

of travel and, in the meanwhile, the outskirts of the Universe would have moved much farther away: we

cannot hope to catch up, even if our spaceships would reach speeds higher than the mentioned c/3 or

c/2 factor. Hence, let us think of exploring our own galaxy: the Milky Way. According to Wikipedia, our

galaxy has a diameter of about 90,000 lightyears. That is an incredible number as well. Even the ‘bulge’

where its spiral arms come together is like 1,000 light years thick. Could we travel around there?

Probably not: our Solar System is not located at this center. It is at a radius of about 27,000 light-years

from our Galactic Center. To be precise, we are living on the inner edge of the Orion Arm⎯so that is just

one of those spiral-shaped concentrations of gas and dust. Oh ! Now that we are talking about our

Galaxy, evidence for dark matter is this: the constant rotational speed appears to contradict the laws of

Keplerian dynamics and suggests that about 90% of the mass of the Milky Way is invisible to telescopes,

neither emitting nor absorbing electromagnetic radiation. It is this conjectural mass that has been

termed ‘dark matter’ but that, according to us, is nothing but anti-matter.

So, what are the nearest places, then? By that, we mean other planetary systems and planets that may

be suitable for life, right? Wikipedia has a nice overview of those, and so we refer you there. There are

two about 4 or 5 lightyears away (so that would take our c/2 spaceships only 8 or 10 years), and then

another dozen or so that are 10 to 20 lightyears further travel. However, preliminary estimates of

surface temperatures show these are either extreme subzero or extremely high temperatures (like a

hundred degrees above or below 0 degrees Celsius). There is only one, Gliese 667 Cc, 22 lightyears

away, that has a nice Mediterranean-like climate (29° C).

I would be willing to go and explore it⎯if only I were younger and, yes, if we would have such c/2

spaceships. I am not ready to be frozen into some much slower ship for a thousand years or so.