Many scientific journals, books, magazines and science web sites state that since Einstein’s theory of gravity doesn’t “fit” into the quantum theory of forces, a new quantum theory of gravity must be found. This essay explodes the prevailing scientific myth that relativity and quantum mechanics are somehow incompatible. The simple fact of the matter is that gravity is not a force at all, so trying to make it “fit” into quantum theory is impossible. This essay demonstrates that relativity and quantum physics are indeed different, but it’s simply a matter of scale. In fact they are perfect reflections of each other.

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3. The picture on the cover of this “commentary” shows a square peg representing quantum mechanics
and a round hole representing general relativity, meaning that the two theories labeled with
Schrödinger’s wave equation and Einstein’s field equation are somehow “incompatible.”
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Introduction
There is a prevailing myth within the scientific community that relativity and quantum mechanics are
incompatible. This statement has been repeated so often and for so long that it has become somewhat
of a mantra. This myth arose largely because general relativity is all about gravity, whereas quantum
physics doesn’t address gravity at all.
The following question was posted on the astronomy.com website: “Why are gravity and quantum
mechanics incompatible with each other? Why can't they be combined into a single formula or
concept?” Physicist James Trefil provided the answer to this question as follows.
“Your questions touch on a fundamental (some would say the fundamental) problem in
theoretical physics. The easiest way to understand the issue is to think about how the two
theories describe the forces of nature. The general theory of relativity is our best concept of
gravity. It states that the presence of mass distorts the space-time grid, like a bowling ball
that sits on a stretched rubber sheet. Particles then move in the shortest paths along that
distorted grid, a process we interpret as the effects of gravity. Thus, relativity, in essence,
gives a geometric interpretation of gravity [emphasis added].”
Oops. Actually, objects in free fall will move along the longest possible paths through spacetime,
called geodesics, not the shortest paths. Dr. Trefil continued answering the question with the following
statement.
“Quantum mechanics, through what physicists call the ’standard model,’ explains the other
three forces of nature. These are the electromagnetic force, which affects electrically
charged particles, the strong nuclear force, which holds particles in nuclei together, and the
weak nuclear force, which governs some radioactive decays [emphasis added].”
Wait a minute. Saying the standard model explains the “other” three forces – the electromagnetic, the
strong nuclear force and the weak nuclear force – implies that gravity is the “fourth” force. It is not.
Albert Einstein’s motivation for spending ten long years working on general relativity was due to his
realization that while a man is falling, he feels no force whatsoever.1
So it seems the underlying premise of relativity being “incompatible” with quantum mechanics stems
from the mistaken idea that gravity is a force. You do feel a force on your body while standing on the
ground or sitting on a chair, but that force isn’t gravity. It’s the electromagnetic force from the surface
you’re standing or sitting on pushing upward on your body. In fact, no force can do this except the
electromagnetic force, which accelerates you away from a geodesic path that otherwise would bring
you to the center of the Earth. The other two forces – the strong and weak nuclear forces – are not felt
outside atomic nuclei.
There is another reason why physicists claim that relativity and quantum mechanics are incompatible.
This stems from a misguided belief in material reductionism (MR), which is very easily disproved.
The premise behind MR is that the whole is equal to the sum of its parts. Take water for example. The
smallest part of water is an oxygen atom joined to two hydrogen atoms forming an H2O molecule. A
strict interpretation of MR would say the H2O molecule should behave just like the oxygen and
1 Einstein remembered this realization as being “the happiest moment in my life.”
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4. hydrogen atoms that make up H2O; a water droplet should have the same properties as an H2O
molecule; and the surface of a pond should resemble a water droplet like the picture shown below.
Pond photo attribution: Courtesy of Johnny Briggs, upsplash.com
Unlike the picture above, the surfaces of real ponds don’t bulge like water droplets, and ripples and
waves don’t form on surfaces of water droplets when wind blows on them. Also, a water droplet has
properties that are nothing like those of individual H2O molecules and vice versa. H2O molecules
totally lack numerous physical properties like melting and boiling temperatures, specific gravity,
refraction index, density, viscosity, pH, electrical conductivity, adhesion, cohesion, surface tension, and
the thermodynamic properties like temperature, specific heat, heat of vaporization, and entropy.
Entropy is a very important property to be discussed further below. The basic premise of MR is simply
false; the truth of the matter is the whole is much greater than the sum of its parts. Some properties
emerge and others disappear (sometimes abruptly) as the scale changes. In short, size matters.2
Features of the Quantum World
At the smallest scales, objects behave in ways that in no way resemble the way human-scale objects
such as tables and chairs do. Physics at the smallest scales is governed by quantum mechanics. So
those who embrace MR ask, “Since the whole is equal to the sum of its parts and tables and chairs are
made of quantum particles, then shouldn’t tables and chairs obey quantum physics?” Quantum objects
have dual complementary particle and wave properties. Both properties are needed to fully define a
quantum object, but both cannot be observed at the same time for the same object. But MR is so fully
ingrained into the physicists’ psyches that some of them even claim tables, chairs and indeed the
universe itself, are all comprised of wave functions which are the linear sums of the wave functions of
their constituent quantum parts.3
In the astronomy.com article, Dr. Trefil points out that according to quantum mechanics, forces result
from the exchange of virtual particles. When two electrons interact (collide) they exchange virtual
photons that results in an electrical force that repels them. The strong force has a “carrier particle”
called a gluon, and the weak force has two of them called the W and Z bosons. Gravity supposedly has
a particle called a graviton to carry the gravitational “force,” but the graviton has never been detected.4
The fact that Nature refuses to cooperate with theoretical physics by coughing up a graviton has further
convinced physicists that general relativity is incompatible with quantum physics.
Once we abandon the twin fallacies that a) gravity is a force, and b) material reductionism is true, then
it becomes clear c) that there is no incompatibility between general relativity and quantum physics, and
2 I would hazard a guess that the properties of our solar system are different than the properties of our galaxy and those in
turn are very different than those of the entire universe, which is why the field of cosmology is such a mess.
3 Nobody has ever seen a wave function because it is a non-physical entity. It is only manifested through the square of its
magnitude, which represents probabilities. Thus the wave function can best be thought of as a precursor of probability.
4 And in my opinion it never will be detected. There’s no need for a force carrier because gravity isn’t a force.
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5. furthermore d) although the two theories are very different, there is a deep hidden connection between
them. Those differences and connections will be examined next.
Further Differences between Quantum and Classical Worlds
Any object on the human scale is vastly more complex than a single electron. An electron has a few
properties that are “burned in” and never change, such as mass, electric charge, and lepton number so it
has virtually no degrees of freedom. The only degree of freedom it does have is the direction of spin,
which can only be “up” or “down” with respect to some direction defined by the device that measures
it. A freshly-minted electron is in a mixed quantum spin state, meaning that its spin can only be known
after it is measured, with the “up” and “down” spin directions each having a 50% probability of being
detected. The Shannon entropy, H, of a system is defined by the probabilities of all possible states by
this equation: H = – Σ pi log2 pi , where pi are the probabilities of all possible states. For the electron’s
spin states each having a probability of 0.5, H = –0.5 log2 0.5 – 0.5 log2 0.5 = 1 bit. Compare this with
the entropy of one cubic centimeter of water at 30C: H = 4.6 × 1022
bits. The number of possible
internal states of a cubic centimeter of water is vastly greater than the two possible spin states of an
electron; it’s an insanely large number N =2H
. Entropy is almost non-existent on the quantum scale, but
it is incredibly abundant on the classical scale.
The only thing known about quantum states before they are measured are their probabilities. In other
words, quantum states are non-computable and so particles don’t actually exist as particles until they
are measured. Prior to measurements, individual quantum particles behave like ghostlike waves.
Classical objects are not ephemeral that way. Your car keys don’t turn into waves and vanish when
you’re not watching them and you can almost always expect to find them right where you left them.
It’s About Time
There is a quantum phenomenon called entanglement, where measuring the state of one of two
maximally-entangled particles will instantaneously influence the measurement of the same state of its
entangled partner. This would seem to violate a fundamental principle of relativity, where signals
cannot travel faster than the speed of light, and instantaneous quantum influences would also appear to
travel backward in time. Quantum objects do not experience time because they have no history;
electrons do not get old. We do experiments on pairs of entangled particles that to us seem to be
separated by space and time, but the particles themselves don’t experience that. They only experience
entanglement. There is a reason why the quantum and classical worlds are very different with respect
to time, and it’s primarily because of an almost total lack of entropy at the quantum scale. Time, as
Einstein quipped, is what clocks measure, and clocks literally measure entropy.5
The Physics of Spacetime
Einstein’s field equations describe gravity in terms of spacetime geometry. But first, what exactly is
spacetime? Mathematically, spacetime is a four-dimensional manifold that conforms to the following
form of the Minkowski equation.
dτ2
=– c2
dt2
+dx2
+dy2
+dz2
,
where dτ is an incremental distance in spacetime, c is the speed of light, dt is an incremental distance in
time, and dx2
+dy2
+dz2
=dr2
, which is the square of an incremental distance in space, dr.
5 The paper The New Thermodynamic Understanding of Clocks describes a number of recent experiments with physical
clocks that show, as Gerard Milburn of University of Queensland in Australia stated, “A clock is a flow meter for
entropy." Einstein noted that time is what is measured by a clock, so time as we experience it is equivalent to the flow
(or buildup) of entropy.
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6. If we divide both sides of the equation by dt2
, we get the following.
(dτ/dt)2
= – c2
+ (dr/dt)2
Light travels through space at the speed of light, or dr/dt = c. The reason light is able to do this is
because the quantum of light, the photon, has no rest mass. The following result for a photon is
obtained from the Minkowski equation.
(dτ/dt)2
= – c2
+c2
=0  dτ=0
In other words, although light definitely moves extremely fast through space, it is essentially “stuck” in
spacetime. The story is different for objects having rest mass. There is a velocity vector through
spacetime, U, with a time component and a space component. It is possible to show that all masses
must travel through space below the speed of light relative to each other and at the same time travel at
the same constant speed, c, through spacetime. The path a mass takes through spacetime depends on
the path it takes relative to other masses in space. The distance, Δτ, along any path a mass travels is
equal to “proper time” or the time measured by a clock traveling along that path. If you want to know
how far you’ve traveled through spacetime, just look at the passage of time on your watch. Since
photons have zero spacetime velocity, they don’t experience any passage of time.
Now, getting back to the physics of spacetime, Einstein’s field equations show how the presence of
mass changes the geometry of spacetime, meaning that the path a mass takes through spacetime is
altered by the presence of other masses. If a mass is not disturbed by any outside force, it will follow a
path called a geodesic, which is the longest possible path, Δτ, between two points in spacetime. It is
also the longest possible interval of proper time between those points as measured by a clock, which
also equals the entropy it encounters along the path. When a mass is disturbed by an outside force, it
essentially takes a “short cut” through spacetime, which reduces the length of its spacetime path, Δτ,
while its speed through spacetime, c, remains constant. All of this is very counter-intuitive yet true.
But exactly how does a mass encounter entropy along its path through spacetime? The answer is
spacetime itself possesses entropy. Physicist Thanu Padmanabhan wrote a paper showing the field
equations of general relativity also have a thermodynamic interpretation, meaning spacetime has
thermodynamic properties of temperature, energy and entropy.6
The reason two massive objects
converge under their mutual gravitational influence is because proximity to one another increases the
local entropy of spacetime. As two masses converge, their combined entropy becomes more than the
sum of the entropy of the separated masses. The apparent mutual attraction is not due to a gravitational
force; it’s because their geodesic paths converge in spacetime in order to increase their entropy. Pulling
the masses apart reduces their entropy, requiring energy from an external force (electromagnetism).
Quantum-Classical Connections
As it turns out instantaneous quantum influences are necessary to prevent faster-than-light signaling,
which would violate causality.7
For the same reason, the outcome of a quantum measurement must be
absolutely random. Not random as in the roll of dice or the spin of a roulette wheel; it must be totally
and absolutely unpredictable. Einstein believed there must b quantum hidden variables that determine
beforehand what those measurements will be. If that were true, it would be possible to design a
communication system involving measurements of entangled particle pairs separated by time and space
that would allow sending messages through space faster than light. Experiments based on Bell’s
theorem proved conclusively that no hidden variables exist. There are other rules, such as the no-
6 Refer to the paper Entropy Density of Space–Time and Gravity
7 Refer to the paper Finite-speed Causal Influence Leads to Superluminal Signaling
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7. cloning rule, and the monogamy rule for maximally-entangled particles, which are also in place to
prevent so-called intelligent beings like us from designing communication systems that would violate
causality and perhaps destroy the universe. The more we examine the “weird” laws of quantum
physics, the more we begin to realize how sensible and needed those rules really are. They have an
important and necessary purpose: to protect causality in the classical world in order to allow it to
emerge from the random, acausal quantum world.
There are other more subtle connections between the two worlds. In 1918, mathematician Emmy
Noether published a landmark theorem that proved physical symmetries have corresponding
conservation laws. These are summarized in the following table.
Symmetry Conserved Quantity
Time Translation, t Energy, e
Space Translation, x Linear momentum, px
Rotation, θ Angular momentum, L
Time translation, space translation and rotation symmetries mean that physical laws don’t change at
different times, in different places or in different directions. In other words, the universe is flat.
Flatness implies maximum entropy and maximum uncertainty. As we saw earlier, mass distorts
spacetime and makes it curve; however, an observer in free fall and not being accelerated by an
external force will not observe any such curvature. Inside a space capsule orbiting the Earth in curved
spacetime, energy, linear momentum, and angular momentum are all conserved. Thus, inside the
capsule, spacetime appears perfectly flat. On the Earth’s surface, a ball thrown upward will change its
vertical momentum as its upward velocity slows down and becomes a downward velocity, and a
spinning gyroscope will precess and change the direction of its angular momentum. These local
violations of the conservation laws occur because stationary observers on the Earth’s surface are being
accelerated upward, making space appear to be asymmetrical. Spatial symmetry will be restored for
observers in a free-falling elevator.
There is a corresponding set of symmetries in the quantum world. According to the Heisenberg
uncertainty principle, there are pairs of measured quantities that are complementary. Decreasing the
uncertainty of one quantity increases the uncertainty of the other. The product of the uncertainties is
greater than or equal to one half the Planck constant, ħ.
Δt Δe ≥ ½ ħ
Δx Δpx ≥ ½ ħ
Δθ ΔL ≥ ½ ħ
Note that the first quantity in each of the above pairs corresponds to a symmetry of spacetime and the
second quantity corresponds to a conserved quantity in Noether’s theorem. There obviously seems to
be a subtle connection between Heisenberg’s uncertainty principle and the symmetries of spacetime.
I should point out that since time equals the accumulation of entropy, which can only increase, time is
not absolutely symmetric. Time translation symmetry is approximately true over small increments of
time, such as the span of human history, but a lack of time translation symmetry would be noticeable
over the life of the universe. On the other hand, astronomical measurements of space have failed to
show any spatial curvature or asymmetry, and I think we can assume that as long as the conservation of
linear and angular momentum hold up, then space is truly perfectly flat.8
8 Cosmologists call this the “flatness problem.” In my opinion this is just another problem like the “measurement
problem,” which seems to bother physicist for no reason.
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8. There are physicists who believe that the number three was an arbitrary choice for the number of
dimensions of space. Current versions of string theory require nine spatial dimensions, while M-theory
requires ten of them, and bosonic string theory calls for 25. Luckily for us, our universe has three
spatial dimensions; otherwise, the Earth wouldn’t be able to stay in orbit around the Sun. But it turns
out that luck isn’t the reason why we live in 3-dimensional space. It’s because of Noether’s theorem
and the conservation of angular momentum.
The angular momentum vector, L, is the cross product of a distance vector, r, and a momentum vector,
p. The diagram below shows this relationship.
Bob, on the right, observes a ball traveling from his right to his left. The moving ball has a mass, m, a
velocity vector, v, and a momentum vector, p = m v. The distance between Bob and the ball is a
distance vector, r. The vectors p and r lie on a flat 2-dimensional plane, shown as a tan surface. The
momentum vector, L, the cross product r × p, lies 90 perpendicular to the r-p plane, pointing in the
third dimension. It turns out that the cross product can only be defined mathematically in three
dimensions. If the number of dimensions, n ≠ 3, then L will point in some direction other than one of
the n dimensions, which is a contradiction.
Thus, Noether’s theorem requires that space have three and only three dimensions. It also requires the
moving object to have mass. The Heisenberg’s uncertainty relationship Δx Δpx is reflected in the
relationship r × p. The quantum world is a shadow of the classical world. Most elementary particles
have a rest masses. Photons and gluons do not, but gluons are always confined within atomic nuclei.
Every elementary particle, except one, has the property of angular momentum, or spin. Fermions have
spins that are odd integer multiples of ½ ħ and bosons have spins that are even integer multiples of ½ ħ.
The lone exception is the Higgs boson having a spin of zero.
The Higgs particle is the force carrier of the Higgs field, which permeates the entire universe. The
Higgs field establishes a non-accelerating and non-rotating frame of reference throughout the universe,
so unless a body with mass is accelerating or rotating then it doesn’t feel any force of acceleration.9
Logically, the Higgs particle must have zero spin because it cannot be spinning relative to its “other
self,” which is the non-rotating Higgs field.
According to Mach’s principle, both linear and rotational inertial forces are due to acceleration relative
to all other massive objects in the universe, and the quantum Higgs boson carries a linear or centripetal
force to any mass that is accelerating or rotating.10
By now it should be abundantly clear to the readers of this essay that there isn’t a trace of
“incompatibility” between general relativity and its shadow reality, quantum mechanics. They work
together in perfect harmony, so they are two sides of the very same coin.
9 This is why gravity cannot be a force; because if it were, an object falling under the influence of gravity would feel a
gravitational force of acceleration.
10 Ernst Mach articulated this principle in 1883. Peter Higgs described it in terms of a quantum field in 1964.
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→
→ →
→ →
→
→ →
→ →→
→
→
→
→
→
→

9. Gödel’s Universe
In 1931, the great Austrian mathematician, logician and philosopher Kurt Gödel published a landmark
proof stating that in any consistent formal mathematical system, there are true statements that are not
provable from the system's axioms. In short, a formal mathematical system can be consistent or
complete but not both.
• Definition of Consistency: There is no statement where both the statement and its negation are
provable from the axioms.
• Definition of Completeness: Any statement or its negation are provable from the axioms.
The classical universe is equivalent to a formal mathematical system based on a set of axioms
expressed as causal laws (the "laws of nature") governing interactions among space, time, matter and
energy. Therefore, the universe itself is either consistent or complete but not both. An inconsistent
universe would be a universe where a cause can give rise to both an effect and its opposite. Clearly this
not the case because experiments based on physical laws are repeatable. We must then conclude that
the universe is consistent and yet it must be incomplete. By definition, the universe is "all there is" and
therefore it must be radically self-referential and complete while being consistent at the same time,
which seems to violate Gödel's proof.
Here is where quantum mechanics comes in, where a quantum superposition of states can produce one
observation or its opposite. For example if the spin states of an electron are in a superposed up and
down state, a spin measurement can produce either an up or down spin state according to a set of
probabilities. The important thing to note is that the results of spin measurements are not subject to any
law of cause and effect; the outcomes of those measurements are completely random and impossible to
predict from axioms, as proven experimentally by violations of Bell's inequality. In other words,
quantum mechanics is inconsistent according to the above definition, but this inconsistency allows it to
be a complete description of quantum reality based on its axioms.
Per statistics and the law of large numbers, a consistent set of axioms (expressed as the classical laws of
nature) will emerge naturally from underlying quantum randomness and inconsistency. Quantum
mechanics is inconsistent yet complete while classical physics is incomplete yet consistent. They subtly
and magically merge together to form our complete and consistent universe depicted below.
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10. A Metaphor
Worlds of Light and Shadow
There are two worlds. One world is visible and filled with light; of planets, stars, and galaxies living in
four-dimensional spacetime. A world of senses; of scented flowers, trees, animals, humans, and
beautiful sunsets. A world filled with space and distances so vast they defy comprehension. A world of
endless time; of birthing, aging, and dying. A world of certainty, where all things have their designated
times and places in “world lines.” A world where objects having mass all move at a constant speed of
light through four-dimensional spacetime, with world lines twisting and bending to the will of gravity,
making surrounding space appear to be flat. A world filled to the brim with entropy; with infinite
possibilities to be realized as amazing actualities in space and time.
The other world is an invisible shadow world of quantum physics; an ageless world completely beyond
time; a world having only a tenuous connection to space. Nothing in this world is certain and nothing
has a definite place in it. A world where objects appear out of nowhere and vanish without a trace;
where objects are defined as separate-but-equal waves and particles, but not both at once. A world of
entangled particles not knowing any distances between themselves and their partners. A world where
entropy is in short supply and gravity is absent.
The invisible shadow world is very different than the world we see. Each world obeys very different
rules, leading some scientists to believe the two worlds are incompatible; but this simply is not true.
The two worlds have different shapes, colors and dimensions; nevertheless, each world is a perfect
reflection of the other and neither one could exist without the other. A world without light is a world
without shadow; a world without shadow is a world without light.
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