A unification of gravity with electromagnetism and quantum

6/23/2016 1
𝑔 𝜇𝜈
𝜕
𝜕𝑥 𝜇
𝜕
𝜕𝑥 𝜈
Ψ =
𝑚2
𝑐0
2
ℏ2
Ψ
𝑔 = −
1
2
𝑐∞ 𝛻𝑐∞
𝑑𝑠2 = 𝑔 𝜇𝜈 𝑑𝑥 𝜇 𝑑𝑥 𝜈
𝑑2
𝑥 𝜇
𝑑𝑠2
= −Γ𝛼𝛽
𝜇 𝑑𝑥 𝛼
𝑑𝑠
𝑑𝑥 𝛽
𝑑𝑠
𝐸0 = 𝑚𝑐𝑐0

Unification of Gravity with
Electromagnetism
Solving the Mystery of Gravity
Xiaofei Huang
Ph.D., Tsinghua University, Beijing China
HuangZeng@yahoo.com
6/23/2016 2

What is Gravity?
• Without knowing what causes
gravity, human beings may never
truly understand how the universe
operates.
6/23/2016 3

Einstein’s Dream
• Gravity and electromagnetism are similar in many ways.
The strengths, for instance, are both inversely proportional
to the square of the distance between two bodies, and
both have infinite range. It was Einstein’s dream in the last
30 years of his life to unify the two forces.
• Einstein replied to a 20-year-old high school dropout
named John Moffat regarding to the subject: “Dear Mr.
Moffat,” the reply began, “Our situation is the following.
We are standing in front of a closed box which we cannot
open, and we try hard to discover about what is and is not
in it.”
• Can we open the box and solve one of the biggest
mysteries of the universe?
6/23/2016 4

Introduction
• General relativity is the most successful theory of gravity so far. The
key idea is that gravity is not an ordinary force, but rather a
property of space-time geometry.
• Its concepts, such as curved spacetime, metric tensor, Gaussian
geometry, and the longest proper time, are not easy for the public
to understand.
• The mathematics used in general relativity is tremendously complex
to comprehend. Ph.D. degrees in differential geometry and physics
are necessary in order to truly master the concepts and formulas of
the theory.
• However, this theory is not compatible with quantum theory,
another foundation of modern physics. The most famous physicists
in the last 100 years, including Einstein himself, have tried and
failed to unify general relativity and quantum theory.
6/23/2016 5

In this presentation, gravity will be reformulated using a
key concept in quantum theory called particle waves. It
will be shown that gravity can be viewed as the refraction
of matter waves. Specifically, gravity emerges when
variations occur in the speed limit of particles. The curved
spacetime is simply a manifestation of the refraction of
matter waves. The warped spacetime is not, as widely
accepted, the cause of gravity, but a result of particle
wave refraction.
This theory is easy to understand and simple in formula. It
is built on the solid foundation of quantum theory,
unifying gravity with electromagnetism under the same
theoretical frame of physics.
6/23/2016 6

The focus of this presentation is on the
mechanism which nature deploys to generate
gravity, rather than some mathematical
equations that just make predictions to match
observations. In philosophy, the mechanism
should be simple, and should be the same as the
one at generating electromagnetism.
6/23/2016 7

PARTICLES AS WAVES
Preliminary Knowledge on
6/23/2016 8

Common Wave Propagation Pattern
• It has been found in the quantum world that the
building blocks of the universe are particles, such as
photons, electrons, neutrinos, neutrons, and protons,
etc. They are simply wave packets that share the
common wave propagation pattern governed by Klein-
Gordon equation.
• The Klein-Gordon equation defines a necessary
condition that all particle waves in nature must satisfy.
• Due to the common wave propagation pattern, all
particles are of the same speed limit at any given
spacetime point, called the common particle speed
limit here.
6/23/2016 9

Particles as Waves of the Same Medium
• The fact that all particle waves satisfy the same
wave propagation pattern and are of the same
speed limit at any given spacetime point is truly
impressing. It is a fact that can only be explained if
every wave is propagating in the same medium.
• Different particles can be viewed as different
excited states of the same medium. In addition to
their common wave-like properties, different
particles may have different additional properties,
such as mass, charge, spin, and color. They lead to
different behaviors and different interactions
among them.
6/23/2016 10

Reconcile the Luminiferous Aether
• Before 1905, it was commonly accepted by the
scientific community that any wave needs some
medium to carry vibrations, including light waves. The
medium carrying light waves is called the luminiferous
aether, suggested by Huygens.
• However, this concept makes it difficult to obtain a
mechanism that explains why the speed of light is
constant. If we think an observer as an object moving
through the aether by pushing away aether around it,
then nobody can figure out a way to explain the
constant speed of light. At the year, Einstein
abandoned the idea of the luminiferous aether.
6/23/2016 11

Reconcile the Luminiferous Aether (II)
• However, neither the Einstein’s postulate of the special
relativity nor Lorentz’s time dilation and length
contraction can solve the puzzle of the constancy of
the speed of light. The former simply hides the
problem into a principle and the latter is too ad hoc.
Both are on the wrong tracks to explain why light
always travels at a constant speed in vacuum.
• The puzzle could only be solved if we think all particles
in nature are waves of the same medium. That is, all of
the them are excited wave packets of the same
luminiferous aether.
6/23/2016 12

Solving the Puzzle of Relativity
• If we believe that all particle waves are propagating in the
same medium and sharing the same propagation pattern,
they will possess a certain symmetry with respect to space
and time. Relativity is nothing more than that symmetry
such that all inertial frames are equal with respect to the
laws of physics.
• In particular, given any spacetime point, one must-have
property of the symmetry is the existence of a common
speed limit for all particles in any inertial frame. If there is
any slightest difference in the speed limit for any two
particles, the symmetry is broken and the principle of
relativity is violated.
6/23/2016 13

Reconcile the Constant Speed of Light
• The common speed limit at the same spacetime point for all
particles is a necessary condition for holding the principle of
relativity. If a particle travels at the speed, it will travel at the
same speed in all inertial frames.
• The constant speed of light is not a true essence of relativity as
Einstein claimed. The speed of light can be a variable without
violating the principle as long as the photon shares exactly the
same speed limit with other particles.
– It has been found recently that structured light is slower than
unstructured light in vacuum. It doesn’t mean the violation of the
principle of relativity. It simply means that structured light is not
traveling at the speed limit.
6/23/2016 14

Reconcile the Constant Speed of Light (II)
• If a photon travels at the speed limit in an inertial
frame, then it will be viewed as traveling at the speed
limit in any other inertial frames. That is, the speed
remains the same in different frames only if the
speed is the speed limit.
• Based on the existing experiments, the speed of light
remains the same in different frames. It implies that
light always travels at the common particle speed
limit. Therefore,
the speed of light =the common particle speed limit
6/23/2016 15

Solving the Puzzle of Gravity
• By assuming a common medium, when the common
particle speed limit is different for different points in space,
gravity naturally emerges as a result of particle wave
refraction. There is no need to have space and time be
magically curved to generate gravity. (Remember that, at
the same point in space, all particles should share exactly
the same speed limit. Otherwise, the principle of relativity
will be violated.)
• Based on this simple assumption, we can make almost
exactly the same predictions as Einstein’s general relativity
on gravitational time dilation, gravitational light bending,
the extra precession of perihelion of Mercury, and
gravitational waves. These two theories approximate each
other in mathematics, but totally different in principle.
6/23/2016 16

GRAVITY AS PARTICLE WAVE
REFRACTION
From wave refraction to gravity
6/23/2016 17

Refraction of Waves
• When a light ray passes from a fast medium to
a slow medium, it bends toward a direction
normal to the boundary between the two
media.
• In general, refraction is the change in direction
of propagation of a wave due to a change in
its speed in a transmission medium.
6/23/2016 18

Examples of Wave Refraction
Step changes (slower speeds) Progressive changes (slower speeds)
6/23/2016 19

The Falling Rate d2x/dt2 is -g
• Assume that the wave
speed changes only along
the vertical x-axis as
𝑐0 1 +
𝑔𝑥
𝑐0
2 , and the initial
velocity component along
the x-axis is zero, then the
falling rate d2x/dt2 of a wave
packet is -g.
• This rate is the same as the
one of a falling object on
the surface of the earth.
x
y
The falling rate of a wave packet is
–g.
6/23/2016 20

The Falling Rate is still -g for a
Contained Case
• If the wave is contained
by two parallel mirrors
that reflect the wave
backward and forward,
then the falling rate is
still –g.
x
y
6/23/2016 21

Why All Objects Fall to Ground
• Particles with mass can be treated as contained cases.
Massless particles, like photons, can be treated as non-
contained cases.
• When the common particle speed limit decreases along the
x-axis as in the case described before, such as when we are
near the surface of the earth, a particle of mass will fall
towards the ground just like the contained cases.
• Using modern optic atomic clocks, it has been detected
that the speed of light slows down towards the surface of
the earth as 𝑐0 1 +
𝑔𝑥
𝑐0
2 , where 𝑥 is the vertical distance
away from the surface of the earth. This is why all objects
fall to the ground with the rate –g.
6/23/2016 22

Why All Objects Fall to the Ground at
the Same Rate
• Different particles of different masses correspond
different wave frequencies because of the energy-mass
equivalence. When the change of the common particle
speed limit is independent of the frequency, the falling
rate along the vertical axis for all particles, massive and
massless, are of the same value with the elapse of the
time.
• This is the basic mechanism of explaining why all
objects fall to ground with the same rate. It also
generalizes Galileo’s principle to include massless
particles such as photons.
6/23/2016 23

Gravity as Particle Wave Refraction
• By assuming that all particles are waves propagated in
the same media, gravity rises when there are variations
of the common particle speed limit. In those cases,
particles no longer move in straight lines. Rather their
trajectories are curved due to particle wave refraction.
• This is a simple mechanism to understand gravity.
There is no need to use advanced concepts such as
curved spacetime, longest proper time, Ricci tensor,
and tremendously complex coupled hyperbolic-elliptic
nonlinear partial differential field equations of Einstein.
Hypothesizing that all particles are waves of a
common medium can solve both the mysteries of
relativity and of gravity.
6/23/2016 24

Keys to Understand Gravity as Particle
Wave Refraction
• All particles are packet waves of the same
medium with exactly the same speed limit.
• Gravity emerges when there are variations in
the common particle speed limit. Therefore,
gravity = particle waves + variations of the
common particle speed limit.
• This is the true essence of gravity. The
remainder of this presentation describes this
idea using mathematical formula.
6/23/2016 25

Why Einstein Missed the Particle
Wave-Based Gravity Theory?
• In 1907, Einstein originally assumed that gravity is caused variations
of the speed of light. However, this was incorrect and he
abandoned the idea later on.
• When Einstein worked on gravity from 1907-1915, quantum theory
was far from mature. Nobody knew at that time that all matter,
including protons, electrons, and atoms, can exhibit wave-like
behavior.
• The concept that matter behaves like a wave was suggested by
Louis de Broglie in 1924. Therefore, matter waves are often referred
to as de Broglie waves.
• The Klein–Gordon equation, the key equation at establishing the
particle wave-based gravity theory, is a relativistic version of the
Schrödinger equation proposed by Oskar Klein and Walter Gordon
in 1926. It is eleven years after Einstein finalized his theory in 1915.
6/23/2016 26

A FORMULATION OF GRAVITY
WITHOUT DIFFERENTIAL GEOMETRY
6/23/2016 27

General Relativity & Fancy Differential Geometry
• General relativity uses differential geometry
with fancy concepts such as curvilinear
coordinates, acceleration frames, Riemann
tensor, Ricci tensor, Christoffel symbols,
covariant/contra-covariant tensor, and
covariant derivatives. It is very hard to
understand, both in physical concepts and in
mathematics, making it inaccessible to the
general public.
6/23/2016 28

No More Fancy Differential Geometry
• In this presentation, only Cartesian coordinates are
used. The objective is not to formulate the laws of
physics in any accelerating frame and any curvilinear
coordinates. Unlike general relativity, different
geometry is not required. It greatly simplifies the
mathematics of understanding gravity and offers better
insights into nature. Anyone who understands
electromagnetism can understand the new theory.
Generalizing it to curvilinear coordinates and
accelerating frames using differential geometry is
straightforward in mathematics. However, it offers no
more insight in physics.
6/23/2016 29

REFRACTION
A Mathematical formulation of
6/23/2016 30

A Universal Motion Equation for
Particle Waves
• It is known in quantum theory that all particles,
both bosons and fermions, satisfy the Klein-
Gordon equation:
□ 𝜓 =
𝑚2
𝑐0
2
ℏ2
𝜓 (1𝑎)
where □ is the d’Alembert operator defined as
□ = −
1
𝑐0
2
𝜕
𝜕𝑡
2
+
𝜕
𝜕𝑥
2
+
𝜕
𝜕𝑦
2
+
𝜕
𝜕𝑧
2
(1)
It is said here that both the operator and the
equation take their standard forms.
6/23/2016 31

Gravity Emerges if c0 is a Variable
• The common particle speed limit is equal to the
constant parameter c0 in the d’Alembert operator.
• When c0 is a variable instead, denoted as c,
gravitational acceleration naturally emerges from
the Klein-Gordon equation as
𝑔 = −𝑐𝛻𝑐
• If we define 𝜙 = 𝑐0 𝑐 − 𝑐0 , where 𝑐0 is the
standard speed of light, when 𝑐 → 𝑐0, the above
equation falls back to the classic equation 𝑔 =
− 𝛻𝜙. 𝜙 therefore represents the gravitational
potential.
6/23/2016 32

From EM Waves to Gravity Waves
• It is known that the propagation of electromagnetic
(EM) waves is also governed by the d’Alembert
operator as
□𝐴 = 𝜇0 𝐽
Here, A is the electromagnetic four potential and J is the four
electric current. This is just one reformulation of Maxwell
equations for electromagnetism.
• If we assume that the propagation of gravity waves
takes the same form as that of EM waves like
□𝐴 𝐺 = 4𝜋𝐺𝐽 𝑚 2
where 𝐴 𝐺 is the gravitational four potential and 𝐽 𝑚 is the four
matter current, then we can restore Newtonian gravity.
6/23/2016 33

Restoring Newton’s Universal Theory of Gravity
• When masses that generate a gravitational
field are motionless, the gravitational field
equation (2) becomes
𝛻2
𝜙 = 4𝜋𝐺𝜌
• This is the Poisson equation for gravity.
Together with 𝑔 = −𝛻𝜙, Newtonian gravity is
completely restored.
6/23/2016 34

Summary of Gravity as Light Refraction
• It has been found that that all particles in nature,
both bosons and fermions, are simply waves
governed by a universal motion equation, called
the Klein-Gordon equation. Consequently, all of
them are of the same speed limit at every point
in space, called the common particle speed limit.
Otherwise, the relativity principle will be violated.
• It has been shown that when the speed limit
changes from place to place, it can generate a
gravitational effect, restoring Newtonian gravity.
6/23/2016 35

GRAVITY AS WAVE REFRACTION
Further mathematical Investigation of
It will be shown in this section that all major results of general relativity can be
restored, such as curved spacetime, geodesic equation, gravitational time
dilation, light bending, extra-precession of the perihelion of Mercury, and
Schwarzschild metric based on this simple theory.
6/23/2016 36

Fine-Tuning the Klein-Gordon Equation
• To match the existing observations on gravitational light
bending, the d’Alembert operator □ in the Klein-Gordon
equation needs to be fine tuned to
□ = −
1
𝑐2
𝜕
𝜕𝑡
2
+
𝑐
𝑐0
2
𝜕
𝜕𝑥
2
+
𝜕
𝜕𝑦
2
+
𝜕
𝜕𝑧
2
(3)
This is called the normal form the operator. In this case, the
Klein-Gordon equation takes its normal form as
−
1
𝑐2
𝜕
𝜕𝑡
2
+
𝑐
𝑐0
2
𝜕
𝜕𝑥
2
+
𝜕
𝜕𝑦
2
+
𝜕
𝜕𝑧
2
Ψ =
𝑚2
𝑐0
2
ℏ2
Ψ (3a)
• The common particle speed limit in this case is 𝑐∞ = 𝑐2
/𝑐0,
which is also the speed of light.
• When 𝑐 → 𝑐0, Eq. (3) falls back to its standard form (1).
6/23/2016 37

An Important Result
• Based on the normal Klein-Gordon equation (3a), at its
classical limit, there is a simple relation between the
gravitational acceleration g and the speed of light 𝑐∞:
𝑔 = −
𝑐3
𝑐0
2 𝛻𝑐 = −
1
2
𝑐∞ 𝛻𝑐∞
• This equation says that g is equal to the half of the
negative gradient of 𝑐∞ multiplied by itself, independent
of the mass of the particle.
• As long as we know the distribution of the speed of light
𝑐∞, we can find out the gravitational acceleration.
6/23/2016 38

𝑔 = −
1
2
𝑐∞ 𝛻𝑐∞is a Universal Result
This equation holds true for any stationary gravitational
fields, regardless of the cause of variations in the speed of
light 𝑐∞. The causes can be regular matter, regular energy,
dark matter, dark energy, the expansion of the universe,
and the non-uniform expansion of the universe. No matter
of the cause, this equation gives a precise relation between
the speed of light and the gravitational acceleration.
This equation covers the cases of Newtonian gravitational
acceleration and of general relativity for explaining the
extra precession of perihelion of Mercury.
6/23/2016 39

Verifying 𝑔 = −
1
2
𝑐∞ 𝛻𝑐∞ is Easy
• Find the rotational velocities of stars in a galaxy using the
21-cm lines of the hydrogen atoms. From them, compute
their gravitational acceleration rates using 𝑔 = 𝑣2/𝑟.
• Compute the distribution of the speed of light 𝑐∞ 𝑥, 𝑦, 𝑧
around the galaxy using the equation 𝑔 = −
1
2
𝑐∞ 𝛻𝑐∞.
• Compute the gravitational lensing effects using 𝑐∞ .
• Using the existing observation to check if the computed
gravitational lensing effects match the observed ones.
6/23/2016 40

𝑔 = −
1
2
𝑐∞ 𝛻𝑐∞ or 𝑔 = −𝑐∞ 𝛻𝑐∞?
• The normal Klein-Gordon equation (3a) can be generalized
to
−
1
𝑐2
𝑖ℏ
𝜕
𝜕𝑡
2
+
𝑐
𝑐0
𝛼
−𝑖ℏ
𝜕
𝜕𝒓
2
𝜓 =
𝑚2 𝑐0
2
ℏ2
𝜓
where the real-valued parameter 𝛼 controls the space contraction
rate.
– When 𝛼 = 2, then 𝑔 = −
1
2
𝑐∞ 𝛻𝑐∞, where 𝑐∞ = 𝑐2/𝑐0. In this
case, the space is curved and Einstein is right about
gravitational light bending.
– When 𝛼 = 0, 𝑔 = −𝑐∞ 𝛻𝑐∞, where 𝑐∞ = 𝑐. In this case, the
space is not curved. Newton is then correct.
• 𝛼 should be fine tuned based on observations.
6/23/2016 41

Importance of the Verification
• If the existing data on gravitational lensing and the
acceleration of stars supports the simple relation 𝑔 =
−
1
2
𝑐∞ 𝛻𝑐∞, we will achieve a milestone in
understanding gravity, of a similar importance to the
discoveries of Galileo, Copernicus, Kepler, Newton, and
Einstein on the subject.
• GR also leads to the same equation, but works only for
the point mass case. Upon verification, it will show that
GR is only an approximation. This equation is more
universal than Newton’s and Einstein’s.
6/23/2016 42

GRAVITATIONAL TIME DILATION,
LIGHT BENDING, AND PERIHELION
PRECESSION OF MERCURY
Mathematical Investigation of
6/23/2016 43

Solution to a Point Mass
• When the relation between the parameter c and the gravitational potential 𝜙 𝐺 has
a definite relation as c = 𝑐0 1 +
𝜙 𝐺
𝑐0
2 , the solution of the field equation (2) for a
static point mass is
−𝑐0
2
1 −
𝐺𝑚
𝑐0
2
𝑟
2
, 1 −
𝐺𝑚
𝑐0
2
𝑟
−2
, 1 −
𝐺𝑚
𝑐0
2
𝑟
−2
𝑟2, 1 −
𝐺𝑚
𝑐0
2
𝑟
−2
𝑟2 sin2 𝜃
• If we rescale the radical r , we can reformulate the metric as
−𝑐0
2
1 +
𝐺𝑚
𝑐0
2
𝑟
−2
, 1 +
𝐺𝑚
𝑐0
2
𝑟
2
, 𝑟2
, 𝑟2
sin2
𝜃
• The Schwarzschild metric of general relativity is its approximation as
−𝑐0
2
1 −
2𝐺𝑚
𝑐0
2
𝑟
, 1 −
2𝐺𝑚
𝑐0
2
𝑟
−1
, 𝑟2
, 𝑟2
sin2
𝜃
• The differences of the two is at the order of 𝐺2 𝑚2/𝑐0
4
𝑟2, beyond the precision of
any existing apparatus to detect.
6/23/2016 44

Usage of the Solution for a Point Mass
• It can be used to explain
1. gravitational time dilation,
2. gravitational light bending,
3. The extra-precession of the perihelion of
Mercury.
6/23/2016 45

GRAVITY AS FORCE
Mathematical Investigation of
6/23/2016 46

Why gravity can be treated as a force?
• Let 𝜙 𝐺 = 𝑐0 𝑐 − 𝑐0 be the gravitational potential. For a slow
moving particle in a weak, static gravitational field, the normal
Klein-Gordon equation (3a) can be approximated as
iℏ
𝜕𝜓
𝜕𝑡
= −
ℏ2
2𝑚
𝛻2 𝜓 + 𝑚𝜙 𝐺 𝜓
• For a particle with charge q in an electric potential 𝜙 𝐸 (the
magnetic strength is zero), the Klein-Gordon equation can be
approximated as
iℏ
𝜕𝜓
𝜕𝑡
= −
ℏ2
2𝑚
𝛻2 𝜓 + 𝑞𝜙 𝐸 𝜓
• If we replace m with q and 𝜙 𝐺 with 𝜙 𝐸 , the former equation falls
back to the later one. That is the fundamental reason that why
gravity can be treated in the same as electric force.
6/23/2016 47

The Importance of Schrödinger Equation
• The equation iℏ
𝜕𝜓
𝜕𝑡
= −
ℏ2
2𝑚
𝛻2 𝜓 + 𝑉𝜓 , in general, is
called the Schrödinger equation. It is as important in
quantum mechanics as Newton’s f=ma is in classical
physics. The latter is the classical limit of the former,
which is more general than the latter.
• The equation describes how the quantum state of a
particle system changes with time. It was formulated in
late 1925, and published in 1926, by the Austrian
physicist Erwin Schrödinger. It is critically important in
understanding the structure of atoms, such as the
hydrogen atom.
6/23/2016 48

REFRACTION
Further Generalization of
6/23/2016 49

A General Form of the d’Alembert Operator
• When the d’Alembert operator takes the normal form (3)
and 𝑐 ≠ 𝑐0 for a point in space in one frame, for another
frame in a relative motion to the first one, it will have the
following general form:
□ = 𝑔 𝜇𝜈
𝜕/𝜕𝑥 𝜇
𝜕/𝜕𝑥 𝜈
4
where 𝑥0 = 𝑡, 𝑥1 = 𝑥, 𝑥2 = 𝑦, 𝑥3 = 𝑧 and 𝜇, 𝜈 = 0, 1,2,3.
Parameters 𝑔
𝜇𝜈
define a symmetric matrix with 10
independent parameters.
• (4) is called the general form of the d’Alembert operator.
With this operator, the Klein-Gordon equation takes its
general form as
𝑔 𝜇𝜈
𝜕
𝜕𝑥 𝜇
𝜕
𝜕𝑥 𝜈
Ψ =
𝑚2 𝑐0
2
ℏ2
Ψ (4a)
6/23/2016 50

Justification of the Generalization
With respect to an inertial frame, when the
parameter c is different from the standard c0 , and
the d’Alembert operator is defined as
□ = −
1
𝑐2
𝜕
𝜕𝑡
2
+
𝑐
𝑐0
2
𝜕
𝜕𝑥
2
+
𝜕
𝜕𝑦
2
+
𝜕
𝜕𝑧
2
then this d’Alembert operator is no longer invariant
under the Lorentz transformation. Therefore, to
another relative moving frame, this operator has to
take the following form
□ = 𝑔 𝜇𝜈
𝜕/𝜕𝑥 𝜇
𝜕/𝜕𝑥 𝜈
6/23/2016 51

Mysteries in Einstein’s Geodesic Equation
Einstein’s geodesic equation is a key equation in general
relativity, defining the motion of a spin-less point mass. It says
that the mass takes the longest proper time between two
events, where the proper time interval is defined as
Δ𝜏 =
𝑃
𝑔 𝜇𝜈 𝑑𝑥 𝜇 𝑑𝑥 𝜈
Where P is the path, and 𝑔 𝜇𝜈 is the metric tensor defining the
geometry of spacetime.
However, there are two mysteries in the Einstein’s formation:
1. How could spacetime be curved based on the metric tensor
𝑔 𝜇𝜈?
2. How could a point particle magically know the geometry of
spacetime defined by the metric tensor 𝑔 𝜇𝜈 and figure out a
path of the longest proper time?
6/23/2016 52

Solving the Mysteries of Geodesic Equation
It can be proved in mathematics that the classical
limit of the general Klein-Gordon equation (4a) for a
spinless point mass falls back to Einstein’s geodesic
equation. Therefore,
1. the curved spacetime is a manifestation of the
universal particle-wave motion equation when it
takes its general form (4a). Its wave propagation
metric 𝑔
𝜇𝜈
determines the distance metric 𝑔 𝜇𝜈 as
the inverse of 𝑔
𝜇𝜈
.
2. Taking the longest proper time for a point mass is
simply a mathematical property of the universal
motion equation at the classical limit.
6/23/2016 53

COMPARISON WITH GENERAL
RELATIVITY
In Search for a Right Interpretation of Gravity
6/23/2016 54
Both of them make almost the same predictions in gravitational
time dilation, light bending, the extra precession of the
perihelion of Mercury, and the propagation of gravitational
waves. They are approximations of each other in mathematics.
Both theories have the Newtonian limit. Which interpretation
brings us closer to the truth?

General Relativity vs the New Theory (I)
• In general relativity, Einstein described gravity as the
spacetime geometry. Space and time owns the
characteristics of an object. It can be bent, curved, and
twisted just like a piece of Jell-O.
• In the new theory, space and time have been restored
with their classical meaning. A universal particle wave
motion equation with variable propagation parameters
manifests itself purely in our perception as that
spacetime is curved. It is akin to looking through a
piece of un-even glass with a distorted view of the
world. It appears to us that the space in this view is
curved. However, this is just our perception, but not
reality.
6/23/2016 55

General Relativity vs the New Theory (II)
• The metric tensor 𝑔 𝜇𝜈 is the central object of general
relativity. It defines the geometry of spacetime as the
source of gravity. It is determined by the matter and
energy content of spacetime.
• In the new theory, the propagation metric 𝑔 𝜇𝜈
is the
central object. It manifests itself as a spacetime
geometry 𝑔 𝜇𝜈 through a universal quantum motion
equation at the classical limit. Specifically, it
determines the line element of spacetime as
𝑑𝑠2
= 𝑔 𝜇𝜈 𝑑𝑥 𝜇
𝑑𝑥 𝜈
, where 𝑔 𝜇𝜈 is the inverse of 𝑔 𝜇𝜈
.
𝑔 𝜇𝜈 is also determined by the matter and energy
content of spacetime.
6/23/2016 56

General Relativity vs the New Theory (III)
• The general Klein-Gordon equation as a motion equation is
more general and more fundamental than Einstein’s
geodesic equation. The later is only a special case of the
former. It is the classical limit of the former for a free, spin-
less, point mass. If the mass is of any size and shape or of any
rotation, the equation may fail to work.
• The general Klein-Gordon equation works at the most
fundamental level of nature. At that level, all particles are
simply waves. The solutions for classical objects of any shape
and spinning can be derived from it, including Einstein’s
geodesic equation for a spinless point mass. Most
importantly, it shows that gravity can be treated as a force,
just like electromagnetic force, with the same footing.
6/23/2016 57

UNIFICATION OF GRAVITY WITH
ELECTROMAGNETISM
Comments of
6/23/2016 58

Unifying Gravity with EM
• Both are based on the same universal motion equation
for particle waves, i.e., the Klein-Gordon equation:
□𝜓 = (𝑚2 𝑐0
2
/ ℏ2)𝜓
• Electromagnetism is generated by coupling a particle
wave with an electromagnetic wave 𝜓 𝐸𝑀 as 𝜓 →
𝜓𝜓 𝐸𝑀 and the coupled wave 𝜓𝜓 𝐸𝑀 still satisfies the
Klein-Gordon equation. It has a perfect implementation
of the inverse-square law.
• Gravity is generated by changing the common particle
speed limit. It implements the inverse-square law only
in approximation. Therefore, it causes the extra-
precession of the perihelion of planets.
6/23/2016 59

Unifying Gravity with EM (II)
• Both gravitational fields and electromagnetic fields
are generated through the same mechanism.
• Both are described by 4-vector potentials.
• The source of EM fields are the electric current J. The
EM 4-vector potentials 𝐴 are generated as
□𝐴 = 𝜇0 𝐽
• One of the sources of gravitational fields are matter
current 𝐽 𝑚 . The gravitational 4-vector potential 𝐴 𝐺
is generated as
□𝐴 𝐺 = 4𝜋𝐺𝐽 𝑚
6/23/2016 60

Summary on the Unification
• The motion of particle waves in a gravitational field and an
electromagnetic field are based on the same quantum
motion equation, the generalized Klein-Gordon equation.
• Both electromagnetic waves and gravitational waves are
based on the same field equation, where the equation for
the latter is just the reformulation of Maxwell equations.
• Both the wave motion equation and the field equation share
exactly the same wave propagation pattern, governed by a
mathematical operator, called the d’Alembert operator.
• Gravity and electromagnetism are thus unified under the
theoretical framework of quantum theory.
6/23/2016 61

FINING TUNING THE FIELD
EQUATION
How are gravitational fields generated?
6/23/2016 62

A Key Hypothesis
• There always exists a local frame with a certain velocity
such that the general d’Alembert Operator (4) has a
diagonalized form as
□ = −𝑔00(𝜙)
𝜕2
𝜕𝑡2
+ 𝑔11(𝜙)
𝜕
𝜕𝑥
2
+
𝜕
𝜕𝑦
2
+
𝜕
𝜕𝑧
2
where 𝑔00
𝜙 defines a fixed relation between the
gravitational potential 𝜙 and the parameter 𝑔00. So is the
relation between 𝑔11 and 𝜙, defined by 𝑔11 𝜙 . In this case,
the gravitational four potential 𝐴 𝐺 has the form of
𝜙 𝐺, 0, 0, 0 .
The above operator is isotropic. That is, the speed limits it
defines are the same in all directions in space.
6/23/2016 63

𝑔00
𝜙 and 𝑔11
𝜙 should be
determined by observations
• If 𝑔 = −
1
2
𝑐∞ 𝛻𝑐∞ and the gravitational time dilation for a point mass M is
1 −
GM
𝑟𝑐0
2, then
𝑔00 𝜙 = 𝑐 1 +
𝜙
𝑐0
2
−2
, 𝑔11 𝜙 = 1 +
𝜙
𝑐0
2
2
• If the gravitational acceleration is 𝑔 = −𝑐∞ 𝛻𝑐∞ instead, then
𝑔00 𝜙 = 𝑐 1 +
𝜙
𝑐0
2
−2
, 𝑔11 𝜙 = 1
• In particular, 𝑔00
𝜙 and 𝑔11
𝜙 can be fine tuned such that the
Schwarzschild metric for a point mass is its solution.
• If the time dilation is 1 +
GM
𝑟𝑐0
2
−1
, then there is no singularity in black
holes in the universe. The existing theories on black holes have to be
reconciled.
6/23/2016 64

Gravitational Field Equation in General
• Based on the key hypothesis, given any gravitational
potential 𝐴 𝐺 and the propagation metric 𝑔 𝜇𝜈 , we can always
find a moving frame such that 𝐴 𝐺 → 𝜙 𝐺, 0, 0, 0 and
[𝑔 𝜇𝜈
] → 𝑑𝑖𝑎𝑔 −𝑔00
(𝜙 𝐺), 𝑔11
(𝜙 𝐺), 𝑔11
(𝜙 𝐺), 𝑔11
(𝜙 𝐺)
• Let L be the Lorentz transformation from the current frame
to the moving frame. It is a function of 𝐴 𝐺. Then we have
[𝑔 𝜇𝜈] = 𝐿(𝐴 𝐺)𝑑𝑖𝑎𝑔 −𝑔00(𝜙 𝐺), 𝑔11(𝜙 𝐺), 𝑔11(𝜙 𝐺), 𝑔11(𝜙 𝐺) 𝐿(𝐴 𝐺) 𝑇
• The above equation indicates that 𝑔 𝜇𝜈
is a function of 𝐴 𝐺,
denoted as 𝑔 𝜇𝜈
𝐴 𝐺 . With this notation, we have the
gravitational field equation (2) as
𝑔 𝜇𝜈
𝐴 𝐺 𝜕𝜇 𝜕𝜈 𝐴 𝐺 = 4𝜋𝐺𝐽 𝑚
6/23/2016 65

The Generality of 𝑔 = −
1
2
𝑐∞ 𝛻𝑐∞
As mentioned before, there always exists a local frame
such that the general Klein-Gordon equation (4a) falls
back to its normal form (3a). Therefore, 𝑔 =
−
1
2
𝑐∞ 𝛻𝑐∞ is also universal because we can always find
a local moving inertial frame such that a remote
gravitational field is stationary with respect to the
moving frame. Transforming the trajectory of a particle
discovered in the moving frame back to the one in the
original frame is simply and straightforward in
mathematics. It can be done by applying the Lorentz
transformation.
6/23/2016 66

Energy-Momentum Relation
What is the energy-momentum relation
when the common particle speed limit is a
variable?
6/23/2016 67

Energy-momentum relation
• In relativity theory, the most important equation is the
energy-momentum relation:
1
𝑐0
2 𝐸2
− 𝑝2
= 𝑚𝑐0
2
where E is the total energy of the particle and p is its
momentum.
• Corresponding to the general Klein-Gordon equation
(4a), the energy-momentum relation for the case of a
variable common particle speed limit is
1
𝑐2 𝐸2 −
𝑐2
𝑐0
2 𝑝2 = 𝑚𝑐0
2
6/23/2016 68

The Rest Energy of a Particle
• According to the new energy-momentum
relation, the rest energy 𝐸0 of a particle
is 𝐸0 = 𝑚𝑐𝑐0, not 𝐸0 = 𝑚𝑐0
2
. The latter is
true only when 𝑐 = 𝑐0.
• The rest energy 𝐸0 defines the gravitational
potential energy for a particle. A particle tends
to move to a place of a lower c. The difference
of the potential energy is
Δ𝐸0 = 𝑚Δ𝑐𝑐0
6/23/2016 69

How to Test the Theory
In addition to 𝑔 = −
1
2
𝑐∞ 𝛻𝑐∞ , the theory makes the
following predictions for a quasi-static field:
– Gravitational waves have spin 1, instead of 2.
– The Schwarzschild radius may not have the form of
rs = 2GM/c2. So, the predication of light bending near
black holes is different from GR.
– The time dilation could be 1-GM/rc2 , instead of
1 − 2𝐺𝑀/𝑟𝑐2 as predicted by GR. When 𝑟 → 0, the
difference of the two is more significant.
6/23/2016 70

How to Test the Theory (II)
At the microscopic scale, the motion equation for a particle is
iℏ
𝜕𝜓
𝜕𝑡
= −
ℏ2
2𝑚
𝛻2
𝜓 + 𝑚𝑐0 𝑐 − 𝑐0 𝜓
It can be verified using the interference patterns of matter waves in the double-
slit experiment, either moving horizontally or vertically under the influence of the
earth’s gravity.
When a spin-1/2 particle is under EM and gravity, our theory predicts that its
wavefunction satisfies the following equation
iℏ
𝜕𝜓
𝜕𝑡
=
1
2𝑚
−𝑖ℏ
𝜕
𝜕𝑟
−
𝑞
𝑐0
𝐴
2
𝜓 − 𝜇𝜎 ⋅ 𝐵𝜓 + 𝑞𝜙 𝑒 𝜓 + 𝑚𝜙𝜓
where (𝜙 𝑒, 𝐴) is the four-vector potential of EM, 𝜓 is a 2-component spinor, the
three components of 𝜎 are the Pauli matrices, 𝐵 is the magnetic field, and 𝜇 =
𝑞ℏ/2𝑚𝑐0 is the Bohr magneton. It is simply the Pauli equation plus the extra-
term 𝑚𝜙𝜓 at the right hand side of the equation, representing the influence of
gravity caused by variations of speed of light.
6/23/2016 71

Summary
• The new theory builds on the solid foundation of quantum
mechanics, solving the incompatibility issue of general
relativity with quantum theory.
• The new theory unifies gravity with electromagnetism. Both
gravitational waves and electromagnetic waves are governed
by the same wave propagation equation, replacing Einstein’s
extremely complex field equation for gravitational waves.
• While general relativity requires that spin-2 particles carry
gravitational force, the new theory requires that spin-1
particles carry gravitational force instead. The latter can be
worked out into renormalizable quantum field theories, but
not the former.
6/23/2016 72

WHAT IS GRAVITY REALLY?!
Gravity is a manifestation of a universal quantum motion
equation
𝑔 𝜇𝜈
𝜕
𝜕𝑥 𝜇
𝜕
𝜕𝑥 𝜈
Ψ =
𝑚2 𝑐0
2
ℏ2
Ψ
Here, the wave propagation metric 𝑔 𝜇𝜈 defines the
gravitational field.
When there is no gravity, the equation falls back to its
standard form
−
1
𝑐0
2
𝜕2
𝜕𝑡2
+
𝜕2
𝜕𝒓2
Ψ =
𝑚2 𝑐0
2
ℏ2
Ψ
(Relativity is simply a symmetry of the equation)
6/23/2016 73

A unification of gravity with electromagnetism and quantum

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