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𝑔𝜇𝜈 𝜕𝜕 𝑥𝜇

𝜕𝜕 𝑥𝜈

Ψ=𝑚2𝑐02

ℏ2Ψ

𝑔=− 12 𝑐∞𝛻𝑐∞

𝑑𝑠2=𝑔𝜇𝜈𝑑𝑥𝜇𝑑𝑥𝜈

𝑑2𝑥𝜇

𝑑 𝑠2=− Γ𝛼𝛽

𝜇 𝑑𝑥𝛼

𝑑𝑠𝑑𝑥 𝛽

𝑑𝑠

𝐸0=𝑚𝑐𝑐0

WHAT IS GRAVITY REALLY?

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Unification of Gravity with Electromagnetism

Solving the Mystery of GravityXiaofei Huang

Ph.D., Tsinghua University, Beijing ChinaHuangZeng@yahoo.com

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What is Gravity?

• Without knowing what causes gravity, human beings may never truly understand how the universe operates.

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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?

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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.

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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.

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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.

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PARTICLES AS WAVES

Preliminary Knowledge on

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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.

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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.

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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.

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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.

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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.

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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.

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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

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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.

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GRAVITY AS PARTICLE WAVE REFRACTION

From wave refraction to gravity

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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.

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Examples of Wave Refraction

Step changes (slower speeds) Progressive changes (slower speeds)

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The Falling Rate d2x/dt2 is -g

• Assume that the wave speed changes only along the vertical x-axis as , 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.

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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

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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 where is the vertical distance away from the surface of the earth. This is why all objects fall to the ground with the rate –g.

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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.

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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.

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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.

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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.

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A FORMULATION OF GRAVITY WITHOUT DIFFERENTIAL GEOMETRY

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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.

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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.

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GRAVITY AS PARTICLE WAVE REFRACTION

A Mathematical formulation of

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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:

where is the d’Alembert operator defined as

It is said here that both the operator and the equation take their standard forms.

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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

• If we define , where is the standard speed of light, when , the above equation falls back to the classic equation . therefore represents the gravitational potential.

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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

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

where is the gravitational four potential and is the four matter current, then we can restore Newtonian gravity.

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Restoring Newton’s Universal Theory of Gravity• When masses that generate a gravitational

field are motionless, the gravitational field equation (2) becomes

• This is the Poisson equation for gravity. Together with , Newtonian gravity is completely restored.

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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.

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GRAVITY AS WAVE REFRACTIONFurther 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.

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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

This is called the normal form the operator. In this case, the Klein-Gordon equation takes its normal form as

• The common particle speed limit in this case is , which is also the speed of light.

• When , Eq. (3) falls back to its standard form (1).

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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

• This equation says that g is equal to the half of the negative gradient of multiplied by itself, independent of the mass of the particle.

• As long as we know the distribution of the speed of light , we can find out the gravitational acceleration.

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is a Universal Result

This equation holds true for any stationary gravitational fields, regardless of the cause of variations in the speed of light . 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.

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Verifying 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 .

• Compute the distribution of the speed of light around the galaxy using the equation .

• Compute the gravitational lensing effects using .• Using the existing observation to check if the computed

gravitational lensing effects match the observed ones.

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or ?

• The normal Klein-Gordon equation (3a) can be generalized to

where the real-valued parameter controls the space contraction rate. – When , then , where . In this case, the space is curved

and Einstein is right about gravitational light bending.– When , , where . In this case, the space is not curved.

Newton is then correct.• should be fine tuned based on observations.

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Importance of the Verification

• If the existing data on gravitational lensing and the acceleration of stars supports the simple relation , 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.

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GRAVITATIONAL TIME DILATION, LIGHT BENDING, AND PERIHELION PRECESSION OF MERCURY

Mathematical Investigation of

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Solution to a Point Mass• When the relation between the parameter c and the

gravitational potential has a definite relation as he solution of the field equation (2) for a static point mass is

• If we rescale the radical r , we can reformulate the metric as

• The Schwarzschild metric of general relativity is its approximation as

• The differences of the two is at the order of , beyond the precision of any existing apparatus to detect.

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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.

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GRAVITY AS FORCE

Mathematical Investigation of

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Why gravity can be treated as a force?

• Let 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

• For a particle with charge q in an electric potential (the magnetic strength is zero), the Klein-Gordon equation can be approximated as

• If we replace m with q and with , 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.

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The Importance of Schrödinger Equation

• The equation , 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.

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GRAVITY AS PARTICLE WAVE REFRACTION

Further Generalization of

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A General Form of the d’Alembert Operator

• When the d’Alembert operator takes the normal form (3) and 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:

where and . Parameters 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

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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

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

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Mysteries in Einstein’s Geodesic EquationEinstein’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

Where P is the path, and 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 ?2. How could a point particle magically know the geometry of

spacetime defined by the metric tensor and figure out a path of the longest proper time?

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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 determines the distance metric as the inverse of .

2. Taking the longest proper time for a point mass is simply a mathematical property of the universal motion equation at the classical limit.

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COMPARISON WITH GENERAL RELATIVITY

In Search for a Right Interpretation of Gravity

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?

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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.

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General Relativity vs the New Theory (II)

• The metric tensor 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 is the central object. It manifests itself as a spacetime geometry through a universal quantum motion equation at the classical limit. Specifically, it determines the line element of spacetime as , where is the inverse of . is also determined by the matter and energy content of spacetime.

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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.

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UNIFICATION OF GRAVITY WITH ELECTROMAGNETISM

Comments of

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Unifying Gravity with EM

• Both are based on the same universal motion equation for particle waves, i.e., the Klein-Gordon equation:

• Electromagnetism is generated by coupling a particle wave with an electromagnetic wave as and the coupled wave

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.

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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 are generated as

• One of the sources of gravitational fields are matter current . The gravitational 4-vector potential is generated as

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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.

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FINING TUNING THE FIELD EQUATION

How are gravitational fields generated?

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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

where defines a fixed relation between the gravitational potential and the parameter . So is the relation between and , defined by In this case, the gravitational four potential has the form of .

The above operator is isotropic. That is, the speed limits it defines are the same in all directions in space.

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and should be determined by observations

• If and the gravitational time dilation for a point mass M is , then

• If the gravitational acceleration is instead, then

• In particular, and can be fine tuned such that the Schwarzschild metric for a point mass is its solution.

• If the time dilation is , then there is no singularity in black holes in the universe. The existing theories on black holes have to be reconciled.

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Gravitational Field Equation in General• Based on the key hypothesis, given any gravitational

potential and the propagation metric , we can always find a moving frame such that and

• Let L be the Lorentz transformation from the current frame to the moving frame. It is a function of Then we have

• The above equation indicates that is a function of , denoted as . With this notation, we have the gravitational field equation (2) as

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The Generality of

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, 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.

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Energy-Momentum Relation

What is the energy-momentum relation when the common particle speed limit is a

variable?

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Energy-momentum relation

• In relativity theory, the most important equation is the energy-momentum relation:

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

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The Rest Energy of a Particle

• According to the new energy-momentum relation, the rest energy of a particle is, not . The latter is true only when .

• The rest energy 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

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How to Test the TheoryIn addition to , 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 as predicted by GR. When , the difference of the two is more significant.

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How to Test the Theory (II)At the microscopic scale, the motion equation for a particle is

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

where is the four-vector potential of EM, is a 2-component spinor, the three components of are the Pauli matrices, is the magnetic field, and is the Bohr magneton. It is simply the Pauli equation plus the extra-term at the right hand side of the equation, representing the influence of gravity caused by variations of speed of light.

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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.

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WHAT IS GRAVITY REALLY?!

Gravity is a manifestation of a universal quantum motion equation

Here, the wave propagation metric defines the gravitational field. When there is no gravity, the equation falls back to its standard form

(Relativity is simply a symmetry of the equation)