Xiaofei Huang, Ph.D., Foster City, CA 94404, HuangZeng@yahoo.com

There are lots of books, videos, and slide shows on relativity with stories of spaceships, trains, lasers, and moving light clocks and light rulers. However, people are still confused by the theory generation after generation. After reading a whole stack of those books, many of them don’t understand why light can propagate in pure vacuum without any medium, why the speed of light is magically constant, why a moving clock is slower and a moving ruler is shorter, and most importantly, how could the laws of physics remain the same for all uniformly moving frames.

Relativity says that light can travel in vacuum without any medium. However, this statement is just the emperor’s new clothes in the theory. It mystifies the propagation of light. It is also bluntly against our intuition. There is no way for any wave to propagate in space without any medium.

Permittivity and permeability are not the properties of space. It makes no sense to assign those physical properties to space. Rather, they are the properties of the medium for propagating electromagnetic waves. All waves need a medium to propagate in reality, including light.

In relativity, it has often been argued that light travels at a constant speed because it propagates in vacuum as waves without the need of any medium. It has been treated as a genius idea at solving the puzzle of the constancy of the speed of light. However, such a reasoning is also the emperor’s new clothes in relativity.

This statement mystifies the constancy of the speed of light, making it unfathomable. It sounds more like a magic than a logic statement. Also, it is logically incorrect to postulate a statement, such as the constancy of the speed of light, to prove the statement itself in a circular way.

What we need here is a true mechanical explanation for the constancy of the speed of light (if it is true in reality) without mystifying it. You don’t understand it

because you are dummies.

There is a critical logic error in relativity theory. To be more specific, the constant speed of light should not be treated as the second key postulate of relativity. The fact is that relativity holds true regardless of whether or not the speed is constant. This postulate is simply redundant for the theory. It is the biggest misconception in the theory, confusing almost everyone including Einstein.

The laws of physics remain the same for all uniformly moving frames.

There are Two Key Postulates in Relativity

Postulate 1: The Principle of Relativity

As measured in any inertial frame of reference, light is always propagated in empty space with a definite velocity c that is independent of the state of motion of the emitting body.

Postulate 2: A constant speed of light

If a particle travels at the common maximal speed limit in one inertial frame, it must travels at the same speed in all inertial frames. That is, its speed is constant in this case.

Two Important Logical Consequences

From the First Postulate

Consequence #2: The Constant Speed

All free objects must share exactly the same maximal speed limit. Otherwise, the principle of relativity is violated.

Consequence #1: A Common Maximum Speed Limit

The Second Postulate is Redundant We can see that the constancy of the speed of an object is a

logical consequence of the first postulate. There is no need to postulate the constancy of the speed of light. Using it as the second postulate to explain relativistic effects such as time dilation, length contraction, and loss of simultaneity could be confusing to the general public. Often times, people mistakenly thought it is the cause of those relativistic effects. However, the truth is that we have those relativistic effects as long as the common maximal speed limit is of a finite value. In other words, we suffer from time dilation and length contraction as long as the common maximal speed limit is of a finite value, regardless of whether or not the speed of light is constant.

The laws of physics remain the same for all uniformly moving frames.

We Only Need the First Postulate

Postulate 1: The Principle of Relativity

As measured in any inertial frame of reference, light is always propagated in empty space with a definite velocity c that is independent of the state of motion of the emitting body.

Postulate 2: A constant speed of light

Constant Speed of Light as Biggest Misconception

The constancy of the speed of light is not only a redundant postulate, but also an inaccurate statement about reality.

In the real world, the photon never moves like a point object with a definite position at any given time instance. Instead, it moves like a wave without a definite position at any given time governed by quantum laws. The light clocks and rulers used in the classical relativity thought experiments never exist in reality. They only exist in our imaginations. Simply put, those thought experiments are wrong in terms of physical reality.

With the same argument, the classical motion equation for light, 𝑐2𝑡2 − 𝑥2 = 0, where the photon is treated as a classical point object, is also incorrect. It should not be used to derive the Lorentz transformation, a key transformation in relativity.

Constant Speed of Light Should be Hypothesis

The constant speed of light should be a hypothesis subject to experimental verification. Up to now, nobody is sure about the constancy of the speed of light. Neither the Maxwell’s equations, nor the Michelson-Morley experiment can tell us the constancy of the speed of light because they are not exactly accurate. Recently, it has been found that spatially structured photons in free space travel slower than the speed of light.

Vacuum isn’t really empty, but is filled with virtual particles. A virtual particle pair can pop out of the vacuum for a very short time interval then disappear. Light will interact with those virtual particles. At present, nobody knows if the interaction with virtual particles in vacuum will slow down light from its maximum speed limit.

Spacetime Metric should not be Treated as

Fundamental Concept in Relativity

It has been used as the fundamental concept both in special relativity and general relativity. However, the concept is valid only at the classical limit. It works only for computing the trajectory of a classical point-like object. It doesn’t work for quantum particles which behave more like waves than point-like objects.

Reconciling Relativity with Quantum Mechanics

Special relativity has been established around 1905 and general relativity is around 1915. Both of them are more than a decade ahead of the establishment of quantum theory. All the concepts used in relativity are classical ones. For example, objects have definite position and velocity at any time instance. Each object has a definite trajectory in spacetime. However, those pictures are inaccurate, often times misleading in the quantum world. You can not simply treat an atomic system as a solar system because each electron of the atom doesn’t have a definite orbit. Rather, it has a cloud of probability. It is desirable to reconcile relativity with quantum theory.

The fundamental reason for relativity remaining as a mystery is because the mechanical explanation for it is missing so far. Without it, relativity remains as a mystery and a magic of nature, hard to be understood by human brains. It is time to offer a mechanical explanation for relativity based on quantum theory.

Common Speed Limit

Based on the standard model, the most successful theory of particle physics, if the Higgs field did not exist, all elementary particles would travel at the speed of light.

Some of elementary particles interact with the Higgs field to have rest mass and slow down to any speed below the speed of light. The ones which do not interact with the Higgs field will always travel at the speed of light.

In quantum field theory, elementary particles are excited states of the underlying physical field, so called field quanta. For example, electron field generates electrons and electromagnetic field generates photons.

A Fundamental Field For All

As mentioned before, to have the principle of relativity, all particles must share exactly the same maximal speed limit at any position of space and at any time. If there is any slightest difference in the maximal speed limit for different particles, the principle of relativity is violated.

To resolve the mystery, the author postulate that all elementary particle fields are different manifestations of a more fundamental field, traditionally called the ether. Different particles correspond to different excited states of this fundamental field, so that they have different spins, charges, colors, and masses.

A Fundamental Postulate

for Relativity and Gravity

at the most fundamental level of nature, everything is made of particle waves and field waves. All of those waves are different excited states of a fundamental medium, traditionally called the ether. Those waves, either free ones or coupled ones, are propagating in space with the same pattern defined by a quadratic partial differential operator 𝑔𝜇𝜈𝜕𝜇𝜕𝜈.

It is postulated by the author that

The Definition of 𝒈𝝁𝝂𝝏𝝁𝝏𝝂

In relativity, 𝑔𝜇𝜈𝜕𝜇𝜕𝜈 is a shorthand notation for

𝜇=0

3

𝜈=0

3

𝑔𝜇𝜈𝜕𝜇𝜕𝜈

where 𝜕𝜇 is a shorthand notation for 𝜕/𝜕𝑥𝜇. As a

convention, 𝑥0 = 𝑡, 𝑥1 = 𝑥, 𝑥2 = 𝑦, and 𝑥3 = 𝑧. 𝑡 is the time coordinate, 𝑥, 𝑦, 𝑧 are spatial coordinates.

Wave Propagation in Gravity-Free VacuumWhen there is no gravity, it has been found in quantum mechanics that the propagation of particle waves and field waves are governed by the following operator:

−1

𝑐2𝜕𝑡2 + 𝜕𝑥

2 + 𝜕𝑦2 + 𝜕𝑧

2

This operator is often times denoted as □, called the d’Alembert operator.In particular, for any free particle, either a boson or a fermion, it satisfies the Klein-Gordon equation as

□𝜓 𝑥, 𝑦, 𝑧, 𝑡 =𝑚2𝑐2

ℏ2𝜓 𝑥, 𝑦, 𝑧, 𝑡

Here, 𝜓 𝑥, 𝑦, 𝑧, 𝑡 is the wave function of the particle describing the state of the particle, 𝑚 is the mass of the particle, 𝑐 is a constant defining the maximum speed limit for all particles, and ℏ is the reduced Planck constant, one of the most important constant in physics.

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The electric field E and the magnetic field B are uniquely defined by the electromagnetic four potential field A.

A Case Study: Electromagnetic WavesElectromagnetic waves can be imagined as a self-propagating oscillating wave of electric and magnetic fields. In general, an electromagnetic field can be represented by a 4 component vector field, called an electromagnetic four-potential field, denoted as A(x, y, z, t). When there is no gravity in a vacuum space, it satisfies the following wave function:

−1

𝑐2𝜕𝑡2 + 𝜕𝑥

2 + 𝜕𝑦2 + 𝜕𝑧

2 𝐴 𝑥, 𝑦, 𝑧, 𝑡 = 0

That is, the propagation of the electromagnetic four-potential field is governed by the d’Alembert operator

−1

𝑐2𝜕𝑡2 + 𝜕𝑥

2 + 𝜕𝑦2 + 𝜕𝑧

2

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Einstein’s Geodesic Equation as the Classical Limit

Using the universal wave propagation operator 𝑔𝜇𝜈𝜕𝜇𝜕𝜈, the Klein-Gordon equation can be simply generalized to

𝑔𝜇𝜈𝜕𝜇𝜕𝜈𝜓 𝑥, 𝑦, 𝑧, 𝑡 =𝑚2𝑐2

ℏ2𝜓 𝑥, 𝑦, 𝑧, 𝑡

It has been proven by the author that the above generalized wave equation falls back to Einstein’s geodesic equation in general relativity at the classical limit. Einstein’s geodesic equation is used to compute the trajectory of a point-like object in a curved spacetime with its metric as 𝑔𝜇𝜈.

The wave equation at the top is more general than Einstein’s geodesic equation because the latter is only the classical limit of the former.

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Essentials of Quantum Waves

the universal wave propagation operator 𝒈𝝁𝝂𝝏𝝁𝝏𝝂manifests itself as the spacetime metric 𝒈𝝁𝝂𝒅𝒙

𝝁𝒅𝒙𝝂 at the classical limit, where the

covariant metric tensor 𝒈𝝁𝝂 is the inverse of the

contravariant metric tensor 𝒈𝝁𝝂. Simply put, the geometry of spacetime is a result of the propagation of quantum particle waves in space and time.

Based on the previous investigation, we can conclude that

As a summary, we have

the universal wave propagation parameters 𝒈𝝁𝝂(𝝁, 𝝂 = 𝟎, 𝟏, 𝟐, 𝟑)

define the geometry of spacetime with the spacetime metric as 𝒈𝜇𝜈.

Quantum Wave Propagation vs Spacetime Metric

The spacetime metric 𝒈𝝁𝝂𝒅𝒙𝝁𝒅𝒙𝝂 is a fundamental

concept in both special and general relativity. From the previous investigation we can see that it is only the classical limit of the universal wave propagation operator 𝒈𝝁𝝂𝝏𝝁𝝏𝝂.

At the most fundamental level of nature, everything are just waves. Therefore, the wave propagation operator is more fundamental than the spacetime metric. The latter belongs to classical physics, valid only when every object can be treated as a point object with a definite position and velocity at any given time.

The universal wave propagation operator 𝒈𝝁𝝂𝝏𝝁𝝏𝝂 is more fundamental

than the spacetime metric 𝒈𝝁𝝂𝒅𝒙𝝁𝒅𝒙𝝂

at understanding relativity .

Einstein suggested that gravitational potential is defined by the spacetime metric 𝑔𝜇𝜈. Whenever

there are variations of 𝑔𝜇𝜈, it causes gravitational

acceleration.

The variations of 𝑔𝜇𝜈 are caused by the presence

of matter and energy and others defined by Einstein’s field equation in general relativity.

Gravitational potential is defined by the universal wave propagation parameters 𝒈𝝁𝝂, instead of the spacetime metric𝒈𝝁𝝂 as

suggested by Einstein.

Since 𝑔𝜇𝜈 is more fundamental than 𝑔𝜇𝜈,

the author hypothesizes that

Demystifying Relativity-The EssenceAt any point of spacetime, the propagation operator 𝑔𝜇𝜈𝜕𝜇𝜕𝜈can be normalized to the standard form as

− Τ1 𝑐2𝜕𝑡2 + 𝜕𝑥

2 + 𝜕𝑦2 + 𝜕𝑧

2

and remains the same after the Lorentz transformation. That is exactly the reason why the laws of physics remain the same regardless of gravity, and the relative motion of an observer.

Mechanical Explanation for Relativity

In particular, to any observer at his local space, every hydrogen atom, water molecule, protein, or any other atoms or molecules remain the same in their structures and properties regardless of the gravity, and motion of the observer. Otherwise, life is impossible.

we have relativity simply because the universal wave propagation operator 𝒈𝝁𝝂𝝏𝝁𝝏𝝂 is normalizable to

− Τ𝟏 𝒄𝟐 𝝏𝒕𝟐 + 𝝏𝒙

𝟐 + 𝝏𝒚𝟐 + 𝝏𝒛

𝟐 and remains

invariant under the Lorentz transformation.

The author concludes that

there is no way for laws of physics to remain the same regardless of gravity and motion if

The universal wave propagation operator is not of the quadratic form 𝒈𝝁𝝂𝝏𝝁𝝏𝝂, or

the propagation operator 𝒈𝝁𝝂𝝏𝝁𝝏𝝂 is not

shared by all particles.

It is an elegant design of nature because

The second condition is the reason for the author to postulate that different elementary particles are propagating in the same medium as different excited states.

𝒈𝝁𝝂𝝏𝝁𝝏𝝂=Relativity, Gravity

Simply put, we have

Important Property of

Universal Wave Propagation Operator

There is a very important property for the universal wave propagation operator in its standard form. It is invariant in form under the Lorentz transformation shown as follows

−1

𝑐2𝜕𝑡2 + 𝜕𝑥

2 + 𝜕𝑦2 + 𝜕𝑧

2

𝑥′ = 𝛾 𝑥 − 𝑣𝑡

𝑦′ = 𝑦

𝑧′ = 𝑧

𝑡′ = 𝛾(𝑡 −𝑣

𝑐2𝑥)

Lorentz Transformation

𝛾 = 1/ 1 −𝑣2

𝑐2

−1

𝑐2𝜕𝑡′2 + 𝜕𝑥′

2 + 𝜕𝑦′2 + 𝜕𝑧′

2

Note that the Lorentz transformation is defined as𝑥′ = 𝛾 𝑥 − 𝑣𝑡

𝑦′ = 𝑦

𝑧′ = 𝑧

𝑡′ = 𝛾(𝑡 −𝑣

𝑐2𝑥)

where 𝛾 = 1/ 1 − 𝑣2/𝑐2, called the Lorentz factor.

The Lorentz transformation defines a linear transformation from the spacetime coordinates 𝑥, 𝑦, 𝑧, 𝑡 of the original frame to the spacetime coordinates 𝑥′, 𝑦′, 𝑧′, 𝑡′ of a new frame. Note that the spatial origin of the new frame is 𝑣𝑡, 0,0 in the original frame. That is, the new frame is moving with a constant velocity 𝑣 with respect to the original one. The two frames are in relative motion with the velocity 𝑣.

the invariance of the universal wave propagation operator in its standard form

−𝟏

𝒄𝟐𝝏𝒕𝟐 + 𝝏𝒙

𝟐 + 𝝏𝒚𝟐 + 𝝏𝒛

𝟐

under the Lorentz transformation is the actual cause for the principle of relativity to hold true in nature. It is a mechanical explanation for why laws of physics remain the same for all uniformly moving frames.

A Mystery Solved!

The author concludes that

The essence of special relativity

The spacetime metric of classical special relativity is −𝑐2𝑑𝑡2 + 𝑑𝑥2 + 𝑑𝑦2 + 𝑑𝑧2

It is a manifestation at the classical limit of the universal wave propagation operator in its standard form

−1

c2∂t2 + ∂x

2 + ∂y2 + ∂z

2

Therefore, the essence of special relativity is the invariance of the universal wave propagation operator in its standard form, not the symmetry of spacetime. The wave propagation is more general andfundamental than the spacetime metric.

The Essence of Special Relativity

The true reason to have the Lorentz transformation is the invariance of the universal wave propagation operator under this transformation such that the relativity principle holds true.

If we use a computer program to simulate any physical process using the universal wave propagation operator at the lowest level, and to simulate the observations of the process in different frames, we always find that the spacetime coordinates of different frames satisfy the Lorentz transformation.

There are many ways to derive the Lorentz transformation from other conditions, such as the constant speed of light (+ homogeneity of space and time + isotropy of space + relativity principle), or the invariance of the space time interval

Δs2 = c2Δt2 − Δx2 − Δy2 −Δz2

However, those are purely mathematical ways that build up logical connections and equivalence among different mathematical statements and equations, not causality in physics. In mathematics, it is also perfectly fine to say that we have the constancy of the speed of light because spacetime satisfies the Lorentz transformation.

Mathematical Equivalence ≠ Physical Causality

The Lorentz transformation is not caused by the constancy of the speed of light. This is an example of the right answer but for the wrong reason in physics. It simply doesn’t matter whether or not the speed of light is constant to have the Lorentz transformation.

The inverse Lorentz transformation can be obtained simply by exchanging the spacetime coordinates as 𝑥 ↔ 𝑥′, 𝑦 ↔ 𝑦′, 𝑧 ↔ 𝑧′, 𝑡 ↔ 𝑡′, and replacing the velocity 𝑣 by −𝑣 because the two frames are moving in an opposite direction relative to each other. Specifically, we have

𝑥 = 𝛾 𝑥′ + 𝑣𝑡′

𝑦 = 𝑦′

𝑧 = 𝑧′

𝑡 = 𝛾(𝑡′ +𝑣

𝑐2𝑥′)

As postulated here, all particles are excited wave packets of the same medium. In reality, no matter how stiff the medium is, the maximal speed limit must be finite. That is the exact reason why light has a speed limit. So are other particles.

This can also be explained using the Klein-Gordon equation mentioned before. Because all particles satisfy the equation as

□𝜓 𝑥, 𝑦, 𝑧, 𝑡 =𝑚2𝑐2

ℏ2𝜓 𝑥, 𝑦, 𝑧, 𝑡

Here, 𝑚 is the mass of the particle, 𝑐 is a constant. When m=0, then the speed of the particle wave is c. When m > 0, from the equation at the classical limit, the speed must be any value less than c, including the value 0. That is the particle can be at rest when it has a mass.

As revealed before, in order to have the laws of physics remain the same, the spacetime coordinates of the moving frame and those of the original one must satisfy the Lorentz transformation as a constraint. This has manifestations on the measurements related to space and time. The examples are relativistic speed addition law, the constant speed phenomenon, time dilation, length contraction, and loss of simultaneity. They are called the relativistic effects.

When time and space are defined by the Lorentz transformation, the classical speed addition law: 𝑢 = 𝑢′ + 𝑣, is no longer valid. Instead, it should be replaced by relativistic speed addition law.

Assume that there is an object with its measured velocity in a moving frame as (𝑢′𝑥 , 𝑢′𝑦, 𝑢′𝑧). Then its velocity in the stationary frame can be discovered using the Lorentz transformation as

(𝑢′𝑥, 𝑢′𝑦 , 𝑢′𝑧)

𝑢𝑥 = (𝑢𝑥′ + 𝑣)/(1 +

𝑣

𝑐2𝑢𝑥′ )

𝑢𝑦 = 𝛾𝑢𝑦′ /(1 +

𝑣

𝑐2𝑢𝑥′ )

𝑢𝑧 = 𝛾𝑢𝑧′ /(1 +

𝑣

𝑐2𝑢𝑧′ )

𝛾 = 1 − 𝑣2/𝑐2

From the relativistic speed addition formula, it is straightforward to prove that if an object is moving at the speed limit c in one frame, then its measured speed in any other frames is always c. That solves the mystery of the constant speed of light. Specifically, if light is traveling at the speed limit, then its speed remains the same in all frames. Otherwise, if light is traveling at a speed less than the speed limit c, then the speed of light can not be constant.

From the formula, we can also see that c is the maximal speed limit for all particles in nature. No matter how long you add up speed for an object by acceleration, c is its maximal speed limit. Its speed can be arbitrarily close to the limit, but never reach the limit.

Whenever we change the state of a clock from stationary to a steady motion, it appears to slow down in ticking rate. If we jump to the frame moving together with the clock, it restores the original ticking rate. This called time dilation.

Time dilation can be derived mathematically from the Lorentz transformation. For a clock stationary in the second frame (x′, y′, z′, t′), let the elapsed time be Δ𝑡′.Since it is stationary in it, we have Δ𝑥′ = 0. From the inverse Lorentz transformation, we have

Δ𝑡 = 𝛾 Δ𝑡′ +𝑣

𝑐2Δ𝑥′ = 𝛾Δ𝑡′

Since 𝛾 > 1 when 𝑣 ≠ 0, we have Δ𝑡 > Δ𝑡′.

Assume there are two identical clocks in relative motion. Then to each clock, the other clock appears to run slower than itself.

Your clock is slower

Your clock is slower

To measure the length of a ruler, either stationary or moving, the key point is to take down the coordinates of the two end points of the ruler at the same time and subtract them to get the measured length.

Your should take down the coordinates of the two end points

of the ruler at the same time!

𝑥1(𝑡) 𝑥2(𝑡)

𝐿𝐴 = 𝑥2 𝑡 − 𝑥1(𝑡)

𝑥3(𝑡′) 𝑥4(𝑡′)

𝐿𝐵 = 𝑥4 𝑡′ − 𝑥3(𝑡′)A

B

Whenever we change the state of a ruler from stationary to a steady motion, it appears to be shortened along the moving direction. If we jump to the frame moving together with the ruler, it restores the original length. This is called length contraction.

Length contraction can also be derived mathematically from the Lorentz transformation. For a ruler stationary in the second frame (x′, y′, z′, t′), let its length along the 𝑥′-axis is Δ𝑥′. Since we measure its length at the same time in the first frame x, y, z, t , we have Δ𝑡 = 0. From the Lorentz transformation, we

have Δ𝑥′ = 𝛾 Δ𝑥 − 𝑣Δ𝑡 = 𝛾Δ𝑥. That leads to Δ𝑥 = Δ𝑥′/𝛾. Since 𝛾 > 1 when 𝑣 ≠ 0, we have Δ𝑥 < Δ𝑥′.

Assume there are two identical rulers in relative motion. Then to each ruler, the other ruler appears to be shorter than itself along the motion direction.

Your ruler is shorter

Your ruler is shorter

If we use a computer program to simulate any clock or ruler using the universal wave propagation operator at the lowest level, we always get time dilation, length contraction, and any other relativistic effects.

Time dilation and length contraction are not caused by the constancy of the speed of light. This is another example of the right answer but for the wrong reason in physics. It simply doesn’t matter whether or not the speed of light is constant to have time dilation and length contraction.

The most important work on relativity theory has been done in history by Galileo, Newton, Michelson, Lorentz, Poincaré, Einstein, and Minkowski. There were also contributions by Voigt, Fizgetald, and many others. This presentation is just an attempt to make it easier for the general public to understand the fascinating theory.