Extension of The Lorentz Transformation: Velocity-Limited Transformation
Feng CHENG
Phone: +861860135**** Email: red_boxer#sohu.com ( # -> @ )
Abstract: In this paper, the
Lorentz transformation is extended to the propagation of sound through air or
water, the propagation of light in a vacuum, and the unlimited velocity of
consciousness. The derivations of all of these transformations are similar to
the derivation of the Lorentz transformation, and all of the examples of these
transformations are simple, common examples in the field of special relativity.
in this study, mathematical formulas are adopted to clarify the reason for
which the special relativity theory is difficult to understand and to illustrate
the unification of the Galilean transformation and the Lorentz transformation. Thus,
we demonstrate that the Lorentz transformation can be extended to meet the need
to conduct observations of high-speed propagation in water and other media in
future applications in science and technology.
Keywords: Lorentz transformation of
sound, Lorentz transformation of light, Lorentz transformation of
consciousness, velocity-limited transformation, measurement
system through listening, observation-based measurement system,
consciousness-based measurement system
1.
velocity-limited
transformation
In a certain situations, two prospectors (A and B) must use a
medium, M, (such as an electromagnetic wave or a sound wave) with a propagation
velocity of VD to detect an event while A is at rest with respect to
the ground and B is undergoing uniform linear motion with a velocity of v with
respect to the ground. The detection must be conducted using only one medium
(such as by listening for a sound wave or by observing an electromagnetic wave,
etc.). Basing on the postulate: the principle of velocity invariance of a
medium, VD remains the same for both prospectors regardless of
whether the object (light source or sound source) that emits M is in motion.
With the relativity principle and a derivation process similar to that of the
Lorentz transformation1-3, an (x, y, z)-coordinate transformation is
easily derived as follows:
t’ = γ(t - vx/VD2)
x’ = γ(x - vt)
y’ = y
z’ = z (1)
where γ = 1/√(1 - (v/VD)2).
This transformation is referred to herein as the “velocity-limited
transformation” and it involves only the magnitude of the velocity and not its
direction, and it appears to be similar to the Lorentz transformation. As in
the derivation of the theorem of velocity addition in the theory of special
relativity4, the velocity addition can be calculated by the
velocity-limited transformation as follow:
u = (v’+ v)/(1 + v’v/VD2) (2)
which will be used below.
Why do we call this transformation a “velocity-limited
transformation”? The time-expansion factor, γ, is a real number; therefore, the
value of the denominator , 1 – (v/VD )2, is
not zero. It is required that 1 – (v/VD )2
> 0 and since v is only a magnitude without a direction, it must be positive
and less than VD (i.e., 0 < v < VD). Thus, it is an
inherent requirement for the velocity-limited transformation that the velocity
of an object does not exceed the propagation velocity of the medium under the
same conditions; in other words, the transformation is velocity-limited.
2.
velocity-limited
transformation of light: Lorentz transformation
The velocity-limited transformation is verified as follows.
When light (a type of electromagnetic wave) is measured (i.e., observed) in a
vacuum or in air and the propagation velocity of light in a vacuum is c (which
is approximately equal to the propagation velocity of light in air), the
Lorentz transformation can be derived by replacing VD with c in the
velocity-limited transformation as defined above formula (1):
t’ = γ(t - vx/c2)
x’ = γ(x - vt)
y’ = y
z’ = z (3)
where γ = 1/√(1 - (v/c)2).
The Lorentz transformation (also called “the velocity-limited
transformation of light” in this paper) has been widely verified experimentally5
but only with light (or electromagnetic waves) as the detection medium. Thus,
the calculation results obtained by the Lorentz transformation are certainly in
good agreement with the measurement results for light. However, the measurement
of light in a vacuum or in air is only one application of the Lorentz
transformation. Yet, the results calculated by the Lorentz transformation are
different from the experimental results in water or other liquids with
refractive indices that are not equal to 1. Therefore, it is necessary to
modify the related velocity-limited transformation such that is consistent with
the experimental results.
3.
velocity-limited
transformation of consciousness: Galilean transformation
To further explain the correctness of the velocity-limited
transformation, the concept of the “velocity of consciousness” is introduced.
The velocity of consciousness does not refer to the velocity of signals through
neurons to process information in the brain but rather to the velocity of
consciousness to simulate or detect the motion of an object in the realm of our
consciousness. The velocity of consciousness is infinite (∞) when simulating or
detecting the motion of an object in the realm of consciousness because our
brains can think of the solar system, the galaxy, or a distance of one billion
light years and even infinitely far within one second. According to the
calculation of velocity, v = s/t = ∞/1= ∞, the velocity of consciousness in
realm of consciousness is infinite; in fact, the velocity of consciousness may
be an arbitrary value(0~∞), but most people will unconsciously choose the
maximum value, infinity. However, the velocity of consciousness cannot be
measured by instruments or equipment of the material world and is only based on
the consensus of human beings.
The velocity-limited transformation can also be applied to
the realm of consciousness: the simulation or imagination in our brains using
our consciousness is analogous to “detection” and is called “the
velocity-limited transformation of consciousness.” It is derived by
substituting ∞ for the velocity of consciousness, VD, into the
velocity-limited transformation formula (1) :
t’ = γ(t - vx/∞2) = γt
x’ = γ(x - vt)
y’ = y
z’ = z (4)
where the time-expansion factor, γ = 1/√(1 - (v/∞)2), is equal
to one (i.e. there is no expansion effect). By substituting γ = 1, the
transformation is derived as follows:
t’ = t
x’ = x - vt
y’ = y
z’ = z (5)
Because, in its un-simplified form,
this transformation is similar to the Lorentz transformation, it is also called
“the Lorentz transformation of consciousness” in this paper. After
simplification, it is equivalent to the Galilean transformation, which means
that the Lorentz transformation of consciousness is the Galilean
transformation; thus, the Galilean transformation is consistent with the common
thinking. The Galilean transformation corresponds to an absolute space-time
view, which is actually a specific case of the relative space-time view when
the velocity of the detection medium is ∞ based on the derivation above.
The relativity theory is considerably difficult to understand
because the velocity of light as a detection medium is limited and finite,
while the velocity of our consciousness is unlimited and infinite.
4.
velocity-limited
transformation of sound
To further verify the correctness of the velocity-limited
transformation, sound propagating the air with small velocity was for
experimental validation. A timer and passive sonar were used to simulate the
listening process (analogous to the observation process with the eyes for light
as the medium). Thus, a “measurement through observation” rather than a
“measurement through listening” was conducted; during the measurement process,
information could only be detected by the ears or the passive sonar instead of
being detected by the eyes or the electromagnetic wave. The velocity-limited
transformation of sound in the air can also be expressed as follows:
t’ = γ(t - vx/VS2)
x’ = γ(x - vt)
y’ = y
z’ = z (6)
where γ = 1/√(1 - (v/VS)2)
and VS is the propagation velocity of sound in the air. This
transformation is similar in form to the Lorentz transformation; therefore, the
“velocity-limited transformation of sound” is also called the “Lorentz
transformation of sound” in this paper. If the detection is made in water, the
propagation velocity of sound in water can be used for VS. Effects,
such as time dilation, length contraction, etc., can also be derived using the
Lorentz transformation of sound. Because of the limitations of the experimental
conditions used in this study, acoustic detection experiments could not be
performed at this time; the experimental results will be supplemented after the
conditions are optimized or other scholars conduct relevant experiments to test
the consistency between the acoustic detection results and the calculation
results from the velocity-limited transformation.
Consider the following simple example: in water, two
submarines (A and B) are moving toward each other, each with a velocity of 0.9VS
with respect to the ground, where VS is the propagation velocity of
sound in water). What is the velocity of A with respect to B? This problem not
trivial; the velocity of A with respect to B is related to the measurement
medium. When the measurement and analysis is made with light (or another type
of electromagnetic wave) or in our consciousness, the relative velocity is 1.8
VS; however, if the measurement is made with only a sound wave, the
result is 0.994 VS as the only detection medium in the measurement
process is the sound wave (rather than another medium faster than sound). The
velocity addition can be calculated using the velocity-limited transformation
as u = (v’ + v)/(1 + v’v/VD2) formula (2), where v’ = v =
0.9VS and VD is the velocity of the detection medium and
varies greatly depending on whether the detection medium is light or sound. The
values above are calculated by substituting VD = 0.75c (the
propagation velocity of light in water) and VD = VS into
the velocity addition equation, thus revealing that different detection media
give rise to different calculation results.
5.
results and conclusions
Coincidentally, most people are accustomed to experiments and
measurements on the ground in air or in a vacuum. The measurements of the life
of a pion in air or observations of celestial bodies (such as via the
aberration of light) take place in air or in a vacuum, which have refractive
indices of ~1 and in which the propagation velocity of the detection medium
(light) is c. In daily life, people are accustomed to conducting observations
with the eyes based on light or electromagnetic wave instead of detecting
sound, which has a relatively slow speed. The velocity-limited transformation of
sound is rarely used, but it can be applied to test the velocity-limited
transformation (moreover, the velocity of sound is easier to achieve than the
velocity of light). However, all experiments regarding the relativity theory
have been conducted with light or electromagnetic waves in a vacuum or in air.
Yet, as technology becomes increasingly advanced in the future and high-speed
experiments are performed in water and other media, it will be necessary to
conduct the calculations by the velocity-limited transformation.
From the velocity-limited transformation of sound (also
called the Lorentz transformation of sound), the velocity-limited
transformation of light and the velocity-limited transformation of
consciousness are inspired by the Lorentz transformation and can also be
referred to as Lorentz transformation of light and sound, respectively. The
velocity-limited transformation is an extension that is only related to the
velocity of the detection medium. To explain the correctness of the
velocity-limited transformation, light in the vacuum was selected as the
detection medium and the Lorentz transformation was derived and subsequently generally
accepted and experimentally well-confirmed5. The velocity of
consciousness may be substituted into the velocity-limited transformation to
derive the Galilean transformation, which supports the unification of the
Galilean transformation and the Lorentz transformation. In the end, experiments
were carried out with sound to further verify the velocity-limited
transformation of sound.
The viewpoint that the velocity of consciousness is infinite
has not been established in other papers, so the application of an infinite
velocity of consciousness in this paper may cause some controversy. However,
physical quantities of the material realm are introduced into the consciousness
realm to explain with mathematical formulas why the special relativity theory
is difficult to understand. In general, people simulate, analyze, and process
in the mind the physical processes occurring in the real world to understand
the physical phenomena. Why do some people understand the special relativity
theory while other people fail? As mentioned above, the velocity of
consciousness can be infinite for most pepole or can be simulated and analyzed
as a fixed finite value such as c for a professional. However, we hope that
more people will understand the special relativity theory and its various
conclusions after reading this article.
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Acknowledgments: We thank NetEase's youdao for help with translating
and reviewing of the manuscript.
