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\begin{document}
\title{The meaning of an infinitely great velocity~~~~~~~~~}
\author[1]{Qing Li}%
\affil[1]{Affiliation not available}%
\vspace{-1em}
\date{\today}
\begingroup
\let\center\flushleft
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\maketitle
\endgroup
\selectlanguage{english}
\begin{abstract}
\textbf{Abstract~} An instantaneous velocity where a moment of the clock
only corresponds to an arbitrary distance or position in space cannot be
implied in Axiom 1, but it indicates that there is only one dimensional
existence, space or time, where a certain moment only corresponds to
itself specifically, not to any other time or any given length of space.
Further , a definition of velocity that consists of two dimensions
representing the relationship between space and time is not valid and
there is only one-dimensional space or time that is independent of each
other in Axiom 1. As a result, the principle of relativity and the
principle of the constant velocity of light are replaced by the
principle of an inertial system and the principle of universal invariant
velocity in Axiom 1. Unlike two dimensions whose magnitude is determined
by the ratio, the magnitude of a single dimension is determined by the
unit values of one dimension, which indicates that an infinitely great
velocity is meaningless. Further, if the two inertial systems are
infinite versus finite in Axiom 3, then this extension of the infinitely
great velocity can be defined as inextensible.%
\end{abstract}%
\sloppy
\textbf{PACS Numbers:} 03.30.+p\textbf{{}}
~
\textbf{1 Introduction}
~
A century ago, Newton and Galileo's absolute view of time and space was
replaced by Einstein's special relativity, in which the Galileo
transformation formula was substituted by the Lorentz transformation
formula. Special relativity based on the principle of relativity and the
principle of the constant velocity of light and space-time
transformation between inertial system observers are characterized by
the observer-independent velocity scale \emph{c} (i.e., the velocity of
light). Twenty years ago, a modified theory of special relativity was
postulated by Amelino-Camelia as doubly special
relativity\textsuperscript{{[}1{]}} (also referred to as deformed
special relativity), which is based on quantum-gravity arguments. Doubly
special relativity, the new relativistic theory in which the space-time
transformations between inertial system observers are characterized by
two observer-independent scales (in addition to the light velocity
scale, there is a second new observer-independent length/momentum scale,
the Planck length/momentum). Further, doubly special relativity predicts
that a value of E\textsubscript{p} [?] 10\textsuperscript{28} eV can be
regarded as the maximum value of energy and momentum for fundamental
particles, while length/momentum remains unchanged in the space-time
inertial frame under the Planck scale.\textbf{}
In mathematics, the mathematical basis for further analysis of
relativity has been provided by new mathematical models, such as
boundary value problems\textsuperscript{{[}2{]}{[}3{]}} and discrete
mathematics\textsuperscript{{[}4{]}{[}5{]}} ( temporal and spatial
discontinuity).{}
There are two concepts that are logically debatable in relativity.
Firstly, the velocity of light is a finite speed, but it is also a
limited speed, which indicates that there are infinitely many different
speed values to choose from between the velocity of light and a velocity
of 0. This concept can be seen in the Lorentz space-time transformation
formula:
\textbf{\emph{X'}}\textsubscript{\textbf{\emph{1}}}
\textbf{\emph{=X}}\textsubscript{\textbf{\emph{1}}}\textbf{\emph{-vl/\{1-(v/c)}}\textsuperscript{\textbf{\emph{2}}}\textbf{\emph{\}}}\textsuperscript{\textbf{\emph{1/2}}}
~and \textbf{\emph{l'}}\textsubscript{\textbf{\emph{1}}}
\textbf{\emph{=l-vx}}\textsubscript{\textbf{\emph{1}}}\textbf{\emph{/\{1-(v/c)}}\textsuperscript{\textbf{\emph{2}}}\textbf{\emph{\}}}\textsuperscript{\textbf{\emph{1/2}}}{}
Secondly, the physical quantities remain unchanged in different inertial
systems; in other words, if we do not assume the two inertial systems
with different motion states, then the two inertial systems cannot be
distinguished. This is called the principle of relativity.
In this paper, based on Axioms 1\textsuperscript{{[}6{]}} and
3\textsuperscript{{[}7{]}}, four perspectives of inertial systems that
differ from Einstein's special relativity are proposed: the principle
of relativity can be replaced by a concept in which an inertial system
is only a specific quantity, relative velocity is meaningless; any
velocity is constant with respect to any other velocity, a definition
of velocity that includes two dimensions (space/time) is not valid and
there is only one-dimensional space or time, and unlike two
dimensions, where the magnitude is determined by the ratio, with one
dimension, it is determined by unit values.
~
\textbf{2 There is no instantaneous or two-dimensional velocity}
~
The relationship between space and time can be expressed in terms of
velocity: \emph{v = s/t, w}here \emph{v} is the velocity, \emph{s} is
the length of space, and \emph{t} is time. There are two implications
regarding the relationship between space and time arising from this
formula: \emph{s} and \emph{t} are dimensions that can be compared,
and \emph{s} and \emph{t} are equivalent. For example, for two
velocities of 3 \emph{} m/s and 4 m/s, 1 second should be equivalent to
3 \emph{} meters for the first velocity, while 1 second should be
equivalent to 4 meters in the second. From Axiom 1, it is known that
each length value is specific (because the unit value differs, each
length value can only be itself), thus the unit second of time is also
specific. If 1 second is equivalent to 3 meters, then 1 second is not
equivalent to 4 meters, so the definition of velocity is meaningless in
Axiom 1. Another example is that 1 second is a finite number in Axiom 1,
so an infinitely great velocity (one second goes into the infinite
distance) is not valid in Axiom 1, because 1 second cannot be equivalent
to an infinite length. Therefore, in Axiom 1, because the conditions for
the two-layered meaning of \emph{v = s/t} cannot be satisfied
simultaneously (that is, the size can be compared and the equivalence
can be met simultaneously), a definition of velocity based on two
dimensions representing the relationship between space and time cannot
be established.
If the two-dimensional property of permissible velocity is true, then
certain conditions must be met. As can be seen in Figure 1, the
properties that can be compared are eliminated. For a velocity composed
of two dimensions, time and space are reduced to dimensions that cannot
be compared, that is, the finite and infinite quantities cannot be
distinguished, nor can the sizes be compared with each other. For
example, 1 meter or 1 second in a general sense can represent any
quantity. For the convenience of the following description, this concept
is defined as two-dimensions-without-size Axiom 1.
Two-dimensions-without-size Axiom 1 is a paradox and meaningless, so the
conclusion can be drawn that, in Axiom 1, only one dimension exists and
space and time are independent of each other. It is now important to
look at some basic properties of Axiom1.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/Tu-1-ppt/Tu-1-ppt}
\end{center}
\end{figure}
Figure 1.~ Ability to compare sizes is eliminated and space and time
turn into one dimension that cannot be compared in size for a velocity
consisting of two dimensions. Further, the finite and infinite
quantities cannot be differentiated and sizes cannot be compared with
each other. This property is achieved by \emph{s=t.}, where \emph{s} is
space and \emph{t} is time. For convenience, this concept is referred to
as two-dimensions-without-size Axiom 1.
~
\textbf{Property 1} There is only one dimension, space or time, and they
are independent of each other. For example, for an event moving at an
infinite distance of 1 second, 1 second is a finite quantity, and space
at an infinite distance is an infinite quantity. The two quantities are
neither equivalent nor dependent on each other. In any other
velocity-describing event, the magnitude of space or time is neither
equivalent (except for each magnitude itself) nor correlated.
\textbf{Property 2} There is no instantaneous velocity at infinity.
Instantaneous velocity is defined as moving to any point of length in
space without time, that is, point 0 corresponds to any point of length
in space, a moment of 1 second corresponds to any point of length in
space, a certain distance in space (for example, 1 meter) corresponds to
any moment in time, and so on.
~
In Axiom 1, the absence of instantaneous velocity has two implications.
First, as mentioned earlier, a single dimension means that there is no
velocity with two dimensions that can be compared. Second, the
independent existence of space and time does not mean that a certain
moment of the clock only corresponds to any distance or position in
space; rather, it means that there is only one dimension, space or time.
Each value of space corresponds only to itself, not to other quantities,
and each value of time corresponds only to itself, not to other
quantities; thus, space or time are independent of each other. For
example, point 0 only corresponds to point 0 and does not correspond to
other quantities (including infinite quantities), while 1 meter only
corresponds to 1 meter and does not correspond to other quantities.
Unlike the concept of simultaneity/non-simultaneity in relativity, this
independence is given a new definition.
~
The independence of the relationship between space and time can also be
illustrated as follows. If we talk about space, it makes no sense for us
to talk about time, and if we talk about time, it makes no sense for us
to talk about space. A given interval of time does not correspond to any
length of space, and a given distance of space does not correspond to
any interval of time. Thus, it can be said that, for two different
locations in space, whether they exhibit simultaneity or
non-simultaneity in time is of no significance; similarly, for two
different intervals in time, whether they are in the same or different
locations in space is also of no significance.
~
The absence of instantaneous velocity does not mean that infinite space
and infinite time do not exist, just that they exist independently. The
absence of instantaneous velocity does not mean that infinite velocity
does not exist, nor does it mean that there is only a finite velocity,
such as the velocity of light. In Axiom 1, the velocity of light is only
a finite speed (300,000 kilometers and 1 second are both finite), thus
it is neither an infinite velocity nor a limit velocity. In Axiom 1, the
single dimension dictates that each value corresponds to itself and does
not correspond to other values, explaining why a clock at some point in
the theory of relativity only corresponds to a certain space with an
equal distance or the position itself, not to the concept of the
distance or the position of others. However, unlike the description of
the theory of relativity, the concept of a single dimension described
does not deny that infinite values exist, and there is aslo no concept
of time shortening or space lengthening here. Details on this will be
described in section 6.
\textbf{~}
\textbf{3 Principle of special relativity and the principle of the
constant velocity of light}
~
Concrete descriptions of the single-dimensional properties of Axiom 1
are provided in this section. By comparing the concept of time and
spacein Einstein's special relativity, the properties of a single
dimension can be more clearly understood.
\textbf{Principle of relativity} Here, inertial frames and relativity
principles are discussed. If \emph{K} is defined as a Cartesian
frame-of-reference system (i.e., an inertial frame), then another
Cartesian frame of reference \emph{K'}, which is moving uniformly in a
straight line with respect to \emph{K}, is also an inertial system(A
rotating Cartesian inertial frame is classified as a non-inertial frame
and is beyond the scope of this paper).There are three implications
here. First, for any coordinate system \emph{K',} all space-time
quantities (i.e., spatiotemporal variables) can be expressed in this
coordinate system, and all quantities are static relative to \emph{K'}.
For example, consider two velocity events \textbf{\emph{s=ct}} or
\textbf{\emph{s=vt}}, both of which can be expressed in \emph{K'}, where
\emph{c} is the velocity of light and \emph{v} is any velocity. If
\emph{K} and \emph{K'} without comparison, then the spatiotemporal
variables relative to \emph{K} at rest cannot be used to distinguish the
motion state from the spatiotemporal variables relative to \emph{K'} at
rest. This is known as the relativity principle. Second, the coordinate
system itself and the quantity expressed in the coordinate system can be
described by different quantitative terms, such as \emph{K'} moving with
velocity \emph{v}\textsubscript{\emph{1}}. Any number of values that
differ from \emph{v}\textsubscript{\emph{1}} can be described along the
\emph{x, y,} and \emph{z} axes of the coordinate system, such as
\emph{s}\textsubscript{\emph{1}}\emph{=ct}\textsubscript{\emph{1}} or
\emph{s}\textsubscript{\emph{2}}\emph{=v}\textsubscript{\emph{2}}\emph{t}\textsubscript{\emph{2}},
where \emph{c} is the velocity of light and
\emph{v}\textsubscript{\emph{2}} \emph{} is any velocity. Third, in
static coordinate system \emph{K} with a velocity of 0, the velocity at
all points is 0. In coordinate system \emph{K'} with a uniform velocity
of \emph{v}, the velocity at all points is \emph{v}. The difference
between K' and K is quantitative, that is, the difference between
\emph{v} and 0. These concepts apply to Axiom
2\textsuperscript{{[}6{]}}.
\textbf{Principle of the constant velocity of light} It has been proven
by Michelson's experiment that the speed of light remains constant in
Cartesian coordinates with uniform linear motion at any velocity. A
moment of a clock corresponds only to a certain distance or position in
space equal to itself and does not correspond to any other distances or
positions. For example, 1 second only corresponds to 300,000 kilometers
(i.e., 1 second is equivalent to 300,000 kilometers) and does not
correspond to other distances.
The implications of the transformation of Cartesian coordinates based on
these two principles are as follows:
In a Cartesian coordinate system that allows instantaneous velocity,
relative velocity is meaningful, which indicates that the quantity of
velocity for given Cartesian coordinates will vary for Cartesian
coordinates with different velocities; that is, the quantity of a given
velocity depends on the motion velocity of the Cartesian coordinates.
Because a certain moment of a clock corresponds to an arbitrary distance
in space, and a certain distance in space corresponds to an arbitrary
time of the clock, the transformation between the two Cartesian
coordinate systems \emph{K'} and \emph{K} is arbitrary. In fact, this
concept is two-dimensions-without-size Axiom 1.
In a Cartesian coordinate system with a constant velocity of light,
the velocity of light is used as the basis for defining space and time
(i.e., light time and light space). For optical space coordinate
\emph{X}\textsubscript{\emph{1}} in frame \emph{K} (stationary
coordinates with velocity 0), the corresponding optical space
coordinates in frame \emph{K'} (a coordinate system with velocity
\emph{v}) is
\textbf{\emph{X'}}\textsubscript{\textbf{\emph{1}}}\textbf{\emph{=
1/(1-v/c)X}}\textsubscript{\textbf{\emph{1}}}\textbf{\emph{(}\emph{X'}}\textsubscript{\textbf{\emph{1}}}\textbf{\emph{\textgreater{}
X}}\textsubscript{\textbf{\emph{1}}}\textbf{\emph{)}\emph{{}}}
This formula is the revised version of the Lorentz transformation.
Unlike the relativistic principle, which holds the coordinates of
\emph{K'} and \emph{K} to be identical, here the coordinates for
\emph{K'} and \emph{K} differ due to the fact that all quantities within
frame \emph{K} are stationary with respect to frame \emph{K}, but they
~are not stationary with respect to frame \emph{K'}.
The formula \textbf{\emph{X'}}\textsubscript{\textbf{\emph{1}}}
\textbf{\emph{=X}}\textsubscript{\textbf{\emph{1}}}\textbf{\emph{-ct}}
cannot be established for a comparison of the coordinates between the
two frames \emph{K'} and \emph{K} because relative velocity is
non-existent in relativity; in other words, a minus sign in the formula
does not exist.\textbf{\emph{{}}}
It is known from that the same proportional extension of \emph{K'}
and \emph{K} coordinates for the two coordinate systems is carried out
as
~\textbf{\emph{X'}}\textsubscript{\textbf{\emph{1}}}\textbf{\emph{:}\emph{1/(1-v/c)X}}\textsubscript{\textbf{\emph{1}}}
,
where \textbf{\emph{X'}}\textsubscript{\textbf{\emph{1}}}\textbf{\emph{=
1/(1-v/c)X}}\textsubscript{\textbf{\emph{1}}}{}
The purpose of this formula is to facilitate a comparison of the
coordinate transformation of the two coordinate systems so that the two
coordinates are compared at the same length value and the same scale of
time.
According to , ~\textbf{\emph{X'}}\textsubscript{\textbf{\emph{1}}}
\textbf{\emph{=1/\{1-(v/c)}}\textsuperscript{\textbf{\emph{2}}}\textbf{\emph{\}}}\textsuperscript{\textbf{\emph{1/2}}}\textbf{\emph{(}\emph{X}}\textsubscript{\textbf{\emph{1}}}\textbf{\emph{-ct}\emph{)}}\textsuperscript{{[}8{]}}
\textbf{\emph{}}
The Lorentz transformation is meaningless; instead,
\textbf{\emph{X'}}\textsubscript{\textbf{\emph{1}}} is given by the
formula
\textbf{\emph{X'}}\textsubscript{\textbf{\emph{1}}}\textbf{\emph{=
1/(1-v/c)X}}\textsubscript{\textbf{\emph{1}}}.
Therefore, the notion that frames \emph{K'} and \emph{K} coincide at
origin 0 is meaningless and frame \emph{K'} does not start at origin 0.
From , because the \emph{K'} and \emph{K} coordinates are different,
the two Lorentz transformation equations
\textbf{\emph{X'}}\textsubscript{\textbf{\emph{1}}}
\textbf{\emph{=1/\{1-(v/c)}}\textsuperscript{\textbf{\emph{2}}}\textbf{\emph{\}}}\textsuperscript{\textbf{\emph{1/2}}}\textbf{\emph{(}\emph{X}}\textsubscript{\textbf{\emph{1}}}\textbf{\emph{-ct}\emph{)}\emph{{}}}
and \textbf{\emph{X'}}\textsubscript{\textbf{\emph{1}}}
\textbf{\emph{=1/\{1-(v/c)}}\textsuperscript{\textbf{\emph{2}}}\textbf{\emph{\}}}\textsuperscript{\textbf{\emph{1/2}}}\textbf{\emph{(}\emph{X}}\textsubscript{\textbf{\emph{1}}}\textbf{\emph{-ct'}\emph{)}\emph{{}}}
are not valid, and they are replaced by the following two equations:
\textbf{\emph{X'}}\textsubscript{\textbf{\emph{1}}}\textbf{\emph{=
ct}}\textsuperscript{\textbf{\emph{'}}}\textsubscript{\textbf{\emph{1}}}
and
\textbf{\emph{X}}\textsubscript{\textbf{\emph{1}}}\textbf{\emph{=ct}}\textsubscript{\textbf{\emph{1}}}
\textbf{\emph{}}
Here \textbf{\emph{X'}}\textsubscript{\textbf{\emph{1}}}\textbf{\emph{=
1/(1-v/c) X}}\textsubscript{\textbf{\emph{1}}} \textbf{\emph{}} and
\textbf{\emph{t}}\textsuperscript{\textbf{\emph{'}}}\textsubscript{\textbf{\emph{1}}}
\textbf{\emph{=1/(1-v/c)t}}\textsubscript{\textbf{\emph{1}}} .
The main characteristics of these two equations that differ from the
Lorentz transformation are that their coordinates are given by
\textbf{\emph{X'}}\textsubscript{\textbf{\emph{2}}}\textbf{\emph{-X'}}\textsubscript{\textbf{\emph{1}}}\textbf{\emph{\textgreater{}X}}\textsubscript{\textbf{\emph{2}}}\textbf{\emph{-X}}\textsubscript{\textbf{\emph{1}}}
(Figure 2).\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/Tu-2-ppt/Tu-2-ppt}
\end{center}
\end{figure}
Figure 2 (1) With stationary Cartesian coordinates,
3\selectlanguage{ngerman}×10\textsuperscript{8} meters is equivalent to 1 second, and the time
beat is given as 1 second (= 3×10\textsuperscript{8} meters). (2) For
Cartesian coordinates with a velocity of 3 m/s, 3×10\textsuperscript{8}
× 1/\selectlanguage{english}(1-3/c\selectlanguage{english})meters is equivalent to 1x1/\selectlanguage{english}(1-3/c\selectlanguage{english})seconds, with the time
beat given as~ 1/\selectlanguage{english}(1-3/c\selectlanguage{english})seconds ~(= 3x1\textsuperscript{8}0
x1/\selectlanguage{english}(1-3/c\selectlanguage{english}) meters). It is concluded from relativity that a velocity of
3 m/s will be given in the form of 3x1\textsuperscript{8}0 x1/\selectlanguage{english}(1-3/c\selectlanguage{english})
meters~ /~ 1x1/\selectlanguage{english}(1-3/c\selectlanguage{english})seconds .
~
~
\textbf{4 Principle of an inertial system and the principle of universal
invariant velocity for Axiom 1}
\textbf{~}
\textbf{Principle of an inertial system for Axiom 1} It is known from
Axiom 1 that relativity is not true. As shown in Figure 3, in Axiom 1,
each inertial system is described by a unit value, such as 2, 3, or 4.
In this system, the properties of the proportional extension of two
inertial frames can then be considered. For example, two inertial frames
2 () 1 are compared, with the following extension ratios being 4 (), 2,
6 (), 3, 8 (), 4, etc. In this comparison of inertial frames, we also
consider a unit extension of 4 () 2 to the same ratio, meaning the next
extension ratio is 8 () 4, followed by 12 () 6, and so on. Comparing 8
() 4 to the same scale unit extension, the next is 16 () 8, followed by
24 () 16, and so on. Although the ratio is 2/1, the two units extend
differently and cannot replace or offset each other because of the
different units (the former is in units of 4 and the latter is in units
of 8). Therefore, relative velocity is meaningless in Axiom 1, which
means that a given quantity, as distinct from the other units of
quantity, can only be itself and not any other quantity; this particular
quantity thus represents only one state, not any other state, and the
Cartesian coordinate system does not apply in Axiom 1. As a result, the
properties of the inertial system of relativity need to be revised.
First, in the principle of an inertial system for Axiom 1, an inertial
system is a specific quantity and only represents a state, so the motion
of all quantities is absolute, and any comparison of the motion of two
quantities is also absolute.
Second, the absoluteness of this motion negates the relativity principle
of relativity theory. Thus, it can be said that a stationary concept is
also meaningless in the principle of an inertial system for Axiom 1. In
the principle of relativity, if \emph{K} is a stationary Cartesian
inertial system (i.e., the coordinates for all space and time variables
are stationary relative to \emph{K}), \emph{K'} is a moving Cartesian
inertial system relative to \emph{K} with velocity \emph{v} so,
following the Axiom 1 inertial system principle, it is meaningless to
talk about all space-time variables as stationary relative to \emph{K'},
and \emph{K} does not exist as an inertial system at rest. It can thus
be said that the Cartesian coordinate system cannot describe the
distribution of quantities in space and time and that all-embracing
variables in space and time that stand stationary relative to the
coordinate system do not exist.
Thirdly, unlike the coordinate system, which must be described with two
different terms as discussed above, the inertial system in Axiom 1 only
has a quantitative term description, such as an inertial system whose
space units are 1, 4, 8, or N (all are multiples of 0). An inertial
frame represents only a specific quantity, that is, only a state.
Fourth, consider the comparison of two inertial systems in Axiom 1, such
as the 4\selectlanguage{english}(\selectlanguage{english})1 proportionate extension, where inertial system 4 extends
in units of 4, and inertial system 1 extends in units of 1. The
extension of the two inertial systems involves an infinite number of
different comparisons except that the extension ratio of 4:1 is fixed.
For example, the extension of inertial system 4 is 4, 8, 12, 16, etc.
(infinitely many different quantities), while the extension of inertial
system 1 is 1, 2, 3, 4, etc. (infinitely many different quantities).
Thus, the difference between the two inertial systems is not a
difference of one quantity, but an infinite number of quantities.
The Cartesian coordinate system which thus describes the difference in a
quantity cannot be applied to describe infinitely many differences. In
addition, in a Cartesian coordinate system moving with velocity
\emph{v}, all of the points moving with velocity \emph{v} are
meaningless because, in Axiom 1, point 0 only represents point 0 and
cannot be endowed with other concepts, such as point 0 moving with a
velocity that is two-dimensional. ~\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/Tu-3-ppt/Tu-3-ppt}
\end{center}
\end{figure}
~
Figure 3~~ In Axiom 1, each inertial frame can be described by unit
quantities, such as 1, 2, 4, etc. Although the ratio is 2:1, the units
are different (b is in 4 units, and c is in 8 units), so the
proportional extension of the units differs and cannot replace or offset
each other.
~
Principle of universal invariant velocity for Axiom 1 In Axiom 1, each
quantity is a specific quantity, that is, it is itself rather than any
other quantity. Thus, we reasonably conclude that, in Axiom 1, any
velocity is itself, not any other velocity (where the essence of the
concept of velocity is single-dimensional space or time). Each velocity
is thus constant relative to other velocities, not just the velocity of
light. This property is defined as the principle of universal velocity
invariance for Axiom 1.
\textbf{~}
\textbf{5 Velocity is one-dimensional}
~
Common misconceptions about velocity can be cleared up using these two
principles.
(1)Velocity is two-dimensional, and there is an instantaneous velocity
going to the infinite distance. Here space and time are independent of
each other, that is, a certain moment of the clock corresponds to any
distance or position in space. For example, 1 second corresponds to any
length, which is the Newtonian absolute space-time view. Because a
moment of a clock corresponds to any distance or position in space, this
means that velocity is variable, that is,, the size of one velocity
depends on how much it corresponds to other velocity, and velocity can
be added or reduced. Therefore, the concept of a specific velocity does
not exist here. The single-dimensional properties and
universal-velocity-invariant properties of Axiom 1 deny the correctness
of this concept. This property is essentially
two-dimensions-without-size Axiom 1, which means that a Galilean
transformation is meaningless.
(2) Velocity is two-dimensional and can be compared in size, in which
there is no instantaneous velocity extending to infinity, and the
velocity of light is a finite magnitude of velocity that is also a limit
velocity. Here, the principle of relativity can be applied. Because the
invariability of the velocity of light (i.e., it remains constant for
Cartesian coordinates at any velocity) has been experimentally
confirmed, the velocity of light has a privileged position as the basis
for defining space and time (i.e., light time and light space), which is
known as relativistic space-time. Here, one second of the clock
corresponds to a space length of only 300,000 kilometers (i.e., 1 second
is equivalent to 300,000 kilometers), two seconds corresponds to 600,000
km (i.e., 2 seconds is equivalent to 600,000 kilometers), and so on.
Here, 1 second does not correspond to other distances, such as 3 meters.
Therefore, the notion of a velocity of 3 m/s (i.e., 1 second is
equivalent to 3 meters) makes no sense with relativity. Speed events of
3 m/s are given in light time and in light space (Figure 3). As a
result, the space-time properties of two inertial systems \emph{K} 'and
\emph{K} (for example, an inertial system with a velocity of 3 m/s is
compared with an inertial system with a velocity of 0) have the
following characteristics. From the observation of \emph{K}', the time
of \emph{K} is prolonged, and the space is shortened. As observed from
K, the time of K' is shortened and the space is elongated (Figure 4).
The single-dimensional nature of Axiom 1 and the infinity of space-time
deny the correctness of this concept.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/Tu-4-ppt/Tu-4-ppt}
\end{center}
\end{figure}
Figure 4 Space-time properties for a comparison of two inertial systems
\emph{K'} and \emph{K}. For example, when inertial system \emph{K'}with
a velocity of 3 m/s is compared with the inertial system \emph{K'}with a
velocity of 0, when observed from \emph{K}, the time of \emph{K} is
lengthened and space is shortened. In contrast, when observed from
\emph{K}, the time of \emph{K}'is shortened and space is elongated .
~
~
Velocity is two-dimensional, and there is an infinite but not
instantaneous velocity. The inertial system principle and universal
velocity invariant principle follow Axiom 1 and do not follow the
relativity principle; in other words, a certain moment of the clock only
corresponds to a specific distance in space and does not correspond to
other distances. For example, infinite time only corresponds to infinite
distance, not to a finite distance (such as a distance of 1 meter), and
a finite clock scale only corresponds to a finite distance, not to
infinite distance. Unlike , here the velocity of light is not the only
basis for defining space and time, allowing for the existence of
arbitrary values for velocity. Two implications arise from this
arbitrary velocity, First, it is meaningful that the space-time is not
equivalent. For example, although 1 second is equivalent to 300,000
kilometers, it is not equivalent to 3 meters at 3 m/s, but a velocity of
3 m/s is meaningful. Second, the magnitude of velocities can be
compared. For example, the velocity of light has the same quantitative
value as the unit of time for 3 m/s. The stationary state of it, unlike
the Cartesian coordinates of relativity, should be given as 0/[?]. The
single-dimensional nature of Axiom 1 denies the correctness of this
concept. The essence of (2) and are still Axiom 2.
There is only one-dimensional space or time, and there is no concept
of velocity, regardless of whether it is infinite or finite. Space and
time are independent of each other here, with a certain moment of the
clock only corresponding to the moment itself, not to other moments or
any distance or position in space. Likewise, a certain distance in space
corresponds only to its own distance, not to any other distance in space
or any time of the clock. Therefore, the inertial system principle of
Axiom 1 and the universal velocity invariant principle are followed
here. Velocity has become single-dimensional space or time and Only the
finite and infinite space, or finite and infinite time can be talked
about.. If the concept of velocity is being referred to, the two values
(distance in space and time in time) are neither equivalent nor
dependent on each other. The essence of (4) is Axioms 1 and 3. For
instance, for a velocity event moving to infinite distance in 1 second,
it can be seen from the above definition that 1 second is not equivalent
to infinite distance, because the concept of a single event of infinite
speed being associated with time and space is meaningless. Rather, 1
second and infinity exist independently as two events: an event of
infinitely great space in one dimension and another event of 1 second in
time in one dimension.{}
\textbf{~}
\textbf{6 Meaning of one-dimensional velocity}
\textbf{~}
~By comparing (3) and (4), we can outline their specific features. For
feature (3), the velocity is determined by the ratio of the two
dimensions. There is an infinitely great velocity, expressed by [?]/dl,
where [?] is infinitely great and dl is infinitesimally small. A state of
zero velocity is denoted as dl/[?], and dl does not equal 0 here (due to
the nature of Axiom 2). Feature (3) follows the inertial system
principle of Axiom 1 and the universal velocity invariant principle ,but
does not follow the relativity principle, so the Cartesian coordinate
system does not apply to (3). For example, a Cartesian coordinate system
with a velocity of 0 (i.e., static) does not exist. Motion is absolute
and there is no static state, so a comparison of two inertial frames is
a comparison of two specific states. For example, let the inertial
system \emph{K'} be the infinite velocity and the inertial system
\emph{K} velocity be 0. Figure 5 shows a comparison of the two inertial
systems. Their spatiotemporal properties are determined by two points
(\emph{a} and \emph{b}). When observed from \emph{K'}, the time
lengthens and space shortens in \emph{K}. In contrast, from the point of
view of \emph{K}, the time in \emph{K'} is shortened and the space is
lengthened. Because Cartesian coordinates do not apply to (3), the
Lorentz transformation does not make sense here. The transformation of
the magnitude of space-time is a universal transformation, which is
determined by the magnitude of \emph{a} and \emph{b}.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/Tu-5-ppt/Tu-5-ppt}
\end{center}
\end{figure}
Figure 5 Comparison between two inertial systems in . For example, the
inertial system \emph{K'} has an infinitely great velocity, and inertial
system \emph{K} has a velocity of 0 (i.e., a stationary state). Their
space-time properties are determined by two points (\emph{a} and
\emph{b}). Observed from \emph{K'}, the time in \emph{K} is lengthened
(\textbf{[?])} and space is shortened (dl), When observed from \emph{K},
the time in \emph{K'} is shortened (dl) and space is elongated
(\textbf{[?])}.
~
Feature (4) can now be considered. The single-dimensional nature means
that its space-time nature is determined by one point, not two (a and
b). Therefore, instead of two dimensions being determined by the ratio,
the size of a single dimension is determined by a one-dimensional unit
value (which varies by unit number). Therefore, it is meaningless to
lengthen or shorten the space-time of two inertial systems in a
two-dimensional state.
How the extension of velocity representing two dimensions differs from
the extension of space or time representing one dimension can now be
discussed. It is suggested by Axiom 2 that the space-time extension of a
velocity can reach infinite distance and that the ratio of the velocity
is arbitrary and either finite or infinite. As shown in Figure 6, in a
comparison of two Cartesian inertial systems moving at different
velocities, their space-time extension can also reach infinite distance.
In Axiom 1, quantitative values extend in units of 0 (i.e., 1 0, 2 0, 3
0, and so on). The extension of two different values (i.e., two inertial
frames) is carried out using an arbitrary integer () 1 that is a
multiple of 0 and is carried out in units. The minimum magnitude is one
0. Unlike , where there is an inertial system with an infinite approach
velocity of 0 (dl/[?]), the nearest 0 inertial system in Axiom 1 is two 0
inertial systems. In Axiom 1, the uniqueness of infinity determines that
the formula~ 1/0=[?]/1=[?] is not true and that only the formula that [?]/0=[?]
is true, meaning the formula 300,000 km /0=[?]/ 300,000 km =[?] is not true.
For each finite length (for example, 1 meter) there is a finity, not an
infinity, so 300,000 kilometers is not sufficient to carry an infinite
amount of burden. The existence of Axiom 1 means that there is only a
finite number of quantities to choose from between 0 and 1 second or
between 0 and 300,000 kilometers, and there must be an infinite number
of quantities to choose from between 0 and [?] (units of seconds or
kilometers). Therefore, the velocity of light is not an ultimate
velocity, and putting the velocity of light into a special superior
position lacks any profound basis in physics.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/Tu-6-ppt/Tu-6-ppt}
\end{center}
\end{figure}
Figure 6 (a) It is suggested by Axiom 2 that the space-time extension of
a velocity can reach infinite distance, the ratio of the velocity is
arbitrary and either finite or infinite. (b) It is known from Axiom 3
that if two inertial systems are compared for finite quantities, then
the extension of the quantities of the two inertial systems must remain
within the finite range and do not reach infinite distance. (c) If two
inertial systems are infinite versus finite, then it is known from Axiom
3 that a change of direction means it is infinitely great and the finite
is not part of infinitely great, so this extension of infinitely great
is defined to be inextensible.
~
Because Axiom 3 is a modification of Axiom 1 (that is, Axiom 3 retains
some of the properties of Axiom 1), if two inertial systems involve a
comparison of finite values, then these two inertial systems extend only
in a finite range and cannot extend to infinity (derived from Axiom 3).
If the two inertial systems are unlimited (i.e., infinite) compared with
limited amount of, so learn from Axiom 3, in which the direction of
change means infinitely great , and the finity is not part of infinitely
great, that then for infinity (infinitely great) has two meanings.
Firstly, it is the largest unit (with an infinitely great unit), i.e.,
there is no bigger or smaller amount, and therefore this extension of
infinitely great is defined as inextensible (Figure 6). Secondly, the
change in direction means that it cannot be added, subtracted,
multiplied or divided, and that it is not a finite component, so it does
not vary with the corresponding value of a finite number. Therefore, the
Lorentz transformation in the two inertial systems of relativity and the
modified Lorentz transformation (corresponding to changes in time and
space length), or other magnitude and value transformations (which apply
to Axioms 1 and 2), are meaningless in Axiom 3. Instead of the
spatiotemporal coordinate transformation or numerical transformation of
the two inertial systems defined in Axioms 1 and 2 (only in the motion
of uniform linear velocity), the spatiotemporal transformation of the
two different inertial systems in Axiom 3 only changes in one direction,
which is a unique quantity-value transformation and represents all
quantity-value transformations (not only in the motion of uniform linear
velocity but also in non-uniform linear velocity).
\textbf{~}
\textbf{7 Conclusions}
~
(1) It is concluded from Axiom 1 that a definition of velocity in
relativity that consists of two dimensions representing the relationship
between space and time is not valid and there is only independent
one-dimensional space or time in Axiom 1. As a result, the principle of
relativity and the principle of the constant velocity of light are
substituted by the principle of the inertial system of Axiom 1 and the
principle of universal invariant velocity of Axiom 1.
(2) Unlike two dimensions whose magnitudes of space and time are
determined by the ratio between the two, the magnitude of a single
dimension is determined by the unit values of one dimension, which
indicates that any velocity (including infinitely great velocity) is
meaningless and there is only infinitely great space in one dimension
and infinitely long time in one dimension.
(3) Because Axiom 3 is a modification of Axiom 1, it retains some
properties of Axiom 1 despite its new properties . Unlike Axiom 1, in
which the transition from finite to infinite is a continuous process, in
Axiom 3, the transition from finite to infinite involves a leap, thus,
if the extensions are within the range of finite quantities for two
inertial systems in Axiom 3, they must only stay in the finite range and
do not reach infinite distance. If these two inertial systems are
infinite versus finite, then it is known from Axiom 3 that the change in
direction means infinite great ,and this extension of infinite great can
be defined as inextensible.
There are some limitations for this study presently. First, there have
been no direct observations made to confirm the conclusions of this
study. Second, due to the difficulty of observing infinity, this
research rests only on logical reasoning, but this does not prevent it
from redefining or approximating the relationship between the physical
quantities in the observable finite range of time and space. In other
words, the conclusion is still applicable to physical quantities within
this range. Thus, one of the greatest benefits of this study may be that
we can redefine the mass--energy equation.
\textbf{~}
\textbf{8 Prospects} ~
~
This paper discusses the concept of inertial systems in Axiom 1 (i.e.
uniform linear motion), so the reader may ask, how does Axiom 1 define
the concept of non-inertial systems (e.g., acceleration or curved
motion)? Because two dimensions do not exist in Axiom 1, neither do many
dimensions, so how does a single dimension define a non-inertial system
(e.g., acceleration)? I will discuss this issue in detail in my next
paper.
~
~~~~~~~~~~~~~~~~~~~~
References
{[}1{]} G. Amelino-Camelia, ``Relativity in spacetimes with
short-distance structure governed by an observer-independent (Planckian)
length scale'' Int. J. Mod. Phys. D., vol. 11, pp. 35--59, 2002.
{[}2{]} M.K. Iqbal, M. Abbas, and I. Wasim, ``New cubic B-spline
approximation for solving third order Emden--Flower type equations,''
Appl .Math. Comput., vol. 331, pp. 319-333, 2018.
{[}3{]} N. Khalid, M. Abbas, and M. K. Iqbal, ``Non-polynomial quintic
spline for solving fourth-order fractional boundary value problems
involving product terms,'' Appl. Math. Comput., vol.349, pp. 393-407,
2019.
{[}4{]} N. Khalid, M. Abbas, M. K. Iqbal, J. Singh, and A.I.M. Ismail,
``A computational approach for solving time fractional differential
equation via spline functions,'' Alex. Eng. J., 2020, in press
{[}5{]} T. Akram, M. Abbas, M. B. Riaz, A. I. Ismail, and N. M. Ali,
``An efficient numerical technique for solving time fractional Burgers
equation,'' Alex. Eng. J., vol. 59, pp. 2201-2220, 2020.
{[}6{]} Qing Li. A geometry consisting of singularities containing only
integers. (preprint Research Square: DOI: 10.21203/rs.3.rs-219046/v1 )
{[}7{]} Qing Li, The meaning of the infinitely great ~(preprint author:
\textbf{}
\textbf{DOI:~}\href{https://doi.org/10.22541/au.160822935.50569408/v1}{\textbf{10.22541/au.160822935.50569408/v1}}\textbf{)}{}
{[}8{]} A. Einstein. ``The meaning of relativity,'' Beijing Science
Press. pp. 22-23 (1979)
~
\textbf{The data availability statement:}
The {[}DATA TYPE{]} data used to support the findings of this study are
included within the article.
~
Author information:
Qing Li
Code Number:050031
402, Unit1, Building 28
West zone of ChangRong Small District
No. 122,YuHua East Road DongYuan Street
YuHua District
ShiJiaZhuang City HeBei Province PR. China. Tel.: +86-13833450232
E-mail: {liqingliyang@126.com}
Backup e-mail: 2895621512@qq.com
Author contributions: Qing Li completed this manuscript in full.
\begin{verbatim}
Funding Acknowledgement: This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors.
\end{verbatim}
\begin{verbatim}
Conflict of interest statement: The author declares no conflict of interest in preparing this article.
\end{verbatim}
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