Using metric views to establish special relativity
The description of physical space is inseparable from measurement. The four-dimensional description of material space should also be related to the four-dimensional measurement. Therefore, it is necessary to study the metric relation of space-time in four dimensional coordinates. The results showed that if you base special relativity on metric relation, not only does it make the elusive theory become easier to understand, and to settle what was the called “struggle of the century”, but also can expand it into states of motion other than translational motion, such as the spin, etc. Make it more widely and deeply applied in the microscopic field.
The main points of using the concept of measurement to establish special relativity are as follows:
(1)Suppose the speed of light in vacuum is constant “ C ” in any coordinate system with relative motion. It’s just a convention. (2) Derivation of Lorentz transformations is in the differential field. (3) To deduce the relation between the vibration frequency of matter wave and the relative velocity of two moving coordinate systems.
(1) The agreement is that the speed of light in vacuum is constant “c” in any coordinate system of relatively uniform motion.
In any 4-dimensional coordinate system, the speed reading is related to the 4-dimensional unite of measure in the coordinate system. For the same moving particle, the velocity readings are different in different coordinate systems with relative motion. If the unit of measure is defined in a coordinate system, then it determines the velocity reading of a moving particle in the coordinate system. If the unit of measure in a coordinate system changes, the velocity reading of the moving particle in the coordinate system will also change. We can think the other way around: Determine the velocity reading of a particular particle in a coordinate system firstly, and then the unit of measure for this coordinate system is defined. As well as, determines the velocity reading of a same particular particle in two or more coordinate systems, and then the units of measure for these coordinate systems are defined. Because of this reverse thinking, a deterministic reading of the velocity of photons can be used to define the units of measure for two or more coordinate systems, and determine the transformation relation of units of measure between each coordinate system.
According to special relativity, the speed of light in vacuum is constant“ c”for any coordinate system in relative motion, think of it as a theorem. But we take this statement as a hypothesis, then to define separately the units of measure of each coordinate system which is in relative motion, and to found the transformation relation of units of measure between these coordinate systems.
That is to say, it is assumed that in any coordinate system of relative motion, the speed of light in vacuum is constant “c”, thus defining the units of measure in each coordinate system. Thus, the relation of transformation of unit between relative moving coordinate systems is established. You can verify that this transformation is exactly the Lorentz transformation. Why is the transformation relation of four-dimensional space-time Lorentz transformation? It is because the speed of light in vacuum is constant. The special theory of relativity holds that the speed of light in vacuum is constant C, which is an objective law. In fact, it’s just a convention.
2. Derivation of Lorentz transformation in the field of differential areas.
(1) According to the expression of the general theory of relativity, the metrics of a coordinate system should be concerned with the distribution of matter. But special relativity deals with a uniform space where matter is evenly distributed. In uniform space, the units of measure in the same coordinate system and in the same direction are the same. (2) The space of relatively uniform motion is an inertial system. The relationship between the
length of a
line and units of measure is linear. (3) 
In any coordinate system of relatively
uniform motion, the speed of light in vacuum is constant
,
The
transformation relation of units of measure between coordinate systems must be
linear. (4)The transformation of measure and the transformation of the reading
of the length are inverse transformations each other.
It
is assumed that the 
system
is fixed and the
system
is a moving local coordinate system. The instantaneous velocity between the two
coordinate
systems is
. The direction of
is along the
x-axis. The speed of light in vacuum is constant. According to four points above, in the
local
coordinate system we can obtain: 
，
，
，
.Since the
velocity of light in vacuum is constant, get the result
.
By comparison coefficient, get the result
, 
,
. Can solve these: 
. For the origin of thesystem, there is the relation: 
. That is 
, 
. And
then by simple calculation, we get: 
, 
. The result of this just is the
Lorentz transform. But we only follow
the four assumptions above. This can explain that in the local scope of instantaneous motion relative to the fixed coordinate system, there transformation relation of the units of measures is Lorentz transform.
Now, the derivation is the same as the derivation in special relativity, but the difference is that the original 4-dimensional reading is generalized to 4-
dimensional differential components. And change the relative velocities
to instantaneous velocities
.
This generalization emphasizes the
instantaneous and local nature of transformation. Between the local coordinate systems, so that all the way down to one point, the Lorentz transformation still holds.
3. To deduce the relation between the vibration energy of matter wave and the relative velocity of two coordinate systems.
To explain the relativity theory from metrics of measure above, it should objectively reflect the conservation and continuity of energy. But that’s not the
case. The Lorentz transformation goes 
，and it actually has to be changed to
，add a variation
component
, this time
mean of
the variation component 
. Due to the , when observed at any time, the
transformation of units of measure between coordinate systems moving at
relatively uniform speed can remain in the original form of Lorentz
transformation , and can remain the speed of light is always . Given this premise, since the
different instantaneous vibration displacements are observed, so that there is
different vibrational energy to be observed too.
Assumed vibrational energy is
, the energies in system is
, the energies in 
system is 
, then
. In
this way, energy is always conserved and continuous when observing a same mass
point at different motion coordinates.
Specific
analysis is as follows: Assumed the system is fixed and thesystem is a moving local
coordinate system. Take the light quantum point for
example. Because the
Lorentz transformation gives us: 
From this equation, the transformation relation of energy between a same light quantum point
in two coordinate systems of relatively uniform velocity can be obtain.
Lorentz
transformation: 
. 
，
，Where
is the relative velocity between two
coordinate systems.
, This formula is：
.
is the velocity of the light quantum point
in thesystem ,which
should be close to 
as
described above. 
.This formula is:
. As this point, the
energy reading transformation relationship between the two coordinate systems
is：
.
The
kinetic energy of the light quantum point as observed from the system :
The kinetic energy of the light quantum point as observed from the system: 
.
The
kinetic energy of a point at rest in the system observed from the
system:
.
， Kinetic energy reading
of system after conversion to system: 
.
Take
is kinetic energy
reading of the light quantum point observed fromsystem. should be the sum of 
and . is the kinetic energy of the
stationary point in system
observed from system.
is the energy
reading from the kinetic energy of the moving point in system transform
to system. 
According
to the principle of conservation and continuity, the kinetic energy reading of
light quantum point is unique observed from
system.
should be equal
. But the actual calculation is 
. It can only change the
Lorentz transformation to
,
,and
.
It
is investigated fromsystem
that the light quantum produces a vibration with an average displacement of
zero. Let the vibrational energy be ,
For this light quantum point there is 
. namely：
.
We’ve got above
. After simplification,
can get:
.
The velocity of light is an
equiprobability vector. Let’s make sure that it’s vectorized in the direction
of 
, then we get 
.
If
observed from system,
this light quantum point already has vibration energy
, Then, the vibration energy of the light
quantum point observed from the system is
So
the simplifies to this: 
.
As
described above, the energy characteristic of light can expressed more
completely. The energy of light should be 
. 
is determined by
the velocity of light
quantum. It’s a fixed value. But is a variable. The speed of a light quantum
does not increase after. If the energy goes up again, what we see is
an increase in . In
theory it can be increased infinitely, thus ensuring the continuity of the
change in light energy.
There is only one coordinate system for a particle, and that is traditionally called a vacuum state.
In
the coordinate system of the vacuum state, for the particle
,
any other
coordinate system, there is relative motion to the vacuum state, so in
any
other coordinate system
. That is, particles always form
a wave.
can vary depending on the
coordinate system we’re looking at. It fits the Doppler effect. The brief
explanation is as follows: Let the vibrational
energy of the light quantum
point is
. Follow the
derivation of above, get:
is the vibrational energy of this light
quantum point observed from the. We have seen above: 
，we can get it by
simplification：
.
If the light source is in
system, observe from system, 
. Change
to
, the minus sign comes up because is negative
when we look
at it from the system.
If the direction of motion is at an angle 
to the direction of the light, so: 
. This
equation is the
Doppler effect:
.
These all are in
a vacuum，near
the center of the particle 
，
， in
and 
are going to change to 
，but is the relative
velocity between coordinate systems，it doesn’t change.
4. To promote Lorentz transformation.
The
speed of light in optically dense medium is smaller than the speed of light in optically
thinner matter. The speed of light quantum
in optically
dense medium.We
get the same Lorentz transformation using the same derivative process, just one
of them is 
, and 
. 
is the
relative velocity
between coordinate systems,
is the velocity of the light
inside the medium.
The
four –dimensional vectors inside the medium is 
The
operator of the wave equation is 
.
In
addition to translational motion, a mass point has spin motion surround a
straight axis that passes through it. Define spin angular velocity is
. If
the spin angular
velocity of light quantum is appointed 
in all coordinates of relative spin motion,
by a similar derivation as mentioned above, it can
be obtained that the metric transformation relation of spin angle also is Lorentz transformation between two relative uniformly spin coordinate. Only
one of them is
, 
. Please refer to other issues of “spin”
Introduction and Contents 引言和目录