In-depth analysis of the curvature of general relativity

                                                                        ——Metric transformation function(2)

The “flat space” and “cured space” are not just two space measured by two units of measure, they also have another physical meaning, that is what the forces are doing. For the same distribution surface of light quantum, if it is measured by the flat metric, and the speed of light is constant and the density is constant, then it is considered that the light quantum system is not affected by any action of external force or internal force.  It corresponds to the plane.

Another situation, if it is measured by the curved metric and the speed of light varies with the spatial density which varies with the spatial position, then it is considered that the light quantum system is subjected to the action of external force or internal force.  

This external force or internal force is called “polymerization stress”.

For the distribution surface of light quantum, if it is measured by the fiat metric,  without considering the polymerization stress, the original virtual state of the distribution surface is obtained. However, if the curved metric is used, considering the action of polymerization stress, the distribution surface obtained is the real state.

1. Strain tensor of light quantum system:

Suppose that surfaceⅠis part of the distribution surface of a light quantum system passing through any point  in space.

Surface Ⅰ: ,, are a set of parameters of orthogonal curvature line network. They are curvature network parameters of curved space.

Plane Ⅱis the same part of the same distribution surface measured by the flat metric past point , it is a virtual surface corresponding to surface Ⅰ. We can also think of the planeⅡas the strain become the curved surfaceⅠunder polymerization stress. 

 Suppose there is a displacement vectoron surface Ⅰ, corresponding to a displacement vectoron plane Ⅱ. ，

.

The reading obtained as measured by the curved metric and flat metric, there are                                                 

，，.

，. , etc. shown as the reading as measured by the curved metric. , etc. shown as

 the reading as measured by the flat metric.

Set 、 shown as the metric tensor of curved metric and flat metric respectively.  is the modulus reading of

 measuring with the curved metric.  is the modulus reading of  measuring with the flat metric.

.

In the curved surfaceⅠ, there is . In the planeⅡthere is

., are the coefficient of differential forms of two surfaces respectively.

。

According to the above, can be regarded as the reading of the modulus of  of the light quantum system on curved

 surfaceⅠunder polymerization stress.  should be the strain tensor before and afire subjected by the polymerization stress of the light quantum on the surface Ⅰ surrounding point .

We can think of light quantum as a continuous elastic medium. According to the mechanics of continuous elastic media, the relationship between stress

 tensor and strain tensor in continuous elastic is as follows: .

The strain tensor on surfaceⅠis .

.

For the Metric transformation function of free particle , take ， is a microscopic quantities, with

values estimated in the range of , and therefore Taylor expansion can be used:

.

The strain tensor on curved surfaceⅠ is .  

For a free particle, the distribution surface of the light quantum system should be the distribution surface of three

 dimension space. Corresponding to the free particle, let curved surfaceⅠmentioned above is shown as surface , and

 the other two surface of three dimensional space are shown as surface  and surface . The strain tensor on

surface  is . The strain tensor of surface and surface  can also be obtained in the same way, they are

， respectively.

2. Stress tensor of light quantum system:

Let’s first assume that: the light quantum system is considered as a continuous elastic medium. In the continuous

 elastic medium of the light quantum, according to Hooke’s theorem,

The relationship between isotropic stress tensor and strain tensor is:

. Where,are Lamet’s constants. In matrix form, the relationship between the stress tensor and the strain tensor is: 

 .  

The stresses represented by this term are the normal direction of the three surfaces.

It is a set of equilibrium field for a particle at rest, the particle is not accelerating. We can think of it as self-

equilibrium at point P, this item can be excluded from the stress tensor formula.

If formal  take this term out of equation, it shows on the premise that the light quantum

 system is a completely elastic mediums, the stress tensor of the distribution surface of the light quantum system due to strain can be the matrix:

,  for the static particle in the material field.

3.  Einstein’s stress tensor:

According to the above, we go from planeⅡto curved surfaceⅠcorrespondingly, that is from the flatness space corresponding to the curvature space. The force applied in this process is the polymerization stress. In the flat space it is applicable to Newton’s mechanics. In curved space it is applicable to Einstein’s mechanical. Therefore, Einstein’s stress tensor must be used to analyze the polymerization stress on the surface. Einstein’s stress tensor formula is:

 . 

Ⅰ. According to differential geometry, there is a curvature tensor in surface:

.

It’s going to be in surface by Gauss’s law:

..

Take,

.

Then get：

。

Because we’re using orthogonal curvature line coordinates,

 when, . .  When,.

When,.  

Use relationships in differential geometry.

For the principal curvature there are. In the surface of，：

。

In the surface of , ，it has nothing to do with .  The parameter is 0，

.  represent the curvature tensoron the surface .

In the same way, In the surface of , it has nothing to do with .  The parameter is 0，：

，

。

In the same way, in the surface of ,：

,

。

In the above formal of surfaces ，,,

the superscripts of ，, when correspond to surfaces，,.

Now back to Einstein’s stress tensor formula for the matter field.

According to the view of general relativity, the stress tensor on the surface of apace in the matter field should be related to the curvature parameters of the surface. Einstein’s stress tensor formula:

. For the material field , the distributed surface of free particles is surface of three dimensional, which should be

 discussed from surface ，,respectively. And the polymerization stress on each surface should also be in three dimensions. Einstein’s stress on surface  is , its three components are ，，..

, , .

, It’s the Einstein stress tensor on surface.

Thus, the stress tensors of the minimal surface of distribution of light quantum systems ，, on orthogonal

directions can be obtained by using the same procedure. They are ，

， respectively.

The Einstein stress tensor is three-dimensional, and equation is the EINSTEIN STRESS TENSOR FOR POINT  ON THE

 SURFACE ，,.

Since the particles are at rest equilibrium, the distribution surface of light quantum in plane is strained by the

 polymerization stress in the matter field. The resulting stress tensor is shown as matrix. However For the same distribution surface, in the curved space, the stress tensor in the material field calculated by Einstein’s stress

tensor formula is shown as matrix. it has now been proved that (1）=（2）, that just is . So it’s on the

 same surface, because this formula is true, it shows that the results of analyzing the stress tensor of particles from the point of view of light quantum systems are in full accord with the stress tensor expression of Einstein.

 Since Einstein’s stress tensor expression is generally accepted to be correct. Then it should be correct too to

 analyze the strain and stress relationship from the viewpoint of light quantum system. This also shows that the assumption that the light quantum system is a continuous elastic medium is also correct. In the process of derivation,

 we apply the metric transformation function, the correctness of the metric transformation function, and it was tested in the derivation are also correct too.

In short, flat space is Deborian space, and curved space is Einstein space.They are the same kind of physical space. Because of Two

 different measurement space formed by two different measurement units.

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