Immersive Translate

一个新的非标准宇宙模型
A new non-standard model of the universe

樊世民
Fan Shimin

摘要：本文首创了一个新的非标准宇宙模型，不仅解释了以往宇宙学三大基本观测现象，对宇宙创生初期的暴涨过程留有存在的空间，还很好地全面地诠释了马赫原理；按照广义相对论原理，又构建了二个引力方程；并对标准宇宙模型阐述了几点不成熟的看法：Robertson-Walker 度规并不完全符合宇宙学原理，也不符合相对论原理。
Abstract:This paper pioneers a new non-standard model of the universe, which not only explains the three basic phenomena of cosmology in the past, leaves room for the explosion process in the early days of the universe, but also explains the Mach principle well and comprehensively. According to the principle of general relativity, two gravitational equations are constructed. He also makes some premature observations about the Standard Universe model: the Robertson-Walker scale does not fully conform to cosmological principles and does not conform to the principles of relativity.

关键词：类光微分几何非标准宇宙学模型引力方程马赫原理
Keywords: light-like differential geometry, non-standard cosmological models, gravitational equations, Mach's principle

自暗能量和暗物质假设提出以来，已历经二十余年，尽管宇宙学已步入精确化时代，但至今仍未有直接证据证实其存在。因为暗能量暗物质不参与电磁、强和弱相互作用，用普通物质制造的仪器也许永远不可能直接观测到其存在。鉴于暗能量和暗物质的特性异常且难以捉摸，加之我对标准宇宙模型持有若干未成熟的见解，本文旨在探索新径，构建一种非标准的宇宙模型。该模型不仅成功解释了哈勃膨胀、微波背景辐射以及轻元素的合成与丰度等观测现象，还深入诠释了马赫原理，并为宇宙创生初期的暴涨过程提供了合理的存在空间。同时，根据广义相对论原理，还引入了两个引力方程。
More than 20 years have passed since the hypothesis of dark energy and dark matter was proposed, and although cosmology has entered the era of precision, there is still no direct evidence to confirm their existence. Because dark energy dark matter does not participate in electromagnetic, strong and weak interactions, it may never be possible for an instrument made of ordinary matter to directly observe its existence. Given the anomalous and elusive nature of dark energy and dark matter, and my somewhat immature insights into the standard model of the universe, this paper aims to explore new ways to construct a non-standard model of the universe. The model not only successfully explains the observed phenomena such as Hubble expansion, microwave background radiation, and the synthesis and abundance of light elements, but also deeply interprets the Mach principle, and provides a reasonable space for the explosion process in the early days of the universe. At the same time, according to the principle of general relativity, two gravitational equations were introduced.

本文分三个部分。第一部分将详细阐述新非标准宇宙模型的构建过程。第二部分将基于广义相对论原理，进一步推导并构建两个新的引力方程。最后，将分享我对标准宇宙模型的一些初步见解与反思。
This article is divided into three parts. The first part will elaborate on the process of building a new non-standard model of the universe. The second part will further derive and construct two new gravitational equations based on the principles of general relativity. Finally, I will share some of my initial insights and reflections on the Standard Universe model.

1.新的非标准宇宙模型
1. New non-standard model of the universe

新的非标准宇宙模型假设：宇宙是一个维度为6、指标数为2的光锥里面的类光超曲面。关于类光子流形的微分几何详见[1]及其它文献，这里只是用[1]中的方法，简要计算出所需要的结论，使用符号的意义和规则也参考[1]。
The new non-standard model of the universe assumes that the universe is an optical-like hypersurface inside a light cone with dimension 6 and index number 2. For more information on the differential geometry of photon-like manifolds, see [1] and other literatures, but here is only a brief calculation of the required conclusions using the method in [1], and the meaning and rules of using symbols are also referred to [1].

在六维时空里，假设曲面：
In six-dimensional space-time, suppose the surface:

$r (X_{1}, X_{2}, X_{3} {,X}_{4} {,X}_{5}, X_{6})$ =

$X_{1}, X_{2}$ 为时间维度， $X_{3} {,X}_{4} {,X}_{5}, X_{6}$ 为空间维度。不难看出：
$X_{1}, X_{2}$ is the temporal dimension, and it $X_{3} {,X}_{4} {,X}_{5}, X_{6}$ is the spatial dimension. It's not hard to see:

超曲面 $r$ 是一个光锥（超曲面）。
sive-translate-walked="d1437900-6a7d-475f-96dc-d1bb61bc1195"> hypersurface r is a light cone (hypersurface).

令 $u_{1} =R$ ; $u_{2} =$ $t$ ; $u_{3} = φ_{1}$ ; $u_{4} = φ_{2}$ ; $u_{5} = φ_{3}$
t $u_{2} =$ t ; $u_{3} = φ_{1}$ ; $u_{4} = φ_{2}$ ; $u_{5} = φ_{3}$

分别计算出 $\frac{∂r}{∂ u_{i}}$ , $\frac{∂r}{∂ u_{i}} * \frac{∂r}{∂ u_{j}}$ , $i$ , $j$ 依次分别等于1,2,3,4,5, 便可计算出超曲面 $r$ 的度规矩阵，即第一基本式：
r be equal $\frac{∂r}{∂ u_{i}} * \frac{∂r}{∂ u_{j}}$ to 1, 2, 3, 4, 5, The degree matrix of the hypersurface r can be calculated , i.e., the first basic formula:

度规矩阵的第一行和第一列均为零，反映了超曲面是一个类光超曲面。6维时空指标数为2里面的类光超曲面是一个的维度为4指标数为1的超曲面。
The first row and the first column of the degree matrix are zeros, reflecting that the hypersurface is a light-like hypersurface. The 6-dimensional spatiotemporal index number is 2, and the light-like hypersurface in it is a hypersurface with a dimension of 4 and an index number of 1.

第一基本形式也可以这样写出：
The first basic form can also be written like this:

$Ⅰ$

降低维度之后：
After lowering the dimension:

或者写成： $Ⅰ_{ij}$ = $g_{ij}$
Or write as: $Ⅰ_{ij}$ = $g_{ij}$

取ξ= $\frac{∂r}{∂ u_{1}}$ =
Take ξ = $\frac{∂r}{∂ u_{1}}$ =

作为类光超曲面上的法向量，先计算 $\frac{∂^{2} r}{{d u_{i}}^{2}}$ , 再计算 $\frac{∂^{2} r}{{d u_{i}}^{2}} *ξ$ ，便可得出超曲面的第二基本式：
As the normal vector on the light-like hypersurface, we can obtain the second basic formula for the hypersurface by calculating $\frac{∂^{2} r}{{d u_{i}}^{2}}$ first and then calculating $\frac{∂^{2} r}{{d u_{i}}^{2}} *ξ$

第二基本形式也可以这样写出：
The second basic form can also be written like this:

$[B]$ = $\frac{-1}{R}$ $[g]$ 或者写成： $B_{ij}$ = $\frac{-1}{R}$ $g_{ij}$ = $Ⅱ_{ij}$
$[B]$ = $\frac{-1}{R}$ $[g]$ or written: $B_{ij}$ = $\frac{-1}{R}$ $g_{ij}$ = $Ⅱ_{ij}$

通过计算 $∂ξ$ ， $∂ξ*∂ξ$ ，可以计算出出超曲面的第三基本式：
By calculating $∂ξ$ , $∂ξ*∂ξ$ the third basic formula for hypersurface can be calculated:

第三基本形式也可以这样写出：
The third basic form can also be written like this:

$[Ⅲ]$ = $\frac{1}{R^{2}}$ $[g]$ 或者写成： $Ⅲ_{ij}$ = $\frac{1}{R^{2}}$ $g_{ij}$
$[Ⅲ]$ = $\frac{1}{R^{2}}$ $[g]$ or written: $Ⅲ_{ij}$ = $\frac{1}{R^{2}}$ $g_{ij}$

可以看出，类光超曲面是一个 $ρ_{1} = \frac{-1}{R}$ 的全脐超曲面。根据[1]，[2]，[3] 里面的证明，全脐类光超曲面的Ricci张量Ric和Ricci标量曲率a的计算方法与黎曼几何的全脐超曲面基本相同，只是其值减少了一半：
As you can see, the light-like hypersurface is a $ρ_{1} = \frac{-1}{R}$ full umbilical hypersurface. According to the proofs in [1], [2], [3], the Ricci tensor Ricci and the Ricci scalar curvature a of the full umbilical hypersurface are calculated in the same way as the full umbilical hypersurface of Riemannian geometry, except that its value is reduced by half:

类光超曲面上面的爱因斯坦张量 $G_{ij}$ :
Einstein's tensor $G_{ij}$ above the light-like hypersurface:

把爱因斯坦张量 $G_{ij}$ 代入爱因斯坦引力方程 $G_{ij}$ = $k_{1} T_{ij}$ ^{[1], [4]}。
Substituting Einstein's tensor $G_{ij}$ into Einstein's gravitational equation $G_{ij}$ = $k_{1} T_{ij}$ ^{[1], [4].}

$k_{1}$ 为比例常数， $T_{ij}$ 为能量动量密度张量。类光超曲面是一个 $ρ_{1} = \frac{-1}{R}$ 的全脐超曲面，它的表面是静态的，所以 $T_{ij}$ ^{[4], [5]}：
$k_{1}$ is the proportionality constant, which is the $T_{ij}$ energy momentum density tensor. A light-like hypersurface is a $ρ_{1} = \frac{-1}{R}$ full-umbilicus hypersurface, and its surface is static, so $T_{ij}$ ^{[4], [5]}:

通过计算，不难得出：
Through calculations, it is not difficult to deduce:

可以看出，假如宇宙物质的平均密度 $ρ$ 永远是正的话，宇宙的平均压强P永远是负的。这也不难理解，由于万有引力的作用，宇宙间的物质永远是相互吸引的，所以平均压强P永远是负的，即便是在宇宙初期也是如此。随着尺度因子 $R$ 在不断变大，平均密度 $ρ$ 在按 $\frac{3}{2 R^{2}}$ 比例在快速地减少，平均压强P的绝对值也在按 $\frac{3}{2 l^{2}}$ 比例在快速地减少。
It can be seen that if the average density $ρ$ of matter in the universe is always positive, the average pressure P of the universe is always negative. It is not difficult to understand that due to the effect of gravity, matter in the universe is always attracted to each other, so the average pressure P is always negative, even at the beginning of the universe. As the scale factor $R$ increases, the average density $ρ$ decreases rapidly and proportionally $\frac{3}{2 R^{2}}$ , and the absolute value of the average pressure P decreases rapidly and proportionally $\frac{3}{2 l^{2}}$ .

假设宇宙以基本均匀的速度在膨胀，即u = $^{dR} /_{dt}$ = = b $R=$ * t =b*t
Suppose the universe is expanding at a fundamentally uniform rate, i.e., u = $^{dR} /_{dt}$ = = b $R=$ * t = b * t = b * t

哈勃常数 H = $^{} /_{R}$ 。根据观察，现在的哈勃时间t₀ = H₀^-1 = 138亿年。
Hubble's constant H = $^{} /_{R}$ . According to observations, the present Hubble time t₀ = H ^0-1 = 13.8 billion years.

假如宇宙在产生后不久,轻元素合成后，宇宙间的粒子包括光子就基本不变了。携带质量的粒子之间的距离平均来讲在不断地变大，这些粒子之间基本上不存在碰撞，也不存在热运动。恒星是在后来的演化过程中通过万有引力慢慢地产生的。决定宇宙温度基本上与这些携带质量的粒子无关，只与光速运动的粒子有关。宇宙中光速运动的粒子是主要光子。
If the universe is created shortly after the light elements are synthesized, the particles in the universe, including photons, will basically remain unchanged. On average, the distances between particles carrying mass are increasing, and there are basically no collisions between these particles and no thermal motion. Stars are slowly created through gravitational attraction in later evolutionary processes. The temperature of the universe is essentially determined by these mass-carrying particles, only by particles moving at the speed of light. Particles moving at the speed of light in the universe are the primary photons.

根据Stefan-Boltzmann 公式 $ρ_{r}$ = $σ T^{4}$ 和 $ρ_{r}$ = $\frac{3}{2 R^{2}}$ , $ρ_{r}$ 指宇宙中热辐射那一部分的能量密度， $R=$ b*t，可以得出，温度的平方 $T^{2}$ 与时间t 成反比：
According to the Stefan-Boltzmann formula $ρ_{r}$ = $σ T^{4}$ and $ρ_{r}$ = , $\frac{3}{2 R^{2}}$ $ρ_{r}$ Refers to the energy density of that part of the universe of thermal radiation, $R=$ b*t, and it can be concluded that the square $T^{2}$ of temperature is inversely proportional to time t

$* t_{1} = * t_{0}$

当现在的宇宙温度即微波背景辐射 $T_{0}$ 被认为是2.7度，时间参数 $t_{0}$ 为138亿年，而氢原子的解耦温度 $T_{1}$ 被认为在3000度左右时，经计算，氢原子解耦时间 $t_{1}$ 大概是几万年左右。
When the current cosmic temperature, the microwave background radiation $T_{0}$ , is considered to be 2.7 degrees, the time parameter $t_{0}$ is 13.8 billion years, and the decoupling temperature $T_{1}$ of the hydrogen atom is considered to be around 3000 degrees, the hydrogen atom decoupling time is calculated $t_{1}$ It's probably about tens of thousands of years.

关于宇宙初期轻元素的合成时间与机制，其计算方法与标准模型大致相同，故此处不再赘述。
As for the synthesis time and mechanism of light elements in the early universe, the calculation method is roughly the same as that of the Standard Model, so I will not repeat it here.

该非标准宇宙模型中的时空是平直的，时间和空间随尺度因子 $R$ 一起膨胀。从运动学上计算，是不产生红移的。红移的产生是因为原子中的电子的重量随着尺度因子 $R$ 变大而变大，从而产生了红移。
Space-time in this non-standard model of the universe is flat, and time and space expand with the scale factor $R$ . Kinematically, there is no redshift. Redshift occurs because the weight of the electrons in an atom increases as the scale factor $R$ increases, resulting in a redshift.

宇宙总的质量M 与平均密度 $ρ$ 、半径的立方成正比。宇宙半径与尺度因子 $R$ 成正比。 M $∝$ $ρ$ * $R$ ³ 。而 $ρ$ = $\frac{3}{2 R^{2}}$ 所以M $∝$ $R$ 。万有引力中的势能是负的，宇宙总的质量的增加可以认为是从势能绝对值的增加中产生的。宇宙中总质能与势能之间基本保持平衡状态。据此推测，在宇宙暴涨的初始时刻，其总质量可能极小甚至为零。持这种观点的物理学家或宇宙学家也不在少数，如霍金，费曼等等，他们在不同的场合下均讲过类似的问题。美国物理学家爱德华·特赖恩甚至还计算出，整个宇宙的总质量在暴涨初只有28克！这篇论文据说发表在自然杂志上面。
The total mass M of the universe is proportional to the cube of the average density $ρ$ and radius. The radius of the universe is proportional to the scale factor $R$ . M $∝$ $ρ$ * $R$ ³ 。 And = $\frac{3}{2 R^{2}}$ so M $∝$ $R$ . $ρ$ The potential energy in gravitational force is negative, and the increase in the total mass of the universe can be considered to be generated from the increase in the absolute value of potential energy. The total mass energy and potential energy in the universe are basically in equilibrium. It is speculated that at the initial moment of the universe's inflation, its total mass may be extremely small or even zero. There are also physicists or cosmologists who hold this view, such as Hawking, Feynman, etc., who have spoken about similar problems on different occasions. United States physicist Edward · Tryon even calculated that the total mass of the entire universe was only 28 grams at the beginning of the explosion! The paper is said to have been published in the journal Nature.

我们已经假设了宇宙在产生后不久，宇宙间的粒子数量包括光子就基本不变了。所以可以认为，不仅宇宙总的质量M随着尺度因子 $R$ 增加而增加，每一个原子中电子的质量m 和原子核的质量也与尺度因子 $R$ 一起增加。从任何一本量子力学的教科书上均可找到，电子变轨时发出光子的能量与电子的质量m成比，每个光子能量与频率成正比，与波长成反比。原子核中的质子和中子的质量比电子大很多，可以不计原子核的质量变化对光谱的影响。最后，电子发出光的波长与宇宙尺度因子 $R$ 成反比：
We have assumed that soon after the creation of the universe, the number of particles in the universe, including photons, remained basically unchanged. Therefore, it can be assumed that not only does the total mass M of the universe increase with the scale factor $R$ , but the mass m of the electrons and the mass of the nucleus of each atom also increase with the scale factor $R$ . As can be found in any textbook of quantum mechanics, the energy of the photons emitted when an electron changes its orbit is proportional to the mass m of the electron, and the energy of each photon is proportional to the frequency and inversely proportional to the wavelength. The mass of protons and neutrons in the nucleus is much larger than that of electrons, and the effect of mass changes in the nucleus on the spectrum can be ignored. Finally, the wavelength at which the electrons emit light is inversely proportional to the cosmic-scale factor $R$ :

$λ_{1} * R_{1}$ = $λ_{0} * R_{0}$

红移z = $\frac{λ_{1} - λ_{0}}{λ_{0}}$ = $\frac{R_{0}}{R_{1}}$ – 1 这个公式与标准宇宙学模型计算的结果相同。
The formula for redshift z = $\frac{λ_{1} - λ_{0}}{λ_{0}}$ = $\frac{R_{0}}{R_{1}}$ – 1 is the same as that calculated by the standard cosmological model.

目前我们仅能通过观测获得当前的哈勃常数H0，而对于宇宙整个演化过程中的哈勃常数，尤其是宇宙初期的值，仍属未知。换句话说，我们其实知道的很少。然而，无论哈勃常数H的具体值如何，本模型均为其提供了合理的存在空间。
At present, we can only obtain the current Hubble constant H0 through observation, and the value of the Hubble constant in the entire evolution of the universe, especially the value at the beginning of the universe, is still unknown. In other words, we don't really know much. However, regardless of the specific value of the Hubble constant H, this model provides a reasonable space for its existence.

2.新的引力方程
2. New gravitational equations

研习广义相对论的人不可能没有学习过微分几何。学习过微分几何的人都知道，描写空间弯曲的方法除了用第一基本形式 $Ⅰ_{ij}$ 、度规张量 $g_{ij}$ 外，还可以用第二基本形式 $Ⅱ_{ij}$ 、张量 $B_{ij}$ ，第三基本形式 $Ⅲ_{ij}$ 来刻画空间的弯曲。简单地说广义相对论原理：物质的分布决定了时空弯曲，时空弯曲反映了物质的分布。爱因斯坦引力方程就是用爱因斯坦张量来反映时空的弯曲。而爱因斯坦张量又是由度规张量 $g_{ij}$ 转化而来的。为什么不直接使用 $g_{ij}$ ，可以看有关这方面的资料。
It is impossible for someone who studies general relativity not to have studied differential geometry. Anyone who has studied differential geometry knows that in addition to the first basic form $Ⅰ_{ij}$ , the gauge tensor $g_{ij}$ , the second basic form $Ⅱ_{ij}$ , the tensor $B_{ij}$ , and the third basic form can be used to describe the curvature of space $Ⅲ_{ij}$ to depict the curvature of space. To put it simply, the principle of general relativity: the distribution of matter determines the curvature of space-time, and the curvature of space-time reflects the distribution of matter. Einstein's gravitational equation uses the Einstein tensor to reflect the curvature of space-time. And the Einstein tensor is a transformation of the gauge tensor $g_{ij}$ . Why not use $g_{ij}$ it directly, you can see the information about this.

仿照爱因斯坦的办法，可以再构建两个引力方程：
Following Einstein's method, two more gravitational equations can be constructed:

适当的改变第三个引力方程 $k_{3}$ 的正负号和大小值，在原条件下，第三个引力方程得出的结果与爱因斯坦引力方程得出的结果无任何异议。下面分析第二个引力方程与爱因斯坦方程在相同条件下的差异。
Appropriately change the plus and minus signs and magnitude values of the third gravitational equation $k_{3}$ , and under the original conditions, the results of the third gravitational equation are not in any dispute with the results obtained by Einstein's gravitational equation. The following is an analysis of the differences between the second gravitational equation and the Einstein equation under the same conditions.

通过第二个引力方程，不难得出：
Using the second gravitational equation, it is not difficult to deduce:

根据Stefan-Boltzmann 公式 $ρ_{r}$ = $σ T^{4}$ 和 $ρ_{r}$ = $\frac{1}{R}$ ,可以得出，温度 $T^{4}$ 与尺度因子 $R$ 成反比，而 $R=$ b*t，所以： $* t_{1} = * t_{0}$ ,因此，氢原子解耦时间 $t_{1}$ 会比用爱因斯坦引力方程计算的结果更短。轻元素的合成时间也是如此。
According to the Stefan-Boltzmann formula $ρ_{r}$ = $σ T^{4}$ and $ρ_{r}$ = $\frac{1}{R}$ , it can be concluded that the temperature $T^{4}$ is inversely proportional to the scale factor $R$ , and $R=$ b*t, so $* t_{1} = * t_{0}$ , hence, the hydrogen atom decoupling time $t_{1}$ It will be shorter than the result calculated with Einstein's gravitational equation. The same goes for the synthesis time of light elements.

由于技术上的原因，氢原子解耦时间 $t$ 和轻元素的合成时间目前仍然是一个不可观测的量。即使未来成为可观测量，也不应轻易因模型与实验不符而淘汰该模型。每个模型从原理上讲，都是一样的，是平等的。
yle="font-family: imitation song; min-height: 10.5pt; font-size: 10.5pt; " lang="null" dim_w='1'> For technical reasons, the decoupling time of hydrogen atoms t and the synthesis time of light elementsIt is still an unobservable quantity. Even if it becomes measurable in the future, it should not be easily phased out because the model does not match the experiment. Each model, in principle, is the same and equal.

此时，宇宙总的质量M便与长度参数 $R^{2}$ 成正比。按照前面讲的，原子中的电子的质量也与尺度因子 $R^{2}$ 成正比。因此：
In this case, the total mass M of the universe is proportional to the length parameter $R^{2}$ . As mentioned earlier, the mass of the electrons in an atom is also proportional to the scale factor $R^{2}$ . Therefore:

$λ_{1} * {R_{1}}^{2}$ = $λ_{0} * {R_{0}}^{2}$

红移z = $\frac{λ_{1} - λ_{0}}{λ_{0}}$ = $\frac{{R_{0}}^{2}}{{R_{1}}^{2}}$ – 1
Redshift z = $\frac{λ_{1} - λ_{0}}{λ_{0}}$ = $\frac{{R_{0}}^{2}}{{R_{1}}^{2}}$ – 1

爱因斯坦引力方程中的比例系数 $k_{1}$ 是结合牛顿引力理论确定的。新的引力方程中的 $k_{3}$ 可以通过与第一引力方程计算的结果相互比较确定。新的引力方程中的 $k_{2}$ 现在还不知道怎么确定。
The proportional coefficient in Einstein's gravitational equation is determined in conjunction with Newton's theory of gravity. $k_{1}$ The new gravitational equation can be determined by comparing the results calculated with $k_{3}$ the first gravitational equation. The new gravitational equation is $k_{2}$ not yet known to be determined.

在相同的边界条件下，三个引力方程得出的结果在大体上保持一致，但具体结论之间存在差异。目前，尚不清楚如何解释这种差异，以及何种条件下三个方程会得出相同的结果。
Under the same boundary conditions, the results of the three gravitational equations are generally consistent, but there are differences between the specific conclusions. At the moment, it is not clear how to explain this difference and under what conditions the three equations will give the same result.

3.几个不成熟的观点：
3. A few immature points:

3.1Robertson-Walker 度规 ${ds}^{2} =- {dt}^{2} + R^{2} (t) [\frac{{dr}^{2}}{1-k r^{2}} + r^{2} ({dθ}^{2} + {sin}^{2} θ {dφ}^{2})]$ 并不完全符合宇宙学原理。宇宙学原理假定，在宇观尺度下，三维空间在任何时刻都是均匀且各向同性的。在微分几何的语境中，这表示三维空间上任意一点的任意方向上的法曲率均相同，即三维空间是全脐的。当 $k$ =1 或 $k$ =0 时三维空间是全脐的，符合宇宙学原理。 $k$ = -1 时三维空间不是全脐的^[6]173。三维弯曲的空间不好想象，以二维弯曲的空间为例，曲率 $k$ = -1 的二维空间是一个类似喇叭形的曲面，曲面上每一点的最大法曲率和最小法曲率包括方向和大小都是不一样的，在最大法曲率方向和最小法曲率方向中间的任何方向上，其法曲率比最大值小，比最小值大，但是它们是不可能相同。仅在每一点上，最大法曲率与最小法曲率的乘积保持恒定。所以 $k$ = -1 的空间不是全脐的，也不符合宇宙学原理。
k ed="d1437900-6a7d-475f-96dc-d1bb61bc1195">k Robertson-Waler gauge ${ds}^{2} =- {dt}^{2} + R^{2} (t) [\frac{{dr}^{2}}{1-k r^{2}} + r^{2} ({dθ}^{2} + {sin}^{2} θ {dφ}^{2})]$ does not fully conform to cosmological principles. The principles of cosmology assume that three-dimensional space is uniform and isotropic at any given moment on a cosmic scale. In the context of differential geometry, this means that the normal curvature is the same in any direction at any point in three-dimensional space, i.e., three-dimensional space is fully umbilicus. When k=1 or k=0 the three-dimensional space is fully umbilicus, in accordance with the cosmological principle. k= -1 when the three-dimensional space is not full umbilicus^[6]173. Three-dimensional curved space is not easy to imagine, taking two-dimensional curved space as an example, the two-dimensional space with curvature k= -1 is a trumpet-like surface, and the maximum normal curvature and minimum normal curvature of each point on the surface are different in the direction and magnitude, in the direction of maximum normal curvatureand the normal curvature of any direction in the middle of the direction of the minimum normal curvature, the normal curvature is smaller than the maximum value and greater than the minimum value, but they cannot be the same. Only at each point, the product of the maximum normal curvature and the minimum normal curvature remains constant. So the space of k= -1 is not fully umbilicus and does not conform to cosmological principles.

3.2.Robertson-Walker 度规也不符合相对论原理。在相对论里面,时间和三维空间都是四维时空的一个维度，可以相互转换，不区分的。但是在计算Robertson-Walker 度规时，一开始就将时间和空间分开了。正确的做法是找出四维时空（或叫超曲面）r（ $t,r,θ,φ$ ）每个参数与五维欧几里得时空（ $t, x_{1}, x_{2,} x_{3}, x_{4}$ ）每个坐标之间的关系：
3.2. The Robertson-Walker gauge also does not conform to the principle of relativity. In the theory of relativity, time and three-dimensional space are both dimensions of four-dimensional space-time, which can be converted into each other without distinction. But when calculating the Robertson-Walker gauge, time and space are separated at the outset. The correct approach is to find out the relationship between each parameter of the four-dimensional space-time (or hypersurface) r( $t,r,θ,φ$ ) and each coordinate of the five-dimensional Euclidean space-time ( $t, x_{1}, x_{2,} x_{3}, x_{4}$ ).

$t$ = $r_{t}$ （ $t,r,θ,φ$ ） … … $x_{4}$ = $r_{4}$ （ $t,r,θ,φ$ ）

分别计算 $dt, {dx}_{1}, {dx}_{2,} d x_{3}, {dx}_{4}$ , 再计算曲面的度规张量:
Calculate $dt, {dx}_{1}, {dx}_{2,} d x_{3}, {dx}_{4}$ separately and then calculate the gauge tensor of the surface:

${ds}^{2} =- {dt}^{2} + {dx}_{1}^{2} +d {x_{2}}^{2} + {dx}_{3}^{2} + {dx}_{4}^{2}$

= $f_{t} (t,r,θ,φ) {dt}^{2} +……+ f_{3} (t,r,θ,φ) {dφ}^{2}$

按照正确的计算方法，是不可能计算出了Robertson-Walker 度规的。在Robertson-Walker 度规 ${ds}^{2} =- {dt}^{2} + R^{2} (t) {dΩ}^{2}$ 中，至少还少一个对时间的微分项：
It would not have been possible to calculate the Robertson-Walker gauge with the correct calculation method. In the Robertson-Walker scale ${ds}^{2} =- {dt}^{2} + R^{2} (t) {dΩ}^{2}$ , there is at least one less differential term for time:

$Ω^{2} * {(t)}^{2} * {dt}^{2}$ ，即 ${ds}^{2} =- {dt}^{2} + R^{2} (t) {dΩ}^{2}$ + $Ω^{2} * {(t)}^{2} * {dt}^{2}$ 。
$Ω^{2} * {(t)}^{2} * {dt}^{2}$ , i.e ${ds}^{2} =- {dt}^{2} + R^{2} (t) {dΩ}^{2}$ . + $Ω^{2} * {(t)}^{2} * {dt}^{2}$ .

当然了，在物理学中，可以假设宇宙的度规遵循Robertson-Walker 度规。
Of course, in physics, it can be assumed that the universe follows the Robertson-Walker scale.

3.3研习广义相对论的人同样都知道，爱因斯坦方程的解是不能确定的，除非再增加一个坐标条件。增加的这个坐标条件，带有很大的随意性，道理也并不充分。每一个物理学家各有各的坐标条件。
3.3 Anyone who studies general relativity also knows that the solution of Einstein's equations cannot be determined unless an additional coordinate condition is added. The increase in this coordinate condition is very arbitrary, and the reason is not sufficient. Every physicist has their own coordinate conditions.

产生这种现象的原因就不分析了，但是，用第二和第三引力方程便不存在这个问题。在标准模型中能够计算结果，是因为引入了时空高度对称性。
The reasons for this phenomenon will not be analyzed, but this problem does not exist with the second and third gravitational equations. The results can be calculated in the Standard Model because of the introduction of a high degree of symmetry in space-time.

3.4新的非标准宇宙模型对马赫原理有了一个较好的诠释。过去，马赫原理并未得到足够的重视，它不仅是物理学中的一个基本原理，也是科学研究的重要方法论。无法观测的物理概念是无意义的。时空与物质紧密相连，不存在脱离物质的时空。物质和时空，物质与物质，紧密地联系在一起，并可以相互转换。每一个粒子的质量与周围物质的分布密切相关。引力即惯性力，在该模型中，惯性力即质量的变化是引力的变化而引起的。
3.4 The new non-standard model of the universe provides a good interpretation of the Mach principle. In the past, the Mach principle was not given enough attention, and it is not only a basic principle in physics, but also an important methodology for scientific research. Unobservable physical concepts are meaningless. Space-time is closely connected with matter, and there is no space-time separated from matter. Matter and space-time, matter and matter, are closely linked and can be converted into each other. The mass of each particle is closely related to the distribution of the surrounding matter. Gravitational force is the inertial force, and in this model, the change in inertial force, i.e., mass, is caused by the change in gravity.

3.5本文在构建模型时未采用暗能量假设。目前，关于暗物质的问题仍未找到确切的解决方案。
3.5 The dark energy assumption is not used in the construction of the model. At the moment, there is still no definite solution to the problem of dark matter.

浩瀚的宇宙广阔无垠，宇宙存在的时间估计也在138亿年以上。宇宙有多大，现在根本不知道。从基本粒子到原子，再到分子、宏观物体、地球、太阳系、银河系、宇宙，其结构复杂得难以想象，我们也知之甚少。但是，人们还是想用一个方程来描述宇宙演化的过程，这多少有点可笑。我也是这可笑的人之一。本文提出的非标准宇宙模型，在解释宇宙膨胀、微波背景辐射及轻元素合成等方面，虽也达到与标准模型相当的水平，但仍属初步探索，存在诸多待完善之处。我们期待各位专家学者的批评指正，共同推动宇宙学的进步与发展。
The vast universe is vast, and the universe is estimated to have existed for more than 13.8 billion years. How big the universe is, now it is simply not known. From elementary particles to atoms, to molecules, macroscopic objects, the Earth, the solar system, the Milky Way, the universe, its structure is unimaginably complex, and we know very little. However, it is somewhat ridiculous that people still want to use an equation to describe the evolution of the universe. I'm one of those ridiculous people. Although the non-standard cosmic model proposed in this paper has reached a level comparable to the standard model in terms of explaining the expansion of the universe, microwave background radiation and light element synthesis, it is still a preliminary exploration and there are many things to be improved. We look forward to the criticism and correction of experts and scholars to jointly promote the progress and development of cosmology.

宇宙学任重而道远。
Cosmology has a long way to go.

参考文献：
References:

[1]Krishan L Duggal, Bayram Sahin. Differential Geometry of Light like Submanifolds[M]. Birkhäuser. Basel. Boston. Berlin [1st Edition 2010]

[2] Dae Ho Jin. LIGHTLIKE HYPERSURFACES OF A SEMI-RIEMANNIAN MANIFOLD OF QUASI-CONSTANT CURVATURE[J]. Commun. Korean Math. Soc. 27 (2012), No. 4, pp. 763–770

http://dx.doi.org/10.4134/CKMS.2012.27.4.763

[3] Cyriaque Atindogbe, et al. LIGHTLIKE OSSERMAN SUBMANIFOLDSOF SEMI-RIEMANNIAN MANIFOLDS[J].

http://arxiv.org/abs/1006.1329v1

[4] 梁灿彬，周彬，等. 微分几何入门与广义相对论[M].北京：科学出版社 2006年现代物理基础丛书 7
[4] Liang Canbin, Zhou B, et al. Introduction to Differential Geometry and General Relativity[M].Beijing:Science Press, 2006, Fundamentals of Modern Physics Series 7

[5]马天.从数学观点看物理世界--几何分析、引力场与相对论[M].北京：科学出版社2012年
[5] Ma Tian. The physical world from a mathematical point of view--geometric analysis, gravitational field and relativity[M].Beijing:Science Press,2012

[6]纪永强.子流形几何[M]. 北京：科学出版社 2004年1月
[6] Ji Yongqiang. Submanifold geometry[M]. Beijing: Science Press, January 2004

fsm1122334455@outlook.com

465654224@qq.com