第十四章 作为旋转射流的扭量光影泡沫；对于暗物质的一种解释
Chapter 14 Twistor-light-silhouette foam as rotating jets; an explanation for the dark matter
Needham 的书《复分析：可视化方法》 § 3.5.5 让我们可视化地看到了在黎曼球面上的斜驶型莫比乌斯变换.
Section 3. V. 5 of Needham's book Visual Complex Analysis lets us visualize the loxodromic type of Möbius transformation on the Riemann sphere.
扎比内·霍森费尔德在她的书《迷失》中写道:
为了解释现有的宇宙学数据,我们必须假设宇宙中包含两种新的、迄今仍无法解释的成分.其一是暗能量.另外一种成分通常归因于粒子暗物质,可以一起描述为流体.不过,像这样的额外流体引起的效应,也可以是因引力对普通物质的反应并不像我们的方程所预测的那样而引起的.(《迷失》第277页.)
In her book Lost in Math: How Beauty Leads Physics Astray, Sabine Hossenfelder wrote:
To explain the existing cosmological data, we have to assume the universe contains two new, heretofore unexplained components. One of them is dark energy. The other component is usually attributed to particle dark matter, collectively described as a fluid. However, an effect just like this extra fluid could also come about because gravity's response to ordinary matter isn't what our equations predict. (Lost in Math: How Beauty Leads Physics Astray, p. 207.)
约翰·惠勒提出了量子泡沫的概念.他写道:
这种量子泡沫不仅由无数个不断生灭的粒子组成,而且包括因涨落变成泡沫状的弯曲时空本身.149《约翰·惠勒自传：京子、黑洞和粒子泡沫》,第140页.)
John Archibald Wheeler invented the idea of "quantum foam". He wrote:
I found myself forced to invent the idea of "quantum foam", made up not merely of particles popping into and out of existence without limit, but of spacetime itself churned into a lather of distorted geometry.
| 当 h' >> h ,存在着宏观量子泡沫. (8.4.4.3-5) |
| When h' >> h , there exists macroscopic quantum foam. (8.4.4.3-5) |
14.1 总的原理
14.1 An overall principle
(1)
根据(8.4.4.1-3),扭量光影泡沫是流体.根据(8.5-3),扭量光影存在的前提是莫比乌斯环型反引力不等于零.
According to (8.4.4.1-3), twistor-light-silhouette foam is a fluid. According to (8.5-3), the premise for the existence of the twistor-light-silhouette is that Möbius-loop-typed antigravitation is not equal to zero.
想象在宇宙空间有一个快速自转的喷泉,扭量光影泡沫就像喷泉的水.鉴于(8.3.2-3),当阻力足够小时,一个旋转的扭量光影变为众多的旋转射流.在气压低的地方,水容易沸腾.类似地,在阻力(例如万有引力)足够弱的地方,一个旋转的扭量光影容易变为众多的旋转射流.基于(8.6.2-3),由于扭量循环,旋转射流是事件粒子.
Imagine that there is a spinning fountain in outer space, and that twistor-light-silhouette foam is like a fountain of water. In view of (8.3.2-3), if the resistance is small enough, the rotational twistor-light-silhouette becomes rotating jets. Where air pressure is low, water tends to boil. similarly, where the resistance (e.g. gravitation) is weak enough, a rotational twistor-light-silhouette is easy to become rotating jets. Based on (8.6.2-3), the rotating jets are event-particles because of the twistor-light-silhouette cycle.
| 将自转的扭量光影泡沫总体的表面上某一点 P 的切向速率记为 v1 ;对于球面而言, P 在赤道上;对于带边界曲面(例如圆锥面)而言, P 是回转半径的末端点. (14.1(1)-1) |
| Denote the tangential speed of a certain point P on the surface of the whole of the spinning twistor-light-silhouete foam by v1 ; for a sphere, P is at the equator; for a surface with boundary, such as a conical shell, P is the end point of the radius of gyration. (14.1(1)-1) |
将扭量光影泡沫总体的平动速率记为v2 .
Denote the speed of translational motion of the whole of the twistor-light-silhouette foam by "v2".
将"of A",即"(A的)",记为", A";将"of B",即"(B的),记为", B".
Denote "of A" by ", A", and "of B" by ", B".
将扭量光影泡沫总体的惯性质量记为"m".
Denote the inertial mass of the whole of the twistor-light-silhouette foam by m.
| 想象在宇宙空间有一个快速自转的喷泉,扭量光影泡沫就像喷泉的水. 下面假设 1. 一个旋转的扭量光影泡沫全部变成了众多旋转射流; 2. 这些旋转射流是事件粒子; 3. 每股旋转射流的运动方向都弯向时空曲率较大处.因此扭量光影泡沫总体的平动速率最终等于旋转射流的速率. 于是有下列关系: 旋转的扭量光影泡沫总体 = 所有的旋转射流 ; Δv1 = v1 ; v2 = v1 ; 扭量光影泡沫总体的平动速率等于一股旋转射流的速率; 扭量光影泡沫总体的平动加速度等于一股旋转射流的加速度. (14.1(1)-2) |
| Imagine that there is a spinning fountain in outer space, and
that the twistor-light-silhouette
foam is like a fountain of
water. Assuming that 1. the rotational twistor-light-silhouette foam has completely become rotating jets; 2. these rotating jets are event-particles; and 3. the direction of the motion of every rotating jet bends towards where the spacetime curvature is larger, and hence the speed of the translational motion of the whole of the twistor-light-silhouette foam is ultimately equal to the speed of a rotating jet. Hence there are the following relations: the whole of the rotational twistor-light-silhouette foam = all of the rotating jets; Δv1 = v1 ; v2 = v1 ; the speed of the translational motion of the whole of the twistor-light-silhouette foam is equal to the speed of a rotating jet; the acceleration of the translational motion of the whole of the twistor-light-silhouette foam is equal to the acceleration of a rotating jet. (14.1(1)-2) |
关于术语"最终等于",见《可视化微分几何和形式：一部五幕数学正剧》第xii页,
For the term "ultimately equal to", see Visual Differential Geometry and Forms: A mathematical drama in five acts. page xviii.
将加速度记为 a .基于(14.1(1)-2),有
a t = Δv2 . (14.1(1)-3)
Denote acceleration by a . Based on (14.1(1)-2), there is the equation
a t = Δv2 . (14.1(1)-3)
| 旋转的扭量光影泡沫总体的平动的加速度的方向指向时空曲率较大处. (14.1(1)-4) |
| The direction of the acceleration of the translational motion of the whole of the rotational twistor-light-silhouette foam points to where the spacetime curvature is larger. (14.1(1)-4) |
(2)
假定你刚刚以同样的切向速率搅动了一个很大的和一个很小的不锈钢圆盆里的水使其旋转,并且在水面上撒了几片木屑.现在使小水盆漂浮在大水盆里的水的上面.可以看到木屑在上面那盆水里的转动切向速率比在下面那盆水里的大.如果水旋转得快,可以 看到旋转射流.
Suppose that you have just spun the water in both a very big stainless steel mixing bowl and a very small stainless steel mixing bowl at the same tangential speed and put a few bits of wood on the water. Now float the small mixing bowl on the water in the big mixing bowl. You can see the bits of wood revolving at a larger tangential speed on the water in the upper mixing bowl than in the lower one. If the water rotates fast, you can see rotating jets.
想像一个航天器 B 围绕着天体 A 运动.
想象 A 的内部运动只有 A 的扭量光影的量子式的,非全即无的自转,自转的赤道面与 A 的赤道面平行.
想象 B 的内部运动只有 B 的扭量光影的量子式的,非全即无的自转,自转的赤道面与 B 的赤道面平行.
Imagine that a spacecraft, B, moves round a celestial body, A.
Imagine that A's internal motion is only the quantumly whole-or-non rotation of its twistor-light-silhouettes, the equatorial plane of the rotation parallel to that of A. Imagine that B's internal motion is only the quantumly whole-or-non rotation of its twistor-light-silhouettes, the equatorial plane of the rotation parallel to that of B.
A的扭量光影的赤道的切向速率是 v1, A .
The tangential speed of the equator of the twistor-light-silhouette of A is v1, A .
当 B 远离 A 时, A 拖曳着 B 的惯性系,使得 B 与 A 具有同样的内部运动.
When B is far away from A, A is dragging the inertial frame of B, which makes B have the same internal motion as A has.
现在 B 漂浮在 A 的扭量光影上面. B 的内部运动的切向速率相对于A的内部运动的切向速率是 v1 ,于是
v1, B = 2 v1, A . (14.1(2)-1)
Now B is floating on the twistor-light-silhouette of A. The tangential speed of B's internal motion with respect to the tangential speed of A's internal motion is v1 , and hence
v1, B = 2 v1, A .
(3)
|
基于(14.1(1)-2)和(10.2-2),可以得到关系式 Δm = m ( Δv2 )2 / c2 . (14.1(3)-1) |
|
Based on (14.1(1)-2) and (10.2-2), one can obtain the relation Δm = m ( Δv2 )2 / c2 . (14.1(3)-1) |
(4)
根据(14.1(3)-1),基于(8.5-3)和(11.3-4),在 μ时间 的期间扭量光影泡沫总体的动能的变化是
ΔE = (1/2) (Δm) (v2)2 ;
ΔE = (1/2) (m (Δv1)2 / c2) (v2)2 . (在 μ时间 的期间 a反引力 ≠ 0). (14.1(4)-1)
(关于 μ时间 , 见(8.6.2-5).)
According to (14.1(3)-1), and based on (8.5-3) and (11.3-4), the change in the kinetic energy of the whole of the twistor-light-silhouette foam during μtime is
ΔE = (1/2) (Δm) (v2)2 ;
ΔE = (1/2) (m (Δv1)2 / c2) (v2)2 ; ( aantigravitational ≠ 0 during μtime ). (14.1(4)-1)
(For μtime , see (8.6.2-5).)
另一方面,从传统力学的观点来看,在单位时间的期间里扭量光影泡沫总体的动能的变化是
ΔE = (1/2) m (Δv2)2 , (在 μ时间 的期间 a反引力 ≠ 0). (14.1(4)-2)
On the other hand, from the point of view of traditional mechanics, the change in the kinetic energy of the whole of the twistor-light-silhouette foam during μtime is
ΔE = (1/2) m (Δv2)2 , ( aantigravitational ≠ 0 during μtime ). (14.1(4)-2)
考虑对应原理,比较(14.1(4)-1)和(14.1(4)-2)可以得到
(1/2) m (Δv2)2 = (1/2) (m (Δv1)2 / c2) (v2)2 , (在 μ时间 的期间 a反引力 ≠ 0).
In view of the correspondence principle, comparing (14.1(4)-1) and (14.1(4)-2), one gets
(1/2) m (Δv2)2 = (1/2) (m (Δv1)2 / c2) (v2)2 , ( aantigravitational ≠ 0 during μtime ).
上式中所有的量都是正的标量,所以有
Δv2 = (Δv1 / c) v2 , (在 μ时间 的期间 a反引力 ≠ 0);
Δv2 = (v2 / c) Δv1 , (在 μ时间 的期间 a反引力 ≠ 0); (14.1(4)-3)
All the quantities in the above equation are positive scalars. Hence there are the equations
Δv2 = (Δv1 / c) v2 , ( aantigravitational ≠ 0 during μtime );
Δv2 = (v2 / c) Δv1 , ( aantigravitational ≠ 0 during μtime ). (14.1(4)-3)
对于A而言,将(14.1(1)-3)代入(14.1(4)-3)得到
a t = Δv2 = (v2 / c) Δv1 ,
即
a · μ时间 = (v2 / c) Δv1 . (在 μ时间 的期间 a反引力 ≠ 0,对于A而言).
For A, substituting (14.1(1)-3) into (14.1(4)-3), one gets
a t = Δv2 = (v2 / c) Δv1 ,
i.e.
a · μtime = (v2 / c) Δv1 . (aantigravitational ≠ 0 during μtime , in the case of A).
鉴于(14.1(1)-2),得到
a · μ时间 = (v1 / c) v1 ,
a = v12 / (c · μ时间) , (在 μ时间 的期间 a反引力 ≠ 0,对于A而言); (14.1(4)-3A)
基于(14.1(1)-4), a 指向时空曲率较大处.关于 μ时间 , 见(8.6.2-5).
In view of (14.1(1)-2), one obtains
a · μtime = (v1 / c) v1 ,
a = v12 / (c · μtime) , (aantigravitational ≠ 0 during μtime , in the case of A); (14.1(4)-3A)
Based on (14.1(1)-4), a points to where the spacetime curvature is larger. For μtime , see (8.6.2-5).
对于B而言,考虑到(14.1(2)-1),由(14.1(4)-3)得到
a = Δv2, B = (v2, B / c) (2 Δv1, A), (在 μ时间 的期间 a反引力 ≠ 0;对于B而言);
即
a = Δv2, B = (2 v2, B / c) Δv1, A , (在 μ时间 的期间 a反引力 ≠ 0;对于B而言); (14.1(4)-3B)
基于(14.1(1)-4), a 指向时空曲率较大处.关于 μ时间 , 见(8.6.2-5).
For B, in view of (14.1(2)-1), from (14.1(4)-3) one gets
a = Δv2, B = (v2, B / c) (2 Δv1, A), (aantigravitational ≠ 0 during μtime , in the case of B) ;
i.e.
a = Δv2, B = (2 v2, B / c) Δv1, A , (aantigravitational ≠ 0 during μtime ; in the case of B); (14.1(4)-3B)
Based on (14.1(1)-4), a points to where the spacetime curvature is larger. For μtime , see (8.6.2-5).
(5)
将航天器 B 远离地球 A 时的平动速率记为 v¥. 将 A 在某纬度处的旋转射流的速率记为 v1, A .
Denote the speed of the spacecraft B when it is far away from Earth A by v¥ . Denote the speed of the rotating jets of the twistor-light-silhouettes of A at a certain latitude by v1, A .
根据 § 14.1(4), B的扭量光影泡沫总体的动能在 μ时间 的期间经历着变化.
According to § 14.1(4), the kinetic energy of the whole of the twistor-light-silhouette foam of B undergoes a change during μtime .
由(14.1(4)-3B)可以得到下列关系式:
Δv2, B / v2, B = ( 2 v2, B / c ) ( Δv1, A ) / v2, B = ( 2 / c ) ( Δv1, A, 初 - Δv1, A, 末 ) ;
From (14.1(4)-3B) one obtains the following relation:
Δv2, B / v2, B = ( 2 v2, B / c ) ( Δv1, A ) / v2, B = ( 2 / c ) ( Δv1, A, initial - Δv1, A, final ) ;
由(14.1(1)-2),有
Δv1, A, 初 - Δv1, A, 末 = v1, A, 初 - v1, A, 末 .
From (14.1(1)-2), there is
Δv1, A, 初 - Δv1, A, 末 = v1, A, initial - v1, A, final .
因此有
Δv2, B / v2, B = ( 2 / c ) ( v1, A, 初 - v1, A, 末 ) , (14.1(5)-1)
Hence there is
Δv2, B / v2, B = ( 2 / c ) ( v1, A, initial - v1, A, final ) . (14.1(5)-1)
假设 v¥ = v2 .于是得到
Δv¥ / v¥ = ( 2 / c ) ( v1, A, 初 - v1, A, 末 ) , (在 μ时间 的期间 a反引力 ≠ 0). (14.1(5)-2)
Suppose that v¥ = v2 . Then one obtains
Δv¥ / v¥ = ( 2 / c ) ( v1, A, initial - v1, A, final ) , (aantigravitational ≠ 0 during μtime). (14.1(5)-2)
根据(14.1(1)-4),鉴于爱因斯坦的自旋圆盘,在 B 的运动方向时空曲率较大.因此 B 的加速度指向 B 运动的方向.
According to (14.1(1)-4) and in view of Einstein's spinning disc, in the direction in which B is moving the spacetime curvature is larger. Hence B's acceleration points to the direction in which B is moving forward.
(6)
如果涡旋变成混沌系统,或者换言之,如果气球变成一个小飞虫团,里面的小飞虫各自混沌飞舞,或者如果桥被洪水冲裹得团团转,那么根据(8.3.2-3), | ∑R |会大得使反引力不能发生.
If a vortex becomes a chaotic system, or, in other words, if a balloon becomes a ball of small flying insects, each of which dances in chaos, or if the bridge is dragged round and round by floods, then according to (8.3.2-3), | ∑R | will be too large for the antigravitation to occur.
参考文献
References
特里斯坦·尼达姆(Tristan Needham)著;刘伟安 译.《可视化微分几何和形式：一部五幕数学正剧》.北京：人民邮电出版社,2024.1 (2024.3重印).
Tristan Needham. Visual Differential Geometry and Forms: A mathematical drama in five acts. Paperback. Princeton University press, Princeton and Oxford. 2021.
14.2 独立转动中的扭量光影泡沫的平动
14.2
Translational motion of twistor-light-silhouette foam in independent rotation由(14.1(4)-3A)和(14.1(1)-4),自由转动的扭量光影泡沫的旋转射流的加速度是
|
a自由转动 = v12 / ( c · μ时间 ); a 指向时空曲率较大处; (在 μ时间 的期间a反引力 ≠ 0); (14.2-1a) |
a自由转动 = 0 ( | v12 / ( c · μ时间 ) | ≤ | ∑R | ); (14.2-1b)
式中 | ∑R | 是不包括a自由转动在内的当地加速度的矢量和的绝对值; a自由转动量子式地变化,具有非全即无的性质;关于 μ时间 的含义,见(8.6.2-5).
From (14.1(4)-3A) and (14.1(1)-4), the acceleration of rotating jet of twistor-light-silhouettes foam in free rotation is
|
aFreeRotation = v12 / ( c · μtime ); a points to where the spacetime curvature is larger; (agravitational ≠ 0 during μtime); (14.2-1a) |
aFreeRotation = 0 ( | v12 / ( c · μtime ) | | ≤ | ∑R | ); (14.2-1b)
where | ∑R | is the absolute value of the vector sum of the local accelerations, excluding aFreeRotation; aFreeRotation varies in a quantum way, and has the nature of "whole, or none"; for the meaning of μtime, see (8.6.2-5).
问题14.2(1) 日全食期间的引力异常
Problem 14.2(1) Gravity variations during a total solar eclipse (Allais effect)
(1)
在日全食期间,太阳、月亮和地球几乎形成一条直线.这时对于月球而言,由太阳导致的摄动加速度和由地球导致的引力加速度的矢量之和 | ∑R | 有可能如此之小以至于根据(14.2-1a)月球的指向地球的反引力可以发生.
During a total solar eclipse the Sun, the Moon and the Earth almost form a line, when for the Moon it is possible for | ∑R |, the vector sum of the perturbative acceleration brought by the Sun and the gravitational acceleration brought by the Earth, to be so small that, according to (14.2-1a), the Moon's antigravitation pointing to the Earth can occur.
月球自转的赤道切向速率是v1 = 4.627 m/s.基于(8.3.2-3)、(14.1(4)-3A)和(11.3-4),取 μ时间 = 1秒,得到
a自由转动 = v12 / (c · 1 s) = 7.14×10-8 m/s2 ,
a自由转动指向地球.
The Moon's equatorial speed of the rotation is v1 = 4.627 m/s. Based on (8.3.2-3), (14.1(4)-3A) and (11.3-4), taking μtime as 1 second, one obtains
aFreeRotation = v12 / (c · 1 s) = 7.14×10-8 m/s2 ,
and aFreeRotation points to the Earth.
(2)
1. 在日食之前,根据等效原理,重力仪所经受的向上的加速度是 g .
1. Before the solar eclipse, according to the principle of equivalence, the upward acceleration the gravimeter on the Earth experienced was g.
2. 在日全食期间,鉴于(8.5-3)和(8.1-1),月球的扭量光影泡沫挟带着反引力以加速度 7.14×10-8 m/s2 降临地球上的重力仪,因此重力仪所经受的向上的加速度近似为
g - 7.14×10-8 m/s2 .
2. During the total solar eclipse, in view of (8.5-3)和(8.1-1), the Moon's twistor-light-silhouette foam with antigravitation descended at an acceleration of 7.14×10-8 m/s2 to the gravimeter on the Earth, and hence the upward acceleration the gravimeter on the Earth experienced was approximately
g - 7.14×10-8 m/s2 .
2.1. 因此在日全食期间,根据等效原理,地球上的重力仪的读数减少的近似值是
7.14×10-8 m/s2 .
2.1 Hence during the total solar eclipse, according to the equivalence principle, the approximate decrease in the reading of the gravimeter on the Earth was
7.14×10-8 m/s2.
观测值是(7.0±2.7)×10-8 m/s2 .[1],[2]
The observed value is (7.0±2.7)×10-8 m/s2 .[1],[2]
参考文献
References
[1] Phys. Rev. D 62, 041101 (2003), "Precise measurement of gravity variations during a total solar eclipse", Qian-shen Wang, Xin-she Yang, Chuan-zhen Wu, Hong-gang Guo, Hong-chen Liu, and Chang-chai Hua.
[1] Phys. Rev. D 62, 041101 (2003), "Precise measurement of gravity variations during a total solar eclipse", Qian-shen Wang, Xin-she Yang, Chuan-zhen Wu, Hong-gang Guo, Hong-chen Liu, and Chang-chai Hua.
[2] Qian-shen Wang et al. http://www.eclipse2006.boun.edu.tr/sss/paper03.pdf
[2] Qian-shen Wang et al. http://www.eclipse2006.boun.edu.tr/sss/paper03.pdf
问题14.2(2) 先锋10号异常
问题14.2(2) The Pioneer anomaly
当先锋10号飞过了至太阳的距离为rP = 40 AU以后,它就具有附加的,异常的,大致是向着太阳的恒定的加速度.
After Pioneer 10 passed the heliocentric distance rP = 40 AU, it experienced an additional anomalous and approximately Sunward constant acceleration.
先锋10号的主体的半径是R = 1.37 m.它的转速是n = 4.8 r/min,即n = (4.8 / 60) r/s.
The radius of the main body of Pioneer 10 was R = 1.37 m. It spun at the rate n = 4.8 r/min, i.e. n = (4.8 / 60) r/s.
在rP以外,先锋10号频繁地位于太阳和一个海外天体之间,几乎形成一条直线.在rP以外太阳的引力很小.因此对于先锋10号而言,由海外天体导致的摄动加速度和由太阳导致的引力加速度的矢量和| ∑R |频繁地变得如此之小以至于根据(14.2-1a)先锋10号的指向太阳的自由转动反引力加速度频繁地发生.
Beyond rP, Pioneer 10 was frequently between the Sun and a TNO (trans-Neptunian object), the three almost forming a line. The Sun's gravitational force was very small beyond rP. Hence for Pioneer 10, | ∑R |, the vector sum of the perturbative acceleration brought by a TNO and the gravitational acceleration brought by the Sun, frequently became so small that, according to (14.2-1a), Pioneer 10's free-rotation antigravitational acceleration pointing to the Sun occurred frequently.
根据力学,圆锥壳相对于自转轴的回转半径是R回转 = (1/2)(1/2) R. 回转半径的末端点的切向速率是v1 = 2 R回转 n, 即
v1 = 2 ( (1/2)(1/2) R ) n = 0.486 94 m/s.
According to mechanics, for a conical shell, the radius of gyration about the rotation-axis is Rgyration = (1/2)(1/2) R. The tangential speed of the end point of the radius of gyration is v1 = 2 Rgyration n, i.e.
v1 = 2 ( (1/2)(1/2) R ) n = 0.486 94 m/s.
基于式(8.3.2-3)、(14.1(4)-3A)、(11.3-4)和(14.1(1)-4),取 μ时间 = 1秒,可以知道,先锋10号的自由转动反引力加速度是
a自由转动 = (v1)2 / (c · 1s) = 7.91×10-10 m/s2 ,
a自由转动 指向太阳.
观测值是(8.84±1.33)×10-10 ms-2.[1]
Based on (8.3.2-3), (14.1(4)-3A), (11.3-4), and (14.1(1)-4), taking μtime as 1 second, one can know that Pioneer 10's free-rotation antigravitational acceleration was
aFreeRotation = (v1)2 / (c · 1 s) = 7.91×10-10 m/s2 ,
and aFreeRotation pointed to the Sun.
The observed value was (8.84±1.33)×10-10 ms-2.[1]
参考文献
[1] Viktor T Toth and Slava G Turyshev. The Pioneer anomaly: seeking an explanation in newly recovered data.2007. http://arxiv.org/PS_cache/gr-qc/pdf/0603/0603016v2.pdf
References
[1] Viktor T Toth and Slava G Turyshev. The Pioneer anomaly: seeking an explanation in newly recovered data.2007. http://arxiv.org/PS_cache/gr-qc/pdf/0603/0603016v2.pdf
问题14.2(3)
对于暗物质的一种解释Problem 14.2(3) An explanation for the dark matter
1. 在一个星系的外围
1. In the outer regions of a galaxy
太阳的质量是 mS = 1.989 1×1030 kg.假设银盘的质量是m = 1×1012 mS ,银盘半径是R = 15.5 千秒差距, 1 千秒差距 = 3.085 678×1019米.因此m = 1.989 1 × 1042 kg, R = 4.782 8 × 1020 m.
The mass of the Sun is mS = 1.989 1×1030 kg. Assume that the mass of the stellar disk of the Milky Way Galaxy is m = 1×1012 mS. The radius of the stellar disk of the Milky Way Galaxy is R = 15.5 kpc. 1 kpc = 3.085 678 × 1019 m. Hence there is m = 1.989 1 × 1042 kg, and R = 4.782 8 × 1020 m.
考虑到(14.1(1)-1),设银盘的回转半径是 R回转, R回转 的末端点是P, P的切向速率是v.假设银盘形如两个面对面放在一起的圆锥体.假设P的轨道是正圆.
In view of (14.1(1)-1), suppose that the radius of gyration of the stellar disk of the Milky Way Galaxy is Rgyration, the end point of Rgyration being P, and that the tangential speed of P is v. Assume that the stellar disk takes the shape of two cones put together face to face. Assume that the orbit of P is circular.
那么根据力学,回转半径是
R回转 = (3/10)1/2 R,
并且有
v = ( G m / R回转 )(1/2) = 7.119×105 m/s.
Then according to mechanics, the radius of gyration is
Rgyration = (3/10)1/2 R,
and there is
v = ( G m / Rgyration )(1/2) = 7.119×105 m/s.
对于接近银心的星球而言,由于它们的向心引力加速度大,根据(14.2-1b), a自由转动 = 0.
In the case of the stars near the Galactic Centre, since their centripetal gravitational accelerations are large, according to (14.2-1b), aFreeRotation = 0.
在银盘外围,星球的向心引力加速度如此之小以至于(14.2-1a)中的条件可以被满足,这使得在银盘外围自由转动反引力可以突然发生,a自由转动 向内指向银心,仿佛在银盘外围有暗物质.
The centripetal gravitational accelerations of the stars in the outer regions of the stellar disk are so small that the condition in (14.2-1a) is satisfied, which lets free-rotation antigravitation of the stellar disk can occur suddenly in the outer regions of the stellar disk, and aFreeRotation points inwards, towards the Galactic Center, as if there is dark matter in the outer regions of the stellar disk.
基于(11.3-4)可以得到
μ时间 = b G m / ( 2 v3 ) = 1.373×1013 s;
Based on (11.3-4), one gets
μtime = b G m / ( 2 v3 ) = 1.373×1013 s;
并且根据(14.2-1a)和(8.6.2-5)得到
a自由转动 = v2 / ( c μ时间 ) = 1.23 × 10-10 m/s2.
and according to (14.2-1a) and (8.6.2-5), one gets
aFreeRotation = v2 / ( c μtime ) = 1.23 × 10-10 m / s2.
这接近于 MOND[1] 给出的加速度 a0 = 1.2 × 10-10 m / s2 .
This is close to the acceleration a0 = 1.2 × 10-10 m/s2 given by MOND[1].
参考文献
References
[1] Wikipedia. Modified Newtonian dynamics. http://en.wikipedia.org/wiki/Modified_Newtonian_dynamics#The_mathematics_of_MOND
[1] Wikipedia. Modified Newtonian dynamics. http://en.wikipedia.org/wiki/Modified_Newtonian_dynamics#The_mathematics_of_MOND
14.3
从动转动中的扭量光影的平动14.3
Translational motion of twistor-light-silhouettes in driven rotation设 A 的赤道切向速率为 v1, A , 航天器 B 以速度 v2 进入 A 的引力场并在 A 的近旁飞过.
设 B 接近和离开 A 的速度矢量的密切渐近线与 A 的赤道面的交角分别是 δi 和 δo .
Suppose that the equatorial speed of A is vA's TLS-TAN, and that a spacecraft B, entering A's gravitational field at a velocity of v2, flies by A. Suppose that δi and δo are the respective angles that the incoming and outgoing osculating asymptotic velocity vectors make with A's equatorial plane.
A 在某一纬度的自转切向速率 vδ 由下面的公式
vδ = v1, A cos δ (14.3-1)
来确定,式中 δ 是 B 当前的轨道与 A 的赤道面的交角.
The tangential speed of A's rotation at a specific latitude, vδ , is determined by the following equation
vδ = v1, A cos δ , (14.3-1)
where δ is the angle that B's orbit at the moment makes with A's equatorial plane.
a传动转动是B的加速度,a传动转动的方向是沿着B的路径. a传动转动的符号与(cos δi - cos δo)的符号相同.当 ( cos δi - cos δo ) > 0 时B将会加速,就仿佛是人在斜坡桥上向下跑动,当 ( cos δi - cos δo ) < 0 时B将会减速,就仿佛是人在斜坡桥上向上跑动.| ∑R |是不包括a传动转动在内的沿着B的路径的合加速度的绝对值.
And aTransmittedRotation is the acceleration of B; aTransmittedRotation is along B's path, and the sign of aTransmittedRotation is the same as that of (cos δi - cos δo). When ( cos δi - cos δo ) > 0, B accelerates as if one ran downwards along a sloping bridge; when ( cos δi - cos δo ) < 0, B decelerates as if one ran upwards along a sloping bridge. | ∑R | is the absolute value of the resultant acceleration which is along B's path, excluding aTransmittedRotation .
问题14.3(1) 星旁飞越(flyby)的异常现象
Problem 14.3(1) The flyby anomaly
根据(14.1(5)-2),
Δv¥ / v¥ = ( 2 / c ) ( v1, A, 初 - v1, A, 末 ) , (在 μ时间 的期间 a反引力 ≠ 0).
According to (14.1(5)-2),
Δv¥ / v¥ = ( 2 / c ) ( v1, A, 初 - v1, A, 末 ) ,(agravitational ≠ 0 during μtime).
设天体 A 是地球, ωE 是地球的自转角速率, RE是地球的平均半径.
Suppose that the celestial body A is the Earth, and that ωE is the Earth's rotational angular speed, and RE the Earth's mean radius.
则
( v1, A, 初 - v1, A, 末 ) = ωE RE ( cos δi - cos δo ),
(在 μ时间 的期间 a反引力 ≠ 0).
Then
( v1, A, initial - v1, A, final ) = ωE RE ( cos δi - cos δo ),
(agravitational ≠ 0 during μtime).
由(14.1(5)-2),有
Δv¥ / v¥ = ( 2 / c ) ( v1, A, 初 - v1, A, 末 ) = ( 2 / c ) ωE RE ( cos δi - cos δo ),
Δv¥ / v¥ = ( 2 / c ) ωE RE ( cos δi - cos δo ).
From (14.1(5)-2), there is
Δv¥ / v¥ = ( 2 / c ) ( v1, A, initial - v1, A, final ) = ( 2 / c ) ωE RE ( cos δi - cos δo ),
Δv¥ / v¥ = ( 2 / c ) ωE RE ( cos δi - cos δo ).
设比例系数K为
K = ( 2 / c ) ωE RE ,
Let the proportionality coefficient K be
K = ( 2 / c ) ωE RE ,
则
Δv¥ / v¥ = K ( cos δi - cos δo ), (习题 14.3(1)-1a);
(在 μ时间 的期间 a反引力 ≠ 0).
Then
Δv¥ / v¥ = K ( cos δi - cos δo ), (Problem 14.3(1)-1a);
(agravitational ≠ 0 during μtime).
并且
Δv¥ / v¥ = 0 , (在 μ时间 的期间 a反引力 = 0). (习题14.3(1)-1b)
and
Δv¥ / v¥ = 0 , (agravitational = 0 during μtime). (Problem 14.3(1)-1b)
根据(14.1(1)-4),鉴于爱因斯坦的自旋圆盘,在 B 的运动方向时空曲率较大.因此 B 的加速度指向 B 运动的方向.
According to (14.1(1)-4) and in view of Einstein's spinning disc, in the direction in which B is moving the spacetime curvature is larger. Hence B's acceleration points to the direction in which B is moving forward.
(习题 14.3(1)-1a)就是由Anderson等在参考文献[问题14.3(1)-1]和[问题14.3(1)-2]中给出的经验公式.
(Problem 14.3(1)-1a) is the empirical formula given in References [问题14.3(1)-1] and [问题14.3(1)-2] by John D. Anderson et al.
参考文献
References
[问题14.3(1)-1] John D. Anderson, James K. Campbell, John E. Ekelund, Jordan Ellis, and James F.Jordan, Anomalous Orbital-Energy Changes Observed during Spacecraft Flybys of Earth, Physical Review Letters, PRL 100, 091102 (2008).
[问题14.3(1)-1] John D. Anderson, James K. Campbell, John E. Ekelund, Jordan Ellis, and James F.Jordan, Anomalous Orbital-Energy Changes Observed during Spacecraft Flybys of Earth, Physical Review Letters, PRL 100, 091102 (2008).
[问题14.3(1)-2] John D. Anderson et al. www2.phys.canterbury.ac.nz/editorial/Anderson2008.pdf
[问题14.3(1)-2] John D. Anderson et al. www2.phys.canterbury.ac.nz/editorial/Anderson2008.pdf