广义相对论课堂2
牛顿引力与SR不相容
自由落体思想及推论
2012.9.13
一、牛顿万有引力定律
不符合Lorentz不变性
不相容于SR
引力万有——惯性系
Einstein1907年底评述
Thorne P.77--
Lorentz不变×
• 瞬时反应、超距作用
（等价于场Ohanian第
2章末节）
– Poisson方程
– 引力势——源
• Lorentz力——场——
速度+B的变换
同是平方反比力
荷只正不负、只吸引不排斥
论文项目
• 引力波带走能量为负——》辐射源自身能
量+——》辐射强度增大
• 标量、矢量、对称张量场MTW §7习题
– 严重缺陷与实验结果不符
– 或不自洽且无精确解
– 1950-60s相对论场论修正回到GR
– Ohanian
– Zeldovich
二、Einstein自由落体思想：
如果一个人自由下落，他将感受
不到自己的重量。
活动：回忆3-5个自由落体情景
——亲身或眼见
失重——太空舱
• 代入——身感+想像—
—思想实验
• 失重——引力消除—
—相对于谁？
– 自由漂浮
– 不漂浮的教室?
– 完美惯性
Reference frame=参考系
• 自由、惯性
• 颠倒了牛顿力学观念
——表格
• “下落”——类似加速包
含了减速
Two
cases
Newton Einstein
引力场
中静止
合力=0 合力=-“
引力”
自由下
落
合力=只 自由=不
受引力 受力
例子—Action!
FFF : z  gt
 1 2
炮弹：z  vt  2 gt

相对：z  vt
1
2
2
理论上优先性
Free fall vs accelerating frame
□
□
□
□
FFF : z  12 gt 2
引力场固定点：z  0
相对：z  - gt
1
2
反向加速！
2
Hartle 6.4
二=》三、Weak Equivalence
Principle
Galileo原理
mI Universal
mG
Free Fall
万有=不依赖于组分（本性）
多种表述wikipedia
•
The trajectory of a point mass in a gravitational field depends only on its initial
position and velocity, and is independent of its composition.
•
All test particles at the alike spacetime point in a given gravitational field will
undergo the same acceleration, independent of their properties, including their rest
mass.[2]
•
All local centers of mass vacuum free fall along identical (parallel-displaced, same
speed) minimum action trajectories independent of all observable properties.
•
The vacuum world line of a body immersed in a gravitational field is independent of
all observable properties.
•
The local effects of motion in a curved space (gravitation) are indistinguishable
from those of an accelerated observer in flat space, without exception.
•
Mass (measured with a balance) and weight (measured with a scale) are locally in
identical ratio for all bodies (the opening page to Newton's "Principia").
质荷比——组分=本性nature
深刻！未尽？量子场论多体
• 引力
• 电磁学
mI a  mG E
mI a  qE
tests
•
•
•
•
•
•
•
•
•
•
•
•
•
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•
•
•
•
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Year Investigator
Sensitivity
500? Philoponus [8]
"small"
1585 Stevin [9]
5x10-2
1590? Galileo [10]
2x10-2
1686 Newton [11]
10-3
1832 Bessel [12]
2x10-5
1910 Southerns [13]
5x10-6
1918 Zeeman [14]
3x10-8
1922 Eötvös [15]
5x10-9
1923 Potter [16]
3x10-6
1935 Renner [17]
2x10-9
1964 Dicke,Roll,Krotkov [18] 3x10-11
1972 Braginsky,Panov [19] 10-12
1976 Shapiro, et al.[20]
10-12
1981 Keiser,Faller [21]
4x10-11
1987 Niebauer, et al.[22]
10-10
1989 Heckel, et al.[23]
10-11
1990 Adelberger, et al.[24] 10-12
1999 Baeßler, et al.[25]
5x10-13
cancelled? MiniSTEP
10-17
2015? MICROSCOPE
10-16
Method
Drop Tower
Drop Tower
Pendulum, Drop Tower
Pendulum
Pendulum
Pendulum
Torsion Balance
Torsion Balance
Pendulum
Torsion Balance
Torsion Balance
Torsion Balance
Lunar Laser Ranging
Fluid Support
Drop Tower
Torsion Balance
Torsion Balance
Torsion Balance
Earth Orbit
Earth Orbit
Eovtos
Dicke
Eöt-Wash group
@Washington Univ. @Seattle
注意精度提升！
弱等效原理
“等效” ≠ 等同
FFF=LIF
1、locally, exactly a point, no frame
2、globally, G cannot be canceled out
• 加速系∞有效----“不等效”！----引力场∞消失
locality!
time！
• local——space+ time!
– tidal effects
– 例题6.1、Problem 6.4
• 3种摆法
四、Two deductions
1、light ray deflection, curved!
2、gravitational time dilation or
clocks slow down
光线偏折
• Two viewpoints切换
– 图左2：习题6.4
– 图右2：
• Curved
– vs 直线
AberationFrench
引力时间膨胀
引力红移
最基本的引力效应
frequency<time=duration
N=VT
引力时间膨胀
• 1、FF viewpoint——
fastest derivation
• （1）gravitational
redshift——from
viewers on Earth
– Doppler effect
– v= g(h/c)
– k=1+v/c—+O(2) 一阶
效应
Cases
Frequency
relative to shift
each other
Static
0
Motion with k
uniform
speed
γ
acceleratin ?
g
讨论twin paradox (Kleppner 12)
• 最早？
• 赵凯华
Doppler效应
源-接收器相对运动
起因不同
Doppler、SR
引力
宇宙学
Physics unclear
• how to measure proper time along any worldline
that belongs to a particle in free fall？
• a nearby geometrodynamic clock =MarzkeWheeler coordinates
Problem 4.6
• By means of the geometrodynamic clock and the
radar-ranging procedure outlined in Fig. 5.5 we
can measure the spacetime interval between
these points.
close enoughlatticework
(2)、auxiliary clock辅助钟推导
• t坐标时，固有时
• 两条世界线——套
• 平移=静态
Physical meaning课外作业阅读笔记
• Signals!
• This shows that if clock II is at the higher potential (ДФ >
0), then dr2 is larger than drx. The signals sent out from
clock I at one-second intervals arrive at clock II with
intervals larger than one second. Clock I, which is
deeper in the gravitational potential, runs slow. This
effect does not depend on the type of signal that is used
(light flashes, cannonballs, messenger rockets, periodic
radio wave, etc.). It is only important that each signal
have exactly the same motion as the preceding one
(same shape of worldline).
Result conditions:
• 1、 valid to lowest order in the
gravitational field strength.
• 2、 if the potential Ф
is time dependent and changes
appreciably in the time needed to send the
signal from one clock to the other, then
fails.
2、viewpoint 2nd
• Nonrel牛顿运动学
enough
h
Vg
c
V
h
g 2
c
c
1V
  1  
2 c 
2
V
a 0
ah ah
 tanh
 tanh 2  2
c
c
c
c
引力场等效加速系
为什么引入加速系？
自学笔记预习、讨论
□
□
□
□
FFF : z  12 gt 2
引力场固定点：z  0
相对：z  - gt
1
2
反向加速！
2
时空弯曲
•
•
•
•
时间弯曲warped/curved
Ohanian: runs slowgeodesicwarp
Similar on Earth
In diagram: since curved depicted in flat
plane