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=Reference frame= 参考系参考系• 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 =只受引力

自由 =不受力

例子— Action!

tv z

gttvz

gtz :FFF2

21

221

相对：

炮弹：

理论上优先性Free fall vs accelerating frame

反向加速！

相对：

引力场固定点：2

21

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gt- z

0z

gtz :FFF

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Hartle 6.4

二 = 》三、 Weak Equivalence Principle

Galileo 原理Universal Free Fall

万有 = 不依赖于组分（本性）GI mm

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E ma m GI

E a mI q

tests• Year Investigator Sensitivity Method• 500? Philoponus [8] "small" Drop Tower• 1585 Stevin [9] 5x10-2 Drop Tower• 1590? Galileo [10] 2x10-2 Pendulum, Drop Tower• 1686 Newton [11] 10-3 Pendulum• 1832 Bessel [12] 2x10-5 Pendulum• 1910 Southerns [13] 5x10-6 Pendulum• 1918 Zeeman [14] 3x10-8 Torsion Balance• 1922 Eötvös [15] 5x10-9 Torsion Balance• 1923 Potter [16] 3x10-6 Pendulum• 1935 Renner [17] 2x10-9 Torsion Balance• 1964 Dicke,Roll,Krotkov [18] 3x10-11 Torsion Balance• 1972 Braginsky,Panov [19] 10-12 Torsion Balance• 1976 Shapiro, et al.[20] 10-12 Lunar Laser Ranging• 1981 Keiser,Faller [21] 4x10-11 Fluid Support• 1987 Niebauer, et al.[22] 10-10 Drop Tower• 1989 Heckel, et al.[23] 10-11 Torsion Balance• 1990 Adelberger, et al.[24] 10-12 Torsion Balance• 1999 Baeßler, et al.[25] 5x10-13 Torsion Balance• cancelled? MiniSTEP 10-17 Earth Orbit• 2015? MICROSCOPE 10-16 Earth Orbit

EovtosDicke

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

AberationFrench

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 relative to each other

Frequency shift

Static 0

Motion with uniform speed

k

γ

accelerating

?

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 =Marzke-Wheeler 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 enoughlatticework

(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

22

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2

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atanh

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2

11

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• Nonrel 牛顿运动学 enough

引力场等效加速系为什么引入加速系？自学笔记预习、讨论

反向加速！

相对：

引力场固定点：2

21

221

gt- z

0z

gtz :FFF

□

□

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时空弯曲• 时间弯曲 warped/curved

• Ohanian: runs slowgeodesicwarp

• Similar on Earth

• In diagram: since curved depicted in flat plane