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 不讲EEP等效原理! 放到第7章讲

一、牛顿万有引力定律 不符合Lorentz不变性 不相容于SR

Einstein1907年底评述 Thorne P.77-- 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） Lorentz力——场——速度+B的变换 Poisson方程 引力势——源 Lorentz力——场——速度+B的变换 缺万有引力定律平方反比力公式

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 Thorne评论：你我引不出什么结果。追到尽头，索求每一点灵感 Wheeler极限

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=只受引力 自由=不受力 free! 太空舱环绕——下落

例子—Action! 对比加速系只是空间局域有效！——

理论上优先性 Free fall vs accelerating frame □ □ □ □ 两种观点各有优缺点

Hartle 6.4

二=》三、Weak Equivalence Principle Galileo原理 Universal Free Fall 万有=不依赖于组分（本性） Galileo——引力加速独立于质量

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"). 第一种——轨迹——经典vs量子

质荷比——组分=本性nature 深刻！未尽？量子场论多体 引力 电磁学 区分测试粒子和源——决定场

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 近代以前：抛物、摆 扭矩 未来：地球轨道 理论与实验

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 Landau:加速系∞仍有效应----“不等效”！----引力场∞消失, also local vs global

locality! time！ local——space+ time! tidal effects 例题6.1、Problem 6.4 3种摆法 对比加速系只是空间局域有效！—— 3种摆法：1钟无潮汐、2种潮汐分别是反映径向依赖距离拉伸和横向依赖方向挤压

四、Two deductions 1、light ray deflection, curved! 2、gravitational time dilation or clocks slow down

光线偏折 Two viewpoints切换 Curved 图左2：习题6.4 图右2： vs 直线AberationFrench

最基本的引力效应 frequency<time=duration N=VT 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 Motion with uniform speed k γ accelerating ? FF viewpoint——Thorne科普书 推导： 重点是第二个信号到达时下面钟才开始下落！加速钟原理！ 实际上推导要求h 足够小，以至于第二个信号正好在第一个信号到达下面钟时同一个universal坐标时。 时空图表达！

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 enoughlatticework Since we cannot get free particles for our clock, we will have to use the next best thing to a free particle: i.e., a particle in free fall in the gravita- tional field. worry that there are time delays in reflection that depend on the structure of matter, but in principle we can replace the reflection of light by a somewhat more complicated process with zero time delay; for example, an observer riding on each particle checks when the worldlines of light signal and particle coincide。。。 What is meant by "constant distance" or "parallel" worldlines? How can we define "parallel" without using a standard of length to check whether the worldlines are keeping their distance? Fortunately, there is a way to construct parallels which does not involve any length measurements.

(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 既然比较引力场不同地点，必须固定！——固有钟 辅助钟是共动系——局部FFF——坐标时！ dt两式子相同，因为静态引力场！ 沿时间方向平移；Clifford Will第32页： Ohanian图标识错误！

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). 第2版3.6 第1班图5.9

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 时空图推导多普勒效应——第二个脉冲过来时光源移动！

引力场等效加速系 为什么引入加速系？ 自学笔记预习、讨论 □ □ □ □ 两种观点各有优缺点

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 2、a fundamental theorem: If g^ix) is the metric as measured with the geometrodynamic clock, then the motion of particles in free fall is along the geodesies of gw. Proof:II is very near to I; as we will see, in this limit the tidal effects can be ignored and the gravitational field can be taken as zero in the freely falling reference frame. Hence the behavior of geometrodynamic clocks in this reference frame is exactly the same as in the absence of gravitational fields. The first clock is at rest and the second has some velocity v with respect to the reference frame; therefore the latter suffers the familiar time dilation by a factor Vl — v2 relative to the former. According to this, the time indicated by the first clock is at a maximum and therefore certainly stationary. 3、It is a corollary of this theorem that spacetime is curved. We know that in a gravitational field the worldlines of particles are certainly not straight lines. Hence these worldlines can only be geodesies if the space- time is curved. 4、Furthermore, the effect of a gravita- tional field on a particle is entirely accounted for by the warping of space- time: gravitation is geometry. 5、The important role played by the equivalence principle in the con- struction of the geometrodynamic clock and in the question of curvature of spacetime must be emphasized. If different particles were to fall at different rates in a gravitational field, it would not be possible to construct a clock that is independent of particle properties. 6，The above conclusion is perfectly general, i.e., all theories of gravitation must be geometric theories. The only difference between diverse theories is how the metric tensor is related to the gravitational field variables, and how the latter are related to the energy-momentum tensor.