广义相对论课堂17 不变性原理、相对论电磁学、Einstein等效原理决定度规理论 2012.11.13

J-D. Bancal, S. Pironio, A. Acín, Y-C. Liang, V. Scarani & N. Gisin NATURE PHYSICS | LETTER Quantum non-locality based on finite-speed causal influences leads to superluminal signalling J-D. Bancal, S. Pironio, A. Acín, Y-C. Liang, V. Scarani & N. Gisin

We derive our results assuming that the speed of causal influences v is defined with respect to a privileged reference frame (or a particular foliation of spacetime into space-like hyperplanes). It should be stressed that although the assumption of a privileged frame is not in line with the spirit of relativity, there is also no empirical evidence implying its absence. In fact, even in a perfectly Lorentz-invariant theory, there can be natural preferred frames owing to the non-Lorentz-invariant distribution of matter—a well-known example of this is the reference frame in which the cosmic microwave background radiation seems to be isotropic (see, for example, ref. 17). Moreover, note that there do exist physical theories that assume a privileged reference frame and are compatible with all observed data, such as Bohmian mechanics18, 19, the collapse theory of Ghirardi, Rimini and Weber20 and its relativistic generalization21. Although both of these theories reproduce all tested (non-relativistic) quantum predictions, they violate the principle of continuity mentioned above (otherwise they would not be compatible with no-signalling as our result implies).

不变性原理 以广义坐标变换为例 Kretchmann 1917

形式不变covariance （Galileo、Lorentz、广义） 协变性原理 方程协变covariant 形式不变covariance （Galileo、Lorentz、广义） 协变性原理

无约束 前2式子说明第1式子非协变； 第2式子x'任意，所以有第3式子； 第3式子说明第2式子协变； 不稀奇——依赖于方程写下的形式 作业——牛顿方程改写成广义协变

物理上 实验结果 动力学量=几何体

方程不变invariance 不但形式而且内容 绝对体～常数 纯粹数学常数、物理学常数 函数（坐标变换函数、Jacobian） 分量（闵氏度规） 动力学体——物理变量——依赖物质状态（条件）：粒子位置、动量、场强、能量密度......

例 牛一律：V=常数？a=0 进化到狭义相对论第一定律 诘问Rosser第434页——定律和具体运动方程 Rahilly 三维空间中光线球面——椭球面 例Galilei 径向自由下落vs抛物 Rahilly 三维空间中光线球面——椭球面 仅x维则无问题

维基百科 The relationship between general covariance and general relativity may be summarized by quoting a standard textbook:[2] Mathematics was not sufficiently refined in 1917 to cleave apart the demands for "no prior geometry" and for a geometric, coordinate-independent formulation of physics. Einstein described both demands by a single phrase, "general covariance." The "no prior geometry" demand actually fathered general relativity, but by doing so anonymously, disguised as "general covariance", it also fathered half a century of confusion.

The essential idea is that coordinates do not exist a priori in nature, but are only artifices used in describing nature, and hence should play no role in the formulation of fundamental physical laws. 类似AB effect——potential

动力学方程 不变性原理决定

对应原理 共动系中：四力、四动量、固有时 四力的时间分量是共动系LB到实验室系

EEP决定度规理论

Einstein Equivalence Principle =EEP WEP LLI=Local Lorentz Invariance LPI=Local Position Invariance 第一条和后两条区别local LPI local实验、Position时空点包含时间

WEP 运动学——力学——》其他物理LLI 点质量——SEP FFF——preferred——对应原理LIF The trajectory of a point mass in a gravitational field depends only on its initial position and velocity, and is independent of its composition. 运动学——力学——》其他物理LLI 点质量——SEP FFF——preferred——对应原理LIF

LLI=Local Lorentz Invariance The outcome of any local non-gravitational experiment is independent of the velocity of the freely-falling reference frame in which it is performed. local 非引力=失重自由下落——FFF SEP 例：电磁——精细结构常数测量 速度——相对性原理 SR

LPI=Local Position Invariance The outcome of any local non-gravitational experiment is independent of where and when in the universe it is performed. 局域非引力同LLI——EEP vs SEP 何地——引力红移实验GRE 何时——物理学常数 上两者合起来——时空position 何地——反证 何时——信号传播过程中变化 提问：看上去GRE异地不同，违反LPI？ 非local实验，至少部分用到异地！

Metric theory 1、Spacetime is endowed with a symmetric metric gμν. 2、测地线 The trajectories of freely falling test bodies are geodesics of that metric. 3、local SR = LLI In local freely falling reference frames, the non-gravitational laws of physics are those written in the language of special relativity.

不同Metric theory 不同在于 1怎么来的——决定引力场的场方程不同 Brans-Dicke ——scalar+tensor Einstein GR—— tensor

局部惯性系 条件一：平直时空 条件二：引力=0——例：极坐标(r=1,0)、匀加速系(ksi^1=0) 三阶导数——组合曲率 例子7.2，djvu157——chapter 2 图直观地看(极点俯视),x=a*theta*cos,… d theta, d phi, sin theta=theta-theta^3/3 局域一点(极点)满足2条件