洛伦兹变换(二)

2016-12-21  鼠标快跑

洛伦兹提出洛伦兹变换是基于以太存在的前提的,然而以太被证实是不存在的,根据光速不变原理,相对于任何惯性参考系,光速都具有相同的数值。爱因斯坦据此提出了狭义相对论。在狭义相对论中,空间和时间并不相互独立,而是一个统一的四维时空整体,不同惯性参考系之间的变换关系式与洛伦兹变换在数学表达式上是一致的,即:

其中xyzt分别是惯性坐标系S下的坐标和时间,x'y'z't'分别是惯性坐标系S'下的坐标和时间。vS'坐标系相对于S坐标系的运动速度,方向沿X轴。

由狭义相对性原理,只需在上述洛伦兹变换中把v变成-vx'y'z't'分别与xyzt互换,就得到洛伦兹变换的反变换式:

洛伦兹变换是高速运动的宏观物体在不同惯性参考系之间进行坐标和时间变换的基本规律。当相对速度v远小于光速c时,洛伦兹变换退化为经典力学中的伽利略变换:

x'=xut y'=y z'=z t'=t

所以,狭义相对论与经典力学并不矛盾,狭义相对论将经典力学扩展到了宏观物体在一切运动速度下的普遍情况,经典力学只是相对论在低速时(v远小于c)的近似情况。一般在处理运动速度不太高的物体时(如天体力学中计算行星的运行轨道),不需考虑到相对论效应,因为用相对论进行处理时计算往往变得非常繁琐,而结果与经典情况相差不大。当处理高速运动的物体时,比如高能加速器中的电子,则必须要考虑相对论效应对结果带来的修正。


Lorenz proposed that Lorenz transformation is based on the premise of the existence of the ether, however, the ether is confirmed to be non-existent, according to the speed of light constant principle, with respect to any inertial reference frame, the speed of light has the same value. Einstein put forward the theory of special relativity. In the theory of relativity, space and time are not independent, but a four-dimensional space-time unity, transform relation and Lorenz transform between different inertial reference system is consistent in mathematical expression, i.e.:

The X, y, Z and T are the coordinates and time of the inertial coordinate system S, X', Y', z'and t' are the coordinates and time of the inertial coordinate system S'respectively. V is the S'coordinate system relative to the S coordinates of the movement speed, the direction along the X axis.

By the special relativity principle, only need to change the V into V, X', Y',, t',, z', y, Z, and X,, t, and, respectively, to obtain the inverse transform of the Lorenz transform:

Lorenz transformation is the basic law of the coordinate and time transformati-on between the macroscopic objects of the high speed motion in different inertial reference systems. When the relative velocity V is much less than the speed of light C, the Lorenz transform is reduced to the classical mechanics:

X'=xut y'=y z'=z t'=t

Therefore, special relativity and the classical mechanics are not contradictory, special relativity will be extended to the general situation of classical mechanics in all velocity under the macroscopic object, classical mechanics theory of relativity only at low speed (V far less than C) approximation of. The general speed is not too high in the processing of motion objects (such as the calculation of planetary orbits in celestial mechanics), without considering the relativistic effect, because the calculation is dealt with by relativity often become very complex, and the situation is similar with the classic. When dealing with high speed moving objects, such as electrons in high energy accelerators, it is necessary to consider the effect of relativistic effects on the correction of the results.


基本公理

狭义相对性原理:一切物理定律(力学定律、电磁学定律以及其他相互作用的动力学定律)在所有惯性系中均有效;或者说,一切物理定律的方程式在洛伦兹变换下保持数学形式不变。

光速不变原理:单向光速是个常数且与光源的运动无关。换言之,在所有惯性系中,真空中的光速不变。

推导过程

洛伦兹变换可以由狭义相对性原理和光速不变原理推导出来。下面根据这两个基本原理,推导坐标的变换式。

设想有两个惯性坐标系S系、S'系,S'系的原点O'相对S系的原点O以速率v沿X轴正方向运动。任意一事件在S系、S'系中的时空坐标分别为(xyzt)、(x'y'z't')。tt'分别是S系和S'系时刻。两惯性坐标系重合时,分别开始计时.

x= 0,则x'+vt' =0。这是变换须满足的一个必要条件,故猜测任意一事件的坐标从S'系到S系的变换为

x=γx'+vt') (1)

式中引入了常数γ,命名为洛伦兹因子。

引入相对性原理,即不同惯性系的物理方程的形式应相同。故上述事件坐标从S系到S'系的变换为

x'=γxvt) (2)

yy'zz'的变换可以直接得出,即

y'=(3)

z'=(4)

把(2)代入(1),解t'

t'=γt +(1-γ2x/γv (5)

在上面推导的基础上,引入光速不变原理,以寻求γ的取值。

由重合的原点OO')发出一束沿X轴正方向的光,设光束的波前坐标为(XYZT)、(X'Y'Z'T')。根据光速不变原理,有

X=cT (6)

X'=cT' (7)

相对论的光速不变原理得出:坐标值X等于光速c乘时刻T,坐标值X'等于光速c乘时刻T'。(1)(2)相乘得

xx'=γ2(xx'-x'vt+xvt'-v2tt') (8)

以波前这一事件作为对象,则(8)写成

XX'=γ2(XX'-X'VT+XVT'-V2TT') (9)

(6)(7)代入(9),化简得洛伦兹因子

γ= (1-(v/c)2)-1/2 (10)

(10)代入(5),化简得

t'=γ(t-vx/c2) (11)

把(2)、(3)、(4)、(11)放在一起,即S系到S'系的洛伦兹变换

x'=γ(xvt),

y'=y

z'=z

t'=γ(t-vx/c2) (12)

根据相对性原理,由(12)得S'系到S系的洛伦兹变换

x=γ(x'+vt'),

y=y'

z=z'

t=γ(t'+vx'/c2) (13)

洛伦兹变换结合动量定理和质量守恒定律,可以得出狭义相对论的所有结论。

爱因斯坦在1905年提出的狭义相对论(一种新的平直时空理论),出发点是两条基本假设:狭义相对性原理和光速不变原理。理论的核心方程式是洛伦兹变换。狭义相对论预言了牛顿经典物理学所没有的一些新效应(相对论效应),如时间膨胀、长度收缩、横向多普勒效应、质速关系、质能关系等,它们已经获得大量实验的直接证明。狭义相对论已经成为现代物理理论的基础之一:一切微观物理理论(如基本粒子理论)和宏观引力理论(如广义相对论)都满足狭义相对论的要求。这些相对论性的动力学理论已经被许多高精度实验所证实 


Basic axiom

The principle of special relativity: all the laws of Physics (mechanics, electromagnetism dynamics law and other laws of interaction) are effective in all inertial systems; or, all the laws of physics equations remain unchanged in mathematical form Lorenz transform.

Principle of constant speed of light: one way speed of light is constant and has nothing to do with the motion of the light source. In other words, the speed of light in a vacuum is constant in all inertial systems.

Derivation process

Lorenz transform can be derived from the special relativity principle and the speed of light invariant principle. According to the following two basic principles, the derivation of the transformation of coordinates.

The assumption is that there are two inertial coordinate system S series, S'series, S' series of the origin O'relative to the origin of the O system S at the rate of V along the X axis of the positive direction of motion. The spatial and temporal coordinates of any event in S system and S'system are (x, y, Z, t), (X', Y', z',, t'). T', t are S series and S' system time. Two the inertial coordinate system coincides with the start time.

If x= 0, then =0 x'+vt'. This is a necessary condition for the transformation to be satisfied, so that the coordinates of any one event can be guessed from the S'system to the S system.

X= gamma (x'+vt') (1)

In the formula, the constant gamma is introduced, named Lorenz factor.

The relativity principle is introduced, that is, the form of the physical equation of different inertial systems should be the same. Therefore, the coordinate transformation from S system to S'system is the

X'= gamma (Xvt) (2)

Y and Y', Z and z' transform can be directly drawn, that is

Y'=y (3)

Z'=z (4)

The (2) into (1), t'solution

T'= gamma T + (1 - 2) x/ gamma V (5)

On the basis of the above derivation, the principle of the speed of light is introduced to search for the value of gamma.

The (X'), which is coincident with the origin O (O'), emits a beam of light along the X axis, and the wave front coordinates (X, Y, Z, T), (, Y',, Z',, T') are located. According to the principle of constant speed of light, there are

X=cT (6)

X'=cT'(7)

The theory of relativity of the speed of light: the coordinate value X is equal to the speed of light C time T, the coordinate value X'is equal to the speed of light C time T'. (1) (2) by multiplying

Xx'= gamma 2 (Xx'x'vt+xvt'v2tt') (8)

To wave front this event as an object, then (8)

XX'= gamma 2 (XX'X'VT+XVT'V2TT') (9)

(6) (7) (9), by simplifying the Lorenz factor

Gamma = (1 - (v/c) 2) 1/2 (10)

(10) by (5), to simplify

T'= gamma (Tvx/c2) (11)

The (2), (3), (4), (11) put together, that is, S system to the S'Lorenz transform

X'= gamma (Xvt),

Y'=y,

Z'=z,

T'= gamma (Tvx/c2) (12)

According to the relativity principle, by (12) the S'system to the S system of the Lorenz transform

X= gamma (x'+vt'),

Y=y',

Z=z',

T= gamma (t'+vx'/c2) (13)

By combining the momentum theorem and the law of mass conservation, the Lorenz transform can obtain all the conclusions of the special relativity.

Einstein's special theory of relativity in 1905 (a new flat space-time theory), the starting point is the two basic assumptions: the special relativity principle and the principle of the speed of light. The core of the theory is the Lorenz transform. Some new effects predicted Newton classical physics have special relativity (relativity), such as time dilation and length contraction, the transverse Doppler effect, the mass velocity relation, mass energy relation, they have a direct proof of a lot of experiments. Special relativity has become one of the basic theories of modern physics. All the micro physical theory (such as the basic particle theory) and the theory of gravity (such as general relativity) meet the requirements of special relativity. These relativistic dynamical theories have been proved by many high precision experiments.



    本站是提供个人知识管理的网络存储空间,所有内容均由用户发布,不代表本站观点。如发现有害或侵权内容,请点击这里 或 拨打24小时举报电话:4000070609 与我们联系。

    猜你喜欢

    0条评论

    发表

    请遵守用户 评论公约

    类似文章 更多
    喜欢该文的人也喜欢 更多