洛伦兹变换（二）

2016-12-21

x'=xut　y'=y　z'=z　t'=t

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.

推导过程

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)

x'=γxvt） (2)

yy'zz'的变换可以直接得出，即

y'=(3)

z'=(4)

t'=γt +(1－γ2x/γv (5)

X=cT (6)

X'=cT' (7)

xx'=γ2(xx'－x'vt+xvt'－v2tt') (8)

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)

x'=γ(xvt)，

y'=y

z'=z

t'=γ(t－vx/c2) (12)

x=γ(x'+vt')，

y=y'

z=z'

t=γ(t'+vx'/c2) (13)

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.

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