Find out how the velocity synthesis formula special relativity is derived

Take, for example, high-speed trains. Let the speed of the car to the ground be v, and the Zheng people on the car will move relative to the train at the speed u along the direction of the train of the town god and the wind, then his speed relative to the ground u is.

v+u)/(1+(vu)/c^2)

If the temporal and spatial coordinates in the Lorentz transform describe the motion of an object, then Einstein's formula for adding velocity is obtained by removing the three spatial coordinate transformations with the time transformation (the first derivative of time is obtained for the three formulas of the Lorentz transformation):

ux'=(ux－v)/(1－vux/c²)

uy'=uy(1－v²/c²)1/2/(1－vux/c²)

uz'=uz(1－v²/c²)1/2/(1－vux/c²)

where (ux',uy',uz') for the object in k'The velocity in the system is along (x',y',z'The component of the axis, (ux, uy, uz) is the corresponding velocity component of the object in the k system, and v is k'The velocity tied to the k-system, where v should be (vx,0,0).

It is recommended to refer to the textbook "Electrodynamics", which is not suitable for copying this large section. If you first learn the Lorentz transformation and accept the fact that the speed of light does not change, it will be easy to derive the relevant transformation formula.

The derivation of the relationship between mass and velocity is as follows: the S' system (where a small ball at rest A', mass M0) moves with velocity V along the x-axis relative to the S system (where a small ball at rest A, mass M0) is positively moving, and the mass of A' relative to the S system is m, and according to the symmetry of the system, the mass of A relative to the S' system is also m.

Assuming that the two small balls collide and become one, the relative velocity of the S' system is U' and the relative velocity of the S system is U, and the law of conservation of momentum holds in both reference frames, the S system: mv=(m+m0)u, and the s' system: -mv=(m+m0)u'.

From the velocity synthesis formula, u'=(u-v) (1-uv c 2), and according to the symmetry of the system, u'=-u, we get: (v u) 2-2v u+(v c) 2=0, solution: v u=1 (1-v 2 c 2), since v >u, v u=1+ (1-v 2 c 2).

So m=m0 (v u-1)=m0 (1-v 2 c 2).

Velocity synthesis formula: v(x)=dx dt= (dx-ut) ( (dt-udx c 2)) =(dx dt-u) (1-(dx dt)u c 2) =(v(x)-u) (1-v(x)u c 2) The same can be done for v(y),v(z).

According to the coordinate transformation relation in the Lorentz transform, x'=γ（x-ut），t'= (t-ux c 2), by velocity is the first derivative of the coordinates versus time.

Get v'=dx'/dt'=d (x-ut) d (t-ux c 2), where time t is the independent variable.

u is a constant, so the equation can be reduced to v'=(dx-udt) (dt-udx c 2), fraction.

Divide the top and bottom by dt at the same time to get v'=(v-u) (1-uv c 2), i.e. the formula for the superposition of velocity in the x-direction. The same can be said for the direction of y and z.

The derivation of velocity begins first with the relativity of time:

In the diagram above, O is the frame of reference and A is the frame of inertia moving with velocity v relative to O. B is a point on the A system.

Suppose: When A moves to the moment of life with O, a photon is emitted from A to B.

In A, the path of the photon is CT', in o look, the distance of the photon is ct , and a has moved the distance of vt in t time t.

The relationship between the three distances is:

ct')²＋vt)²＝ct)²；Pythagorean theorem).

After a simple equation transformation, the Lorentz transform is obtained: t'＝t√(1－v²/c²)

This is the formula for the time transformation obtained by standing on the O system and observing the time on the A system.

It should be noted that the motion is relative, when O is the frame of reference, the velocity of A is V, and conversely, when A is the frame of reference, then the velocity of O is V, so the two velocities are equal and do not need to be converted.

In order to distinguish which system is used as the frame of reference, we temporarily denote the observation time as the (which is regarded as stationary) frame of reference as t-static, and the time of the system under test as t-motion. In this way, the above conclusion is written: t dynamic t static (1 v c), which has the advantage of not confusing the time of the frame of reference with the time of the observed system.

If another system b, b has a certain velocity u relative to system a, what is the time to observe b on system o?

Obviously, the observed velocity of B by A is based on the time system of A, and the velocity of B measured by A is based on the relativity of A at rest and B as motion. Therefore, in turn, the velocity at which b observes a must be u. It's a simple truth, just like when you see people in the distance become smaller, people in the distance look at you and you also get smaller.

The ratio of getting smaller is the same.

Since the velocity distance is time, the conversion of time determines the transformation of velocity.

As above: a observes the time of b in the same form: t dynamic t static (1 v c), except that v here is changed to u, because it is the relative velocity between a and b.

From the above analysis, it can be seen that from the speed of the O system to the speed of the B system, the time needs to be transformed twice. The conversion of distances requires not only two Lorentz transformations, but also the effect of the optical path difference generated in the middle after one transformation.

At present, there are many such derivation processes, and the results are not the same, I have seen some derivation processes, and I feel that most of them ignore certain factors. At present, the accepted derivation results are relatively close to the experimental results: VX'＝(vx－ux)/(1－ux×vx/c²)

And note that u is relative to the velocity of the system a, so u is positive and negative, u is positive and negative, and the calculated results are different.

The specific derivation process is too complicated, and there are too many calculus formulas involved in the middle, which cannot be typed, so they are not written out one by one. The relevant article can be consulted. But don't blindly assume that those derivatives are correct, and if you have studied calculus, you can also try to derive them yourself.

The ultra-simple, Lorentz transformation of the special theory of relativity is just that.