Lorentz transformation

Lorentz transformation . Transformation of the coordinates of space and time, which allows the description of electromagnetic phenomena to pass from one fixed system to another equipped with constant speed.

Explanation

Suppose two inertial reference systems A and B where the length and time scales are the same. The system B moves with a speed v with respect to A along the coincident axes x and X in such a way that for the instant t = T = 0 the origins of the coordinate axes are at the same point.

Suppose, also, that for the instant t = T = 0 at the coordinate origin there was a flash of light, then after a certain time t in the system A the light will reach the points that are in the sphere of radius ct, analogously, also in system B after a time T the light will travel the distance cT. In other words, for the system A the points of the luminous sphere will satisfy the equation.

And in system B, the equation

This follows from Einstein’s postulates .

Considering that space and time are homogeneous, we assume that there is a linear relationship between the coordinates and the time of the different systems. Then between the x and x coordinates the following dependency is possible:

From this it follows that point X = 0 (the reference origin of system B) moves with speed v with respect to system A and at time t = T = 0 the points x = 0 and X = 0 coincide. The gamma magnitude is for now an unknown coefficient that for v much less than c must be made equal to unity, as in the Galileo transformations ; gamma, apparently, is a function of v and c.

The y, Y and z, Z coordinates must not vary during the movement of the systems along the x axis, or

Y = y, Z = z (IV)

As also happens when making the transformation of Galileo.

The time T in system B will depend linearly on time t and the coordinate x in system A; therefore we will assume that:

T = at + bx (V)

Where a and b are unknown constants that being v much less than c, they must take the values: a = 1 and b = 0.

Substituting III, IV and V in II we will obtain:

It is required to choose the values ​​of the gamma coefficients, a and b in such a way that VI is equal to I. Obviously for this the following equalities must be satisfied

Operating with these and the previous equations we obtain the following values ​​for said coefficients:

In this way we arrive at the transformation of the coordinates of system A with respect to those of system B which are known today as the Lorentz transformation.

 

 

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