# I m confused in 2 query 1- describe lorentz transformation equation and 2- derive the lorentz transformation between 2 inertial systems in which 1 is in motion is both ques is same or diff if same help me in solving this if different also help me plz dont post any links becaz in links ans is not properly and diagram r not there means i m not able to see diagrams plz help me

*S*with Cartesian coordinates

*(x,y,z,t)*and a moving frame of reference w.r.t.

*S*with velocity

*v*in

*x*-direction. The frame of reference coincide at .

*Derivation of lorentz in my opinion is not required until higher grade under grad studies. It can be approached in various methods, one involves modifying galilean transformation equations, another is bit shorter but needs understanding of tensor arithmetic.*

## An expanding sphere of light

Deriving the Lorentz transformations from first principles, is one of a flash of light originating at the origin of two frames of reference S and S’ which are moving relative to each other with a velocity . We set up our model so that at time the origins of the two frames of reference are in the same place.

The flash of light will expand as a sphere, moving with a velocity in both frames of reference, in accordance with Einstein’s 2nd principle of relativity. For reference frame S we can write that the square of the radius of the sphere is so

For the reference frame S’ we can write that

These two equations must be equal, as it is the same sphere of light and therefore the sphere must have the same radius in the two reference frames. Let us see if we can transform from one to the other using the Galilean transforms, which are

Expanding the brackets of the right hand side gives

As we can see, the two expressions are not equal as the left hand side has the extra terms . This means that a Galilean transformations does not work. The extra terms involve a combination of and , which suggests that both the equations linking and *and* and need to be modified, not just the equation for as is the case in the Galilean transformations.

## Modifying the Galilean transformations

Let us assume that the transformations can be written as

We need to find the values of and which correctly transform the equations for the expanding sphere of light. We do this by substituting equations (3) and (4) into equation (2). Before we do this, we note that the origin of the primed frame is a point that moves with speed as seen in the unprimed frame S. Therefore its location in the unprimed frame S at time is just . So we can write equation (3) as

Re-writing equation (3)

Now we substitute this expression and equation (4) into equation (2)

Equating coefficients:

From equations (5) and (6) we can write

and

Multiplying equations (8) and (9) and squaring equation (7) we get

so

so

Thus we can write

Using equation (8) we can write

so

Taking the negative square root we can write

From equation (9) we can write

which leads to

and so

which is the same as .

If we define

we can write

Thus we can finally write our transformations as

These are known as the Lorentz transformations.

Regards.

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