Inside Story

# GALILEAN AND LORENTZ TRANSFORMATION

Today we are going to talk about a very important concept that is the Galilean and Lorentz Transformation and this concept of transformation is very important for the mathematical explanation of **relativistic physics**. When Einstein put his **special theory of relativity** in September 1905. Then various new concepts get born out of this theory, that is the **time dilatation**, **length contraction**, **mass-energy equivalence**, **relativistic mass.** And all these new concepts can only be explained by using mathematics.

So today we are going to take a look at this foundation mathematics of special theory of relativity, without which special theory of relativity is somehow incomplete. So Let’s starts our talk with **Galilean transformation**.

## GALILEAN TRANSFORMATION

Consider the two frames of reference S and S’ having the coordinates of any point P is (x,y,z) and (x’, y’,z’), initially both frames of reference are at the same position i.e. coincide with each other, where t is zero for both references when they are at the initial stage.

Then if the second frame S’ starts moving in the direction of the x-axis with some **constant velocity** v with respect to the first frame of reference S. (The first frame is a fixed frame of reference).

Here you have to remember that this velocity is very very very small in comparison to the **speed of light**. This is a day-to-day life velocity. Galilean Transformation is applicable only for small moving frame velocities.

If the second frame is moving away from the fixed frame S with v velocity then the time elapsed for any observer which is present at fixed frame S will be “t” but for the observer which is in the moving frame will be t’.

**vt. x’ = x – vt.**if second frame S’ starts moving towards the fixed frame S, then total distance x is given as-

**x = x’ + vt’**

**y’ = y**and

**z’ =z**, and if velocity is very small then t and t’ will also same and there will be no difference in time.

**x’ = x – vt, y’ = y, z’ = z,**and

**t’ = t**

This relationship is called Transformation, and this Transformation is given by

**Galileo**so it is called

**Galilean transformation**. Now let’s talk about

**Lorentz Transformation.**

## LORENTZ TRANSFORMATION

We have talked about the Galilean transformation above, which is applicable only for small velocity. In which S’ is moving with small velocity v wrt fixed frame S. But what would happen if second frame S’ starts moving with the approx of the **speed of light**.

If it is moving with the speed of light then the difference between the t for the fixed frame and t’ for moving frame is significant because time becomes very very slow for moving frame w.r.t to fixed rest frame, which is called **time dilation**. Now let’s talk about the **Lorentz Transformation.**

### DERIVATION FOR LORENTZ TRANSFORMATIONS

Consider the two frames of reference S and S’ having the coordinates of any point P is **(x,y,z,t)** and **(x’, y’,z’,t’)**, initially both frames of reference are at the same position ie. coincide to each other, where t is zero for both references when they are at the initial stage.

Then if the second frame S’ starts moving in the direction of the x-axis with some constant velocity v with respect to the first frame of reference S. And this velocity v is approximately equal to the speed of light, (the first frame is a fixed frame of reference).

Now think that from a light source, a beam of light is to be focused on the point P from the origin of the fixed frame of reference S. And the same light beam is also focused on the point P from the origin of the moving frame of reference S’.

And we know from the above data that the distance of point P from first frame S is x, and from the second frame S’ is x’.

Then these distances can also be expressed as: , from the first frame of reference S, and , from the second frame of reference S’, which is moving.

and from the galilean transformation we know that, , and

But in Lorentz Transformation this relation is written as:

Where k is some constant.

Multiplying these two equations we get,

Putting the value of x and x’ in the multiplication, we get,

And finally we get

and this value of constant k is called **Lorentz factor** which is denoted by γ.

And So,

This is a very important result because it is very helpful to derive the expression for **time dilation**, **relativistic mass**, **length contraction**. So this is all about **Galilean**** and Lorentz transformation.**