# Lorentz Transformation: A Simplified Overview

Okay…. there are already a lot of articles and books have been written on this topic, but still, I need to cover this for two main reasons. First, my upcoming articles will be based on this topic and I don’t want my readers to have trouble while finding adequate material to understand this topic. Second, as I have also studied many articles and books about this topic, so while reading them I always felt that there is always the ‘student’s side’ of the explanation was missing from the text. So, in this article I will try to include that.

Lorentz transform plays a critical and fundamental role in the space
of relativity. Without it you cannot derive the expressions for the popular
theories like length contraction, time-dilation and popular equation of mass-energy
relation, e = mc^2. So, by this I hope you can understand the gravity this
topic holds.

The Lorentz transformations are named after the Dutch physicist **Hendrick
Lorentz**.

Before, we start the derivation of the transformation, **we need to keep
two postulates in our memory**:

**1)** These transformations will follow **the principle of relativity**. The principle
of relativity states that, all laws of classical mechanics are valid in any two
coordinate systems moving uniformly relative to each other

**2)** The universal **constancy of the speed of light** in vacuum will not be
violated

So, now you are ready to jump into the juicy derivation part….

**Derivation of Lorentz transformations**

Suppose, there are two co-ordinate systems, S and S’. The space and
time coordinate of S are x, y, z and t, and for S’ they are x’, y’, z’ and t’.

Now, suppose, in the course of motion of S’ relative to S in the
x-direction with a speed v, a flash of light is emitted at the origins of S and
S’ when they were just coincident. Let’s identify that instant as **t=0=t’**.

So, for each observer, the flash of light would **spread out like a
spherical wave with speed c according to two postulates,** so for each wave,

Time = Distance / speed

**t = √ (x**²** + y**²** + z**²**) / c**

**c**²** t**²** = x**²** + y**²** + z**² (1)

Distance = √ (x² + y² + z²), because it has 3 axes.

**Equation 1 is called a Spacetime interval equation**.

Now, as the spacetime interval is invariant, we can write,

**x**²** + y**²** + z**²**-**** c**²** t**²** =
x’**²** + y’**²** + z’**²**–**** c**²** t’**²** = 0** (2)

**I will write another article to prove equation 2, so, stay tuned for
that…**

Since the relative motion occurs in the x-direction, the y and z
coordinates should remain unchanged,

**y = y’**

**z = z’**

Now, after substituting this in equation 2, we get,

**x**²** ****-**** c**²** t**²** = x’**²** ****–**** c**²**t’**² (3)

Now, from the figure, at O’, x’ should be 0 but x is vt and at O, x is
0 but x’ is -vt. So, we can write,

**x’ = k (x – vt)
**(4)

**x = k’ (x’ – vt’) **(5)

where, k and k’ are the constants and added for **precise measurement**.

From eq. (4) and (5), we eliminate x’,

**t’ = k [ t – (x/v)(1–1/kk’)] **(6)

Now, from eq. (3), (4) and (6),

**x**²**-**** c**²** t**²** ****≡**** (k**²**)(x**²** – 2xvt + v**²** t**²**) – c**²** k**²** [t**²** – (2xt/v)(1 – 1/kk’) + (x**²**/v**²**)(1 – 1/kk’)**²**] = 0
**(7)

By equation the coefficients of t^2, as eq. (7) is an identity and true
values for all values of x and t, we get,

**– c**²** = k**²** v**²** – k**²** c**²** **

**k = 1 / √ (1 ****–**** v**²** / c**²** ) **(8)

and this equation is known as **Lorentz Factor**

and by equating coefficient of xt,

**0 = –2k**²**v + 2(c**²** k**²** / v) (1 – 1/k k’) **(9)

Now, using Eq. 8 and 9 in Eq. 4
and 6, we get ,

**x’ = x – vt / (√ 1 – v**²**/c**²**) **(10)

**y’ = y
**(11)

**z’ = z **(12)

**t’ = (t – vx/c**²**) / (√ 1 – v**²**/c**²**) **(13)

Eq. 10, 11, 12 and 13 are known as** Lorentz transformation or Einstein –
Lorentz Transformation**.

Also, we can obtain the r**everse transformation** if we replace x’ with x
and t’ with t,

**x = x’ – vt’ / (√ 1 – v**²**/c**²**) **

**y = y’
**

**z = z’ **

**t = (t’ – vx’/c**²**) / (√ 1 – v**²**/c**²**) **

Now from Lorentz and its reverse transforms, if **v << c **then** we can ignore the fraction involving v and c as its value will be much lower than one and subtracting it from 1 will give a result which will be approximately equal to 1 **and thus we
can also obtain the following equations which are also known as **Galilean transformation**,

**x’ = x – vt**

**y’ = y**

**z’ = z**

**t’ = t**

It’s interesting, isn’t ???

In my upcoming articles, I will write about **spacetime interval and how its invariant,
4-vectors, tensors** etc. are very important to understand relativity and
cosmology. So, I will be writing about these gems, very soon…

Once these articles will be available links will be provided here...

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