# Understanding the Invariancy of Space-Time Interval Equations with Mathematical Proof!!!

In the last article, I explained about **Lorentz Transformation** in a very simple way and in this article, we are going use that beast.

In this article, I going to explain the **invariancy of Spacetime
interval equations in a mathematical way**.

But before you continue, you have to read my articles on **Introduction to Space-Time** and **Lorentz Transformation** to understand this article in a better
way.

Let’s get started….

As I have explained **in my article on Introduction to Space-time**, as it
sounds, made up of two critical words, **space and time**. **Space**, as per the
**traditional physics**, represents the** 3-dimensional coordinate system of our
universe**. Scientists **before Einstein** believed that our universe is **limited to
these 3 coordinates (x, y and z-axis)** and the universe according to this
convention is called **Newtonian**. But, in **1905**, Einstein showed in his** Special
Theory of relativity** that **time is also an integral factor**. Well, it was not
before **1908**, when one of the Einstein’s own professor and renowned physicist
**Herman Minkowski** proposed the concept of the **union of space and time** as a way to
**reformulate Special Relativity and by this way, the concept of space-time was
born**.

To derive the equations to prove its invariancy, we need to understand
the above picture. Here in this picture, there are two coordinate systems, **S
and S’**. **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’**.

Using this picture, **I derived the Lorentz transformation equations in
my Introduction to Lorentz Transformation article** and in this article, we are going
to f**ind the space-time interval equation using it**.

**Proof of Invariancy**

As we all know,

Time = Distance / Velocity

**t = √ (x² + 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.

Now, to prove invariancy, we need to show,

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

Why??? Because invariancy means in every condition, the space-time
interval equation in both S and S’ will be equal. Which means that we can write
the space-time interval equation for S and S’ plane and equate them as we did
in equation 2.

Now, proving LHS = RHS,

Where, LHS = **x****²**** + y****²**** + z****²** **– c****²**** t****²**** ** (space-time interval equation in S plane)

And, RHS = **x’****²**** + y’****²**** + z’****²** **– c****²**** t’****²**** **

Now, using Lorentz Transformations in RHS,

**[x – vt / (√ 1 – v****²****/c****²****)]****² ****+ y****²**** + z****²** **– c****²**** [(t – vx/c****²****) / (√ 1 – v****²****/c****²****)]****²**** **(3)

Since,

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

**y’ = y **

**z’ = z
**

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

Now if you solve equation 3, you will get,

**x****²**** + y****²**** + z****²** **– c****²**** t****²**

which is our LHS.

Thus, LHS = RHS and** invariancy of the space-time interval is proved.**

**Same can also be proved by inverse Lorentz Transforms and putting
those in LHS and then deriving RHS from it. The choice is yours always!!!**

So, that’s its guys…that’s all about Space-time intervals and its
invariancy.

In my upcoming articles, I will write about** 4-vectors, tensors** etc. are very important to understand
relativity and cosmology in general. 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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