Some Hidden Truths in This Relative World

Rabin, one of my good friends, asked at his Facebook Wall two weeks back that if anybody moves with a velocity more than that of light, will it enter into the negative time zone? Will it reach its destination today after starting its journey tomorrow?  Frankly speaking, I had no clear answer to it as my knowledge was limited on the subject. And that had created a weird feeling inside me. Hence, I did some R&D on the subject for last couple of days and reached to a conclusion, which I will try to explain as following.

Let’s start with the famous and well-known equation of Einstein, E = MC² , where E = Energy, M = Mass and C = Velocity of Light.

It tells us that energy and mass of any particle are equivalent. In a deeper sense, the internal energy of any particle is represented by its mass.

Now, if the particle moves with a velocity V, then a Kinetic Energy of  ½ MV²   will be added to its existing internal energy. Then the equation of total energy will be

E = MC²  + ½ MV² 

Hence, if a particle has velocity, then the equation of Energy should also contain the term velocity in it.

Hence, a better equation of Energy may be expressed as

 Now, let’s test the above equation with different conditions.

i.               If V<<C, then V²/C² ≈ 0, and E = MC²

ii.              If V becomes substantial, then by applying Binomial Theorem, we get,

 E = MC²  ( 1- V²/C²)^-1/2 = MC²  (1+1/2V²/C²) = MC²  + ½ MV² 

Hence, E = MC² is an approximation and valid only for low or zero velocity.  

Now, let’s go into further depth and consider another two conditions as below.   

i.               If V ≈  C, then     = Infinite

It means infinite energy is required to bring the velocity of a particle near the velocity of light, whereas the energy of the Universe is finite. Hence, it’s next to impossible for any particle having mass M to attain the velocity near the light velocity.

ii.              If V = C, then    = Undefined

It means the amount of energy required to bring the velocity of any particle with mass M equal to the light velocity, is undefined or meaningless.

Hence, we can conclude from above that, no particle with mass M in this Universe can attain the velocity near or equal to the velocity of light, thus the question of exceeding the velocity of light by any particle having mass does not arise at all.

Now, let’s look into the question of exceeding the light velocity with another interesting equation i.e. the relativistic time equation or equation of Time Dilation.

The equation is, ∆ t’ =  , where ∆ t’ = Time change in the particle with velocity, ∆ t = Time change in a rest body.

The equation tells us that the change in time in a particle with velocity becomes slower than that of a rest body with increase of the velocity, which is called Time Dilation in other terms.

Now, if V becomes more than C, then ∆ t’ will be imaginary. Hence, the condition of V > C is not possible any way.

Let’s go into further depth to see some magic in the world of quantum.

We know the equation of momentum is the product of mass and velocity. But it’s also an approximation. Let me take the opportunity to define the term Momentum here. It’s the ability of any particle to push an object and make it move. It does not matter whether the particle is having mass or not, in getting that ability.

A more fulfilling equation of Momentum for the particle with Mass and substantial Velocity can be expressed as below

Now, we already know the equation of Energy, which is 

If we interchange V in above two equations, we get the equation of energy as following.

E² = (PC)² + (MC²)²

This is the most fulfilling equation of Energy in all respect.

Now, let’s assume three conditions.

i.               If, P = 0, then E = MC²

It means that if a particle has zero momentum, must have mass.

ii.              If, M = 0, then E = PC

It means that if a particle has zero mass, must have momentum.

Hence, the photon or light particle has momentum though it has no mass.

Also, photon can attain the speed of light, as it has no mass.

iii.            If, P = 0 & M = 0, then E = 0

It means that if any particle has neither mass nor momentum, then energy won’t exist, which is an impossible state.

Now, let me summarize the conclusions in a nutshell as below:

1.     No particle having mass M in this Universe can attain the velocity near or equal to the velocity of light.

2.     Any particle at rest i.e. with zero momentum must have mass.

3.     If a particle has zero mass must have momentum.

4.     Photon, the light particle, can attain the speed of light since it has no mass.

5.     Photon has momentum though it has no mass.

6.     A particle having neither momentum nor mass does not exist in this Universe.

Then what are you waiting for?

Let’s all immerse into the world of quantum.

Believe me, you will get the Real Taste of the Relative Truths in that world.

Photon with Momentum