Simple proof of Hubble's Law

I proved the Hubble's law when I was in 11th standard. I was only 16 and had just been introduced to calculus. I was actually thinking about dark matter, and how it influences the Universe; and I concluded, that if the Universe is expanding, then the relative velocity of any 2 objects in the Universe is directly proportional to the distance between them.

Moreover, the proportionality constant comes out to be the inverse of the Age of the Universe! I was really overjoyed with what I got, only to be disappointed after sometime when I found out that this law already exists, and is called Hubble's Law.

But, Sir Hubble derived this law from experimental evidences, while I did it totally on a theoretical basis. So, I think this proof might have some significance.

PROOF:
Consider the Universe at the time of the BIG BANG. It was a singularity, with all the mass concentrated at a single point with almost infinite density. I feel, the first thing that got created was space-time. Everything else came after that.

Just like spreading of electric field is done by electromagnetic waves, space-time spreads through gravitational waves. They travel with the speed of light.

I have also proved that the speed of light is not constant, but depends on the age of the Universe. But, for now, lets make matters simple by considering it to be constant.

So, we have a picture of space-time expanding(stretching would also be appropriate) with the speed of light.

After a time t1:
Age of the Universe=t1
Radius= c*t1
Suppose there are 2 objects on the same diameter of the Universe, but on opposite sides of the center. One at a distance x1 from the center, and the other at x2.

After some time, say the age of the Universe is t2:
Age: t2
Radius= c*t2
Distances of thee objects become x1' and x2' respectively.


Now, if we see this process as a whole,

                                                                     c*t1 became c*t2
                                                           and,      x1  became  x1'

Therefore,  x1'= x1(t2/t1)
Similarly,    x2'=x2(t2/t1)

We have,  x1' = x1 + dx1
 and          x2' = x2 + dx2

Adding the above 2 equations, we get,

x1' + x2' = x1+ x2 + dx1 + dx2

 Let X = x1 + x2 = Distance between the 2 objects.
 so,  dX = dx1 + dx2

 Therefore, x1' + x2' = X + dX

x1(t2/t1) + x2(t2/t1) = X +dX

(t2/t1)(x1 +x2) = X + dX

Since, t2=t1 + dt,

X(dt/t1) = dX

Therefore, dX/dt = X/t1

dX/dt is he relative velocity of the 2 objects.

So, V(relative) = X/t1

It clearly shows that the relative velocity is proportional to the distance between them, and inversely proportional to the age of the Universe.

Note:
This form of the law gives correct results only when t1 and t2 are almost equal.

There are many many more factors to be considered, I will try to consider them and improve the law and its proof over time.

Cheers,
Sushant Mahajan,
2nd year Engineering Physics,
ITBHU.