Einstein and Newton- The Dark Connection!

The Newton Story

In 1687 Newton proposed the universal law of gravitation relating the force between two masses. The force was attractive in nature and varied as inverse square of the distance between the masses. Newton did not understand why the force varies as inverse-square the understanding of this came later with relation to space being three dimensional.

In Newton's theroy there was no mechanism of interaction between the particles and it was essentilly an action at a distance theory. Newton was surprised when he found out that the force of gravity on a particle outside a spherical object was independent of distribution of the mass. The force behaved as if the whole mass is concentrated at the center (known as

Birkhoff's theorem). We now know that there are only two forces that follow this property namely: the inverse square law and when the force is proportional to the distance. Incidently these two are the only two central forces that are stable against radial perturbations (known as Bertrand's theorem). The trajectories of particles in this general force field are elliptical for bound orbits. We know from observations that bodies moving in the solar system follow bound

and stable orbits. Thus the most general force law consistant with this observation can be written as a combination of these two forces

Since we observe gravity always as an attractive force, the coefficient of the second term if positive (meaning a repulsive force) has to be very small. If we divide the equation (1) by the mass of the test particle we get the following equation

..........(1)

At this stage we leap some 240 years into the picture where our friend Einstein awaits!

The Einstein story

Newton's laws were heralded as s great success to man's intellect. It explained with great precission the motion of all the heavenly bodies. One of the instances where the observations didn't match with the observations was the motion of the planet Mercury. It was observed that the orbit of Mercury was precessing at a rate which was slightly more than what was predicted from Newtonian mechanics. The difference between theory and observations was 43 arc seconds/century. It was first thought that this was the effect of some hitherto unknown object which was christined as Vulcan (a similar thing has happened for the orbit of Uranus which led to the discovery of Neptune). In this case however no such object was found. Einstein proposed a new theory of gravity which accounted for this discrepancy. This theory required a revision of our understanding of gravity. In Einstein's theory of general relativity space and time were coupled and played a dynamic role in motion of objects. When Einsteins equations were applied to study the evolution of universe they gave unstable solutions for a static universe (which is expected because with only attractive force either everything is coming

closer or moving apart). In order to get a static and stable solution Einstein introduced a repulsive term in his equations known as the cosmological constant L. With this term the

equations for an homogeneous and isotroopic universe take the following form

....(2)

If we substitute

and

we find that equation(1) and equation (2) take the same form! So, its interesting that two of our greatest minds were so close in this regards even though they were centuries apart from each other.

Images:

http://healthcenterinternationalresearches.webs.com/einstein_theorising.jpg

http://www.hypatiamaze.org/isaac/newton1.jpg

Godel and Goodstein theorems

I have started reading Penrose's book the Emperor's new mind and came across these interesting theorems in the preface about Godel's and Goodstein's theorem. Here I briefly mention the two following the treatment from the book:

Godel's theorem:

Suppose that we are given a computational procedure P for establishing mathematical assertions of a particularly well defined type such as the Fermat's last theorem. Then if we are prepared to accept the successful derivation of some assertion by use of rules of P provides us with an unassailable demonstration of the truth of that assertion- then we must also accept as unassailably true some other assertion G(P) which is beyond the scope of the rules of P.

Goodstein's theorem:

Consider any positive number, let us say 3. First, we express this as a sum of distinct powers of 2:

3 = 2^1 + 1.

We now apply a succession of simple operations to this expression, these alternating between:

1. increasing the 'base' by 1.
2. subtract 1.

It may seem that the numbers would be ever increasing, but the theorem tells us that no matter what positive number we start with, we always end up with zero!

The reader is encouraged to verify this by taking some number. An example starting with 3 can be seen here: click

What is rather more extraordinary is that Goodstein's theroem is actually a Godel theorem for the procedure of mathematical induction. Recall that mathematical induction provides a way of proving that some mathematical statement S(n) holds for all n = 1,2,3 ... The procedure is to show that if it holds for n =1, then it also holds for n+1. If P stands for the procedure of mathematical induction, then we take G(P) to be Goodstein's theorem.