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
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
If we substitute
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.