1 Newton and Einstein

1.1 Newton

Far, far above us, the sun, the moon, the stars, and the planets appear to move perpetually in some kind of periodic motion. It is a grand thing :) to occasionally pause from the busyness of life and wonder about these heavenly bodies. Several hundred years ago, in the 17th century, Newton proposed that the motions of the planets could be understood from a simple force law

Eqn 1
F=ma\mathbf{F} = m\mathbf{a}

together with the law of gravitational force that exists between any two masses MM and mm:

Eqn 2
F=−GMmr2r^\mathbf{F} = -G \frac{Mm}{r^2}\hat{\mathbf{r}}

where r^\hat{\mathbf{r}} is the unit vector pointing from MM to mm, and GG the gravitational constant.

Newton's theory could explain the motions of the planets, and yet, as a physical theory, to be judged on the basis of its physical postulates, the concepts it made use of, and the mental pictures it created, it was found to be unsatisfactory.

On the other hand, let us ponder the following basic conceptual point. Given a mass MM, it is very desirable to have a theory in which the gravitational effect of MM on other bodies is characterized by MM alone (at least this should be possible in situations where the effects of the other bodies in the interactions on MM are negligible). In Newton's theory, this can be achived by speaking of a gravitational field created by MM. But, in order to determine the motion of a body due to MM, the gravitational force due to MM is needed, and, even though MM would exert different forces on bodies having different masses, they would all follow the same path. This feels very contrived.

Let us pause a little here. The object of mechanics is to determine the paths, or curves in spacetime, of physical bodies. It is an experimental fact that as far as the gravitational effect of a given mass MM on other bodies is concerned, the bodies subject to the force all follow the same curves in spacetime. The mathematical theory of curved space had been worked out by Gauss and Riemann, and in fact the possibility that the geometry of space could be influenced by its content matter had been proposed by mathematicians. So, it seemed there was little choice but to link mass directly to the theory of curved spacetime.

1.2 Einstein

In 1915 Einstein proposed the following equation:

Eqn 3
Gμν=−κTμν,μ,ν=0,1,2,3G_{\mu\nu} = -\kappa T_{\mu\nu}, \qquad \mu,\nu=0, 1, 2, 3

where κ=8πG/c4\kappa = 8\pi G/c^4, GμνG_{\mu\nu} the Einstein tensor, and TμνT_{\mu\nu} the energy-momentum tensor. According to the equation, the mass of a given body determines the geometry of spacetime, and the paths of other bodies simply follow that geometry. In this way, the concept of force was entirely eliminated.

The Einstein's equation contains within itself a completely new proposal of our understanding of space and time and gravitational interactions. Einstein's theory not only explains all that Newton's theory could explain, it also resolves the remaining difficulties encountered with the latter and opens up vast, unexplored possibilities. We now use it to study our universe, including its origin and its evolution over time.

2 Next

The mystery of gravitational force is not completely solved. Gravity itself is a weak force, and in the realm of elementary particles so much has been achieved by ignoring it. But, beside the fact that if we keep probing into smaller and smaller scale, gravity will once again become significant (at the Planck scale), gravity's role and its relation with quantum mechanics, the theory we currently use to understand elementary particles and their interactions, also need to be clarified.

There're many interesting, deep questions for us to think about.