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THE DISTORTED GEOMETRY
 
 
THE DISTORTED GEOMETRY

Einstein’s historical journey began in 1907, when he was sitting in his patent office patiently. Suddenly, a strange idea came to him - ‘gravity is same as acceleration’ and he jumped to his feet (as he was ought to - an Einstein must jump on such crazy and simple ideas). Einstein used to call this idea his best intellectual experience. You would not find anything special in this, unless you were a philosopher. But, this simple statement is the powerhouse of relativity. Ages ago, there were two terms - inertial mass and gravitational mass. Inertial mass is a quantity which refers to the measure of object’s resistance to the force applied to the object. It is related to the formula F=ma. Gravitational mass refers to the measure of the body’s interaction with the gravitational field. What Einstein meant was that there was no difference between these two masses (still at present, no one has been able to find a difference; even Newton had to treat gravitational mass as inertial

mass while writing the equations of motion in gravity). This clearly meant that gravitational field and accelerated motion were the same thing. So, we could say that a law for generalization of special relativity would yield the theory of gravitation or a theory of gravitation would provide a rule for accelerated motion. That’s what Einstein did. For some time, he worked in an accelerated frame and soon discovered the results like bending of light in gravity. You can refer to the book ‘On Shoulders Of Giants’ by Stephen Hawking for the original papers of Einstein.
 

He soon realized that he needed something more general to solve this problem because till then, he had not considered the general framework. He needed a system where the presence of matter could be incorporated in the theory. For about four years, he kept searching for such a theory, but, was in vain. All was because years ago, he used to sit by a beautiful lake and ponder over the troubles with Physics while his friend Marcel Grossmann would take notes on hyperbolic geometry for him. Einstein contacted his old friend seeking for a help in this problem, which was driving him mad. Grossman navigated a big library and finally got the clue. Years ago, Bernhard Riemann developed a theory on surface geometry. If laws could be represented in terms of tensors and metric of space, everything would be fine. Tensors can represent any quantity which can be physically relevant. Riemann had tried to formulate such a theory where all laws could be represented in terms of the properties of the space, but he had no underlying principle for this. Einstein had the underlying principle – the equivalence of gravitational field and acceleration – and he, then, had the perfect tool. Now, we will see how he formulated the law of gravitation. We will just see the underlying ideas but you can study the theory from here – http://people.hofstra.edu/Stefan_Waner/diff_geom/tc.html (click here).

We have already introduced the concept of tensors in previous sections. Tensors follow covariance and contra variance laws; so, an equation written completely in terms of linear combinations of tensors of a particular order (or type) would be invariant. Thus, tensor geometry provides the mathematical treatment for equivalence principle. Now, the tensor called ‘stress tensor’ gives the energy of a system. This tensor gives the presence of matter. Then comes the fundamental idea of Einstein. The presence of matter causes the space-time to distort and this distortion affects the surrounding matter. Let us see some elements which play important roles in curved space geometry and hence, lead to Einstein field equation.

A very important term for a curved space is that of geodesic. Geodesic is a path for which the integral òds is the least. Equation of geodesic can be obtained in terms of metric tensor from the notion of variation principle. Now, ds = Ö(gabdxadxb). Let it be parameterized by l. So,

ds = [Ö{gab(dxa/dl)(dxb/dl)}]dl

Now, dòds = 0 for the path or dò [Ö{gab(dxa/dl)(dxb/dl)}]dl = 0.

We obtain the equation for geodesic on surface - for the path characterized by x1, x2, x3, … xn coordinates. The path’s component along xa is: 

                                d2xa/ds2 + Gbac (dxb/ds)(dxc/ds) = 0.

where Gbac is called the Christoffel symbol:

Gbac = (1/2)gad [dgbd/dxc + dgcd/dxb - dgbc/dxd]

For a flat space, gij is constant. So, Christoffel symbol vanishes and we get d2xa/ds2 = 0, equation of a straight line. Christoffel symbol satisfies Gbac = Gcab (symmetry). The concept of differentiation is rather different in curved spaces. A vector Au has the derivative Au;v defined as:

Au;v = dAu/dxv - AwGuwv.

This is called covariant derivative and is of equal importance as the normal derivative on a plane surface. The metric tensor g is constant under covariant differentiation.
 

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