General Theory of Relativity - Introduction

 

The Special Theory of Relativity appeared in 1905. It took Einstein another 10 years to work out his General Theory of Relativity, which appeared in 1916. The General Theory is often referred to as one of the most beautiful theories in physics. Einstein worked very hard on this theory and made some mistakes. He had to learn mathematical techniques which were not as yet commonly employed in physics. The General Theory is a theory of gravity. It does not explain why gravity exists, but it does give another way of looking at gravity.

We have already seen that the laws of classical physics must be modified for objects moving close to the speed of light relative to other objects. The General Theory implies that there is a limit in which the Newtonian theory of gravity is incorrect. We already know that there are some problems with Newtonian theory. Suppose you move your hand. This should change the force on a distant star instantaneously according to Newtonian theory; however, from the theory of relativity we expect that the change in force cannot be transmitted at a speed faster than the speed of light. Newton’s Law of Universal Gravitation can only be an approximation (and, usually a very good approximation) to a correct description of gravity.

Einstein wanted to incorporate gravity into his theory of special relativity. He came up with the following thought experiments that motivated the theory. Suppose you are in an elevator and someone cuts the rope. Since inertial mass equals gravitational mass, the elevator and all objects in it fall at the same rate. From within the elevator it appears that you are weightless. On the other hand, take a spaceship in outer space and accelerate it uniformly. From within the ship it will appear that a gravitational field is present since all objects fall at the same rate, even though there is no "true" gravitational field present. This led Einstein to the second postulate of the General theory: The effects of a uniform gravitational field and uniform acceleration are equivalent. Note that this is equivalent to postulating that inertial mass equals gravitational mass since this fact is necessary to equate the two (if inertial mass does not equal gravitational mass, then different objects in the elevator fall at different rates and gravity would not be eliminated). The second postulate is known as the Principle of Equivalence.

The first postulate of the general theory is more subtle. The first postulate states that The laws of physics take the same form in all reference frames (no longer a restriction to inertial frames). You must be able to write a law in a form that is manifestly covariant. That is the law must be written using objects such as 4-vectors and tensors that guarantees that the equation takes the same form for an arbitrary transformation of the coordinates (space and time) from one reference frame to another. The first law is known as the Principle of General Covariance. As is outlined on the notes in the course pack on relativity, these laws have some very interesting consequences.

You should not think that, just because you can simulate gravity by going into an accelerating reference frame, gravity and acceleration are entirely equivalent. It is impossible to go into a single accelerating reference frame to eliminate the gravitational field of the Earth. This can be done only locally by going into a free-falling reference frame. At different locations relative to the Earth, the frames would differ in their direction and acceleration. Only for a uniform (constant) gravitational field (which does not exist) could you eliminate gravity globally (everywhere) by going into a single, accelerating inertial frame. Mass is the source of gravity.

The General Theory describes how the mass and energy distribution in the Universe creates the structure of space-time and how objects move in this curved space-time. The mathematics is very complicated and, even today, there are only a handful of solutions of Einstein’s equations.