General Theory of Relativity:

Newtonian theory of Gravity envisages an instantaneous force acting over arbitrarily long distances. This conflicts with Special Relativity, where physical influences cannot travel at infinitely large speeds. A change in the force due to local change in the mass distribution cannot be instantaneously communicated to remote points. Einstein took up to modify the Newtonian law.

It was well known that the mass of a particle required in Newton's second law, the inertial mass, was the same as that which entered the law of Gravitation. This has a peculiar consequence. Given a large gravitating body, all the other particles when released from a given point with given velocity would trace out the same trajectory, regardless of their masses. This fact about Gravitation had been demonstrated by Galileo in his famous (perhaps apocryphal) experiment at the Tower of Pisa. It had been checked to a high degree of accuracy by the Hungarian experimenter Eotvos the nineteenth century.

Einstein promoted this observation, the equivalence of Gravitational and inertial masses to a physical principle, the Equivalence Principle. As we have seen, the Principle means that in the vicinity of a large mass, the trajectories of particles are independent of their inertia and are determined purely by their initial velocities. The trajectories can then be thought of as a property of the space-time itself rather than belonging to the test particles. The space-time around the Sun can be thought of as a rubber sheet stretched by the weight of the sun. This is schematically shown in fig. 3 This was experimentally confirmed by Eddington during the total solar eclipse of 1916. The rays arriving from distant stars and grazing the photosphere were deflected by a few seconds of an arc.

Einstein's observation upto this point amounted to elegant geometric reformulation of known facts. But too guess the full form of the Gravitational law, he used the following device. Suppose we know a law in the absence of Gravity. This should be Lorentz covariant.

  

Figure 3: A sheet of rubber stretched by a ball placed on it serves as a model for the gravitation field of the ball.

That is to say the equations will involve vector fields which will transform linearly if we make a change from one inertial frame to another. In a curved space-time an inertial frame can be defined only locally, in a small region of space and time. Consider a closed cabin falling freely in the local Gravitational field. Since all material bodies inside this cabin fall with the same acceleration, they will appear to be floating freely. No effect of gravity will be discernible. If the cabin is too large or the experiment lasts too long, the curvature effects will shown up. Thus, a small cabin will indeed work like an inertial frame provided it is falling freely in the local Gravitational field. At another point we will need another frame, again falling freely there, whose motion will be quite different from the first one.

We could patch together the choices made by different observers and come up with a coordinate system covering a large part of the space-time. This cannot itself be an inertial frame of reference, being patched from several distinct ones. But the situation is the same as for an arbitrary curved space with tangent planes at various points patched together to map out a region. To make the reference system smooth, we may smooth out the jerky joining points of transition between one local frame and another. This would give rise to a picture as in fig. 4.

In a curved space one does not choose one rigid, Cartesian system of coordinates, but any convenient, curvilinear coordinate system. An example is shown in fig. 4.

  

Figure 4: A curved surface with a coordinate system with local "inertial frames" shown for "observers" P and Q.

Einstein insisted that in the presence of Gravity, the same should be permitted for physical frames of reference. The laws should be such as to be covariant under arbitrary changes of the curvilinear coordinates. The complete freedom in the choice of a coordinate system has the implication that the local inertial observers at different points, may use completely different Lorentz frames to describe their physics. This is indicated by different co-ordinate systems in the tangent planes at points P and Q in the fig. 4.

  

Figure 5: The surface of fig. 4 with a different coordinate system.

In fig. 5 we have shown a new coordinate system on the space of fig. 4. Furthermore, under a change of co-ordinates, the two frames suffer different transformations at the two different points, as indicates by the new frames at the same points P and Q in fig. 5.

Thus the freedom in the choice of the Cartesian Lorentz frames is greatly expanded to a freedom in choice of Lorentz frames, at each point independently. This principle was called by Einstein the Principle of General Covariance. This symmetry principle served as a tool for guessing correct laws of motion for particles and the laws of Electromagnetism in the presence of Gravity. Further, the same principle was exploited by Einstein to guess the correct law of motion for the Gravitational field itself.

In Einstein's own words he had generalised the Special Relativity of inertial frames under Lorentz transformations to General Relativity of arbitrary curvilinear coordinate systems. In the decades since, a different view is taken of the Principle of General Covariance. Couched in the language of a symmetry principle, (operations that do not affect Physics) it really is a prescription for guessing the form of a physical interaction (very much affecting Physics), viz., Gravity. But the details of this discussion cannot be included here.

In the 1930's T. Kaluza and independently Oskar Klein, proposed a theory in which Electromagnetism as well as nuclear interactions appear as a consequence of General Covariance in space-times with a larger number of dimensions. The extra dimensions of space would be too tiny for us to directly experience. The program however did not work, and under closer investigation in the 1980's it was shown to have fundamental disagreement with some known facts of the nuclear world.