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Tensors and Relativity: Chapter 5

# The story so far

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We are at last ready to embark on our central task, namely, that of extending Special Relativity to a theory which incorporates gravitation. In this section we will consider the physical principles which guided Einstein  in his search for the general theory. There is a school of thought that considers this an unnecessary process, but rather argues that it is sufficient to first state the theory and then investigate its consequences. There seems little doubt, however, that consideration of these physical principles helps gives insight into the theory and promotes understanding. The mere fact that they were important to Einstein would seem sufficient to justify their inclusion. If nothing else, it will help us understand how one of the greatest achievements of the human mind came about.

Many physical theories today start by specifying a Lagrangian from which everything flows and we could adopt the same attitude with General Relativity. Although this is a very beautiful way about going about things, in taking that approach we would miss out on gaining some understanding of how the framework of General Relativity is different from that of Newtonian theory and Special Relativity. Moreover if we discover limitations in the theory, then there is more chance of rescuing it by investigating the physical basis of the theory rather than simply tinkering with the mathematics- an unfortunate trait of much of modern theoretical physics these days!

But before we embark on the most exciting journey of discovery, we must first remind ourselves of where we have got to. So far we have only discussed Special Relativity. Here forces have only played a background role and we have never introduced gravitation explicitly as a possible force. One aspect of Special Relativity is the existence of a global inertial frame , all of whose coordinate points are always at rest relative to the origin, and all of whose clocks run at the same rate relative to the origin's clock. From Einstein's postulates we were led to the idea of the spacetime interval which gave an invariant geometrical meaning to certain physical statements. We discovered that the mathematical function that calculates the spacetime interval  is the metric , and so the metric of Special Relativity is defined physically by lengths of rods and the readings of clocks. This closeness to experiment is of course its strength.

Let us now ask the following question:

• Is is possible to construct a frame in which all clocks run at the same rate?

This is a crucial question and we will show that in a non- uniform gravitational field the answer, experimentally , is NO.

• So: Gravitational fields are incompatible with global Special Relativity i.e. it is impossible to construct a global inertial frame.

We shall see that in small regions of spacetime [ regions small enough that non- uniformities of the gravitational field are too small to measure] one can always construct a ``Local Intertial Frame''  [LIF]. In this sense we will have to build Special Relativity into a more general theory.

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